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xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="physics">Physics</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a> </li> </ul> <hr /> <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a> <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a> <ul> <li> Axiomatizations <ul> <li> <a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net">local net</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem </li> <li> <a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a> </li> </ul> </li> </ul> </li> <li> Tools <ul> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> Structural phenomena <ul> <li> <a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a> </li> </ul> </li> <li> Types of quantum field thories <ul> <li> <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a> </li> <li> examples <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/membrane">membrane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </div></div></div> <h4 id="fields_and_quanta">Fields and quanta</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> and <a class="existingWikiWord" href="/nlab/show/fundamental+particle">particles</a> in <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a> and in the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>: <a class="existingWikiWord" href="/nlab/show/force">force</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge</a> <a class="existingWikiWord" href="/nlab/show/bosons">bosons</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/photon">photon</a> - <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> (<a class="existingWikiWord" href="/nlab/show/abelian+group">abelian</a> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/W-boson">W</a>, <a class="existingWikiWord" href="/nlab/show/Z-boson">Z</a>, <a class="existingWikiWord" href="/nlab/show/B-boson">B-boson</a> - <a class="existingWikiWord" href="/nlab/show/electroweak+field">electroweak field</a> (<a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/gluon">gluon</a> - <a class="existingWikiWord" href="/nlab/show/strong+nuclear+force">strong nuclear force</a> (<a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/graviton">graviton</a> - <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/infraparticle">infraparticle</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/scalar+field">scalar</a> <a class="existingWikiWord" href="/nlab/show/bosons">bosons</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Higgs+boson">Higgs boson</a>, <a class="existingWikiWord" href="/nlab/show/inflaton">inflaton</a> (<a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a>)</li> </ul> <a class="existingWikiWord" href="/nlab/show/matter">matter</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> (<a class="existingWikiWord" href="/nlab/show/spinors">spinors</a>, <a class="existingWikiWord" href="/nlab/show/Dirac+fields">Dirac fields</a>) <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/flavor+%28particle+physics%29">flavors</a> of <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental</a> <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> in the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>:</th><th></th><th></th><th></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generation+of+fermions">generation of fermions</a></td><td style="text-align: left;">1st generation</td><td style="text-align: left;">2nd generation</td><td style="text-align: left;">3d generation</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quarks">quarks</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">up-type</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/up+quark">up quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/charm+quark">charm quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/top+quark">top quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;">down-type</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/down+quark">down quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strange+quark">strange quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bottom+quark">bottom quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/leptons">leptons</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">charged</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/electron">electron</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/muon">muon</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tauon">tauon</a></td></tr> <tr><td style="text-align: left;">neutral</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/electron+neutrino">electron neutrino</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/muon+neutrino">muon neutrino</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tau+neutrino">tau neutrino</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bound+states">bound states</a>:</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/mesons">mesons</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/light+mesons">light mesons</a>: <a