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href="https://de.wikipedia.org/wiki/Zitterbewegung" title="Zitterbewegung – German" lang="de" hreflang="de" data-title="Zitterbewegung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Zitterbewegung" title="Zitterbewegung – Spanish" lang="es" hreflang="es" data-title="Zitterbewegung" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Zitterbewegung" title="Zitterbewegung – French" lang="fr" hreflang="fr" data-title="Zitterbewegung" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B9%98%ED%84%B0%EB%B2%A0%EB%B2%A0%EA%B6%81" title="치터베베궁 – Korean" lang="ko" hreflang="ko" data-title="치터베베궁" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%A0%D7%95%D7%A2%D7%AA_%D7%A8%D7%A2%D7%93%D7%95%D7%93" title="תנועת רעדוד – Hebrew" lang="he" hreflang="he" data-title="תנועת רעדוד" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%84%E3%82%A3%E3%83%83%E3%82%BF%E3%83%BC%E3%83%99%E3%83%B4%E3%82%A7%E3%83%BC%E3%82%B0%E3%83%B3%E3%82%AF" title="ツィッターベヴェーグンク – Japanese" lang="ja" hreflang="ja" data-title="ツィッターベヴェーグンク" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Zitterbewegung" title="Zitterbewegung – Polish" lang="pl" hreflang="pl" data-title="Zitterbewegung" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Zitterbewegung" title="Zitterbewegung – Portuguese" lang="pt" hreflang="pt" data-title="Zitterbewegung" 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/Zitterbewegung" title="Zitterbewegung – Russian" lang="ru" hreflang="ru" data-title="Zitterbewegung" data-language-autonym="Русский" data-language-local-name="Russian" 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July 2021</a> <span class="mw-diff-edit"><a href="/w/index.php?title=Zitterbewegung&action=edit&oldid=1036077288" title="Zitterbewegung">edit</a></span><span class="mw-diff-timestamp" data-timestamp="2021-07-29T10:21:29Z"></span></strong></div><div id="mw-diff-otitle2"><a href="/wiki/Special:Contributions/2A02:1812:143A:7700:19EA:1D3F:B619:5890" class="mw-userlink mw-anonuserlink" title="Special:Contributions/2A02:1812:143A:7700:19EA:1D3F:B619:5890" data-mw-revid="1036077288"><bdi>2a02:1812:143a:7700:19ea:1d3f:b619:5890</bdi></a> <span class="mw-usertoollinks">(<a href="/w/index.php?title=User_talk:2A02:1812:143A:7700:19EA:1D3F:B619:5890&action=edit&redlink=1" class="new mw-usertoollinks-talk" title="User talk:2A02:1812:143A:7700:19EA:1D3F:B619:5890 (page does not exist)">talk</a>)</span><div class="mw-diff-usermetadata"></div></div><div id="mw-diff-otitle3"> <span class="comment comment--without-parentheses"><span class="autocomment"><a href="#Interpretation">→<bdi dir="ltr">Interpretation</bdi></a></span></span></div><div id="mw-diff-otitle5"></div><div id="mw-diff-otitle4"><a href="/w/index.php?title=Zitterbewegung&diff=prev&oldid=1036077288" title="Zitterbewegung" id="differences-prevlink">← Previous edit</a></div></td> <td colspan="2" class="diff-ntitle diff-side-added"><div id="mw-diff-ntitle1"><strong><a href="/w/index.php?title=Zitterbewegung&oldid=1036077435" title="Zitterbewegung">Revision as of 10:23, 29 July 2021</a> <span class="mw-diff-edit"><a href="/w/index.php?title=Zitterbewegung&action=edit&oldid=1036077435" title="Zitterbewegung">edit</a></span><span class="mw-diff-timestamp" data-timestamp="2021-07-29T10:23:02Z"></span> <span class="mw-diff-undo"><a href="/w/index.php?title=Zitterbewegung&action=edit&undoafter=1036077288&undo=1036077435" title=""Undo" reverts this edit and opens the edit form in preview mode. It allows adding a reason in the summary.">undo</a></span></strong></div><div id="mw-diff-ntitle2"><a href="/wiki/Special:Contributions/2A02:1812:143A:7700:19EA:1D3F:B619:5890" class="mw-userlink mw-anonuserlink" title="Special:Contributions/2A02:1812:143A:7700:19EA:1D3F:B619:5890" data-mw-revid="1036077435"><bdi>2a02:1812:143a:7700:19ea:1d3f:b619:5890</bdi></a> <span class="mw-usertoollinks">(<a href="/w/index.php?title=User_talk:2A02:1812:143A:7700:19EA:1D3F:B619:5890&action=edit&redlink=1" class="new mw-usertoollinks-talk" title="User talk:2A02:1812:143A:7700:19EA:1D3F:B619:5890 (page does not exist)">talk</a>)</span><div class="mw-diff-usermetadata"></div></div><div id="mw-diff-ntitle3"> <span class="comment comment--without-parentheses"><span class="autocomment"><a href="#Interpretation_as_an_artifact">→<bdi dir="ltr">Interpretation as an artifact</bdi></a></span></span></div><div id="mw-diff-ntitle5"></div><div id="mw-diff-ntitle4"><a href="/w/index.php?title=Zitterbewegung&diff=next&oldid=1036077435" title="Zitterbewegung" id="differences-nextlink">Next edit →</a></div></td> </tr><tr> <td colspan="2" class="diff-lineno">Line 58:</td> <td colspan="2" class="diff-lineno">Line 58:</td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>In [[quantum electrodynamics]] the negative-energy states are replaced by [[positron]] states, and the zitterbewegung is understood as the result of interaction of the electron with spontaneously forming and annihilating electron-positron [[Pair production|pairs]].