Instanton - Wikipedia

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class="vector-toc-numb">2</span> <span>Quantum mechanics</span> </div> </a> <button aria-controls="toc-Quantum_mechanics-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Quantum mechanics subsection</span> </button> <ul id="toc-Quantum_mechanics-sublist" class="vector-toc-list"> <li id="toc-Motivation_of_considering_instantons" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Motivation_of_considering_instantons"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Motivation of considering instantons</span> </div> </a> <ul id="toc-Motivation_of_considering_instantons-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-WKB_approximation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#WKB_approximation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>WKB approximation</span> </div> </a> <ul id="toc-WKB_approximation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Path_integral_interpretation_via_instantons" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Path_integral_interpretation_via_instantons"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Path integral interpretation via instantons</span> </div> </a> <ul id="toc-Path_integral_interpretation_via_instantons-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Explicit_formula_for_double-well_potential" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Explicit_formula_for_double-well_potential"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Explicit formula for double-well potential</span> </div> </a> <ul id="toc-Explicit_formula_for_double-well_potential-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Results" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Results"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Results</span> </div> </a> <ul id="toc-Results-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Periodic_instantons" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Periodic_instantons"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Periodic instantons</span> </div> </a> <ul id="toc-Periodic_instantons-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Instantons_in_reaction_rate_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Instantons_in_reaction_rate_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>Instantons in reaction rate theory</span> </div> </a> <ul id="toc-Instantons_in_reaction_rate_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inverted_double-well_formula" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inverted_double-well_formula"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.8</span> <span>Inverted double-well formula</span> </div> </a> <ul id="toc-Inverted_double-well_formula-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Quantum_field_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Quantum_field_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Quantum field theory</span> </div> </a> <ul id="toc-Quantum_field_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Yang–Mills_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Yang–Mills_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Yang–Mills theory</span> </div> </a> <ul id="toc-Yang–Mills_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Various_numbers_of_dimensions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Various_numbers_of_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Various numbers of dimensions</span> </div> </a> <ul id="toc-Various_numbers_of_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-4d_supersymmetric_gauge_theories" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#4d_supersymmetric_gauge_theories"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>4d supersymmetric gauge theories</span> </div> </a> <ul id="toc-4d_supersymmetric_gauge_theories-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Explicit_solutions_on_R4" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Explicit_solutions_on_R4"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Explicit solutions on R<sup>4</sup></span> </div> </a> <ul id="toc-Explicit_solutions_on_R4-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References_and_notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References_and_notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References and notes</span> </div> </a> <ul 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hreflang="ko" data-title="순간자" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Istantone" title="Istantone – Italian" lang="it" hreflang="it" data-title="Istantone" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Instanton" title="Instanton – Portuguese" lang="pt" hreflang="pt" data-title="Instanton" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%98%D0%BD%D1%81%D1%82%D0%B0%D0%BD%D1%82%D0%BE%D0%BD" title="Инстантон – Russian" lang="ru" hreflang="ru" 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img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:292px;max-width:292px"><div class="trow"><div class="tsingle" style="width:143px;max-width:143px"><div class="thumbimage" style="height:119px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:-y-(x%5E2%2By%5E2%2B1)_plot;_BPST_instanton.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/-y-%28x%5E2%2By%5E2%2B1%29_plot%3B_BPST_instanton.png/141px--y-%28x%5E2%2By%5E2%2B1%29_plot%3B_BPST_instanton.png" decoding="async" width="141" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/-y-%28x%5E2%2By%5E2%2B1%29_plot%3B_BPST_instanton.png/212px--y-%28x%5E2%2By%5E2%2B1%29_plot%3B_BPST_instanton.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/92/-y-%28x%5E2%2By%5E2%2B1%29_plot%3B_BPST_instanton.png/282px--y-%28x%5E2%2By%5E2%2B1%29_plot%3B_BPST_instanton.png 2x" data-file-width="451" data-file-height="382" /></a></span></div></div><div class="tsingle" style="width:145px;max-width:145px"><div class="thumbimage" style="height:119px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:X-(x%5E2%2By%5E2%2B1)_plot;_BPST_instanton.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/X-%28x%5E2%2By%5E2%2B1%29_plot%3B_BPST_instanton.png/143px-X-%28x%5E2%2By%5E2%2B1%29_plot%3B_BPST_instanton.png" decoding="async" width="143" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/X-%28x%5E2%2By%5E2%2B1%29_plot%3B_BPST_instanton.png/215px-X-%28x%5E2%2By%5E2%2B1%29_plot%3B_BPST_instanton.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7b/X-%28x%5E2%2By%5E2%2B1%29_plot%3B_BPST_instanton.png/286px-X-%28x%5E2%2By%5E2%2B1%29_plot%3B_BPST_instanton.png 2x" data-file-width="459" data-file-height="386" /></a></span></div></div></div><div class="trow"><div class="tsingle" style="width:158px;max-width:158px"><div class="thumbimage" style="height:129px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Curvature_of_BPST_Instanton.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Curvature_of_BPST_Instanton.png/156px-Curvature_of_BPST_Instanton.png" decoding="async" width="156" height="130" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Curvature_of_BPST_Instanton.png/234px-Curvature_of_BPST_Instanton.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Curvature_of_BPST_Instanton.png/312px-Curvature_of_BPST_Instanton.png 2x" data-file-width="459" data-file-height="382" /></a></span></div></div><div class="tsingle" style="width:130px;max-width:130px"><div class="thumbimage" style="height:129px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:BPST_on_sphere.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/BPST_on_sphere.png/128px-BPST_on_sphere.png" decoding="async" width="128" height="130" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/BPST_on_sphere.png/192px-BPST_on_sphere.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/BPST_on_sphere.png/256px-BPST_on_sphere.png 2x" data-file-width="1133" data-file-height="1148" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">The <i>dx<sup>1</sup>⊗σ<sub>3</sub></i> coefficient of a <a href="/wiki/BPST_instanton" title="BPST instanton">BPST instanton</a> on the <i>(x<sup>1</sup>,x<sup>2</sup>)</i>-slice of <b>R</b><sup>4</sup> where <i>σ<sub>3</sub></i> is the third <a href="/wiki/Pauli_matrix" class="mw-redirect" title="Pauli matrix">Pauli matrix</a> (top left). The <i>dx<sup>2</sup>⊗σ<sub>3</sub></i> coefficient (top right). These coefficients determine the restriction of the BPST instanton <i>A</i> with <i>g=2,ρ=1,z=0</i> to this slice. The corresponding field strength centered around <i>z=0</i> (bottom left). A visual representation of the field strength of a BPST instanton with center <i>z</i> on the <a href="/wiki/Compactification_(mathematics)" title="Compactification (mathematics)">compactification</a> <i>S<sup>4</sup></i> of <b>R</b><sup>4</sup> (bottom right). The BPST instanton is a classical instanton solution to the <a href="/wiki/Yang%E2%80%93Mills_equations" title="Yang–Mills equations">Yang–Mills equations</a> on <b>R</b><sup>4</sup>.</div></div></div></div> <p>An <b>instanton</b> (or <b>pseudoparticle</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>) is a notion appearing in theoretical and <a href="/wiki/Mathematical_physics" title="Mathematical physics">mathematical physics</a>. An instanton is a classical solution to <a href="/wiki/Equations_of_motion" title="Equations of motion">equations of motion</a> with a finite, <a href="/wiki/Vacuum_state" class="mw-redirect" title="Vacuum state">non-zero action</a>, either in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> or in <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>. More precisely, it is a solution to the equations of motion of the <a href="/wiki/Classical_field_theory" title="Classical field theory">classical field theory</a> on a <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean</a> <a href="/wiki/Spacetime" title="Spacetime">spacetime</a>.<sup id="cite_ref-:0_4-0" class="reference"><a href="#cite_note-:0-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>In such quantum theories, solutions to the equations of motion may be thought of as <a href="/wiki/Critical_point_(mathematics)" title="Critical point (mathematics)">critical points</a> of the <a href="/wiki/Action_(physics)" title="Action (physics)">action</a>. The critical points of the action may be <a href="/wiki/Maxima_and_minima" class="mw-redirect" title="Maxima and minima">local maxima</a> of the action, <a href="/wiki/Maxima_and_minima" class="mw-redirect" title="Maxima and minima">local minima</a>, or <a href="/wiki/Saddle_point" title="Saddle point">saddle points</a>. Instantons are important in <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> because: </p> <ul><li>they appear in the <a href="/wiki/Functional_integration" title="Functional integration">path integral</a> as the leading quantum corrections to the classical behavior of a system, and</li> <li>they can be used to study the tunneling behavior in various systems such as a <a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills theory</a>.</li></ul> <p>Relevant to <a href="/wiki/Dynamics_(mechanics)" class="mw-redirect" title="Dynamics (mechanics)">dynamics</a>, families of instantons permit that instantons, i.e. different critical points of the equation of motion, be related to one another. In physics instantons are particularly important because the condensation of instantons (and noise-induced anti-instantons) is believed to be the explanation of the <a href="/wiki/Supersymmetric_theory_of_stochastic_dynamics#Classification_of_stochastic_dynamics" title="Supersymmetric theory of stochastic dynamics">noise-induced chaotic phase</a> known as <a href="/wiki/Self-organized_criticality" title="Self-organized criticality">self-organized criticality</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Mathematics">Mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Instanton&action=edit&section=1" title="Edit section: Mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Yang%E2%80%93Mills_equations" title="Yang–Mills equations">Yang–Mills equations</a> and <a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory (mathematics)</a></div> <p>Mathematically, a <i>Yang–Mills instanton</i> is a self-dual or anti-self-dual <a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">connection</a> in a <a href="/wiki/Principal_bundle" title="Principal bundle">principal bundle</a> over a four-dimensional <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a> that plays the role of physical <a href="/wiki/Space-time" class="mw-redirect" title="Space-time">space-time</a> in <a href="/wiki/Non-abelian_group" title="Non-abelian group">non-abelian</a> <a href="/wiki/Gauge_theory" title="Gauge theory">gauge theory</a>. Instantons are topologically nontrivial solutions of <a href="/wiki/Yang%E2%80%93Mills_equation" class="mw-redirect" title="Yang–Mills equation">Yang–Mills equations</a> that absolutely minimize the energy functional within their topological type.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The first such solutions were discovered in the case of four-dimensional Euclidean space compactified to the <a href="/wiki/Hypersphere" class="mw-redirect" title="Hypersphere">four-dimensional sphere</a>, and turned out to be localized in space-time, prompting the names <i>pseudoparticle</i> and <i>instanton</i>. </p><p>Yang–Mills instantons have been explicitly constructed in many cases by means of <a href="/wiki/Twistor_theory" title="Twistor theory">twistor theory</a>, which relates them to algebraic <a href="/wiki/Vector_bundle" title="Vector bundle">vector bundles</a> on <a href="/wiki/Algebraic_surface" title="Algebraic surface">algebraic surfaces</a>, and via the <a href="/wiki/ADHM_construction" title="ADHM construction">ADHM construction</a>, or hyperkähler reduction (see <a href="/wiki/Hyperk%C3%A4hler_manifold" title="Hyperkähler manifold">hyperkähler manifold</a>), a geometric invariant theory procedure. The groundbreaking work of <a href="/wiki/Simon_Donaldson" title="Simon Donaldson">Simon Donaldson</a>, for which he was later awarded the <a href="/wiki/Fields_medal" class="mw-redirect" title="Fields medal">Fields medal</a>, used the <a href="/wiki/Yang%E2%80%93Mills_equations#Moduli_space_of_Yang-Mills_connections" title="Yang–Mills equations">moduli space of instantons</a> over a given four-dimensional differentiable manifold as a new invariant of the manifold that depends on its <a href="/wiki/Differentiable_structure" class="mw-redirect" title="Differentiable structure">differentiable structure</a> and applied it to the construction of <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphic</a> but not <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphic</a> four-manifolds. Many methods developed in studying instantons have also been applied to <a href="/wiki/%27t_Hooft%E2%80%93Polyakov_monopole" title="'t Hooft–Polyakov monopole">monopoles</a>. This is because magnetic monopoles arise as solutions of a dimensional reduction of the Yang–Mills equations.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Quantum_mechanics">Quantum mechanics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Instanton&action=edit&section=2" title="Edit section: Quantum mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <i>instanton</i> can be used to calculate the transition probability for a quantum mechanical particle tunneling through a potential barrier. One example of a system with an <i>instanton</i> effect is a particle in a <a href="/wiki/Double-well_potential" title="Double-well potential">double-well potential</a>. In contrast to a classical particle, there is non-vanishing probability that it crosses a region of potential energy higher than its own energy.<sup id="cite_ref-:0_4-1" class="reference"><a href="#cite_note-:0-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Motivation_of_considering_instantons">Motivation of considering instantons</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Instanton&action=edit&section=3" title="Edit section: Motivation of considering instantons"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider the quantum mechanics of a single particle motion inside the double-well potential <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x)={1 \over 4}(x^{2}-1)^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(x)={1 \over 4}(x^{2}-1)^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a0a40e66ddf92f97839321aec71d772eb1f1ec6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.92ex; height:5.176ex;" alt="{\displaystyle V(x)={1 \over 4}(x^{2}-1)^{2}.