class="existingWikiWord" href="/nlab/show/pion">pion</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/%CF%81-meson">ρ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/%CF%89-meson">ω-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/f1-meson">f1-meson</a> <a class="existingWikiWord" href="/nlab/show/a1-meson">a1-meson</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strange+quark">strange</a>-mesons: <a class="existingWikiWord" href="/nlab/show/%CF%95-meson">ϕ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mover><mi>s</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">s \bar s</annotation></semantics></math>), <a class="existingWikiWord" href="/nlab/show/kaon">kaon</a>, <a class="existingWikiWord" href="/nlab/show/K%2A-meson">K*-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">u s</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">d s</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/eta-meson">eta-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>u</mi><mo>+</mo><mi>d</mi><mi>d</mi><mo>+</mo><mi>s</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">u u + d d + s s</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/charm+quark">charmed</a> <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy mesons</a>: <a class="existingWikiWord" href="/nlab/show/D-meson">D-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>c</mi></mrow><annotation encoding="application/x-tex"> u c</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">d c</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">s c</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/J%2F%CF%88-meson">J/ψ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mover><mi>c</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">c \bar c</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bottom+quark">bottom</a> <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy mesons</a>: <a class="existingWikiWord" href="/nlab/show/B-meson">B-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">q b</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/%CF%92-meson">ϒ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mover><mi>b</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">b \bar b</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/baryons">baryons</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nucleons">nucleons</a>: <a class="existingWikiWord" href="/nlab/show/proton">proton</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mi>u</mi><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(u u d)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/neutron">neutron</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mi>d</mi><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(u d d)</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> (also: <a class="existingWikiWord" href="/nlab/show/antiparticles">antiparticles</a>) <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective particles</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Goldstone+bosons">Goldstone bosons</a></li> </ul> <a class="existingWikiWord" href="/nlab/show/hadrons">hadrons</a> (<a class="existingWikiWord" href="/nlab/show/bound+states">bound states</a> of the above <a class="existingWikiWord" href="/nlab/show/quarks">quarks</a>) <ul> <li> <a class="existingWikiWord" href="/nlab/show/meson">meson</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/scalar+meson">scalar meson</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/%CF%83-meson">σ-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/pion">pion</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/kaon">kaon</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D-meson">D-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/B-meson">B-meson</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/vector+meson">vector meson</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/%CF%89-meson">ω-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%CF%81-meson">ρ-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/f1-meson">f1-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/a1-meson">a1-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/b1-meson">b1-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/h1-meson">h1-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/K%2A-meson">K*-meson</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/tensor+meson">tensor meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quarkonium">quarkonium</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/charmonium">charmonium</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%CF%92-meson">ϒ-meson</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/exotic+meson">exotic meson</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/XYZ+meson">XYZ meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tetraquark">tetraquark</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/baryon">baryon</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/nucleon">nucleon</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/proton">proton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/neutron">neutron</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/chemical+element">chemical