<ref>Zhi-Yong, W., & Cai-Dong, X. (2008). Zitterbewegung in quantum field theory. Chinese Physics B, 17(11), 4170.</ref></div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>In [[quantum electrodynamics]] the negative-energy states are replaced by [[positron]] states, and the zitterbewegung is understood as the result of interaction of the electron with spontaneously forming and annihilating electron-positron [[Pair production|pairs]].<ref>Zhi-Yong, W., & Cai-Dong, X. (2008). Zitterbewegung in quantum field theory. Chinese Physics B, 17(11), 4170.</ref></div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>More recently, it has been noted that it seem to be an artifact. Zitterbewegung is due to the "small components" of the dirac 4-spinor, due to a little bit of antiparticle mixed up in the particle wavefunction for a nonrelativistic motion. It doesn't appear in the correct second quantized theory, or rather, it is resolved by using Feynman propagators and doing QED. Nevertheless, it is an interesting way to understand certain QED effects heuristically from a single particle picture.</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>More recently, it has been noted that it seem to be an artifact. Zitterbewegung is due to the "small components" of the dirac 4-spinor, due to a little bit of antiparticle mixed up in the particle wavefunction for a nonrelativistic motion. It doesn't appear in the correct second quantized theory, or rather, it is resolved by using Feynman propagators and doing QED. Nevertheless, it is an interesting way to understand certain QED effects heuristically from a single particle picture.<ins class="diffchange diffchange-inline"> <ref> https://physics.stackexchange.com/questions/28672/is-zitterbewegung-an-artefact-of-single-particle-theory </ref></ins></div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>== Experimental simulation ==</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>== Experimental simulation ==</div></td> </tr> </table><hr class='diff-hr' id='mw-oldid' /> <h2 class='diff-currentversion-title'>Revision as of 10:23, 29 July 2021</h2> <div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="/wiki/Physics" title="Physics">physics</a>, the <b>zitterbewegung</b> ("jittery motion" in <a href="/wiki/German_language" title="German language">German</a>) is the predicted rapid oscillatory motion of elementary particles that obey <a href="/wiki/Relativistic_wave_equations" title="Relativistic wave equations">relativistic wave equations</a>. The existence of such motion was first discussed by <a href="/wiki/Gregory_Breit" title="Gregory Breit">Gregory Breit</a> in 1928<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> as a result of his analysis of the <a href="/wiki/Wave_packet" title="Wave packet">wave packet</a> solutions of the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a> for relativistic <a href="/wiki/Electron" title="Electron">electrons</a> in free space, in which an <a href="/wiki/Interference_(wave_propagation)#Quantum_interference" class="mw-redirect" title="Interference (wave propagation)">interference</a> between positive and negative <a href="/wiki/Energy_state" class="mw-redirect" title="Energy state">energy states</a> produces what appears to be a fluctuation (up to the speed of light) of the position of an electron around the median, with an <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a> of <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">2<i>mc</i><sup>2</sup></span><span class="sr-only">/</span><span class="den"><i>ℏ</i></span></span>⁠</span></span>, or approximately <span class="nowrap"><span data-sort-value="7021160000000000000♠"></span>1.6<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>21</sup></span> radians per second. For the <a href="/wiki/Hydrogen-like_atom" title="Hydrogen-like atom">hydrogen atom</a>, zitterbewegung can be invoked as a heuristic way to derive the <a href="/wiki/Fine_structure#Darwin_term" title="Fine structure">Darwin term</a>, a small correction of the energy level of the <a href="/wiki/Atomic_orbital" title="Atomic orbital">s-orbitals</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Theory">Theory</h2></div> <div class="mw-heading mw-heading3"><h3 id="Free_fermion">Free fermion</h3></div> <p>The time-dependent <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a> is written 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 H\psi (\mathbf {x} ,t)=i\hbar {\frac {\partial \psi }{\partial t}}(\mathbf {x} ,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ψ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\psi (\mathbf {x} ,t)=i\hbar {\frac {\partial \psi }{\partial t}}(\mathbf {x} ,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62338d3632aed9883de29ec51277e2d359736a0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.639ex; height:5.676ex;" alt="{\displaystyle H\psi (\mathbf {x} ,t)=i\hbar {\frac {\partial \psi }{\partial t}}(\mathbf {x} ,t)}"></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 \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> is the (reduced) <a href="/wiki/Planck_constant" title="Planck constant">Planck constant</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 \psi (\mathbf {x} ,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {x} ,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0d619b826c3569f03532d5d67084d378a43b5bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.607ex; height:2.843ex;" alt="{\displaystyle \psi (\mathbf {x} ,t)}"></span> is the <a href="/wiki/Wave_function" title="Wave function">wave function</a> (<a href="/wiki/Bispinor" title="Bispinor">bispinor</a>) of a <a href="/wiki/Fermion" title="Fermion">fermionic</a> particle <a href="/wiki/Spin_1/2_particles" class="mw-redirect" title="Spin 1/2 particles">spin-½</a>, and <span class="texhtml mvar" style="font-style:italic;">H</span> is the Dirac <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> of a <a href="/wiki/Free_particle" title="Free particle">free particle</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 H=\beta mc^{2}+\sum _{j=1}^{3}\alpha _{j}p_{j}c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mi>β</mi> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=\beta mc^{2}+\sum _{j=1}^{3}\alpha _{j}p_{j}c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70ae43f97f7bf1f49fd4808600e1dd02dbb4c4e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.661ex; height:7.509ex;" alt="{\displaystyle H=\beta mc^{2}+\sum _{j=1}^{3}\alpha _{j}p_{j}c}"></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="{\textstyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3296968aacae7d971a6d649f6b32b5969ed6d4d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\textstyle m}"></span> is the mass of the particle, <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="{\textstyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d411ca19645ddd4fff0704de95ec770681093bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\textstyle c}"></span> is the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</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="{\textstyle p_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle p_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3bbf2b50e15fe11ec7c0f2b7ac5a0a05f030fe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:2.169ex; height:2.343ex;" alt="{\textstyle p_{j}}"></span> is the <a href="/wiki/Momentum_operator" title="Momentum operator">momentum operator</a>, 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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></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 \alpha _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/293a364991ab1ee55c25b0f60fd9e52af7b7dbde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.397ex; height:2.343ex;" alt="{\displaystyle \alpha _{j}}"></span> are matrices related to the <a href="/wiki/Gamma_matrices" title="Gamma matrices">Gamma matrices</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="{\textstyle \gamma _{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \gamma _{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eaae1d7c3f0ae7227091468c9bc57845dac270" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.428ex; height:2.343ex;" alt="{\textstyle \gamma _{\mu }}"></span>, 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="{\textstyle \beta =\gamma _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \beta =\gamma _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14bad78459a33c225a3af52426e82039f2b52999" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.689ex; height:2.676ex;" alt="{\textstyle \beta =\gamma _{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="{\textstyle \alpha _{j}=\gamma _{0}\gamma _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \alpha _{j}=\gamma _{0}\gamma _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71e60bb70b16f40e624afa43aa24aa7e02d209ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.868ex; height:2.343ex;" alt="{\textstyle \alpha _{j}=\gamma _{0}\gamma _{j}}"></span>. </p><p>In the <a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg picture</a>, the time dependence of an arbitrary observable <span class="texhtml mvar" style="font-style:italic;">Q</span> obeys the 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 -i\hbar {\frac {\partial Q}{\partial t}}=\left[H,Q\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>Q</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mi>H</mi> <mo>,</mo> <mi>Q</mi> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i\hbar {\frac {\partial Q}{\partial t}}=\left[H,Q\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fae5fb9f65448e69c7101a8a9fc00b360f7b44a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.272ex; height:5.509ex;" alt="{\displaystyle -i\hbar {\frac {\partial Q}{\partial t}}=\left[H,Q\right].