}"></span> The potential energy takes its minimal value at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\pm 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae75026eda1ba1ac961f5003150288e9d2f3049b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.399ex; height:2.176ex;" alt="{\displaystyle x=\pm 1}"></span>, and these are called classical minima because the particle tends to lie in one of them in classical mechanics. There are two lowest energy states in classical mechanics. </p><p>In quantum mechanics, we solve the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\hbar ^{2} \over 2m}{\partial ^{2} \over \partial x^{2}}\psi (x)+V(x)\psi (x)=E\psi (x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\hbar ^{2} \over 2m}{\partial ^{2} \over \partial x^{2}}\psi (x)+V(x)\psi (x)=E\psi (x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/679e2afd97b7e2f52c70d44e98aa407931c7fa9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:37.629ex; height:6.009ex;" alt="{\displaystyle -{\hbar ^{2} \over 2m}{\partial ^{2} \over \partial x^{2}}\psi (x)+V(x)\psi (x)=E\psi (x),}"></span></dd></dl> <p>to identify the energy eigenstates. If we do this, we will find only the unique lowest-energy state instead of two states. The ground-state wave function localizes at both of the classical minima <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\pm 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae75026eda1ba1ac961f5003150288e9d2f3049b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.399ex; height:2.176ex;" alt="{\displaystyle x=\pm 1}"></span> instead of only one of them because of the quantum interference or quantum tunneling. </p><p>Instantons are the tool to understand why this happens within the semi-classical approximation of the path-integral formulation in Euclidean time. We will first see this by using the WKB approximation that approximately computes the wave function itself, and will move on to introduce instantons by using the path integral formulation. </p> <div class="mw-heading mw-heading3"><h3 id="WKB_approximation">WKB approximation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Instanton&action=edit&section=4" title="Edit section: WKB approximation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One way to calculate this probability is by means of the semi-classical <a href="/wiki/WKB_approximation" title="WKB approximation">WKB approximation</a>, which requires the value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de68de3a92517953436c93b5a76461d49160cc41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }"></span> to be small. The <a href="/wiki/Schr%C3%B6dinger_equation#Time-independent_equation" title="Schrödinger equation">time independent Schrödinger equation</a> for the particle reads </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{2}\psi }{dx^{2}}}={\frac {2m(V(x)-E)}{\hbar ^{2}}}\psi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>m</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>ψ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}\psi }{dx^{2}}}={\frac {2m(V(x)-E)}{\hbar ^{2}}}\psi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cfbf16e47817ccecae9e3ecad59bb1d55fce1a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:25.271ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}\psi }{dx^{2}}}={\frac {2m(V(x)-E)}{\hbar ^{2}}}\psi .}"></span></dd></dl> <p>If the potential were constant, the solution would be a plane wave, up to a proportionality factor, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi =\exp(-\mathrm {i} kx)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>k</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =\exp(-\mathrm {i} kx)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e96c8abff11d53c02f2015ae8e14fbb26b35276" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.356ex; height:2.843ex;" alt="{\displaystyle \psi =\exp(-\mathrm {i} kx)\,}"></span></dd></dl> <p>with </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\frac {\sqrt {2m(E-V)}}{\hbar }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> <mi>m</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo>−</mo> <mi>V</mi> <mo stretchy="false">)</mo> </msqrt> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\frac {\sqrt {2m(E-V)}}{\hbar }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c29daa2a6b168ffb4fd3b3c2300e6580a53c4510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.532ex; height:6.343ex;" alt="{\displaystyle k={\frac {\sqrt {2m(E-V)}}{\hbar }}.}"></span></dd></dl> <p>This means that if the energy of the particle is smaller than the potential energy, one obtains an exponentially decreasing function. The associated tunneling amplitude is proportional to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-{\frac {1}{\hbar }}\int _{a}^{b}{\sqrt {2m(V(x)-E)}}\,dx},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>m</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>E</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-{\frac {1}{\hbar }}\int _{a}^{b}{\sqrt {2m(V(x)-E)}}\,dx},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/263e4e93535b07d79e315f74993353d5e3158f1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.801ex; height:3.843ex;" alt="{\displaystyle e^{-{\frac {1}{\hbar }}\int _{a}^{b}{\sqrt {2m(V(x)-E)}}\,dx},}"></span></dd></dl> <p>where <i>a</i> and <i>b</i> are the beginning and endpoint of the tunneling trajectory. </p> <div class="mw-heading mw-heading3"><h3 id="Path_integral_interpretation_via_instantons">Path integral interpretation via instantons</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Instanton&action=edit&section=5" title="Edit section: Path integral interpretation via instantons"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Alternatively, the use of <a href="/wiki/Path_integral_formulation" title="Path integral formulation">path integrals</a> allows an <i>instanton</i> interpretation and the same result can be obtained with this approach. In path integral formulation, the transition amplitude can be expressed as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(a,b;t)=\langle x=a|e^{-{\frac {i\mathbb {H} t}{\hbar }}}|x=b\rangle =\int d[x(t)]e^{\frac {iS[x(t)]}{\hbar }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>;</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mi>t</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>=</mo> <mi>b</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>∫</mo> <mi>d</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi>S</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(a,b;t)=\langle x=a|e^{-{\frac {i\mathbb {H} t}{\hbar }}}|x=b\rangle =\int d[x(t)]e^{\frac {iS[x(t)]}{\hbar }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/445df67a4dea1212f753d5a7137b2ce057831992" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:51.02ex; height:6.009ex;" alt="{\displaystyle K(a,b;t)=\langle x=a|e^{-{\frac {i\mathbb {H} t}{\hbar }}}|x=b\rangle =\int d[x(t)]e^{\frac {iS[x(t)]}{\hbar }}.}"></span></dd></dl> <p>Following the process of <a href="/wiki/Wick_rotation" title="Wick rotation">Wick rotation</a> (analytic continuation) to Euclidean spacetime (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle it\rightarrow \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi>t</mi> <mo stretchy="false">→</mo> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle it\rightarrow \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/277e15dd6aedc0207debc1926ede372b1c17213d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.458ex; height:2.176ex;" alt="{\displaystyle it\rightarrow \tau }"></span>), one gets </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{E}(a,b;\tau )=\langle x=a|e^{-{\frac {\mathbb {H} \tau }{\hbar }}}|x=b\rangle =\int d[x(\tau )]e^{-{\frac {S_{E}[x(\tau )]}{\hbar }}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>;</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mi>τ</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>=</mo> <mi>b</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>∫</mo> <mi>d</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{E}(a,b;\tau )=\langle x=a|e^{-{\frac {\mathbb {H} \tau }{\hbar }}}|x=b\rangle =\int d[x(\tau )]e^{-{\frac {S_{E}[x(\tau )]}{\hbar }}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dacb66c25d8be36c7b74a4a2cd1eedb33b3a104" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:55.023ex; height:6.176ex;" alt="{\displaystyle K_{E}(a,b;\tau )=\langle x=a|e^{-{\frac {\mathbb {H} \tau }{\hbar }}}|x=b\rangle =\int d[x(\tau )]e^{-{\frac {S_{E}[x(\tau )]}{\hbar }}},}"></span></dd></dl> <p>with the Euclidean action </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{E}=\int _{\tau _{a}}^{\tau _{b}}\left({\frac {1}{2}}m\left({\frac {dx}{d\tau }}\right)^{2}+V(x)\right)d\tau .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>τ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{E}=\int _{\tau _{a}}^{\tau _{b}}\left({\frac {1}{2}}m\left({\frac {dx}{d\tau }}\right)^{2}+V(x)\right)d\tau .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfa39c915727823253b31a5aab24c142eb8f64f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.346ex; height:7.509ex;" alt="{\displaystyle S_{E}=\int _{\tau _{a}}^{\tau _{b}}\left({\frac {1}{2}}m\left({\frac {dx}{d\tau }}\right)^{2}+V(x)\right)d\tau .}"></span></dd></dl> <p>The potential energy changes sign <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x)\rightarrow -V(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(x)\rightarrow -V(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/349bab255e0b3db6a86fa91d6d8e192c7cb16fca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.275ex; height:2.843ex;" alt="{\displaystyle V(x)\rightarrow -V(x)}"></span> under the Wick rotation and the minima transform into maxima, thereby <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ab3e825c2bf9c80d11d12e070a4626d48e03c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.926ex; height:2.843ex;" alt="{\displaystyle V(x)}"></span> exhibits two "hills" of maximal energy. </p><p>Let us now consider the local minimum of the Euclidean action <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{E}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{E}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2b2456cdf534f97d00648dcfc145a7f74aa11d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.913ex; height:2.509ex;" alt="{\displaystyle S_{E}}"></span> with the double-well potential <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x)={1 \over 4}(x^{2}-1)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(x)={1 \over 4}(x^{2}-1)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5c3cdbbecbf2950e930bfc6e5560e5b51d93177" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.274ex; height:5.176ex;" alt="{\displaystyle V(x)={1 \over 4}(x^{2}-1)^{2}}"></span>, and we set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6100c5ebd48c6fd848709f2be624465203eb173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=1}"></span> just for simplicity of computation. Since we want to know how the two classically lowest energy states <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\pm 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae75026eda1ba1ac961f5003150288e9d2f3049b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.399ex; height:2.176ex;" alt="{\displaystyle x=\pm 1}"></span> are connected, let us set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/231103d8099e125875dd690668e93a56aa10bd99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.299ex; height:2.343ex;" alt="{\displaystyle a=-1}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f55bc77dec8088791b5c1ed51e634cc1b431fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b=1}"></span>. For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/231103d8099e125875dd690668e93a56aa10bd99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.299ex; height:2.343ex;" alt="{\displaystyle a=-1}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f55bc77dec8088791b5c1ed51e634cc1b431fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b=1}"></span>, we can rewrite the Euclidean action as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{E}=\int _{\tau _{a}}^{\tau _{b}}d\tau {1 \over 2}\left({dx \over d\tau }-{\sqrt {2V(x)}}\right)^{2}+{\sqrt {2}}\int _{\tau _{a}}^{\tau _{b}}d\tau {dx \over d\tau }{\sqrt {V(x)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> </msubsup> <mi>d</mi> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> </msubsup> <mi>d</mi> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{E}=\int _{\tau _{a}}^{\tau _{b}}d\tau {1 \over 2}\left({dx \over d\tau }-{\sqrt {2V(x)}}\right)^{2}+{\sqrt {2}}\int _{\tau _{a}}^{\tau _{b}}d\tau {dx \over d\tau }{\sqrt {V(x)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f64f7aa6e73ab95aa90fd24cd2cbe1cff996876" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:57.991ex; height:6.676ex;" alt="{\displaystyle S_{E}=\int _{\tau _{a}}^{\tau _{b}}d\tau {1 \over 2}\left({dx \over d\tau }-{\sqrt {2V(x)}}\right)^{2}+{\sqrt {2}}\int _{\tau _{a}}^{\tau _{b}}d\tau {dx \over d\tau }{\sqrt {V(x)}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad =\int _{\tau _{a}}^{\tau _{b}}d\tau {1 \over 2}\left({dx \over d\tau }-{\sqrt {2V(x)}}\right)^{2}+\int _{-1}^{1}dx{1 \over {\sqrt {2}}}(1-x^{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> </msubsup> <mi>d</mi> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mi>d</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad =\int _{\tau _{a}}^{\tau _{b}}d\tau {1 \over 2}\left({dx \over d\tau }-{\sqrt {2V(x)}}\right)^{2}+\int _{-1}^{1}dx{1 \over {\sqrt {2}}}(1-x^{2}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a24ce6d95eae49d837bb07170902db88d308fdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:55.017ex; height:6.843ex;" alt="{\displaystyle \quad =\int _{\tau _{a}}^{\tau _{b}}d\tau {1 \over 2}\left({dx \over d\tau }-{\sqrt {2V(x)}}\right)^{2}+\int _{-1}^{1}dx{1 \over {\sqrt {2}}}(1-x^{2}).}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \geq {2{\sqrt {2}} \over 3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \geq {2{\sqrt {2}} \over 3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c73ef4e0e133c9ae87cd03f4e48218619a2d082a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.52ex; height:5.843ex;" alt="{\displaystyle \quad \geq {2{\sqrt {2}} \over 3}.}"></span></dd></dl> <p>The above inequality is saturated by the solution of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {dx \over d\tau }={\sqrt {2V(x)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {dx \over d\tau }={\sqrt {2V(x)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e564ed0584d771884b94937f85c9d36be186486" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.893ex; height:5.509ex;" alt="{\displaystyle {dx \over d\tau }={\sqrt {2V(x)}}}"></span> with the condition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(\tau _{a})=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(\tau _{a})=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f561eabdff948e208fe9745a410b89af1e2edcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.326ex; height:2.843ex;" alt="{\displaystyle x(\tau _{a})=-1}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(\tau _{b})=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(\tau _{b})=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30ce93d75051319c98d8cf8e8ceecd72550dd687" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.354ex; height:2.843ex;" alt="{\displaystyle x(\tau _{b})=1}"></span>. Such solutions exist, and the solution takes the simple form when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{a}=-\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{a}=-\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70051f520f53cf75eb176f337aff634debb64ab3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.348ex; height:2.343ex;" alt="{\displaystyle \tau _{a}=-\infty }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{b}=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{b}=\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f97255dc35eae1430c265f06b6d20abcea449637" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.376ex; height:2.009ex;" alt="{\displaystyle \tau _{b}=\infty }"></span>. The explicit formula for the instanton solution is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(\tau )=\tanh \left({1 \over {\sqrt {2}}}(\tau -\tau _{0})\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tanh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(\tau )=\tanh \left({1 \over {\sqrt {2}}}(\tau -\tau _{0})\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeb6f82f836ef40ab9da44ccf2b582127ac2544a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:28.016ex; height:6.509ex;" alt="{\displaystyle x(\tau )=\tanh \left({1 \over {\sqrt {2}}}(\tau -\tau _{0})\right).