element</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/carbon">carbon</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/nitrogen">nitrogen</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lambda+baryon">Lambda baryon</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/pentaquark">pentaquark</a> </li> </ul> </li> </ul> <a class="existingWikiWord" href="/nlab/show/solitons">solitons</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrmion</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/caloron">caloron</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a> </li> </ul> in <a class="existingWikiWord" href="/nlab/show/grand+unified+theory">grand unified theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/leptoquark">leptoquark</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Z%27-boson">Z'-boson</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/MSSM">minimally extended supersymmetric standard model</a> <a class="existingWikiWord" href="/nlab/show/superpartners">superpartners</a> bosinos: <ul> <li> <a class="existingWikiWord" href="/nlab/show/gravitino">gravitino</a> - <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> of <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> (<a class="existingWikiWord" href="/nlab/show/Rarita-Schwinger+field">Rarita-Schwinger field</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/gaugino">gaugino</a> - <a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gluino">gluino</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/sfermions">sfermions</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/squark">squark</a></li> </ul> <a class="existingWikiWord" href="/nlab/show/dark+matter">dark matter</a> candidates <ul> <li> <a class="existingWikiWord" href="/nlab/show/WIMP">WIMP</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/axion">axion</a> </li> </ul> Exotica <ul> <li> <a class="existingWikiWord" href="/nlab/show/preon">preon</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/graviphoton">graviphoton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/dilaton">dilaton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/monopole">monopole</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/dual+graviton">dual graviton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/giant+graviton">giant graviton</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/auxiliary+fields">auxiliary fields</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/ghost+field">ghost field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/antifield">antifield</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/antighost+field">antighost field</a>, <a class="existingWikiWord" href="/nlab/show/Nakanishi-Lautrup+field">Nakanishi-Lautrup field</a> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#NeutrinoMasses'>Neutrino masses</a></li> <li><a href='#history_and_prediction'>History and prediction</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> The neutrino is one of the fundamental <a class="existingWikiWord" href="/nlab/show/particles">particles</a>/<a class="existingWikiWord" href="/nlab/show/matter">matter</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> in the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>. <h2 id="NeutrinoMasses">Neutrino masses</h2> The origin and nature of the observed tiny but non-vanishing <a class="existingWikiWord" href="/nlab/show/neutrino">neutrino</a> <a class="existingWikiWord" href="/nlab/show/masses">masses</a> remains one of the open problems of the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>. Review includes <a href="#FisherKayserMcFarland99">Fisher-Kayser-McFarland 99</a>. On neutrino <a class="existingWikiWord" href="/nlab/show/masses">masses</a> and the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> as an <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective field theory</a>: <a href="#Weinberg09">Weinberg 09, p. 15</a>: <blockquote> I also noted at the same time that interactions between a pair of lepton doublets and a pair of scalar doublets can generate a neutrino mass, which is suppressed only by a factor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>M</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">M^{-1}</annotation></semantics></math>, and that therefore with a reasonable estimate of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> could produce observable neutrino oscillations. The subsequent confirmation of neutrino oscillations lends support to the view of the Standard Model as an effective field theory, with M somewhere in the neighborhood of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>10</mn> <mn>16</mn></msup><mi>GeV</mi></mrow><annotation encoding="application/x-tex">10^{16} GeV</annotation></semantics></math>. </blockquote> In fact this scale is about the <a class="existingWikiWord" href="/nlab/show/GUT+scale">GUT scale</a> for <a class="existingWikiWord" href="/nlab/show/grand+unified+theories">grand unified