}"></span></dd></dl> <p>In particular, the time-dependence of the <a href="/wiki/Position_operator" title="Position operator">position operator</a> 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 {\partial x_{k}(t)}{\partial t}}={\frac {i}{\hbar }}\left[H,x_{k}\right]=c\alpha _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mi>H</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mi>c</mi> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial x_{k}(t)}{\partial t}}={\frac {i}{\hbar }}\left[H,x_{k}\right]=c\alpha _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a4ef7d2df54f3d62abfae88bdbddab7685d114" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.341ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial x_{k}(t)}{\partial t}}={\frac {i}{\hbar }}\left[H,x_{k}\right]=c\alpha _{k}}"></span>.</dd></dl> <p>where <span class="texhtml"><i>x<sub>k</sub></i>(<i>t</i>)</span> is the position operator at time <span class="texhtml mvar" style="font-style:italic;">t</span>. </p><p>The above equation shows that the operator <span class="texhtml mvar" style="font-style:italic;">α<sub>k</sub></span> can be interpreted as the <span class="texhtml mvar" style="font-style:italic;">k</span>-th component of a "velocity operator". </p><p>Note that this implies that </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 \left\langle \left({\frac {\partial x_{k}(t)}{\partial t}}\right)^{2}\right\rangle =c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⟩</mo> </mrow> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle \left({\frac {\partial x_{k}(t)}{\partial t}}\right)^{2}\right\rangle =c^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78ba1862ad8028591fa06ab38f00aea4c7858cb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.603ex; height:7.509ex;" alt="{\displaystyle \left\langle \left({\frac {\partial x_{k}(t)}{\partial t}}\right)^{2}\right\rangle =c^{2}}"></span>,</dd></dl> <p>as if the "root mean square speed" in every direction of space is the speed of light. </p><p>To add time-dependence to <span class="texhtml mvar" style="font-style:italic;">α<sub>k</sub></span>, one implements the Heisenberg picture, which says </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 \alpha _{k}(t)=e^{\frac {iHt}{\hbar }}\alpha _{k}e^{-{\frac {iHt}{\hbar }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi>H</mi> <mi>t</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </msup> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi>H</mi> <mi>t</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{k}(t)=e^{\frac {iHt}{\hbar }}\alpha _{k}e^{-{\frac {iHt}{\hbar }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a71c44e899a85e3898a49bca376b975f62f56c56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.737ex; height:4.176ex;" alt="{\displaystyle \alpha _{k}(t)=e^{\frac {iHt}{\hbar }}\alpha _{k}e^{-{\frac {iHt}{\hbar }}}}"></span>.</dd></dl> <p>The time-dependence of the velocity operator 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 \hbar {\frac {\partial \alpha _{k}(t)}{\partial t}}=i\left[H,\alpha _{k}\right]=2\left(i\gamma _{k}m-\sigma _{kl}p^{l}\right)=2i\left(p_{k}-\alpha _{k}H\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>i</mi> <mrow> <mo>[</mo> <mrow> <mi>H</mi> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mi>m</mi> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mi>H</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar {\frac {\partial \alpha _{k}(t)}{\partial t}}=i\left[H,\alpha _{k}\right]=2\left(i\gamma _{k}m-\sigma _{kl}p^{l}\right)=2i\left(p_{k}-\alpha _{k}H\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0861b1a62360f4eb20af79407d70a4cb03154826" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:56.492ex; height:5.843ex;" alt="{\displaystyle \hbar {\frac {\partial \alpha _{k}(t)}{\partial t}}=i\left[H,\alpha _{k}\right]=2\left(i\gamma _{k}m-\sigma _{kl}p^{l}\right)=2i\left(p_{k}-\alpha _{k}H\right)}"></span>,</dd></dl> <p>where </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 \sigma _{kl}\equiv {\frac {i}{2}}\left[\gamma _{k},\gamma _{l}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> </msub> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{kl}\equiv {\frac {i}{2}}\left[\gamma _{k},\gamma _{l}\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e30fb903c017cedae72c7cb47d457903775cb62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.972ex; height:5.176ex;" alt="{\displaystyle \sigma _{kl}\equiv {\frac {i}{2}}\left[\gamma _{k},\gamma _{l}\right].