Editing Quantum field theory (section)

====Gauge symmetry====
{{Main|Gauge theory}}

If the following transformation to the fields is performed at every spacetime point {{math|''x''}} (a local transformation), then the QED Lagrangian remains unchanged, or invariant:
:<math>\psi(x) \to e^{i\alpha(x)}\psi(x),\quad A_\mu(x) \to A_\mu(x) + ie^{-1} e^{-i\alpha(x)}\partial_\mu e^{i\alpha(x)},</math>
where {{math|''α''(''x'')}} is any function of spacetime coordinates. If a theory's Lagrangian (or more precisely the [[action (physics)|action]]) is invariant under a certain local transformation, then the transformation is referred to as a [[gauge symmetry]] of the theory.{{r|peskin|page1=482–483}} Gauge symmetries form a [[group (mathematics)|group]] at every spacetime point. In the case of QED, the successive application of two different local symmetry transformations <math>e^{i\alpha(x)}</math> and <math>e^{i\alpha'(x)}</math> is yet another symmetry transformation <math>e^{i[\alpha(x)+\alpha'(x)]}</math>. For any {{math|''α''(''x'')}}, <math>e^{i\alpha(x)}</math> is an element of the {{math|[[U(1)]]}} group, thus QED is said to have {{math|U(1)}} gauge symmetry.{{r|peskin|page1=496}} The photon field {{math|''A<sub>μ</sub>''}} may be referred to as the {{math|U(1)}} [[gauge boson]].

{{math|U(1)}} is an [[Abelian group]], meaning that the result is the same regardless of the order in which its elements are applied. QFTs can also be built on [[non-Abelian group]]s, giving rise to [[Yang–Mills theory|non-Abelian gauge theories]] (also known as Yang–Mills theories).{{r|peskin|page1=489}} [[Quantum chromodynamics]], which describes the strong interaction, is a non-Abelian gauge theory with an {{math|[[special unitary group|SU(3)]]}} gauge symmetry. It contains three Dirac fields {{math|''ψ<sup>i</sup>'', ''i'' {{=}} 1,2,3}} representing [[quark]] fields as well as eight vector fields {{math|''A<sup>a,μ</sup>'', ''a'' {{=}} 1,...,8}} representing [[gluon]] fields, which are the {{math|SU(3)}} gauge bosons.{{r|peskin|page1=547}} The QCD Lagrangian density is:{{r|peskin|page1=490–491}}
:<math>\mathcal{L} = i\bar\psi^i \gamma^\mu (D_\mu)^{ij} \psi^j - \frac 14 F_{\mu\nu}^aF^{a,\mu\nu} - m\bar\psi^i \psi^i,</math>
where {{math|''D<sub>μ</sub>''}} is the gauge [[covariant derivative]]:
:<math>D_\mu = \partial_\mu - igA_\mu^a t^a,</math>
where {{math|''g''}} is the coupling constant, {{math|''t<sup>a</sup>''}} are the eight [[Lie algebra|generators]] of {{math|SU(3)}} in the [[fundamental representation]] ({{math|3×3}} matrices),
:<math>F_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + gf^{abc}A_\mu^b A_\nu^c,</math>
and {{math|''f<sup>abc</sup>''}} are the [[structure constants]] of {{math|SU(3)}}. Repeated indices {{math|''i'',''j'',''a''}} are implicitly summed over following Einstein notation. This Lagrangian is invariant under the transformation:
:<math>\psi^i(x) \to U^{ij}(x)\psi^j(x),\quad A_\mu^a(x) t^a \to U(x)\left[A_\mu^a(x) t^a + ig^{-1} \partial_\mu\right]U^\dagger(x),</math>
where {{math|''U''(''x'')}} is an element of {{math|SU(3)}} at every spacetime point {{math|''x''}}:
:<math>U(x) = e^{i\alpha(x)^a t^a}.</math>

The preceding discussion of symmetries is on the level of the Lagrangian. In other words, these are "classical" symmetries. After quantization, some theories will no longer exhibit their classical symmetries, a phenomenon called [[anomaly (physics)|anomaly]]. For instance, in the path integral formulation, despite the invariance of the Lagrangian density <math>\mathcal{L}[\phi,\partial_\mu\phi]</math> under a certain local transformation of the fields, the [[measure (mathematics)|measure]] <math display="inline">\int\mathcal D\phi</math> of the path integral may change.{{r|zee|page1=243}} For a theory describing nature to be consistent, it must not contain any anomaly in its gauge symmetry. The Standard Model of elementary particles is a gauge theory based on the group {{math|SU(3) × SU(2) × U(1)}}, in which all anomalies exactly cancel.{{r|peskin|page1=705–707}}

The theoretical foundation of [[general relativity]], the [[equivalence principle]], can also be understood as a form of gauge symmetry, making general relativity a gauge theory based on the [[Lorentz group]].<ref>Veltman, M. J. G. (1976). ''Methods in Field Theory, Proceedings of the Les Houches Summer School, Les Houches, France, 1975''.</ref>

[[Noether's theorem]] states that every continuous symmetry, ''i.e.'' the parameter in the symmetry transformation being continuous rather than discrete, leads to a corresponding [[conservation law]].{{r|peskin|zee|page1=17–18|page2=73}} For example, the {{math|U(1)}} symmetry of QED implies [[charge conservation]].<ref>{{cite journal |last1=Brading |first1=Katherine A.|author1-link=Katherine Brading |date=March 2002 |title=Which symmetry? Noether, Weyl, and conservation of electric charge |journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics |volume=33 |issue=1 |pages=3–22 |doi=10.1016/S1355-2198(01)00033-8 |bibcode=2002SHPMP..33....3B |citeseerx=10.1.1.569.106 }}</ref>

Gauge-transformations do not relate distinct quantum states. Rather, it relates two equivalent mathematical descriptions of the same quantum state. As an example, the photon field {{math|''A<sup>μ</sup>''}}, being a [[four-vector]], has four apparent degrees of freedom, but the actual state of a photon is described by its two degrees of freedom corresponding to the [[photon polarization|polarization]]. The remaining two degrees of freedom are said to be "redundant" — apparently different ways of writing {{math|''A<sup>μ</sup>''}} can be related to each other by a gauge transformation and in fact describe the same state of the photon field. In this sense, gauge invariance is not a "real" symmetry, but a reflection of the "redundancy" of the chosen mathematical description.{{r|zee|page1=168}}

To account for the gauge redundancy in the path integral formulation, one must perform the so-called [[Faddeev–Popov ghost|Faddeev–Popov]] [[gauge fixing]] procedure. In non-Abelian gauge theories, such a procedure introduces new fields called "ghosts". Particles corresponding to the ghost fields are called ghost particles, which cannot be detected externally.{{r|peskin|page1=512–515}} A more rigorous generalization of the Faddeev–Popov procedure is given by [[BRST quantization]].{{r|peskin|page1=517}}