theories</a>, see for instance (<a href="#Mohapatra06">Mohapatra 06</a>) for more on this relation, and see at <a class="existingWikiWord" href="/nlab/show/seesaw+mechanism">seesaw mechanism</a>. Detailed matching of parameters of non-supersymmetric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(10)</annotation></semantics></math>-GUT to <a class="existingWikiWord" href="/nlab/show/neutrino">neutrino</a> <a class="existingWikiWord" href="/nlab/show/masses">masses</a> is discussed in <a href="#OhlssonPernow19">Ohlsson-Pernow 19</a> <h2 id="history_and_prediction">History and prediction</h2> The history of the prediction of the neutrino is interesting and may (or may not) contain some general lessons for theoretical physics/<a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>. Before 1930 <a class="existingWikiWord" href="/nlab/show/experiments">experiments</a> detected an apparent violation of the <a class="existingWikiWord" href="/nlab/show/conservation+law">conservation law</a> for <a class="existingWikiWord" href="/nlab/show/energy">energy</a> in processes of <a class="existingWikiWord" href="/nlab/show/beta+decay">beta decay</a>. Back then <a class="existingWikiWord" href="/nlab/show/Niels+Bohr">Niels Bohr</a> proposed that, therefore, apparently the fundamental <a class="existingWikiWord" href="/nlab/show/conservation+laws">conservation laws</a> in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> may be violated and possibly only hold <a class="existingWikiWord" href="/nlab/show/probability+theory">statistically</a> over many quantum mechanical processes, but possibly not microscopically. Opposed to this was the suggestion by <a class="existingWikiWord" href="/nlab/show/Wolfgang+Pauli">Wolfgang Pauli</a>, who insisted that the <a class="existingWikiWord" href="/nlab/show/conservation+laws">conservation laws</a> ought to hold true, and that therefore there must be an undetected new <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a> created in <a class="existingWikiWord" href="/nlab/show/beta+decay">beta decay</a>, which carries away the apparently missing <a class="existingWikiWord" href="/nlab/show/energy">energy</a>. Pauli with his argument and suggestion turned out to be right. The missing particle – called the neutrino by <a class="existingWikiWord" href="/nlab/show/Enrico+Fermi">Enrico Fermi</a> in 1933 – was finally directly detected in 1956, hence 26 years after its proposal. (Compare maybe to the <a class="existingWikiWord" href="/nlab/show/Higgs+boson">Higgs boson</a> for which detection came 50 years after prediction.) Notice that back then, predicting unobserved and possibly practically unobservable <a class="existingWikiWord" href="/nlab/show/fundamental+particles">fundamental particles</a> was not taken as lightly as in some circles it is these days (e.g. in <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> and/or <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>). In (<a href="#AP06">AP 06, p. 7</a>) Pauli is quoted on this as having said: <blockquote> I’ve done a terrible thing today, something which no theoretical physicist should ever do. I have suggested something that can never be verified experimentally. </blockquote> But concerning the motivation for this predictions, notice that back around 1915 <a class="existingWikiWord" href="/nlab/show/Emmy+Noether">Emmy Noether</a> had proven the theorem now named after her – <a class="existingWikiWord" href="/nlab/show/Noether%27s+theorem">Noether's first theorem</a> – which asserts that <a class="existingWikiWord" href="/nlab/show/symmetries">symmetries</a> of <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> correspond to <a class="existingWikiWord" href="/nlab/show/conservation+laws">conservation laws</a>. In particular if a system (notably the <a class="existingWikiWord" href="/nlab/show/forces">forces</a> and <a class="existingWikiWord" href="/nlab/show/interactions">interactions</a> of <a class="existingWikiWord" href="/nlab/show/fundamental+particles">fundamental particles</a>) does not change over <a class="existingWikiWord" href="/nlab/show/time">time</a>, then <a class="existingWikiWord" href="/nlab/show/Noether%27s+theorem">Noether's first theorem</a> asserts that the <a class="existingWikiWord" href="/nlab/show/energy">energy</a> of the system must be conserved. The theorem applies as soon as the <a class="existingWikiWord" href="/nlab/show/physics">physics</a> is described by a <a class="existingWikiWord" href="/nlab/show/local+Lagrangian">local Lagrangian</a>/<a class="existingWikiWord" href="/nlab/show/local+action+functional">local action functional</a> and the <a class="existingWikiWord" href="/nlab/show/principle+of+extremal+action">principle of extremal action</a>. This was in principle well established in 1930, even though maybe not as widely appreciated as it could have been. The generalization of this to <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> and <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> was maybe fully understood only later. But in view of this theorem, Bohr’s