}"></span></dd></dl> <p>Now, because both <span class="texhtml mvar" style="font-style:italic;">p<sub>k</sub></span> and <span class="texhtml mvar" style="font-style:italic;">H</span> are time-independent, the above equation can easily be integrated twice to find the explicit time-dependence of the position operator. </p><p>First: </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 \alpha _{k}(t)=\left(\alpha _{k}(0)-cp_{k}H^{-1}\right)e^{-{\frac {2iHt}{\hbar }}}+cp_{k}H^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>c</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>i</mi> <mi>H</mi> <mi>t</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </mrow> </msup> <mo>+</mo> <mi>c</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{k}(t)=\left(\alpha _{k}(0)-cp_{k}H^{-1}\right)e^{-{\frac {2iHt}{\hbar }}}+cp_{k}H^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/952482e7d512680719f8507667b19377bc293075" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.697ex; height:4.343ex;" alt="{\displaystyle \alpha _{k}(t)=\left(\alpha _{k}(0)-cp_{k}H^{-1}\right)e^{-{\frac {2iHt}{\hbar }}}+cp_{k}H^{-1}}"></span>,</dd></dl> <p>and finally </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_{k}(t)=x_{k}(0)+c^{2}p_{k}H^{-1}t+{\tfrac {1}{2}}i\hbar cH^{-1}\left(\alpha _{k}(0)-cp_{k}H^{-1}\right)\left(e^{-{\frac {2iHt}{\hbar }}}-1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mi>c</mi> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>c</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>i</mi> <mi>H</mi> <mi>t</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{k}(t)=x_{k}(0)+c^{2}p_{k}H^{-1}t+{\tfrac {1}{2}}i\hbar cH^{-1}\left(\alpha _{k}(0)-cp_{k}H^{-1}\right)\left(e^{-{\frac {2iHt}{\hbar }}}-1\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0d72bf3c61170428c67cec0a569569beed5ac57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:70.685ex; height:6.176ex;" alt="{\displaystyle x_{k}(t)=x_{k}(0)+c^{2}p_{k}H^{-1}t+{\tfrac {1}{2}}i\hbar cH^{-1}\left(\alpha _{k}(0)-cp_{k}H^{-1}\right)\left(e^{-{\frac {2iHt}{\hbar }}}-1\right)}"></span>.</dd></dl> <p>The resulting expression consists of an initial position, a motion proportional to time, and an oscillation term with an amplitude equal to the <a href="/wiki/Compton_wavelength" title="Compton wavelength">Compton wavelength</a>. That oscillation term is the so-called zitterbewegung. </p> <div class="mw-heading mw-heading3"><h3 id="Interpretation_as_an_artifact">Interpretation as an artifact</h3></div> <p>In quantum mechanics, the zitterbewegung term vanishes on taking expectation values for wave-packets that are made up entirely of positive- (or entirely of negative-) energy waves. The standard relativistic velocity can be recovered by taking a <a href="/wiki/Foldy%E2%80%93Wouthuysen_transformation" title="Foldy–Wouthuysen transformation">Foldy–Wouthuysen transformation</a>, when the positive and negative components are decoupled. Thus, we arrive at the interpretation of the zitterbewegung as being caused by interference between positive- and negative-energy wave components. </p><p>In <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a> the negative-energy states are replaced by <a href="/wiki/Positron" title="Positron">positron</a> states, and the zitterbewegung is understood as the result of interaction of the electron with spontaneously forming and annihilating electron-positron <a href="/wiki/Pair_production" title="Pair production">pairs</a>.<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> </p><p>More recently, it has been noted that it seem to be an artifact. Zitterbewegung is due to the "small components" of the dirac 4-spinor, due to a little bit of antiparticle mixed up in the particle wavefunction for a nonrelativistic motion. It doesn't appear in the correct second quantized theory, or rather, it is resolved by using Feynman propagators and doing QED. Nevertheless, it is an interesting way to understand certain QED effects heuristically from a single particle picture. <sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Experimental_simulation">Experimental simulation</h2></div> <p>Zitterbewegung of a free relativistic particle has never been observed directly, although some authors believe they have found evidence in favor of its existence.