}"></span></dd></dl> <p>Here <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56dad457e274b970f5d98b9dc40bef7f895c7f6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.07ex; height:2.009ex;" alt="{\displaystyle \tau _{0}}"></span> is an arbitrary constant. Since this solution jumps from one classical vacuum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fefa55268918f98da2e0dcc19ea86d78f84ac56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.399ex; height:2.343ex;" alt="{\displaystyle x=-1}"></span> to another classical vacuum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee42176e76ae6b56d68c42ced807e08b962a2b54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=1}"></span> instantaneously around <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =\tau _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau =\tau _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/657474fd98e04748af121852a656305536fd4944" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.371ex; height:2.009ex;" alt="{\displaystyle \tau =\tau _{0}}"></span>, it is called an instanton. </p> <div class="mw-heading mw-heading3"><h3 id="Explicit_formula_for_double-well_potential">Explicit formula for double-well potential</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Instanton&action=edit&section=6" title="Edit section: Explicit formula for double-well potential"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The explicit formula for the eigenenergies of the Schrödinger equation with <a href="/wiki/Double-well_potential" title="Double-well potential">double-well potential</a> has been given by Müller–Kirsten<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> with derivation by both a perturbation method (plus boundary conditions) applied to the Schrödinger equation, and explicit derivation from the path integral (and WKB). The result is the following. Defining parameters of the Schrödinger equation and the potential by the equations </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{2}y(z)}{dz^{2}}}+[E-V(z)]y(z)=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mo stretchy="false">[</mo> <mi>E</mi> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}y(z)}{dz^{2}}}+[E-V(z)]y(z)=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79f7f61dbb1a8b0d748bd8339cfb8e53e90b4b52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:29.557ex; height:6.176ex;" alt="{\displaystyle {\frac {d^{2}y(z)}{dz^{2}}}+[E-V(z)]y(z)=0,}"></span></dd></dl> <p>and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(z)=-{\frac {1}{4}}z^{2}h^{4}+{\frac {1}{2}}c^{2}z^{4},\;\;\;c^{2}>0,\;h^{4}>0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>></mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(z)=-{\frac {1}{4}}z^{2}h^{4}+{\frac {1}{2}}c^{2}z^{4},\;\;\;c^{2}>0,\;h^{4}>0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e63c81517b071ab25a719ed0f0446d53c844a11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:43.444ex; height:5.176ex;" alt="{\displaystyle V(z)=-{\frac {1}{4}}z^{2}h^{4}+{\frac {1}{2}}c^{2}z^{4},\;\;\;c^{2}>0,\;h^{4}>0,}"></span></dd></dl> <p>the eigenvalues for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{0}=1,3,5,...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{0}=1,3,5,...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73157d0bc26018ef1dd0709bb87836a1a667a0d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.494ex; height:2.509ex;" alt="{\displaystyle q_{0}=1,3,5,...}"></span> are found to be: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\pm }(q_{0},h^{2})=-{\frac {h^{8}}{2^{5}c^{2}}}+{\frac {1}{\sqrt {2}}}q_{0}h^{2}-{\frac {c^{2}(3q_{0}^{2}+1)}{2h^{4}}}-{\frac {{\sqrt {2}}c^{4}q_{0}}{8h^{10}}}(17q_{0}^{2}+19)+O({\frac {1}{h^{16}}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>3</mn> <msubsup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mn>8</mn> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>17</mn> <msubsup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>19</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> </mfrac> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\pm }(q_{0},h^{2})=-{\frac {h^{8}}{2^{5}c^{2}}}+{\frac {1}{\sqrt {2}}}q_{0}h^{2}-{\frac {c^{2}(3q_{0}^{2}+1)}{2h^{4}}}-{\frac {{\sqrt {2}}c^{4}q_{0}}{8h^{10}}}(17q_{0}^{2}+19)+O({\frac {1}{h^{16}}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b1cfc0b877a51c334506fa000a0d18f5745d58b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:79.514ex; height:7.009ex;" alt="{\displaystyle E_{\pm }(q_{0},h^{2})=-{\frac {h^{8}}{2^{5}c^{2}}}+{\frac {1}{\sqrt {2}}}q_{0}h^{2}-{\frac {c^{2}(3q_{0}^{2}+1)}{2h^{4}}}-{\frac {{\sqrt {2}}c^{4}q_{0}}{8h^{10}}}(17q_{0}^{2}+19)+O({\frac {1}{h^{16}}})}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mp {\frac {2^{q_{0}+1}h^{2}(h^{6}/2c^{2})^{q_{0}/2}}{{\sqrt {\pi }}2^{q_{0}/4}[(q_{0}-1)/2]!}}e^{-h^{6}/6{\sqrt {2}}c^{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∓</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">]</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mp {\frac {2^{q_{0}+1}h^{2}(h^{6}/2c^{2})^{q_{0}/2}}{{\sqrt {\pi }}2^{q_{0}/4}[(q_{0}-1)/2]!}}e^{-h^{6}/6{\sqrt {2}}c^{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5afc9d83c9e870a9edf97a7e7ff501c6f27e76d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.082ex; height:7.343ex;" alt="{\displaystyle \mp {\frac {2^{q_{0}+1}h^{2}(h^{6}/2c^{2})^{q_{0}/2}}{{\sqrt {\pi }}2^{q_{0}/4}[(q_{0}-1)/2]!}}e^{-h^{6}/6{\sqrt {2}}c^{2}}.}"></span></dd></dl> <p>Clearly these eigenvalues are asymptotically (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h^{2}\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h^{2}\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08db9b9712c39b89a734217c38c43a71417d8475" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.331ex; height:2.676ex;" alt="{\displaystyle h^{2}\rightarrow \infty }"></span>) degenerate as expected as a consequence of the harmonic part of the potential. </p> <div class="mw-heading mw-heading3"><h3 id="Results">Results</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Instanton&action=edit&section=7" title="Edit section: Results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Results obtained from the mathematically well-defined Euclidean <a href="/wiki/Line_integral" title="Line integral">path integral</a> may be Wick-rotated back and give the same physical results as would be obtained by appropriate treatment of the (potentially divergent) Minkowskian path integral. As can be seen from this example, calculating the transition probability for the particle to tunnel through a classically forbidden region (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ab3e825c2bf9c80d11d12e070a4626d48e03c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.926ex; height:2.843ex;" alt="{\displaystyle V(x)}"></span>) with the Minkowskian path integral corresponds to calculating the transition probability to tunnel through a classically allowed region (with potential −<i>V</i>(<i>X</i>)) in the Euclidean path integral (pictorially speaking – in the Euclidean picture – this transition corresponds to a particle rolling from one hill of a double-well potential standing on its head to the other hill). This classical solution of the Euclidean equations of motion is often named "kink solution" and is an example of an <i>instanton</i>. In this example, the two "vacua" (i.e. ground states) of the <a href="/wiki/Double-well_potential" title="Double-well potential">double-well potential</a>, turn into hills in the Euclideanized version of the problem. </p><p>Thus, the <i>instanton</i> field solution of the (Euclidean, i. e., with imaginary time) (1 + 1)-dimensional field theory – first quantized quantum mechanical description – allows to be interpreted as a tunneling effect between the two vacua (ground states – higher states require periodic instantons) of the physical (1-dimensional space + real time) Minkowskian system. In the case of the double-well potential written </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(\phi )={\frac {m^{4}}{2g^{2}}}\left(1-{\frac {g^{2}\phi ^{2}}{m^{2}}}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>2</mn> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(\phi )={\frac {m^{4}}{2g^{2}}}\left(1-{\frac {g^{2}\phi ^{2}}{m^{2}}}\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5511f3ad4accb09189655e8c916afc36eeca6073" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.178ex; height:6.843ex;" alt="{\displaystyle V(\phi )={\frac {m^{4}}{2g^{2}}}\left(1-{\frac {g^{2}\phi ^{2}}{m^{2}}}\right)^{2}}"></span></dd></dl> <p>the instanton, i.e. solution of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{2}\phi }{d\tau ^{2}}}=V'(\phi ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ϕ</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>V</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}\phi }{d\tau ^{2}}}=V'(\phi ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/478270e43ba5fe5cdc39cabe9a7a917d26bed54d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.036ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}\phi }{d\tau ^{2}}}=V'(\phi ),}"></span></dd></dl> <p>(i.e. with energy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{cl}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>l</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{cl}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cefb4a05d15394bff2e377d80760e965514ceef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.411ex; height:2.509ex;" alt="{\displaystyle E_{cl}=0}"></span>), is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{c}(\tau )={\frac {m}{g}}\tanh \left[m(\tau -\tau _{0})\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>g</mi> </mfrac> </mrow> <mi>tanh</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mi>m</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{c}(\tau )={\frac {m}{g}}\tanh \left[m(\tau -\tau _{0})\right],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5535c7b6fe1007f2ad79974993540d93939d4bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.258ex; height:5.176ex;" alt="{\displaystyle \phi _{c}(\tau )={\frac {m}{g}}\tanh \left[m(\tau -\tau _{0})\right],}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =it}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mi>i</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau =it}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11688c247c390010802ff9dfd85cd239334669b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.942ex; height:2.176ex;" alt="{\displaystyle \tau =it}"></span> is the Euclidean time. </p><p><i>Note</i> that a naïve perturbation theory around one of those two vacua alone (of the Minkowskian description) would never show this <i>non-perturbative tunneling effect</i>, dramatically changing the picture of the vacuum structure of this quantum mechanical system. In fact the naive perturbation theory has to be supplemented by boundary conditions, and these supply the nonperturbative effect, as is evident from the above explicit formula and analogous calculations for other potentials such as a cosine potential (cf. <a href="/wiki/Mathieu_function" title="Mathieu function">Mathieu function</a>) or other periodic potentials (cf. e.g. <a href="/wiki/Lam%C3%A9_function" title="Lamé function">Lamé function</a> and <a href="/wiki/Spheroidal_wave_function" title="Spheroidal wave function">spheroidal wave function</a>) and irrespective of whether one uses the Schrödinger equation or the <a href="/wiki/Functional_integration" title="Functional integration">path integral</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Therefore, the perturbative approach may not completely describe the vacuum structure of a physical system. This may have important consequences, for example, in the theory of <a href="/wiki/Axion" title="Axion">"axions"</a> where the non-trivial QCD vacuum effects (like the <i>instantons</i>) spoil the <a href="/wiki/Peccei%E2%80%93Quinn_theory" title="Peccei–Quinn theory">Peccei–Quinn symmetry</a> explicitly and transform massless <a href="/wiki/Nambu%E2%80%93Goldstone_boson" class="mw-redirect" title="Nambu–Goldstone boson">Nambu–Goldstone bosons</a> into massive <a href="/wiki/Chiral_symmetry_breaking" title="Chiral symmetry breaking">pseudo-Nambu–Goldstone ones</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Periodic_instantons">Periodic instantons</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Instanton&action=edit&section=8" title="Edit section: Periodic instantons"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Periodic_instantons" title="Periodic instantons">Periodic instantons</a></div> <p>In one-dimensional field theory or quantum mechanics one defines as "instanton" a field configuration which is a solution of the classical (Newton-like) equation of motion with Euclidean time and finite Euclidean action. In the context of <a href="/wiki/Soliton" title="Soliton">soliton</a> theory the corresponding solution is known as a <a href="/wiki/Sine-Gordon_equation#Soliton_solutions" title="Sine-Gordon equation">kink</a>. In view of their analogy with the behaviour of classical particles such configurations or solutions, as well as others, are collectively known as <a href="/wiki/Pseudoparticles" class="mw-redirect" title="Pseudoparticles">pseudoparticles</a> or pseudoclassical configurations. The "instanton" (kink) solution is accompanied by another solution known as "anti-instanton" (anti-kink), and instanton and anti-instanton are distinguished by "topological charges" +1 and −1 respectively, but have the same Euclidean action. </p><p>"Periodic instantons" are a generalization of instantons.<sup id="cite_ref-Harald_J.W._Müller-Kirsten_2012_9-0" class="reference"><a href="#cite_note-Harald_J.W._Müller-Kirsten_2012-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> In explicit form they are expressible in terms of <a href="/wiki/Jacobian_elliptic_functions" class="mw-redirect" title="Jacobian elliptic functions">Jacobian elliptic functions</a> which are periodic functions (effectively generalisations of trigonometrical functions). In the limit of infinite period these periodic instantons – frequently known as "bounces", "bubbles" or the like – reduce to instantons. </p><p>The stability of these pseudoclassical configurations can be investigated by expanding the Lagrangian defining the theory around the pseudoparticle configuration and then investigating the equation of small fluctuations around it. For all versions of quartic potentials (double-well, inverted double-well) and periodic (Mathieu) potentials these equations were discovered to be Lamé equations, see <a href="/wiki/Lam%C3%A9_function" title="Lamé function">Lamé function</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> The eigenvalues of these equations are known and permit in the case of instability the calculation of decay rates by evaluation of the path integral.<sup id="cite_ref-Harald_J.W._Müller-Kirsten_2012_9-1" class="reference"><a href="#cite_note-Harald_J.W._Müller-Kirsten_2012-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Instantons_in_reaction_rate_theory">Instantons in reaction rate theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Instanton&action=edit&section=9" title="Edit section: Instantons in reaction rate theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the context of reaction rate theory, periodic instantons are used to calculate the rate of tunneling of atoms in chemical reactions. The progress of a chemical reaction can be described as the movement of a pseudoparticle on a high dimensional <a href="/wiki/Potential_energy_surface" title="Potential energy surface">potential energy surface</a> (PES). The thermal rate constant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> can then be related to the imaginary part of the free energy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> by<sup id="cite_ref-:inst_chapter_11-0" class="reference"><a href="#cite_note-:inst_chapter-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k(\beta )=-{\frac {2}{\hbar }}{\text{Im}}\mathrm {F} ={\frac {2}{\beta \hbar }}{\text{Im}}\ {\text{ln}}(Z_{k})\approx {\frac {2}{\hbar \beta }}{\frac {{\text{Im}}Z_{k}}{{\text{Re}}Z_{k}}},\ \ {\text{Re}}Z_{k}\gg {\text{Im}}Z_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Im</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mi>β</mi> <mi class="MJX-variant">ℏ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Im</mtext> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>ln</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mi class="MJX-variant">ℏ</mi> <mi>β</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Im</mtext> </mrow> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Re</mtext> </mrow> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>Re</mtext> </mrow> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>≫</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Im</mtext> </mrow> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k(\beta )=-{\frac {2}{\hbar }}{\text{Im}}\mathrm {F} ={\frac {2}{\beta \hbar }}{\text{Im}}\ {\text{ln}}(Z_{k})\approx {\frac {2}{\hbar \beta }}{\frac {{\text{Im}}Z_{k}}{{\text{Re}}Z_{k}}},\ \ {\text{Re}}Z_{k}\gg {\text{Im}}Z_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7ee0726e6968254c28a483713e15013ab6fa19a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:61.59ex; height:5.676ex;" alt="{\displaystyle k(\beta )=-{\frac {2}{\hbar }}{\text{Im}}\mathrm {F} ={\frac {2}{\beta \hbar }}{\text{Im}}\ {\text{ln}}(Z_{k})\approx {\frac {2}{\hbar \beta }}{\frac {{\text{Im}}Z_{k}}{{\text{Re}}Z_{k}}},\ \ {\text{Re}}Z_{k}\gg {\text{Im}}Z_{k}}"></span> </p><p>whereby <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29a05237076c50ce9cf9a75c02ff57abefac0de4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.676ex; height:2.509ex;" alt="{\displaystyle Z_{k}}"></span> is the canonical partition function, which is calculated by taking the trace of the Boltzmann operator in the position representation. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{k}={\text{Tr}}(e^{-\beta {\hat {H}}})=\int d\mathbf {x} \left\langle \mathbf {x} \left|e^{-\beta {\hat {H}}}\right|\mathbf {x} \right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Tr</mtext> </mrow> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow> <mo>⟨</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow> <mo>|</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </msup> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{k}={\text{Tr}}(e^{-\beta {\hat {H}}})=\int d\mathbf {x} \left\langle \mathbf {x} \left|e^{-\beta {\hat {H}}}\right|\mathbf {x} \right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae7abfb71cb76bf6ab2ebff308c1a6671c1a4740" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.588ex; height:5.676ex;" alt="{\displaystyle Z_{k}={\text{Tr}}(e^{-\beta {\hat {H}}})=\int d\mathbf {x} \left\langle \mathbf {x} \left|e^{-\beta {\hat {H}}}\right|\mathbf {x} \right\rangle }"></span> </p><p>Using a Wick rotation and identifying the Euclidean time with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar \beta =1/(k_{b}T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mi>β</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar \beta =1/(k_{b}T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/619aaf5c3e50ffe3354cf7c7498a860c3aec4580" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.656ex; height:2.843ex;" alt="{\displaystyle \hbar \beta =1/(k_{b}T)}"></span>, one obtains a path integral representation for the partition function in mass-weighted coordinates:<sup id="cite_ref-:inst_review_12-0" class="reference"><a href="#cite_note-:inst_review-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{k}=\oint {\mathcal {D}}\mathbf {x} (\tau )e^{-S_{E}[\mathbf {x} (\tau )]/\hbar },\ \ \ S_{E}=\int _{0}^{\beta \hbar }\left({\frac {\dot {\mathbf {x} }}{2}}^{2}+V(\mathbf {x} (\tau ))\right)d\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi class="MJX-variant">ℏ</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>˙</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{k}=\oint {\mathcal {D}}\mathbf {x} (\tau )e^{-S_{E}[\mathbf {x} (\tau )]/\hbar },\ \ \ S_{E}=\int _{0}^{\beta \hbar }\left({\frac {\dot {\mathbf {x} }}{2}}^{2}+V(\mathbf {x} (\tau ))\right)d\tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/320bd30e74b81500988d366e65f36a6866b179f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:59.265ex; height:6.509ex;" alt="{\displaystyle Z_{k}=\oint {\mathcal {D}}\mathbf {x} (\tau )e^{-S_{E}[\mathbf {x} (\tau )]/\hbar },\ \ \ S_{E}=\int _{0}^{\beta \hbar }\left({\frac {\dot {\mathbf {x} }}{2}}^{2}+V(\mathbf {x} (\tau ))\right)d\tau }"></span> </p><p>The path integral is then approximated via a steepest descent integration, which takes into account only the contributions from the classical solutions and quadratic fluctuations around them. This yields for the rate constant expression in mass-weighted coordinates </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k(\beta )={\frac {2}{\beta \hbar }}\left({\frac {{\text{det}}\left[-{\frac {\partial ^{2}}{\partial \tau ^{2}}}+\mathbf {V} ''(x_{\text{RS}}(\tau ))\right]}{{\text{det}}\left[-{\frac {\partial ^{2}}{\partial \tau ^{2}}}+\mathbf {V} ''(x_{\text{Inst}}(\tau ))\right]}}\right)^{\frac {1}{2}}{\exp \left({\frac {-S_{E}[x_{\text{inst}}(\tau )+S_{E}[x_{\text{RS}}(\tau )]}{\hbar }}\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mi>β</mi> <mi class="MJX-variant">ℏ</mi> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>det</mtext> </mrow> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>RS</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>det</mtext> </mrow> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>Inst</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>inst</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>RS</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k(\beta )={\frac {2}{\beta \hbar }}\left({\frac {{\text{det}}\left[-{\frac {\partial ^{2}}{\partial \tau ^{2}}}+\mathbf {V} ''(x_{\text{RS}}(\tau ))\right]}{{\text{det}}\left[-{\frac {\partial ^{2}}{\partial \tau ^{2}}}+\mathbf {V} ''(x_{\text{Inst}}(\tau ))\right]}}\right)^{\frac {1}{2}}{\exp \left({\frac {-S_{E}[x_{\text{inst}}(\tau )+S_{E}[x_{\text{RS}}(\tau )]}{\hbar }}\right)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b278362ef03f52e564b9e4f0ecc5e4150467f06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:78.001ex; height:11.343ex;" alt="{\displaystyle k(\beta )={\frac {2}{\beta \hbar }}\left({\frac {{\text{det}}\left[-{\frac {\partial ^{2}}{\partial \tau ^{2}}}+\mathbf {V} ''(x_{\text{RS}}(\tau ))\right]}{{\text{det}}\left[-{\frac {\partial ^{2}}{\partial \tau ^{2}}}+\mathbf {V} ''(x_{\text{Inst}}(\tau ))\right]}}\right)^{\frac {1}{2}}{\exp \left({\frac {-S_{E}[x_{\text{inst}}(\tau )+S_{E}[x_{\text{RS}}(\tau )]}{\hbar }}\right)}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{\text{Inst}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Inst</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{\text{Inst}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ff8a108999cbecfe5bd31ea5b2282a22b6180a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.438ex; height:2.009ex;" alt="{\displaystyle \mathbf {x} _{\text{Inst}}}"></span> is a periodic instanton and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{\text{RS}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>RS</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{\text{RS}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6c2a334714ffbb87f2e1a08720ad3c70ecdcbc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.767ex; height:2.009ex;" alt="{\displaystyle \mathbf {x} _{\text{RS}}}"></span> is the trivial solution of the pseudoparticle at rest which represents the reactant state configuration. </p> <div class="mw-heading mw-heading3"><h3 id="Inverted_double-well_formula">Inverted double-well formula</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Instanton&action=edit&section=10" title="Edit section: Inverted double-well formula"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As for the double-well potential one can derive the eigenvalues for the inverted double-well potential. In this case, however, the eigenvalues are complex. Defining parameters by the equations </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{2}y}{dz^{2}}}+[E-V(z)]y(z)=0,\;\;\;V(z)={\frac {1}{4}}h^{4}z^{2}-{\frac {1}{2}}c^{2}z^{4},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mo stretchy="false">[</mo> <mi>E</mi> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>V</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}y}{dz^{2}}}+[E-V(z)]y(z)=0,\;\;\;V(z)={\frac {1}{4}}h^{4}z^{2}-{\frac {1}{2}}c^{2}z^{4},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/332765930b5f62fd956e0144d2081ee2617209bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:52.993ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}y}{dz^{2}}}+[E-V(z)]y(z)=0,\;\;\;V(z)={\frac {1}{4}}h^{4}z^{2}-{\frac {1}{2}}c^{2}z^{4},}"></span></dd></dl> <p>the eigenvalues as given by Müller-Kirsten are, for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{0}=1,3,5,...,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{0}=1,3,5,...,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e9fd0333494e2a8e5b1d2718f2404b75f99b795" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.528ex; height:2.509ex;" alt="{\displaystyle q_{0}=1,3,5,...,}"></span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\frac {1}{2}}q_{0}h^{2}-{\frac {3c^{2}}{4h^{4}}}(q_{0}^{2}+1)-{\frac {q_{0}c^{4}}{h^{10}}}(4q_{0}^{2}+29)+O({\frac {1}{h^{16}}})\pm i{\frac {2^{q_{0}}h^{2}(h^{6}/2c^{2})^{q_{0}/2}}{(2\pi )^{1/2}[(q_{0}-1)/2]!}}e^{-h^{6}/6c^{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>4</mn> <msubsup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>29</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>±</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msup> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">]</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\frac {1}{2}}q_{0}h^{2}-{\frac {3c^{2}}{4h^{4}}}(q_{0}^{2}+1)-{\frac {q_{0}c^{4}}{h^{10}}}(4q_{0}^{2}+29)+O({\frac {1}{h^{16}}})\pm i{\frac {2^{q_{0}}h^{2}(h^{6}/2c^{2})^{q_{0}/2}}{(2\pi )^{1/2}[(q_{0}-1)/2]!}}e^{-h^{6}/6c^{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e416151fa2f50ce104325672d8225ae2793bb80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:86.966ex; height:7.009ex;" alt="{\displaystyle E={\frac {1}{2}}q_{0}h^{2}-{\frac {3c^{2}}{4h^{4}}}(q_{0}^{2}+1)-{\frac {q_{0}c^{4}}{h^{10}}}(4q_{0}^{2}+29)+O({\frac {1}{h^{16}}})\pm i{\frac {2^{q_{0}}h^{2}(h^{6}/2c^{2})^{q_{0}/2}}{(2\pi )^{1/2}[(q_{0}-1)/2]!}}e^{-h^{6}/6c^{2}}.}"></span></dd></dl> <p>The imaginary part of this expression agrees with the well known result of Bender and Wu.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> In their notation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar =1,q_{0}=2K+1,h^{6}/2c^{2}=\epsilon .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>K</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>ϵ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar =1,q_{0}=2K+1,h^{6}/2c^{2}=\epsilon .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18e9625e802231ad3a80018fd0a94153b7c54bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.525ex; height:3.176ex;" alt="{\displaystyle \hbar =1,q_{0}=2K+1,h^{6}/2c^{2}=\epsilon .}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Quantum_field_theory">Quantum field theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Instanton&action=edit&section=11" title="Edit section: Quantum field theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table align="right" width="250px"> <tbody><tr> <th style="color:#black; background:#dddddd; font-size:100%; text-align:center;" colspan="2">Hypersphere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01e57c690f890937838c10ba57853ff21bf30ec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{3}}"></span> </th></tr> <tr> <td> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Hypersphere_coord.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Hypersphere_coord.PNG/250px-Hypersphere_coord.PNG" decoding="async" width="250" height="332" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/3/32/Hypersphere_coord.PNG 1.5x" data-file-width="320" data-file-height="425" /></a><figcaption>Hypersphere <a href="/wiki/Stereographic_projection" title="Stereographic projection">Stereographic projection</a> <hr /> Parallels (red), <a href="/wiki/Meridian_(perimetry,_visual_field)" title="Meridian (perimetry, visual field)">meridians</a> (blue) and hypermeridians (green).<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup></figcaption></figure> </td></tr></tbody></table> <p>In studying <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> (QFT), the vacuum structure of a theory may draw attention to instantons. Just as a double-well quantum mechanical system illustrates, a naïve vacuum may not be the true vacuum of a field theory. Moreover, the true vacuum of a field theory may be an "overlap" of several topologically inequivalent sectors, so called "<a href="/wiki/Topological" class="mw-redirect" title="Topological">topological</a> <a href="/wiki/Vacua" class="mw-redirect" title="Vacua">vacua</a>". </p><p>A well understood and illustrative example of an <i>instanton</i> and its interpretation can be found in the context of a QFT with a <a href="/wiki/Non-abelian_group" title="Non-abelian group">non-abelian gauge group</a>,<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>note 2<span class="cite-bracket">]</span></a></sup> a <a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills theory</a>. For a Yang–Mills theory these inequivalent sectors can be (in an appropriate gauge) classified by the third <a href="/wiki/Homotopy_group" title="Homotopy group">homotopy group</a> of <a href="/wiki/SU(2)" class="mw-redirect" title="SU(2)">SU(2)</a> (whose group manifold is the <a href="/wiki/3-sphere" title="3-sphere">3-sphere</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01e57c690f890937838c10ba57853ff21bf30ec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{3}}"></span>). A certain topological vacuum (a "sector" of the true vacuum) is labelled by an <a href="/wiki/Topological_invariant" class="mw-redirect" title="Topological invariant">unaltered transform</a>, the <a href="/wiki/Pontryagin_index" class="mw-redirect" title="Pontryagin index">Pontryagin index</a>. As the third homotopy group of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01e57c690f890937838c10ba57853ff21bf30ec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{3}}"></span> has been found to be the set of <a href="/wiki/Integer" title="Integer">integers</a>, </p> <dl><dd><a href="/wiki/Homotopy_group" title="Homotopy group"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4a078131a0c973b388cf10c0ac6d7598fe21578" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.379ex; height:2.009ex;" alt="{\displaystyle \pi _{3}}"></span></a><a href="/wiki/3-sphere" title="3-sphere"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S^{3})=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S^{3})=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f5c0c479286ed52bd66553654bfb124247e2939" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.838ex; height:3.176ex;" alt="{\displaystyle (S^{3})=}"></span></a><a href="/wiki/Integer" title="Integer"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/140fa8f1431a045315775b65ff511be09fd0698c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.937ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} \,}"></span></a></dd></dl> <p>there are infinitely many topologically inequivalent vacua, denoted by <a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a9067e980b685ba4df6708890bd5d7af98ea40d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.968ex; height:2.843ex;" alt="{\displaystyle N\rangle }"></span></a>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> is their corresponding Pontryagin index. An <i>instanton</i> is a field configuration fulfilling the classical equations of motion in Euclidean spacetime, which is interpreted as a tunneling effect between these different topological vacua. It is again labelled by an integer number, its Pontryagin index, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span>. One can imagine an <i>instanton</i> with index <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> to quantify tunneling between topological vacua <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |N\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>N</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |N\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2166be15e80d454ae0c47209fcac7180aef3ede3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.615ex; height:2.843ex;" alt="{\displaystyle |N\rangle }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |N+Q\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>N</mi> <mo>+</mo> <mi>Q</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |N+Q\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0b6780126d41f4ebf1ede10da1819599f46b7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.294ex; height:2.843ex;" alt="{\displaystyle |N+Q\rangle }"></span>. If <i>Q</i> = 1, the configuration is named <a href="/wiki/BPST_instanton" title="BPST instanton">BPST instanton</a> after its discoverers <a href="/wiki/Alexander_Belavin" title="Alexander Belavin">Alexander Belavin</a>, <a href="/wiki/Alexander_Markovich_Polyakov" title="Alexander Markovich Polyakov">Alexander Polyakov</a>, <a href="/wiki/Albert_S._Schwarz" class="mw-redirect" title="Albert S. Schwarz">Albert S. Schwarz</a> and <a href="/w/index.php?title=Yu._S._Tyupkin&action=edit&redlink=1" class="new" title="Yu. S. Tyupkin (page does not exist)">Yu. S. Tyupkin</a>. The true vacuum of the theory is labelled by an "angle" theta and is an overlap of the topological sectors: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\theta \rangle =\sum _{N=-\infty }^{N=+\infty }e^{i\theta N}|N\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>θ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>=</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> <mi>N</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>N</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\theta \rangle =\sum _{N=-\infty }^{N=+\infty }e^{i\theta N}|N\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3496e54d42475d368246bbf5da57c41b2925b238" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.162ex; height:7.509ex;" alt="{\displaystyle |\theta \rangle =\sum _{N=-\infty }^{N=+\infty }e^{i\theta N}|N\rangle .