suggestion that energy conservation fails would have implied that fundamental physics is not governed by an <a class="existingWikiWord" href="/nlab/show/action+principle">action principle</a>, something arguably more dramatic than some violation of some energy conservation might seem. Today Pauli is widely acknowledged for holding up the conservation laws against Bohr’s proposal, see for instance (<a href="#AP06">AP 06, p. 6</a>), where it says: <blockquote> Pauli’s belief in the absolute credibility of symmetry principles led him to defend conservation laws even when at that time the empirical evidence was doubtful. His prediction of the neutrino is a great example. </blockquote> But on the other hand, it seems that Pauli’s respect for the conservation laws was not informed by Noether’s theorem, but rested rather on an intuitive feeling, for in (<a href="#AP06">AP 06, p.5</a>) Pauli is quoted as late as 1953 thus: <blockquote> I am very much in favour of the general principle to bring empirical conservation laws and invariance properties in connection with mathematical groups of transformations of the laws of nature. </blockquote> While this does support the correct answer, it seems to be a rather weak way of stating it, given that Noether’s theorem establishes this “connection” as a theorem, already back in 1915. Similarly, in (<a href="#AP06">AP 06, p. 6</a>) Pauli is quoted as reacting to Bohr’s proposal by saying: <blockquote> I am myself fairly convinced <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[...]</annotation></semantics></math> that Bohr with his corresponding deliberations concerning a violation of energy conservation is entirely on the wrong track! <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[...]</annotation></semantics></math> The idea of a violation of the conservation of energy in β-decay is and remains, in my opinion, cheap and very clumsy philosophy. </blockquote> If Pauli had really been relying on symmetry and the Noether theorem, he could have said “provably wrong” instead of just “cheap and clumsy”. Especially since “clumsy” suggests “possible, even if not enjoyable”, where in fact it is impossible unless the whole foundations of physics are changed. <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/mass+term">mass term</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/sterile+neutrino">sterile neutrino</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/ANITA+experiment">ANITA experiment</a> </li> </ul> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/flavor+%28particle+physics%29">flavors</a> of <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental</a> <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> in the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>:</th><th></th><th></th><th></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generation+of+fermions">generation of fermions</a></td><td style="text-align: left;">1st generation</td><td style="text-align: left;">2nd generation</td><td style="text-align: left;">3d generation</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quarks">quarks</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">up-type</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/up+quark">up quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/charm+quark">charm quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/top+quark">top quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;">down-type</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/down+quark">down quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strange+quark">strange quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bottom+quark">bottom quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/leptons">leptons</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">charged</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/electron">electron</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/muon">muon</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tauon">tauon</a></td></tr> <tr><td style="text-align: left;">neutral</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/electron+neutrino">electron neutrino</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/muon+neutrino">muon neutrino</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tau+neutrino">tau neutrino</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bound+states">bound states</a>:</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/mesons">mesons</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/light+mesons">light mesons</a>: <a class="existingWikiWord" href="/nlab/show/pion">pion</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/%CF%81-meson">ρ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/%CF%89-meson">ω-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/f1-meson">f1-meson</a> <a class="existingWikiWord" href="/nlab/show/a1-meson">a1-meson</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strange+quark">strange</a>-mesons: <a