<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> It has also been simulated twice in model systems that provide condensed-matter analogues of the relativistic phenomenon. The first example, in 2010, placed a trapped ion in an environment such that the non-relativistic Schrödinger equation for the ion had the same mathematical form as the Dirac equation (although the physical situation is different).<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><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> Then, in 2013, it was simulated in a setup with <a href="/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">Bose–Einstein condensates</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>Other proposals for condensed-matter analogues include semiconductor nanostructures, <a href="/wiki/Graphene" title="Graphene">graphene</a> and <a href="/wiki/Topological_insulators" class="mw-redirect" title="Topological insulators">topological insulators</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><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><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <ul><li><a href="/wiki/Casimir_effect" title="Casimir effect">Casimir effect</a></li> <li><a href="/wiki/Lamb_shift" title="Lamb shift">Lamb shift</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <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"> <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"><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="CITEREFBreit1928" class="citation journal cs1"><a href="/wiki/Gregory_Breit" title="Gregory Breit">Breit, Gregory</a> (1928). <a rel="nofollow" class="external text" href="http://www.pnas.org/cgi/doi/10.1073/pnas.14.7.553">"An Interpretation of Dirac's Theory of the Electron"</a>. <i>Proceedings of the National Academy of Sciences</i>. <b>14</b> (7): 553–559. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.14.7.553">10.1073/pnas.14.7.553</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/0027-8424">0027-8424</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1085609">1085609</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16587362">16587362</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+National+Academy+of+Sciences&rft.atitle=An+Interpretation+of+Dirac%27s+Theory+of+the+Electron&rft.volume=14&rft.issue=7&rft.pages=553-559&rft.date=1928&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1085609%23id-name%3DPMC&rft.issn=0027-8424&rft_id=info%3Apmid%2F16587362&rft_id=info%3Adoi%2F10.1073%2Fpnas.14.7.553&rft.aulast=Breit&rft.aufirst=Gregory&rft_id=http%3A%2F%2Fwww.pnas.org%2Fcgi%2Fdoi%2F10.1073%2Fpnas.14.7.553&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZitterbewegung" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreiner1995" class="citation book cs1">Greiner, Walter (1995). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-3-642-88082-7"><i>Relativistic Quantum Mechanics</i></a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-88082-7">10.1007/978-3-642-88082-7</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-99535-7" title="Special:BookSources/978-3-540-99535-7"><bdi>978-3-540-99535-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativistic+Quantum+Mechanics&rft.date=1995&rft_id=info%3Adoi%2F10.1007%2F978-3-642-88082-7&rft.isbn=978-3-540-99535-7&rft.aulast=Greiner&rft.aufirst=Walter&rft_id=https%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%2F978-3-642-88082-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZitterbewegung" class="Z3988"></span></span> </li> <li 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"Zitterbewegung, chirality, and minimal conductivity in graphene". <i><a href="/wiki/The_European_Physical_Journal_B" class="mw-redirect" title="The European Physical Journal B">The European Physical Journal B</a></i>. <b>51</b> (2): 157–160. <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/cond-mat/0512337">cond-mat/0512337</a></span>. <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%2Fepjb%2Fe2006-00203-1">10.1140/epjb/e2006-00203-1</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+B&rft.atitle=Zitterbewegung%2C+chirality%2C+and+minimal+conductivity+in+graphene&rft.volume=51&rft.issue=2&rft.pages=157-160&rft.date=2006&rft_id=info%3Aarxiv%2Fcond-mat%2F0512337&rft_id=info%3Adoi%2F10.1140%2Fepjb%2Fe2006-00203-1&rft.aulast=Katsnelson&rft.aufirst=M.