}"></span></dd></dl> <p><a href="/wiki/Gerardus_%27t_Hooft" class="mw-redirect" title="Gerardus 't Hooft">Gerard 't Hooft</a> first performed the field theoretic computation of the effects of the BPST instanton in a theory coupled to fermions in <a rel="nofollow" class="external autonumber" href="http://inspirehep.net/search?p=PHRVA,D14,3432">[1]</a>. He showed that zero modes of the Dirac equation in the instanton background lead to a non-perturbative multi-fermion interaction in the low energy effective action. </p> <div class="mw-heading mw-heading2"><h2 id="Yang–Mills_theory"><span id="Yang.E2.80.93Mills_theory"></span>Yang–Mills theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Instanton&action=edit&section=12" title="Edit section: Yang–Mills theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The classical Yang–Mills action on a <a href="/wiki/Principal_bundle" title="Principal bundle">principal bundle</a> with structure group <i>G</i>, base <i>M</i>, <a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">connection</a> <i>A</i>, and <a href="/wiki/Curvature" title="Curvature">curvature</a> (Yang–Mills field tensor) <i>F</i> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{YM}=\int _{M}\left|F\right|^{2}d\mathrm {vol} _{M},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>M</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <msup> <mrow> <mo>|</mo> <mi>F</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{YM}=\int _{M}\left|F\right|^{2}d\mathrm {vol} _{M},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c90fa201267fd3f7597e75e82ea13f797a5a7c5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.323ex; height:5.676ex;" alt="{\displaystyle S_{YM}=\int _{M}\left|F\right|^{2}d\mathrm {vol} _{M},}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\mathrm {vol} _{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\mathrm {vol} _{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0408c2b2ea95505c407c0c8849d0a7a7fbc95909" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.212ex; height:2.509ex;" alt="{\displaystyle d\mathrm {vol} _{M}}"></span> is the <a href="/wiki/Volume_form" title="Volume form">volume form</a> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>. If the inner product on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}"></span>, the <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> in which <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> takes values, is given by the <a href="/wiki/Killing_form" title="Killing form">Killing form</a> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}"></span>, then this may be denoted as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{M}\mathrm {Tr} (F\wedge *F)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> <mi mathvariant="normal">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>F</mi> <mo>∧</mo> <mo>∗</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{M}\mathrm {Tr} (F\wedge *F)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b977d368740dd2f99c02a7adec231475bde81881" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.264ex; height:5.676ex;" alt="{\displaystyle \int _{M}\mathrm {Tr} (F\wedge *F)}"></span>, since </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\wedge *F=\langle F,F\rangle d\mathrm {vol} _{M}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>∧</mo> <mo>∗</mo> <mi>F</mi> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>F</mi> <mo>,</mo> <mi>F</mi> <mo fence="false" stretchy="false">⟩</mo> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\wedge *F=\langle F,F\rangle d\mathrm {vol} _{M}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5da8bf68485a1e5c72e5533e418e9791fda690c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.509ex; height:2.843ex;" alt="{\displaystyle F\wedge *F=\langle F,F\rangle d\mathrm {vol} _{M}.}"></span></dd></dl> <p>For example, in the case of the <a href="/wiki/Gauge_group" class="mw-redirect" title="Gauge group">gauge group</a> <a href="/wiki/U(1)" class="mw-redirect" title="U(1)">U(1)</a>, <i>F</i> will be the electromagnetic field <a href="/wiki/Tensor" title="Tensor">tensor</a>. From the <a href="/wiki/Action_(physics)" title="Action (physics)">principle of stationary action</a>, the Yang–Mills equations follow. They are </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} F=0,\quad \mathrm {d} {*F}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>F</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mi>F</mi> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} F=0,\quad \mathrm {d} {*F}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5e2bc61aa238d9c1b32c1d6ee8ce6e50292dc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.754ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} F=0,\quad \mathrm {d} {*F}=0.}"></span></dd></dl> <p>The first of these is an identity, because d<i>F</i> = d<sup>2</sup><i>A</i> = 0, but the second is a second-order <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equation</a> for the connection <i>A</i>, and if the Minkowski current vector does not vanish, the zero on the rhs. of the second equation is replaced by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {J} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7686846b1a6b756cb514954000004ab5e7b2a5ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.381ex; height:2.176ex;" alt="{\displaystyle \mathbf {J} }"></span>. But notice how similar these equations are; they differ by a <a href="/wiki/Hodge_star" class="mw-redirect" title="Hodge star">Hodge star</a>. Thus a solution to the simpler first order (non-linear) equation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {*F}=\pm F\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mi>F</mi> </mrow> <mo>=</mo> <mo>±</mo> <mi>F</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {*F}=\pm F\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf36c502b6ce1c415bbd5e4d69073824e04dd122" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.938ex; height:2.176ex;" alt="{\displaystyle {*F}=\pm F\,}"></span></dd></dl> <p>is automatically also a solution of the Yang–Mills equation. This simplification occurs on 4 manifolds with :<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bac386d8f227fb823cede9b3e33d706cad3ed306" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle s=1}"></span> so that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *^{2}=+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mo>∗</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *^{2}=+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3732a42e8e878ae619ca67bef4b25e16f24e9d6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.286ex; height:2.843ex;" alt="{\displaystyle *^{2}=+1}"></span> on 2-forms. Such solutions usually exist, although their precise character depends on the dimension and topology of the base space M, the principal bundle P, and the gauge group G. </p><p>In nonabelian Yang–Mills theories, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DF=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>F</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle DF=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f6a8128b61ebc06c7d093b2fb7dad61ab75e021" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.926ex; height:2.176ex;" alt="{\displaystyle DF=0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D*F=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>∗</mo> <mi>F</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D*F=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c722d420a364c9850644f2dfc1fcaf8c67eb932" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.121ex; height:2.176ex;" alt="{\displaystyle D*F=0}"></span> where D is the <a href="/wiki/Exterior_covariant_derivative" title="Exterior covariant derivative">exterior covariant derivative</a>. Furthermore, the <a href="/wiki/Bianchi_identity" class="mw-redirect" title="Bianchi identity">Bianchi identity</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DF=dF+A\wedge F-F\wedge A=d(dA+A\wedge A)+A\wedge (dA+A\wedge A)-(dA+A\wedge A)\wedge A=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>F</mi> <mo>=</mo> <mi>d</mi> <mi>F</mi> <mo>+</mo> <mi>A</mi> <mo>∧</mo> <mi>F</mi> <mo>−</mo> <mi>F</mi> <mo>∧</mo> <mi>A</mi> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mi>A</mi> <mo>+</mo> <mi>A</mi> <mo>∧</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>A</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mi>A</mi> <mo>+</mo> <mi>A</mi> <mo>∧</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mi>A</mi> <mo>+</mo> <mi>A</mi> <mo>∧</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>A</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle DF=dF+A\wedge F-F\wedge A=d(dA+A\wedge A)+A\wedge (dA+A\wedge A)-(dA+A\wedge A)\wedge A=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45c0d7f93e23f9b1332a83d3de7f060f1fa9d493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:91.474ex; height:2.843ex;" alt="{\displaystyle DF=dF+A\wedge F-F\wedge A=d(dA+A\wedge A)+A\wedge (dA+A\wedge A)-(dA+A\wedge A)\wedge A=0}"></span></dd></dl> <p>is satisfied. </p><p>In <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, an <i>instanton</i> is a <a href="/wiki/Topology" title="Topology">topologically</a> nontrivial field configuration in four-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> (considered as the <a href="/wiki/Wick_rotation" title="Wick rotation">Wick rotation</a> of <a href="/wiki/Minkowski_spacetime" class="mw-redirect" title="Minkowski spacetime">Minkowski spacetime</a>). Specifically, it refers to a <a href="/wiki/Yang%E2%80%93Mills" class="mw-redirect" title="Yang–Mills">Yang–Mills</a> <a href="/wiki/Gauge_field" class="mw-redirect" title="Gauge field">gauge field</a> <i>A</i> which approaches <a href="/wiki/Pure_gauge" class="mw-redirect" title="Pure gauge">pure gauge</a> at <a href="/wiki/Point_at_infinity" title="Point at infinity">spatial infinity</a>. This means the field strength </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =d\mathbf {A} +\mathbf {A} \wedge \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =d\mathbf {A} +\mathbf {A} \wedge \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e51a1732f05d68ff6c7d09c867bb675b883e73a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.479ex; height:2.343ex;" alt="{\displaystyle \mathbf {F} =d\mathbf {A} +\mathbf {A} \wedge \mathbf {A} }"></span></dd></dl> <p>vanishes at infinity. The name <i>instanton</i> derives from the fact that these fields are localized in space and (Euclidean) time – in other words, at a specific instant. </p><p>The case of instantons on the <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">two-dimensional space</a> may be easier to visualise because it admits the simplest case of the gauge <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>, namely U(1), that is an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a>. In this case the field <i>A</i> can be visualised as simply a <a href="/wiki/Vector_field" title="Vector field">vector field</a>. An instanton is a configuration where, for example, the arrows point away from a central point (i.e., a "hedgehog" state). In Euclidean <a href="/wiki/Four-dimensional_space" title="Four-dimensional space">four dimensions</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4abb9b9dab94f7b25a4210364f0f9032704bfb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{4}}"></span>, abelian instantons are impossible. </p><p>The field configuration of an instanton is very different from that of the <a href="/wiki/Vacuum_state" class="mw-redirect" title="Vacuum state">vacuum</a>. Because of this instantons cannot be studied by using <a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagrams</a>, which only include <a href="/wiki/Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">perturbative</a> effects. Instantons are fundamentally <a href="/wiki/Non-perturbative" title="Non-perturbative">non-perturbative</a>. </p><p>The Yang–Mills energy is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}\int _{\mathbb {R} ^{4}}\operatorname {Tr} [*\mathbf {F} \wedge \mathbf {F} ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </msub> <mi>Tr</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mo>∗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\int _{\mathbb {R} ^{4}}\operatorname {Tr} [*\mathbf {F} \wedge \mathbf {F} ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7cd36eddca40093ac18dd44dab83963abd5154b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.31ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{2}}\int _{\mathbb {R} ^{4}}\operatorname {Tr} [*\mathbf {F} \wedge \mathbf {F} ]}"></span></dd></dl> <p>where ∗ is the <a href="/wiki/Hodge_dual" class="mw-redirect" title="Hodge dual">Hodge dual</a>. If we insist that the solutions to the Yang–Mills equations have finite <a href="/wiki/Energy" title="Energy">energy</a>, then the <a href="/wiki/Curvature" title="Curvature">curvature</a> of the solution at infinity (taken as a <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a>) has to be zero. This means that the <a href="/wiki/Chern%E2%80%93Simons" class="mw-redirect" title="Chern–Simons">Chern–Simons</a> invariant can be defined at the 3-space boundary. This is equivalent, via <a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes' theorem</a>, to taking the <a href="/wiki/Integral" title="Integral">integral</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbb {R} ^{4}}\operatorname {Tr} [\mathbf {F} \wedge \mathbf {F} ].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </msub> <mi>Tr</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {R} ^{4}}\operatorname {Tr} [\mathbf {F} \wedge \mathbf {F} ].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d1eb0d11eb6c2a4b450d6f4b9402e3774145412" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.408ex; height:5.676ex;" alt="{\displaystyle \int _{\mathbb {R} ^{4}}\operatorname {Tr} [\mathbf {F} \wedge \mathbf {F} ].}"></span></dd></dl> <p>This is a homotopy invariant and it tells us which <a href="/wiki/Homotopy_class" class="mw-redirect" title="Homotopy class">homotopy class</a> the instanton belongs to. </p><p>Since the integral of a nonnegative <a href="/wiki/Integrand" class="mw-redirect" title="Integrand">integrand</a> is always nonnegative, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq {\frac {1}{2}}\int _{\mathbb {R} ^{4}}\operatorname {Tr} [(*\mathbf {F} +e^{-i\theta }\mathbf {F} )\wedge (\mathbf {F} +e^{i\theta }*\mathbf {F} )]=\int _{\mathbb {R} ^{4}}\operatorname {Tr} [*\mathbf {F} \wedge \mathbf {F} +\cos \theta \mathbf {F} \wedge \mathbf {F} ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </msub> <mi>Tr</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mo>∗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>∗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </msub> <mi>Tr</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mo>∗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq {\frac {1}{2}}\int _{\mathbb {R} ^{4}}\operatorname {Tr} [(*\mathbf {F} +e^{-i\theta }\mathbf {F} )\wedge (\mathbf {F} +e^{i\theta }*\mathbf {F} )]=\int _{\mathbb {R} ^{4}}\operatorname {Tr} [*\mathbf {F} \wedge \mathbf {F} +\cos \theta \mathbf {F} \wedge \mathbf {F} ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07f23e17d996d990175c6f52601a6b6f6383cedd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:74.416ex; height:5.676ex;" alt="{\displaystyle 0\leq {\frac {1}{2}}\int _{\mathbb {R} ^{4}}\operatorname {Tr} [(*\mathbf {F} +e^{-i\theta }\mathbf {F} )\wedge (\mathbf {F} +e^{i\theta }*\mathbf {F} )]=\int _{\mathbb {R} ^{4}}\operatorname {Tr} [*\mathbf {F} \wedge \mathbf {F} +\cos \theta \mathbf {F} \wedge \mathbf {F} ]}"></span></dd></dl> <p>for all real θ. So, this means </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}\int _{\mathbb {R} ^{4}}\operatorname {Tr} [*\mathbf {F} \wedge \mathbf {F} ]\geq {\frac {1}{2}}\left|\int _{\mathbb {R} ^{4}}\operatorname {Tr} [\mathbf {F} \wedge \mathbf {F} ]\right|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </msub> <mi>Tr</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mo>∗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo stretchy="false">]</mo> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>|</mo> <mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </msub> <mi>Tr</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo stretchy="false">]</mo> </mrow> <mo>|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\int _{\mathbb {R} ^{4}}\operatorname {Tr} [*\mathbf {F} \wedge \mathbf {F} ]\geq {\frac {1}{2}}\left|\int _{\mathbb {R} ^{4}}\operatorname {Tr} [\mathbf {F} \wedge \mathbf {F} ]\right|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2d19c2b8ea997dcbdb2ff2e8a67d2def8342828" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.883ex; height:5.843ex;" alt="{\displaystyle {\frac {1}{2}}\int _{\mathbb {R} ^{4}}\operatorname {Tr} [*\mathbf {F} \wedge \mathbf {F} ]\geq {\frac {1}{2}}\left|\int _{\mathbb {R} ^{4}}\operatorname {Tr} [\mathbf {F} \wedge \mathbf {F} ]\right|.