class="existingWikiWord" href="/nlab/show/%CF%95-meson">ϕ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mover><mi>s</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">s \bar s</annotation></semantics></math>), <a class="existingWikiWord" href="/nlab/show/kaon">kaon</a>, <a class="existingWikiWord" href="/nlab/show/K%2A-meson">K*-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">u s</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">d s</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/eta-meson">eta-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>u</mi><mo>+</mo><mi>d</mi><mi>d</mi><mo>+</mo><mi>s</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">u u + d d + s s</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/charm+quark">charmed</a> <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy mesons</a>: <a class="existingWikiWord" href="/nlab/show/D-meson">D-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>c</mi></mrow><annotation encoding="application/x-tex"> u c</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">d c</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">s c</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/J%2F%CF%88-meson">J/ψ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mover><mi>c</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">c \bar c</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bottom+quark">bottom</a> <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy mesons</a>: <a class="existingWikiWord" href="/nlab/show/B-meson">B-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">q b</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/%CF%92-meson">ϒ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mover><mi>b</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">b \bar b</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/baryons">baryons</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nucleons">nucleons</a>: <a class="existingWikiWord" href="/nlab/show/proton">proton</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mi>u</mi><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(u u d)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/neutron">neutron</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mi>d</mi><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(u d d)</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> <h2 id="references">References</h2> A discussion of <a class="existingWikiWord" href="/nlab/show/Wolfgang+Pauli">Wolfgang Pauli</a>‘s thoughts leading him to the prediction of the neutrino: <ul> <li id="AP06"> Atmanspacher, Primas, Pauli’s ideas…, 2006 (<a href="http://www.igpp.de/english/tda/pdf/paulijcs8.pdf">pdf</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Francesco+Vissani">Francesco Vissani</a>, First steps towards understanding neutrinos [<a href="https://arxiv.org/abs/2310.07834">arXiv:2310.07834</a>] </li> </ul> Review: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Goran+Senjanovic">Goran Senjanovic</a>, LHC and the neutrino paradigm, talk at CERN EP seminar 2011 (<a href="https://indico.cern.ch/event/148714">cern:148714</a>, <a href="https://youtu.be/VVoMUHokn2U">recording</a>) </li> <li> Athar et al. Status and Perspectives of Neutrino Physics, <a href="https://www.iupapneutrinopanel.org/">IUPAP Neutrino Panel</a> Report (<a href="https://arxiv.org/abs/2111.07586">arXiv:2111.07586</a>) </li> <li> M. Sajjad Athar, A. Fatima, S. K. Singh, Neutrinos and their interactions with matter [<a href="https://arxiv.org/abs/2206.13792">arXiv:2206.13792</a>] </li> <li> Theory of Neutrino Physics – Snowmass TF11 (aka NF08) Topical Group Report [<a href="https://arxiv.org/abs/2209.07983">arXiv:2209.07983</a>] </li> <li> Ken-ichi Hikasa, CPT, Majorana fermions, and particle physics beyond the Standard Model [<a href="https://arxiv.org/abs/2405.19774">arXiv:2405.19774</a>] </li> </ul> See also <ul> <li>Wikipedia, <a href="http://en.wikipedia.org/wiki/Neutrino">Neutrino</a></li> </ul> On the neutrino mass problem: <ul> <li id="FisherKayserMcFarland99"> Peter Fisher, Boris Kayser, Kevin S. McFarland, Neutrino Mass and Oscillation Ann. Rev. Nucl. Part. Sci.49:481-528, 1999 (<a href="https://arxiv.org/abs/hep-ph/9906244">arXiv:hep-ph/9906244</a>) </li> <li id="Mohapatra06"> R. N. Mohapatra, Models of Neutrino Masses: A Brief Overview, 2006 (<a href="http://lss.fnal.gov/conf/C0606131/2.2mohapat.