+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZitterbewegung" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDóraCayssolSimonMoessner2012" class="citation journal cs1">Dóra, Balász; Cayssol, Jérôme; Simon, Ference; Moessner, Roderich (2012). "Optically engineering the topological properties of a spin Hall insulator". <i><a href="/wiki/Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a></i>. <b>108</b> (5): 056602. <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/1105.5963">1105.5963</a></span>. <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%2FPhysRevLett.108.056602">10.1103/PhysRevLett.108.056602</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/22400947">22400947</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+Letters&rft.atitle=Optically+engineering+the+topological+properties+of+a+spin+Hall+insulator&rft.volume=108&rft.issue=5&rft.pages=056602&rft.date=2012&rft_id=info%3Aarxiv%2F1105.5963&rft_id=info%3Apmid%2F22400947&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.108.056602&rft.aulast=D%C3%B3ra&rft.aufirst=Bal%C3%A1sz&rft.au=Cayssol%2C+J%C3%A9r%C3%B4me&rft.au=Simon%2C+Ference&rft.au=Moessner%2C+Roderich&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZitterbewegung" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShiZhangCheng2013" class="citation journal cs1">Shi, Likun; Zhang, Shoucheng; Cheng, Kai (2013). "Anomalous Electron Trajectory in Topological Insulators". <i><a href="/wiki/Physical_Review_B" title="Physical Review B">Physical Review B</a></i>. <b>87</b> (16). <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/1109.4771">1109.4771</a></span>. <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%2FPhysRevB.87.161115">10.1103/PhysRevB.87.161115</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+B&rft.atitle=Anomalous+Electron+Trajectory+in+Topological+Insulators&rft.volume=87&rft.issue=16&rft.date=2013&rft_id=info%3Aarxiv%2F1109.4771&rft_id=info%3Adoi%2F10.1103%2FPhysRevB.87.161115&rft.aulast=Shi&rft.aufirst=Likun&rft.au=Zhang%2C+Shoucheng&rft.au=Cheng%2C+Kai&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZitterbewegung" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchrödinger1930" class="citation book cs1 cs1-prop-foreign-lang-source">Schrödinger, E. (1930). <i>Über die kräftefreie Bewegung in der relativistischen Quantenmechanik</i> [<i>On the free movement in relativistic quantum mechanics</i>] (in German). pp. 418–428. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/881393652">881393652</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%C3%9Cber+die+kr%C3%A4ftefreie+Bewegung+in+der+relativistischen+Quantenmechanik&rft.pages=418-428&rft.date=1930&rft_id=info%3Aoclcnum%2F881393652&rft.aulast=Schr%C3%B6dinger&rft.aufirst=E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZitterbewegung" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchrödinger1931" class="citation book cs1 cs1-prop-foreign-lang-source">Schrödinger, E. (1931). <i>Zur Quantendynamik des Elektrons</i> [<i>Quantum Dynamics of the Electron</i>] (in German). pp. 63–72.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Zur+Quantendynamik+des+Elektrons&rft.pages=63-72&rft.date=1931&rft.aulast=Schr%C3%B6dinger&rft.aufirst=E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZitterbewegung" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMessiah1962" class="citation book cs1">Messiah, A. (1962). <a rel="nofollow" class="external text" href="https://archive.org/details/QuantumMechanicsVolumeIi">"XX, Section 37"</a> <span class="cs1-format">(pdf)</span>. <i>Quantum Mechanics</i>. Vol. II. pp. 950–952. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780471597681" title="Special:BookSources/9780471597681"><bdi>9780471597681</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=XX%2C+Section+37&rft.btitle=Quantum+Mechanics&rft.pages=950-952&rft.date=1962&rft.isbn=9780471597681&rft.aulast=Messiah&rft.aufirst=A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2FQuantumMechanicsVolumeIi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZitterbewegung" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div> <ul><li><a rel="nofollow" class="external text" href="https://www.newscientist.com/channel/fundamentals/mg19526112.300-the-word-zitterbewebung.html">Zitterbewegung in New Scientist</a></li> <li><a rel="nofollow" class="external text" href="http://geocalc.clas.asu.edu/pdf/ZBWinQM15**.pdf">Geometric Algebra in Quantum Mechanics</a></li></ul></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Zitterbewegung&oldid=1036077435">https://en.wikipedia.org/w/index.php?title=Zitterbewegung&oldid=1036077435</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Quantum_field_theory" title="Category:Quantum field theory">Quantum field theory</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden category: <ul><li><a href="/wiki/Category:CS1_German-language_sources_(de)" title="Category:CS1 German-language sources (de)">CS1 German-language sources (de)</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 29 July 2021, at 10:23<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright"><span class="nowraplinks">This version of the page has been <a href="/wiki/Help:Page_history" title="Help:Page history">revised</a>. 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