}"></span></dd></dl> <p>If this bound is saturated, then the solution is a <a href="/wiki/Bogomol%27nyi_Prasad_Sommerfield_bound" class="mw-redirect" title="Bogomol'nyi Prasad Sommerfield bound">BPS</a> state. For such states, either ∗<i>F</i> = <i>F</i> or ∗<i>F</i> = − <i>F</i> depending on the sign of the <a href="/wiki/Homotopy_invariant" class="mw-redirect" title="Homotopy invariant">homotopy invariant</a>. </p><p>In the Standard Model instantons are expected to be present both in the <a href="/wiki/Electroweak_interaction" title="Electroweak interaction">electroweak sector</a> and the chromodynamic sector, however, their existence has not yet been experimentally confirmed.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Instanton effects are important in understanding the formation of condensates in the vacuum of <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">quantum chromodynamics</a> (QCD) and in explaining the mass of the so-called 'eta-prime particle', a <a href="/wiki/Goldstone_boson" title="Goldstone boson">Goldstone-boson</a><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>note 3<span class="cite-bracket">]</span></a></sup> which has acquired mass through the <a href="/wiki/Chiral_anomaly" title="Chiral anomaly">axial current anomaly</a> of QCD. Note that there is sometimes also a corresponding <a href="/wiki/Soliton" title="Soliton">soliton</a> in a theory with one additional space dimension. Recent research on <i>instantons</i> links them to topics such as <a href="/wiki/D-branes" class="mw-redirect" title="D-branes">D-branes</a> and <a href="/wiki/Black_holes" class="mw-redirect" title="Black holes">Black holes</a> and, of course, the vacuum structure of QCD. For example, in oriented <a href="/wiki/String_theory" title="String theory">string theories</a>, a Dp brane is a gauge theory instanton in the world volume (<i>p</i> + 5)-dimensional <i>U</i>(<i>N</i>) gauge theory on a stack of <i>N</i> D(<i>p</i> + 4)-branes. </p> <div class="mw-heading mw-heading2"><h2 id="Various_numbers_of_dimensions">Various numbers of dimensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Instanton&action=edit&section=13" title="Edit section: Various numbers of dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Instantons play a central role in the nonperturbative dynamics of gauge theories. The kind of physical excitation that yields an instanton depends on the number of dimensions of the spacetime, but, surprisingly, the formalism for dealing with these instantons is relatively dimension-independent. </p><p>In 4-dimensional gauge theories, as described in the previous section, instantons are gauge bundles with a nontrivial <a href="/wiki/Differential_form" title="Differential form">four-form</a> <a href="/wiki/Characteristic_class" title="Characteristic class">characteristic class</a>. If the gauge symmetry is a <a href="/wiki/Unitary_group" title="Unitary group">unitary group</a> or <a href="/wiki/Special_unitary_group" title="Special unitary group">special unitary group</a> then this characteristic class is the second <a href="/wiki/Chern_class" title="Chern class">Chern class</a>, which vanishes in the case of the gauge group U(1). If the gauge symmetry is an orthogonal group then this class is the first <a href="/wiki/Pontrjagin_class" class="mw-redirect" title="Pontrjagin class">Pontrjagin class</a>. </p><p>In 3-dimensional gauge theories with <a href="/wiki/Higgs_field" class="mw-redirect" title="Higgs field">Higgs fields</a>, <a href="/wiki/%27t_Hooft%E2%80%93Polyakov_monopole" title="'t Hooft–Polyakov monopole">'t Hooft–Polyakov monopoles</a> play the role of instantons. In his 1977 paper <a rel="nofollow" class="external text" href="http://inspirehep.net/record/112352">Quark Confinement and Topology of Gauge Groups</a>, <a href="/wiki/Alexander_Markovich_Polyakov" title="Alexander Markovich Polyakov">Alexander Polyakov</a> demonstrated that instanton effects in 3-dimensional <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">QED</a> coupled to a <a href="/wiki/Scalar_field" title="Scalar field">scalar field</a> lead to a mass for the <a href="/wiki/Photon" title="Photon">photon</a>. </p><p>In 2-dimensional abelian gauge theories <a href="/w/index.php?title=Worldsheet_instanton&action=edit&redlink=1" class="new" title="Worldsheet instanton (page does not exist)">worldsheet instantons</a> are magnetic <a href="/wiki/Vortex" title="Vortex">vortices</a>. They are responsible for many nonperturbative effects in string theory, playing a central role in <a href="/wiki/Mirror_symmetry_(string_theory)" title="Mirror symmetry (string theory)">mirror symmetry</a>. </p><p>In 1-dimensional <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, instantons describe <a href="/wiki/Quantum_tunneling" class="mw-redirect" title="Quantum tunneling">tunneling</a>, which is invisible in perturbation theory. </p> <div class="mw-heading mw-heading2"><h2 id="4d_supersymmetric_gauge_theories">4d supersymmetric gauge theories</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Instanton&action=edit&section=14" title="Edit section: 4d supersymmetric gauge theories"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Supersymmetric gauge theories often obey <a href="/wiki/Supersymmetry_nonrenormalization_theorems" title="Supersymmetry nonrenormalization theorems">nonrenormalization theorems</a>, which restrict the kinds of quantum corrections which are allowed. Many of these theorems only apply to corrections calculable in <a href="/wiki/Perturbation_theory" title="Perturbation theory">perturbation theory</a> and so instantons, which are not seen in perturbation theory, provide the only corrections to these quantities. </p><p>Field theoretic techniques for instanton calculations in supersymmetric theories were extensively studied in the 1980s by multiple authors. Because supersymmetry guarantees the cancellation of fermionic vs. bosonic non-zero modes in the instanton background, the involved 't Hooft computation of the instanton saddle point reduces to an integration over zero modes. </p><p>In <i>N</i> = 1 supersymmetric gauge theories instantons can modify the <a href="/wiki/Superpotential" title="Superpotential">superpotential</a>, sometimes lifting all of the vacua. In 1984, <a href="/wiki/Ian_Affleck" title="Ian Affleck">Ian Affleck</a>, <a href="/wiki/Michael_Dine" title="Michael Dine">Michael Dine</a> and <a href="/wiki/Nathan_Seiberg" title="Nathan Seiberg">Nathan Seiberg</a> calculated the instanton corrections to the superpotential in their paper <a rel="nofollow" class="external text" href="http://inspirehep.net/record/15868">Dynamical Supersymmetry Breaking in Supersymmetric QCD</a>. More precisely, they were only able to perform the calculation when the theory contains one less flavor of <a href="/wiki/Chiral_superfield" class="mw-redirect" title="Chiral superfield">chiral matter</a> than the number of colors in the special unitary gauge group, because in the presence of fewer flavors an unbroken nonabelian gauge symmetry leads to an infrared divergence and in the case of more flavors the contribution is equal to zero. For this special choice of chiral matter, the vacuum expectation values of the matter scalar fields can be chosen to completely break the gauge symmetry at weak coupling, allowing a reliable semi-classical saddle point calculation to proceed. By then considering perturbations by various mass terms they were able to calculate the superpotential in the presence of arbitrary numbers of colors and flavors, valid even when the theory is no longer weakly coupled. </p><p>In <i>N</i> = 2 supersymmetric gauge theories the superpotential receives no quantum corrections. However the correction to the metric of the <a href="/wiki/Moduli_space" title="Moduli space">moduli space</a> of vacua from instantons was calculated in a series of papers. First, the one instanton correction was calculated by <a href="/wiki/Nathan_Seiberg" title="Nathan Seiberg">Nathan Seiberg</a> in <a rel="nofollow" class="external text" href="http://inspirehep.net/record/23243">Supersymmetry and Nonperturbative beta Functions</a>. The full set of corrections for SU(2) Yang–Mills theory was calculated by <a href="/wiki/Nathan_Seiberg" title="Nathan Seiberg">Nathan Seiberg</a> and <a href="/wiki/Edward_Witten" title="Edward Witten">Edward Witten</a> in "<a rel="nofollow" class="external text" href="http://inspirehep.net/record/374836">Electric – magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang–Mills theory</a>," in the process creating a subject that is today known as <a href="/wiki/Seiberg%E2%80%93Witten_theory" title="Seiberg–Witten theory">Seiberg–Witten theory</a>. They extended their calculation to SU(2) gauge theories with fundamental matter in <a rel="nofollow" class="external text" href="http://inspirehep.net/record/375702">Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD</a>. These results were later extended for various gauge groups and matter contents, and the direct gauge theory derivation was also obtained in most cases. For gauge theories with gauge group U(N) the Seiberg–Witten geometry has been derived from gauge theory using <a href="/w/index.php?title=Nekrasov_partition_functions&action=edit&redlink=1" class="new" title="Nekrasov partition functions (page does not exist)">Nekrasov partition functions</a> in 2003 by <a href="/wiki/Nikita_Nekrasov" title="Nikita Nekrasov">Nikita Nekrasov</a> and <a href="/wiki/Andrei_Okounkov" title="Andrei Okounkov">Andrei Okounkov</a> and independently by <a href="/wiki/Hiraku_Nakajima" title="Hiraku Nakajima">Hiraku Nakajima</a> and <a href="/w/index.php?title=Kota_Yoshioka&action=edit&redlink=1" class="new" title="Kota Yoshioka (page does not exist)">Kota Yoshioka</a>. </p><p>In <i>N</i> = 4 supersymmetric gauge theories the instantons do not lead to quantum corrections for the metric on the moduli space of vacua. </p> <div class="mw-heading mw-heading2"><h2 id="Explicit_solutions_on_R4">Explicit solutions on R<sup>4</sup></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Instanton&action=edit&section=15" title="Edit section: Explicit solutions on R4"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <a href="/wiki/Ansatz" title="Ansatz">ansatz</a> provided by <a href="/wiki/Ed_Corrigan" title="Ed Corrigan">Corrigan</a> and <a href="/wiki/David_Fairlie" title="David Fairlie">Fairlie</a> provides a solution to the anti-self dual Yang–Mills equations with gauge group SU(2) from any <a href="/wiki/Harmonic_function" title="Harmonic function">harmonic function</a> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4abb9b9dab94f7b25a4210364f0f9032704bfb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{4}}"></span>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-dunajski_19-0" class="reference"><a href="#cite_note-dunajski-19"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> The ansatz gives explicit expressions for the gauge field and can be used to construct solutions with arbitrarily large instanton number. </p><p>Defining the antisymmetric <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {su}}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">u</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {su}}(2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/369267d5eae15a7478ebb52c71fcec1449a10dc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.244ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {su}}(2)}"></span>-valued objects <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b2a5225e32ad6a094f47f8bb48dbb742e303797" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.422ex; height:2.343ex;" alt="{\displaystyle \sigma _{\mu \nu }}"></span> as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{ij}=\epsilon _{ijk}T_{k}\,,\sigma _{i4}=-\sigma _{4i}=T_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{ij}=\epsilon _{ijk}T_{k}\,,\sigma _{i4}=-\sigma _{4i}=T_{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bed522589d3e54ce9d4b56997e5de69a97e7c0d5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.755ex; height:2.843ex;" alt="{\displaystyle \sigma _{ij}=\epsilon _{ijk}T_{k}\,,\sigma _{i4}=-\sigma _{4i}=T_{i},}"></span> where Greek indices run from 1 to 4, Latin indices run from 1 to 3, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8dd1c50cb9436474f83624c3f679ccf3eebbfef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.157ex; height:2.509ex;" alt="{\displaystyle T_{i}}"></span> is a basis of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {su}}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">u</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {su}}(2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/369267d5eae15a7478ebb52c71fcec1449a10dc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.244ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {su}}(2)}"></span> satisfying <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [T_{i},T_{j}]=-\epsilon _{ijk}T_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mo>−</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [T_{i},T_{j}]=-\epsilon _{ijk}T_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d24f7c5f1fc7318e7e2fa76050820f5fc4aca9aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.383ex; height:3.009ex;" alt="{\displaystyle [T_{i},T_{j}]=-\epsilon _{ijk}T_{k}}"></span>. Then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\mu }=\sigma _{\mu \nu }{\frac {\partial _{\nu }\rho }{\rho }}=\sigma _{\mu \nu }\partial _{\nu }\log(\rho )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mi>ρ</mi> </mrow> <mi>ρ</mi> </mfrac> </mrow> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\mu }=\sigma _{\mu \nu }{\frac {\partial _{\nu }\rho }{\rho }}=\sigma _{\mu \nu }\partial _{\nu }\log(\rho )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86e54b6017e5f11ecfeb59da90f507037dbadc53" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.092ex; height:6.009ex;" alt="{\displaystyle A_{\mu }=\sigma _{\mu \nu }{\frac {\partial _{\nu }\rho }{\rho }}=\sigma _{\mu \nu }\partial _{\nu }\log(\rho )}"></span> is a solution as long as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho :\mathbb {R} ^{4}\rightarrow \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho :\mathbb {R} ^{4}\rightarrow \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03804250a23bd9e28f7233833c385176428622c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.164ex; height:3.176ex;" alt="{\displaystyle \rho :\mathbb {R} ^{4}\rightarrow \mathbb {R} }"></span> is harmonic. </p><p>In four dimensions, the <a href="/wiki/Fundamental_solution" title="Fundamental solution">fundamental solution</a> to <a href="/wiki/Laplace%27s_equation" title="Laplace's equation">Laplace's equation</a> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x-y|^{-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x-y|^{-2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e080f2b8feff132da3a65ff335548ca5eb8152a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.952ex; height:3.343ex;" alt="{\displaystyle |x-y|^{-2}}"></span> for any fixed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>. Superposing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf2b9cbfe9051fd4e7b50c8028866d497eac35b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.066ex; height:2.343ex;" alt="{\displaystyle N+1}"></span> of these gives <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span>-soliton solutions of the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (x)=\sum _{p=1}^{N}{\frac {\lambda _{p}}{|x-x_{p}|^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (x)=\sum _{p=1}^{N}{\frac {\lambda _{p}}{|x-x_{p}|^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23796fe09c59f3f7aed3f1d04077a6c58eb16223" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:21.571ex; height:7.676ex;" alt="{\displaystyle \rho (x)=\sum _{p=1}^{N}{\frac {\lambda _{p}}{|x-x_{p}|^{2}}}.