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/MohapatraNeutrinoMasses.pdf" title="pdf">pdf</a>) </li> <li> Boris Kayser, Are Neutrinos Their Own Antiparticles?, J. Phys. Conf. Ser. 173:012013, 2009 (<a href="https://arxiv.org/abs/0903.0899">arXiv:0903.0899</a>) </li> <li> Guido Fantini, Andrea Gallo Rosso, <a class="existingWikiWord" href="/nlab/show/Francesco+Vissani">Francesco Vissani</a>, Vanessa Zema, The formalism of neutrino oscillations: an introduction, in: The State of the Art of Neutrino Physics World Scientific (2018) [<a href="https://arxiv.org/abs/1802.05781">arXiv:1802.05781</a>, <a href="https://doi.org/10.1142/10600">doi:10.1142/10600</a>] </li> </ul> What it would mean if neutrinos were <a class="existingWikiWord" href="/nlab/show/Dirac+spinors">Dirac spinors</a>: <ul> <li>Salvador Centelles Chuliá, Theory and phenomenology of Dirac neutrinos (<a href="https://arxiv.org/abs/2110.15755">arXiv:2110.15755</a>)</li> </ul> With further emphasis on the <a class="existingWikiWord" href="/nlab/show/flavor+problem">flavor problem</a>: <ul> <li id="Xing19">Zhi-zhong Xing, Flavor structures of charged fermions and massive neutrinos, Physics Reports Volume 854, 20 April 2020, Pages 1-147 (<a href="https://arxiv.org/abs/1909.09610">arXiv:1909.09610</a>, <a href="https://doi.org/10.1016/j.physrep.2020.02.001">doi:10.1016/j.physrep.2020.02.001</a>)</li> </ul> Comments on neutrinos masses as a hint for the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> being an <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective field theory</a> are in <ul> <li id="Weinberg09"><a class="existingWikiWord" href="/nlab/show/Steven+Weinberg">Steven Weinberg</a>, Effective Field Theory, Past and Future (<a href="http://arxiv.org/abs/0908.1964">arXiv:0908.1964</a>)</li> </ul> See also <ul> <li> <a class="existingWikiWord" href="/nlab/show/Pierre+Ramond">Pierre Ramond</a>, Neutrinos and Particle Physics Models (<a href="https://arxiv.org/abs/1902.01741">arXiv:1902.01741</a>) </li> <li> M.C. Gonzalez-Garcia, Neutrino Masses and Mixing (<a href="https://arxiv.org/abs/1902.04583">arXiv:1902.04583</a>) </li> <li> Shao-Feng Ge, Jing-Yu Zhu, The Normal Neutrino Mass Hierarchy is Exactly What We Need! (<a href="https://arxiv.org/abs/1910.02666">arXiv:1910.02666</a>) </li> <li> S. Bilenky, Neutrinos: Majorana or Dirac? (<a href="https://arxiv.org/abs/2008.02110">arXiv:2008.02110</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Francesco+Vissani">Francesco Vissani</a>, What is matter according to particle physics and why try to observe its creation in lab?, Universe 7 3 (2021) 61 [<a href="https://arxiv.org/abs/2103.02642">arXiv:2103.02642</a>, <a href="https://doi.org/10.3390/universe7030061">doi:10.3390/universe7030061</a>] </li> </ul> More on this in relation to <a class="existingWikiWord" href="/nlab/show/GUT">GUT</a> models: <ul> <li> <a href="#Mohapatra06">Mohapatra 06</a> </li> <li id="Witten17"> <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, p. 2 of Symmetry and Emergence (<a href="https://arxiv.org/abs/1710.01791">arXiv:1710.01791</a>) </li> <li id="OhlssonPernow19"> Tommy Ohlsson, Marcus Pernow, Fits to Non-Supersymmetric SO(10) Models with Type I and II Seesaw Mechanisms Using Renormalization Group Evolution (<a href="https://arxiv.org/abs/1903.08241">arXiv:1903.08241</a>) </li> </ul> Discussion of neutrinos as <a class="existingWikiWord" href="/nlab/show/dark+matter">dark matter</a>-candidates: <ul> <li id="NeutrinoWhite16"> A White Paper on keV Sterile Neutrino Dark Matter, Journal of Cosmology and Astroparticle Physics, Volume 2017, January 2017 (<a href="https://arxiv.org/abs/1602.04816">arXiv:1602.04816</a>) </li> <li> D. J. Buettner, P. D. Morley, Why not Neutrinos as the Dark Matter? A Critical Review, KATRIN and New Research Directions, <a href="https://indico.phy.ornl.gov/event/142/#:~:text=The%207th%20Symposium%20on%20Neutrinos,NC%2C%2028801%2C%20US">7th Symposium on Neutrinos and Dark Matter in Nuclear Physics (NDM22)</a> [<a href="https://arxiv.org/abs/2208.06460">arXiv:2208.06460</a>] </li> </ul> Discussion of <a class="existingWikiWord" href="/nlab/show/neutrino">neutrino</a> <a class="existingWikiWord" href="/nlab/show/masses">masses</a> in <a class="existingWikiWord" href="/nlab/show/leptoquark">leptoquark</a>-<a class="existingWikiWord" href="/nlab/show/model+%28in+theoretical+physics%29">models</a> for <a class="existingWikiWord" href="/nlab/show/flavour+anomalies">flavour anomalies</a>: <ul> <li id="BGV19">Innes Bigaran, John Gargalionis, Raymond R. Volkas, A near-minimal leptoquark model for reconciling flavour anomalies and generating radiative neutrino masses (<a href="https://arxiv.org/abs/1906.01870">arXiv:1906.01870</a>)</li> </ul> Discussion of neutrino masses in <a class="existingWikiWord" href="/nlab/show/M-theory+on+G%E2%82%82-manifolds">M-theory on G₂-manifolds</a>: <ul> <li>Eric Gonzalez, <a class="existingWikiWord" href="/nlab/show/Gordon+Kane">Gordon Kane</a>, Khoa Nguyen, Neutrino mass matrices from localization in M-theory on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> orbifold (<a href="https://arxiv.org/abs/2107.12893">arXiv:2107.12893</a>)</li> </ul> On neutrino masses in <a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">type IIA string theory</a> <a class="existingWikiWord" href="/nlab/show/orientifold">orientifold</a> models: <ul> <li>Gonzalo F. Casas, <a class="existingWikiWord" href="/nlab/show/Luis+E.+Ib%C3%A1%C3%B1ez">Luis E. Ibáñez</a>, <a class="existingWikiWord" href="/nlab/show/Fernando+Marchesano">Fernando Marchesano</a>, On small Dirac Neutrino Masses in String Theory [<a href="https://arxiv.org/abs/2406.14609">arXiv:2406.14609</a>]</li> </ul> </body></html> </div> <div class="revisedby"> Last revised on July 18, 2024 at 12:41:38. 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