}"></span> All solutions of instanton number 1 or 2 are of this form, but for larger instanton number there are solutions not of this form. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Instanton&action=edit&section=16" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Instanton_fluid" title="Instanton fluid">Instanton fluid</a> – Non-perturbative path integral approximation</li> <li><a href="/wiki/Caloron" title="Caloron">Caloron</a> – Finite temperature instanton</li> <li><a href="/wiki/Sidney_Coleman" title="Sidney Coleman">Sidney Coleman</a> – American physicist (1937–2007)</li> <li><a href="/wiki/Holstein%E2%80%93Herring_method#Physical_Interpretation" title="Holstein–Herring method">Holstein–Herring method</a> – effective means of getting the exchange energy splittings of asymptotically degenerate energy states in molecular systems<span style="display:none" class="category-wikidata-fallback-annotation">Pages displaying wikidata descriptions as a fallback</span></li> <li><a href="/wiki/Gravitational_instanton" title="Gravitational instanton">Gravitational instanton</a> – Four-dimensional complete Riemannian manifold satisfying the vacuum Einstein equations</li> <li><a href="/wiki/Semiclassical_transition_state_theory" title="Semiclassical transition state theory">Semiclassical transition state theory</a> – Chemical reaction rate theory</li> <li><a href="/wiki/Yang%E2%80%93Mills_equations" title="Yang–Mills equations">Yang–Mills equations</a> – Partial differential equations whose solutions are instantons</li> <li><a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory (mathematics)</a> – Study of vector bundles, principal bundles, and fibre bundles</li></ul> <div class="mw-heading mw-heading2"><h2 id="References_and_notes">References and notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Instanton&action=edit&section=17" title="Edit section: References and notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>Notes</dt></dl> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width"> <ol class="references"> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Because this projection is <a href="/wiki/Conformal_map" title="Conformal map">conformal</a>, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles: the curves that intersect <0,0,0,1> have infinite radius (= straight line).</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">See also: <a href="/wiki/Non-abelian_gauge_theory" class="mw-redirect" title="Non-abelian gauge theory">Non-abelian gauge theory</a></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">See also: <a href="/wiki/Chiral_symmetry_breaking" title="Chiral symmetry breaking">Pseudo-Goldstone boson</a></span> </li> </ol></div> <dl><dt>Citations</dt></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Instantons in Gauge Theories. Edited by Mikhail A. Shifman. World Scientific, 1994.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Interactions Between Charged Particles in a Magnetic Field. By Hrachya Nersisyan, Christian Toepffer, Günter Zwicknagel. Springer, Apr 19, 2007. Pg 23</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Large-Order Behaviour of Perturbation Theory. Edited by J.C. Le Guillou, J. Zinn-Justin. Elsevier, Dec 2, 2012. Pg. 170.</span> </li> <li id="cite_note-:0-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFVaĭnshteĭnZakharovNovikovShifman1982" class="citation journal cs1">Vaĭnshteĭn, A. I.; Zakharov, Valentin I.; Novikov, Viktor A.; Shifman, Mikhail A. (1982-04-30). <a rel="nofollow" class="external text" href="https://iopscience.iop.org/article/10.1070/PU1982v025n04ABEH004533/meta">"ABC of instantons"</a>. <i>Soviet Physics Uspekhi</i>. <b>25</b> (4): 195. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1070%2FPU1982v025n04ABEH004533">10.1070/PU1982v025n04ABEH004533</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0038-5670">0038-5670</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Soviet+Physics+Uspekhi&rft.atitle=ABC+of+instantons&rft.volume=25&rft.issue=4&rft.pages=195&rft.date=1982-04-30&rft_id=info%3Adoi%2F10.1070%2FPU1982v025n04ABEH004533&rft.issn=0038-5670&rft.aulast=Va%C4%ADnshte%C4%ADn&rft.aufirst=A.+I.&rft.au=Zakharov%2C+Valentin+I.&rft.au=Novikov%2C+Viktor+A.&rft.au=Shifman%2C+Mikhail+A.&rft_id=https%3A%2F%2Fiopscience.iop.org%2Farticle%2F10.1070%2FPU1982v025n04ABEH004533%2Fmeta&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInstanton" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/Yang-Mills+instanton">"Yang-Mills instanton in nLab"</a>. <i>ncatlab.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-04-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ncatlab.org&rft.atitle=Yang-Mills+instanton+in+nLab&rft_id=https%3A%2F%2Fncatlab.org%2Fnlab%2Fshow%2FYang-Mills%2Binstanton&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInstanton" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">See, for instance, <a href="/wiki/Nigel_Hitchin" title="Nigel Hitchin">Nigel Hitchin</a>'s paper "Self-Duality Equations on Riemann Surface".</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">H.J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed. (World Scientific, 2012), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4397-73-5" title="Special:BookSources/978-981-4397-73-5">978-981-4397-73-5</a>; formula (18.175b), p. 525.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">H.J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed., World Scientific, 2012, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4397-73-5" title="Special:BookSources/978-981-4397-73-5">978-981-4397-73-5</a>.</span> </li> <li id="cite_note-Harald_J.W._Müller-Kirsten_2012-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-Harald_J.W._Müller-Kirsten_2012_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Harald_J.W._Müller-Kirsten_2012_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Harald J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed., World Scientific (Singapore, 2012).</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiangMüller-KirstenTchrakian1992" class="citation journal cs1">Liang, Jiu-Qing; Müller-Kirsten, H.J.W.; Tchrakian, D.H. (1992). "Solitons, bounces and sphalerons on a circle". <i>Physics Letters B</i>. <b>282</b> (1–2). Elsevier BV: 105–110. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1992PhLB..282..105L">1992PhLB..282..105L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0370-2693%2892%2990486-n">10.1016/0370-2693(92)90486-n</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0370-2693">0370-2693</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Letters+B&rft.atitle=Solitons%2C+bounces+and+sphalerons+on+a+circle&rft.volume=282&rft.issue=1%E2%80%932&rft.pages=105-110&rft.date=1992&rft.issn=0370-2693&rft_id=info%3Adoi%2F10.1016%2F0370-2693%2892%2990486-n&rft_id=info%3Abibcode%2F1992PhLB..282..105L&rft.aulast=Liang&rft.aufirst=Jiu-Qing&rft.au=M%C3%BCller-Kirsten%2C+H.J.W.&rft.au=Tchrakian%2C+D.H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInstanton" class="Z3988"></span></span> </li> <li id="cite_note-:inst_chapter-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-:inst_chapter_11-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZaverkinKästner2020" class="citation book cs1">Zaverkin, Viktor; <a href="/w/index.php?title=Johannes_K%C3%A4stner&action=edit&redlink=1" class="new" title="Johannes Kästner (page does not exist)">Kästner, Johannes</a> (2020). "Instanton Theory to Calculate Tunnelling Rates and Tunnelling Splittings". <a rel="nofollow" class="external text" href="https://doi.org/10.1039/9781839160370"><i>Tunnelling in Molecules: Nuclear Quantum Effects from Bio to Physical Chemistry</i></a>. London: Royal Society of Chemistry. p. 245-260. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-83916-037-0" title="Special:BookSources/978-1-83916-037-0"><bdi>978-1-83916-037-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Instanton+Theory+to+Calculate+Tunnelling+Rates+and+Tunnelling+Splittings&rft.btitle=Tunnelling+in+Molecules%3A+Nuclear+Quantum+Effects+from+Bio+to+Physical+Chemistry&rft.place=London&rft.pages=245-260&rft.pub=Royal+Society+of+Chemistry&rft.date=2020&rft.isbn=978-1-83916-037-0&rft.aulast=Zaverkin&rft.aufirst=Viktor&rft.au=K%C3%A4stner%2C+Johannes&rft_id=https%3A%2F%2Fdoi.org%2F10.1039%2F9781839160370&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInstanton" class="Z3988"></span></span> </li> <li id="cite_note-:inst_review-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-:inst_review_12-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKästner2014" class="citation journal cs1">Kästner, Johannes (2014). <a rel="nofollow" class="external text" href="https://doi.org/10.1002/wcms.1165">"Theory and Simulation of Atom Tunneling in Chemical Reactions"</a>. <i>WIREs Comput. Mol. Sci</i>. <b>4</b>: 158. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fwcms.1165">10.1002/wcms.1165</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=WIREs+Comput.+Mol.+Sci.&rft.atitle=Theory+and+Simulation+of+Atom+Tunneling+in+Chemical+Reactions&rft.volume=4&rft.pages=158&rft.date=2014&rft_id=info%3Adoi%2F10.1002%2Fwcms.1165&rft.aulast=K%C3%A4stner&rft.aufirst=Johannes&rft_id=https%3A%2F%2Fdoi.org%2F10.1002%2Fwcms.1165&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInstanton" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBenderWu1973" class="citation journal cs1">Bender, Carl M.; Wu, Tai Tsun (1973-03-15). "Anharmonic Oscillator. II. A Study of Perturbation Theory in Large Order". <i>Physical Review D</i>. <b>7</b> (6). American Physical Society (APS): 1620–1636. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1973PhRvD...7.1620B">1973PhRvD...7.1620B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevd.7.1620">10.1103/physrevd.7.1620</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0556-2821">0556-2821</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+D&rft.atitle=Anharmonic+Oscillator.+II.+A+Study+of+Perturbation+Theory+in+Large+Order&rft.volume=7&rft.issue=6&rft.pages=1620-1636&rft.date=1973-03-15&rft.issn=0556-2821&rft_id=info%3Adoi%2F10.1103%2Fphysrevd.7.1620&rft_id=info%3Abibcode%2F1973PhRvD...7.1620B&rft.aulast=Bender&rft.aufirst=Carl+M.&rft.au=Wu%2C+Tai+Tsun&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInstanton" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAmorosoKarSchott2021" class="citation journal cs1">Amoroso, Simone; Kar, Deepak; Schott, Matthias (2021). "How to discover QCD Instantons at the LHC". <i>The European Physical Journal C</i>. <b>81</b> (7): 624. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2012.09120">2012.09120</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2021EPJC...81..624A">2021EPJC...81..624A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1140%2Fepjc%2Fs10052-021-09412-1">10.1140/epjc/s10052-021-09412-1</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:229220708">229220708</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+European+Physical+Journal+C&rft.atitle=How+to+discover+QCD+Instantons+at+the+LHC&rft.volume=81&rft.issue=7&rft.pages=624&rft.date=2021&rft_id=info%3Aarxiv%2F2012.09120&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A229220708%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1140%2Fepjc%2Fs10052-021-09412-1&rft_id=info%3Abibcode%2F2021EPJC...81..624A&rft.aulast=Amoroso&rft.aufirst=Simone&rft.au=Kar%2C+Deepak&rft.au=Schott%2C+Matthias&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInstanton" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCorriganFairlie1977" class="citation journal cs1">Corrigan, E.; Fairlie, D.B. (March 1977). "Scalar field theory and exact solutions to a classical SU (2) gauge theory". <i>Physics Letters B</i>. <b>67</b> (1): 69–71. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0370-2693%2877%2990808-5">10.1016/0370-2693(77)90808-5</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Letters+B&rft.atitle=Scalar+field+theory+and+exact+solutions+to+a+classical+SU+%282%29+gauge+theory&rft.volume=67&rft.issue=1&rft.pages=69-71&rft.date=1977-03&rft_id=info%3Adoi%2F10.1016%2F0370-2693%2877%2990808-5&rft.aulast=Corrigan&rft.aufirst=E.&rft.au=Fairlie%2C+D.B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInstanton" class="Z3988"></span></span> </li> <li id="cite_note-dunajski-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-dunajski_19-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDunajski2010" class="citation book cs1">Dunajski, Maciej (2010). <i>Solitons, instantons, and twistors</i>. Oxford: Oxford University Press. p. 123. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780198570639" title="Special:BookSources/9780198570639"><bdi>9780198570639</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Solitons%2C+instantons%2C+and+twistors&rft.place=Oxford&rft.pages=123&rft.pub=Oxford+University+Press&rft.date=2010&rft.isbn=9780198570639&rft.aulast=Dunajski&rft.aufirst=Maciej&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInstanton" class="Z3988"></span></span> </li> </ol></div></div> <dl><dt>General</dt></dl> <ul><li><i>Instantons in Gauge Theories</i>, a compilation of articles on instantons, edited by <a href="/wiki/Mikhail_A._Shifman" class="mw-redirect" title="Mikhail A. Shifman">Mikhail A. Shifman</a>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F2281">10.1142/2281</a></li> <li><i>Solitons and Instantons</i>, R. Rajaraman (Amsterdam: North Holland, 1987), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-87047-4" title="Special:BookSources/0-444-87047-4">0-444-87047-4</a></li> <li><i>The Uses of Instantons</i>, by <a href="/wiki/Sidney_Coleman" title="Sidney Coleman">Sidney Coleman</a> in <i>Proc. Int. School of Subnuclear Physics</i>, (Erice, 1977); and in <i>Aspects of Symmetry</i> p. 265, Sidney Coleman, Cambridge University Press, 1985, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-31827-0" title="Special:BookSources/0-521-31827-0">0-521-31827-0</a>; and in <i>Instantons in Gauge Theories</i></li> <li><i>Solitons, Instantons and Twistors</i>. M. Dunajski, Oxford University Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-857063-9" title="Special:BookSources/978-0-19-857063-9">978-0-19-857063-9</a>.</li> <li><i><a rel="nofollow" class="external text" href="https://books.google.com/books/about/The_Geometry_of_Four_manifolds.html?id=LbHmMtrebi4C">The Geometry of Four-Manifolds</a></i>, S.K. Donaldson, P.B. Kronheimer, Oxford University Press, 1990, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-853553-8" title="Special:BookSources/0-19-853553-8">0-19-853553-8</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Instanton&action=edit&section=18" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wiktionary-logo-en-v2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/16px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/24px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/32px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> The dictionary definition of <a href="https://en.wiktionary.org/wiki/instanton" class="extiw" title="wiktionary:instanton"><i>instanton</i></a> at Wiktionary</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl 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.mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="String_theory" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="text-align:center;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:String_theory_topics" title="Template:String theory topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:String_theory_topics" title="Template talk:String theory topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:String_theory_topics" title="Special:EditPage/Template:String theory topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="String_theory" style="font-size:114%;margin:0 4em"><a href="/wiki/String_theory" title="String theory">String theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/String_(physics)" title="String (physics)">Strings</a></li> <li><a href="/wiki/Cosmic_string" title="Cosmic string">Cosmic strings</a></li> <li><a href="/wiki/History_of_string_theory" title="History of string theory">History of string theory</a> <ul><li><a href="/wiki/First_superstring_revolution" class="mw-redirect" title="First superstring revolution">First superstring revolution</a></li> <li><a href="/wiki/Second_superstring_revolution" class="mw-redirect" title="Second superstring revolution">Second superstring revolution</a></li></ul></li> <li><a href="/wiki/String_theory_landscape" title="String theory landscape">String theory landscape</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nambu%E2%80%93Goto_action" title="Nambu–Goto action">Nambu–Goto action</a></li> <li><a href="/wiki/Polyakov_action" title="Polyakov action">Polyakov action</a></li> <li><a href="/wiki/Bosonic_string_theory" title="Bosonic string theory">Bosonic string theory</a></li> <li><a href="/wiki/Superstring_theory" title="Superstring theory">Superstring theory</a> <ul><li><a href="/wiki/Type_I_string_theory" title="Type I string theory">Type I string</a></li> <li><a href="/wiki/Type_II_string_theory" title="Type II string theory">Type II string</a> <ul><li><a href="/wiki/Type_II_string_theory" title="Type II string theory">Type IIA string</a></li> <li><a href="/wiki/Type_II_string_theory" title="Type II string theory">Type IIB string</a></li></ul></li> <li><a href="/wiki/Heterotic_string_theory" title="Heterotic string theory">Heterotic string</a></li></ul></li> <li><a href="/wiki/N%3D2_superstring" class="mw-redirect" title="N=2 superstring">N=2 superstring</a></li> <li><a href="/wiki/F-theory" title="F-theory">F-theory</a></li> <li><a href="/wiki/String_field_theory" title="String field theory">String field theory</a></li> <li><a href="/wiki/Matrix_string_theory" title="Matrix string theory">Matrix string theory</a></li> <li><a href="/wiki/Non-critical_string_theory" title="Non-critical string theory">Non-critical string theory</a></li> <li><a href="/wiki/Non-linear_sigma_model" title="Non-linear sigma model">Non-linear sigma model</a></li> <li><a href="/wiki/Tachyon_condensation" title="Tachyon condensation">Tachyon condensation</a></li> <li><a href="/wiki/RNS_formalism" title="RNS formalism">RNS formalism</a></li> <li><a href="/wiki/GS_formalism" title="GS formalism">GS formalism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/String_duality" title="String duality">String duality</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/T-duality" title="T-duality">T-duality</a></li> <li><a href="/wiki/S-duality" title="S-duality">S-duality</a></li> <li><a href="/wiki/U-duality" title="U-duality">U-duality</a></li> <li><a href="/wiki/Montonen%E2%80%93Olive_duality" title="Montonen–Olive duality">Montonen–Olive duality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Particles and fields</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Graviton" title="Graviton">Graviton</a></li> <li><a href="/wiki/Dilaton" title="Dilaton">Dilaton</a></li> <li><a href="/wiki/Tachyon" title="Tachyon">Tachyon</a></li> <li><a href="/wiki/Ramond%E2%80%93Ramond_field" title="Ramond–Ramond field">Ramond–Ramond field</a></li> <li><a href="/wiki/Kalb%E2%80%93Ramond_field" title="Kalb–Ramond field">Kalb–Ramond field</a></li> <li><a href="/wiki/Magnetic_monopole" title="Magnetic monopole">Magnetic monopole</a></li> <li><a href="/wiki/Dual_graviton" title="Dual graviton">Dual graviton</a></li> <li><a href="/wiki/Dual_photon" title="Dual photon">Dual photon</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Brane" title="Brane">Branes</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/D-brane" title="D-brane">D-brane</a></li> <li><a href="/wiki/NS5-brane" title="NS5-brane">NS5-brane</a></li> <li><a href="/wiki/M2-brane" title="M2-brane">M2-brane</a></li> <li><a href="/wiki/M5-brane" title="M5-brane">M5-brane</a></li> <li><a href="/wiki/S-brane" title="S-brane">S-brane</a></li> <li><a href="/wiki/Black_brane" title="Black brane">Black brane</a></li> <li><a href="/wiki/Black_hole" title="Black hole">Black holes</a></li> <li><a href="/wiki/Black_string" class="mw-redirect" title="Black string">Black string</a></li> <li><a href="/wiki/Brane_cosmology" title="Brane cosmology">Brane cosmology</a></li> <li><a href="/wiki/Quiver_diagram" title="Quiver diagram">Quiver diagram</a></li> <li><a href="/wiki/Hanany%E2%80%93Witten_transition" title="Hanany–Witten transition">Hanany–Witten transition</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Conformal_field_theory" title="Conformal field theory">Conformal field theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Virasoro_algebra" title="Virasoro algebra">Virasoro algebra</a></li> <li><a href="/wiki/Mirror_symmetry_(string_theory)" title="Mirror symmetry (string theory)">Mirror symmetry</a></li> <li><a href="/wiki/Conformal_anomaly" title="Conformal anomaly">Conformal anomaly</a></li> <li><a href="/wiki/Conformal_symmetry" title="Conformal symmetry">Conformal algebra</a></li> <li><a href="/wiki/Superconformal_algebra" title="Superconformal algebra">Superconformal algebra</a></li> <li><a href="/wiki/Vertex_operator_algebra" title="Vertex operator algebra">Vertex operator algebra</a></li> <li><a href="/wiki/Loop_algebra" title="Loop algebra">Loop algebra</a></li> <li><a href="/wiki/Kac%E2%80%93Moody_algebra" title="Kac–Moody algebra">Kac–Moody algebra</a></li> <li><a href="/wiki/Wess%E2%80%93Zumino%E2%80%93Witten_model" title="Wess–Zumino–Witten model">Wess–Zumino–Witten model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Gauge_theory" title="Gauge theory">Gauge theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Anomaly_(physics)" title="Anomaly (physics)">Anomalies</a></li> <li><a class="mw-selflink selflink">Instantons</a></li> <li><a href="/wiki/Chern%E2%80%93Simons_form" title="Chern–Simons form">Chern–Simons form</a></li> <li><a href="/wiki/Bogomol%27nyi%E2%80%93Prasad%E2%80%93Sommerfield_bound" title="Bogomol'nyi–Prasad–Sommerfield bound">Bogomol'nyi–Prasad–Sommerfield bound</a></li> <li><a href="/wiki/Exceptional_Lie_group" class="mw-redirect" title="Exceptional Lie group">Exceptional Lie groups</a> (<a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a>, <a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F<sub>4</sub></a>, <a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a>, <a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a>, <a href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E<sub>8</sub></a>)</li> <li><a href="/wiki/ADE_classification" title="ADE classification">ADE classification</a></li> <li><a href="/wiki/Dirac_string" title="Dirac string">Dirac string</a></li> <li><a href="/wiki/P-form_electrodynamics" title="P-form electrodynamics"><i>p</i>-form electrodynamics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Geometry</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Worldsheet" title="Worldsheet">Worldsheet</a></li> <li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li> <li><a href="/wiki/Compactification_(physics)" title="Compactification (physics)">Compactification</a></li> <li><a href="/wiki/Why_10_dimensions" class="mw-redirect" title="Why 10 dimensions">Why 10 dimensions</a>?</li> <li><a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler manifold</a></li> <li><a href="/wiki/Ricci-flat_manifold" title="Ricci-flat manifold">Ricci-flat manifold</a> <ul><li><a href="/wiki/Calabi%E2%80%93Yau_manifold" title="Calabi–Yau manifold">Calabi–Yau manifold</a></li> <li><a href="/wiki/Hyperk%C3%A4hler_manifold" title="Hyperkähler manifold">Hyperkähler manifold</a> <ul><li><a href="/wiki/K3_surface" title="K3 surface">K3 surface</a></li></ul></li> <li><a href="/wiki/G2_manifold" title="G2 manifold">G<sub>2</sub> manifold</a></li> <li><a href="/wiki/Spin(7)-manifold" title="Spin(7)-manifold">Spin(7)-manifold</a></li></ul></li> <li><a href="/wiki/Generalized_complex_structure" title="Generalized complex structure">Generalized complex manifold</a></li> <li><a href="/wiki/Orbifold" title="Orbifold">Orbifold</a></li> <li><a href="/wiki/Conifold" title="Conifold">Conifold</a></li> <li><a href="/wiki/Orientifold" title="Orientifold">Orientifold</a></li> <li><a href="/wiki/Moduli_space" title="Moduli space">Moduli space</a></li> <li><a href="/wiki/Ho%C5%99ava%E2%80%93Witten_theory" title="Hořava–Witten theory">Hořava–Witten theory</a></li> <li><a href="/wiki/K-theory_(physics)" title="K-theory (physics)">K-theory (physics)</a></li> <li><a href="/wiki/Twisted_K-theory" title="Twisted K-theory">Twisted K-theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Supergravity" title="Supergravity">Supergravity</a></li> <li><a href="/wiki/Eleven-dimensional_supergravity" title="Eleven-dimensional supergravity">Eleven-dimensional supergravity</a></li> <li><a href="/wiki/Type_I_supergravity" title="Type I supergravity">Type I supergravity</a></li> <li><a href="/wiki/Type_IIA_supergravity" title="Type IIA supergravity">Type IIA supergravity</a></li> <li><a href="/wiki/Type_IIB_supergravity" title="Type IIB supergravity">Type IIB supergravity</a></li> <li><a href="/wiki/Superspace" title="Superspace">Superspace</a></li> <li><a href="/wiki/Lie_superalgebra" title="Lie superalgebra">Lie superalgebra</a></li> <li><a href="/wiki/Lie_supergroup" class="mw-redirect" title="Lie supergroup">Lie supergroup</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Holography" title="Holography">Holography</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Holographic_principle" title="Holographic principle">Holographic principle</a></li> <li><a href="/wiki/AdS/CFT_correspondence" title="AdS/CFT correspondence">AdS/CFT correspondence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/M-theory" title="M-theory">M-theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Matrix_theory_(physics)" title="Matrix theory (physics)">Matrix theory</a></li> <li><a href="/wiki/Introduction_to_M-theory" title="Introduction to M-theory">Introduction to M-theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">String theorists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mina_Aganagi%C4%87" title="Mina Aganagić">Aganagić</a></li> <li><a href="/wiki/Nima_Arkani-Hamed" title="Nima Arkani-Hamed">Arkani-Hamed</a></li> <li><a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li> <li><a href="/wiki/Tom_Banks_(physicist)" title="Tom Banks (physicist)">Banks</a></li> <li><a href="/wiki/David_Berenstein" title="David Berenstein">Berenstein</a></li> <li><a href="/wiki/Raphael_Bousso" title="Raphael Bousso">Bousso</a></li> <li><a href="/wiki/Thomas_Curtright" title="Thomas Curtright">Curtright</a></li> <li><a href="/wiki/Robbert_Dijkgraaf" title="Robbert Dijkgraaf">Dijkgraaf</a></li> <li><a href="/wiki/Jacques_Distler" title="Jacques Distler">Distler</a></li> <li><a href="/wiki/Michael_R._Douglas" title="Michael R. Douglas">Douglas</a></li> <li><a href="/wiki/Michael_Duff_(physicist)" title="Michael Duff (physicist)">Duff</a></li> <li><a href="/wiki/Gia_Dvali" class="mw-redirect" title="Gia Dvali">Dvali</a></li> <li><a href="/wiki/Sergio_Ferrara" title="Sergio Ferrara">Ferrara</a></li> <li><a href="/wiki/Willy_Fischler" title="Willy Fischler">Fischler</a></li> <li><a href="/wiki/Daniel_Friedan" title="Daniel Friedan">Friedan</a></li> <li><a href="/wiki/Sylvester_James_Gates" title="Sylvester James Gates">Gates</a></li> <li><a href="/wiki/Ferdinando_Gliozzi" title="Ferdinando Gliozzi">Gliozzi</a></li> <li><a href="/wiki/Rajesh_Gopakumar" title="Rajesh Gopakumar">Gopakumar</a></li> <li><a href="/wiki/Michael_Green_(physicist)" title="Michael Green (physicist)">Green</a></li> <li><a href="/wiki/Brian_Greene" title="Brian Greene">Greene</a></li> <li><a href="/wiki/David_Gross" title="David Gross">Gross</a></li> <li><a href="/wiki/Steven_Gubser" title="Steven Gubser">Gubser</a></li> <li><a href="/wiki/Sergei_Gukov" title="Sergei Gukov">Gukov</a></li> <li><a href="/wiki/Alan_Guth" title="Alan Guth">Guth</a></li> <li><a href="/wiki/Andrew_J._Hanson" title="Andrew J. Hanson">Hanson</a></li> <li><a href="/wiki/Jeffrey_A._Harvey" title="Jeffrey A. Harvey">Harvey</a></li> <li><a href="/wiki/Gerard_%27t_Hooft" title="Gerard 't Hooft">'t Hooft</a></li> <li><a href="/wiki/Petr_Ho%C5%99ava_(theorist)" class="mw-redirect" title="Petr Hořava (theorist)">Hořava</a></li> <li><a href="/wiki/Gary_Gibbons" title="Gary Gibbons">Gibbons</a></li> <li><a href="/wiki/Shamit_Kachru" title="Shamit Kachru">Kachru</a></li> <li><a href="/wiki/Michio_Kaku" title="Michio Kaku">Kaku</a></li> <li><a href="/wiki/Renata_Kallosh" title="Renata Kallosh">Kallosh</a></li> <li><a href="/wiki/Theodor_Kaluza" title="Theodor Kaluza">Kaluza</a></li> <li><a href="/wiki/Anton_Kapustin" title="Anton Kapustin">Kapustin</a></li> <li><a href="/wiki/Igor_Klebanov" title="Igor Klebanov">Klebanov</a></li> <li><a href="/wiki/Vadim_Knizhnik" title="Vadim Knizhnik">Knizhnik</a></li> <li><a href="/wiki/Maxim_Kontsevich" title="Maxim Kontsevich">Kontsevich</a></li> <li><a href="/wiki/Oskar_Klein" title="Oskar Klein">Klein</a></li> <li><a href="/wiki/Andrei_Linde" title="Andrei Linde">Linde</a></li> <li><a href="/wiki/Juan_Mart%C3%ADn_Maldacena" class="mw-redirect" title="Juan Martín Maldacena">Maldacena</a></li> <li><a href="/wiki/Stanley_Mandelstam" title="Stanley Mandelstam">Mandelstam</a></li> <li><a href="/wiki/Donald_Marolf" title="Donald Marolf">Marolf</a></li> <li><a href="/wiki/Emil_Martinec" title="Emil Martinec">Martinec</a></li> <li><a href="/wiki/Shiraz_Minwalla" title="Shiraz Minwalla">Minwalla</a></li> <li><a href="/wiki/Greg_Moore_(physicist)" title="Greg Moore (physicist)">Moore</a></li> <li><a href="/wiki/Lubo%C5%A1_Motl" title="Luboš Motl">Motl</a></li> <li><a href="/wiki/Sunil_Mukhi" title="Sunil Mukhi">Mukhi</a></li> <li><a href="/wiki/Robert_Myers_(physicist)" title="Robert Myers (physicist)">Myers</a></li> <li><a href="/wiki/Dimitri_Nanopoulos" title="Dimitri Nanopoulos">Nanopoulos</a></li> <li><a href="/wiki/Hora%C8%9Biu_N%C4%83stase" title="Horațiu Năstase">Năstase</a></li> <li><a href="/wiki/Nikita_Nekrasov" title="Nikita Nekrasov">Nekrasov</a></li> <li><a href="/wiki/Andr%C3%A9_Neveu" title="André Neveu">Neveu</a></li> <li><a href="/wiki/Holger_Bech_Nielsen" title="Holger Bech Nielsen">Nielsen</a></li> <li><a href="/wiki/Peter_van_Nieuwenhuizen" title="Peter van Nieuwenhuizen">van Nieuwenhuizen</a></li> <li><a href="/wiki/Sergei_Novikov_(mathematician)" title="Sergei Novikov (mathematician)">Novikov</a></li> <li><a href="/wiki/David_Olive" title="David Olive">Olive</a></li> <li><a href="/wiki/Hirosi_Ooguri" title="Hirosi Ooguri">Ooguri</a></li> <li><a href="/wiki/Burt_Ovrut" title="Burt Ovrut">Ovrut</a></li> <li><a href="/wiki/Joseph_Polchinski" title="Joseph Polchinski">Polchinski</a></li> <li><a href="/wiki/Alexander_Markovich_Polyakov" title="Alexander Markovich Polyakov">Polyakov</a></li> <li><a href="/wiki/Arvind_Rajaraman" title="Arvind Rajaraman">Rajaraman</a></li> <li><a href="/wiki/Pierre_Ramond" title="Pierre Ramond">Ramond</a></li> <li><a href="/wiki/Lisa_Randall" title="Lisa Randall">Randall</a></li> <li><a href="/wiki/Seifallah_Randjbar-Daemi" title="Seifallah Randjbar-Daemi">Randjbar-Daemi</a></li> <li><a href="/wiki/Martin_Ro%C4%8Dek" title="Martin Roček">Roček</a></li> <li><a href="/wiki/Ryan_Rohm" title="Ryan Rohm">Rohm</a></li> <li><a href="/wiki/Augusto_Sagnotti" title="Augusto Sagnotti">Sagnotti</a></li> <li><a href="/wiki/Jo%C3%ABl_Scherk" title="Joël Scherk">Scherk</a></li> <li><a href="/wiki/John_Henry_Schwarz" title="John Henry Schwarz">Schwarz</a></li> <li><a href="/wiki/Nathan_Seiberg" title="Nathan Seiberg">Seiberg</a></li> <li><a href="/wiki/Ashoke_Sen" title="Ashoke Sen">Sen</a></li> <li><a href="/wiki/Stephen_Shenker" title="Stephen Shenker">Shenker</a></li> <li><a href="/wiki/Warren_Siegel" title="Warren Siegel">Siegel</a></li> <li><a href="/wiki/Eva_Silverstein" title="Eva Silverstein">Silverstein</a></li> <li><a href="/wiki/%C4%90%C3%A0m_Thanh_S%C6%A1n" title="Đàm Thanh Sơn">Sơn</a></li> <li><a href="/wiki/Matthias_Staudacher" title="Matthias Staudacher">Staudacher</a></li> <li><a href="/wiki/Paul_Steinhardt" title="Paul Steinhardt">Steinhardt</a></li> <li><a href="/wiki/Andrew_Strominger" title="Andrew Strominger">Strominger</a></li> <li><a href="/wiki/Raman_Sundrum" title="Raman Sundrum">Sundrum</a></li> <li><a href="/wiki/Leonard_Susskind" title="Leonard Susskind">Susskind</a></li> <li><a href="/wiki/Paul_Townsend" title="Paul Townsend">Townsend</a></li> <li><a href="/wiki/Sandip_Trivedi" title="Sandip Trivedi">Trivedi</a></li> <li><a href="/wiki/Neil_Turok" title="Neil Turok">Turok</a></li> <li><a href="/wiki/Cumrun_Vafa" title="Cumrun Vafa">Vafa</a></li> <li><a href="/wiki/Gabriele_Veneziano" title="Gabriele Veneziano">Veneziano</a></li> <li><a href="/wiki/Erik_Verlinde" title="Erik Verlinde">Verlinde</a></li> <li><a href="/wiki/Herman_Verlinde" title="Herman Verlinde">Verlinde</a></li> <li><a href="/wiki/Julius_Wess" title="Julius Wess">Wess</a></li> <li><a href="/wiki/Edward_Witten" title="Edward Witten">Witten</a></li> <li><a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Yau</a></li> <li><a href="/wiki/Tamiaki_Yoneya" title="Tamiaki Yoneya">Yoneya</a></li> <li><a href="/wiki/Alexander_Zamolodchikov" title="Alexander Zamolodchikov">Zamolodchikov</a></li> <li><a href="/wiki/Alexei_Zamolodchikov" title="Alexei Zamolodchikov">Zamolodchikov</a></li> <li><a href="/wiki/Eric_Zaslow" title="Eric Zaslow">Zaslow</a></li> <li><a href="/wiki/Bruno_Zumino" title="Bruno Zumino">Zumino</a></li> <li><a href="/wiki/Barton_Zwiebach" title="Barton Zwiebach">Zwiebach</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" 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