Editing Quantum field theory (section)

===Other theories===
The quantization and renormalization procedures outlined in the preceding sections are performed for the free theory and [[quartic interaction|{{math|''ϕ''<sup>4</sup>}} theory]] of the real scalar field. A similar process can be done for other types of fields, including the [[complex numbers|complex]] scalar field, the [[vector field]], and the [[Dirac field]], as well as other types of interaction terms, including the electromagnetic interaction and the [[Yukawa interaction]].

As an example, [[quantum electrodynamics]] contains a Dirac field {{math|''ψ''}} representing the [[electron]] field and a vector field {{math|''A<sup>μ</sup>''}} representing the electromagnetic field ([[photon]] field). (Despite its name, the quantum electromagnetic "field" actually corresponds to the classical [[electromagnetic four-potential]], rather than the classical electric and magnetic fields.) The full QED Lagrangian density is:
:<math>\mathcal{L} = \bar\psi\left(i\gamma^\mu\partial_\mu - m\right)\psi - \frac 14 F_{\mu\nu}F^{\mu\nu} - e\bar\psi\gamma^\mu\psi A_\mu,</math>
where {{math|''γ<sup>μ</sup>''}} are [[Dirac matrices]], <math>\bar\psi = \psi^\dagger\gamma^0</math>, and <math>F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu</math> is the [[electromagnetic field strength]]. The parameters in this theory are the (bare) electron mass {{math|''m''}} and the (bare) [[elementary charge]] {{math|''e''}}. The first and second terms in the Lagrangian density correspond to the free Dirac field and free vector fields, respectively. The last term describes the interaction between the electron and photon fields, which is treated as a perturbation from the free theories.{{r|peskin|page1=78}}
[[File:Electron-positron-annihilation.svg|thumb]]

Shown above is an example of a tree-level Feynman diagram in QED. It describes an electron and a positron annihilating, creating an [[off-shell]] photon, and then decaying into a new pair of electron and positron. Time runs from left to right. Arrows pointing forward in time represent the propagation of electrons, while those pointing backward in time represent the propagation of positrons. A wavy line represents the propagation of a photon. Each vertex in QED Feynman diagrams must have an incoming and an outgoing fermion (positron/electron) leg as well as a photon leg.

====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}}

====Spontaneous symmetry-breaking====
{{Main|Spontaneous symmetry breaking}}

[[Spontaneous symmetry breaking]] is a mechanism whereby the symmetry of the Lagrangian is violated by the system described by it.{{r|peskin|page1=347}}

To illustrate the mechanism, consider a linear [[sigma model]] containing {{math|''N''}} real scalar fields, described by the Lagrangian density:
:<math>\mathcal{L} = \frac 12 \left(\partial_\mu\phi^i\right)\left(\partial^\mu\phi^i\right) + \frac 12 \mu^2 \phi^i\phi^i - \frac{\lambda}{4} \left(\phi^i\phi^i\right)^2,</math>
where {{math|''μ''}} and {{math|''λ''}} are real parameters. The theory admits an {{math|[[orthogonal group|O(''N'')]]}} global symmetry:
:<math>\phi^i \to R^{ij}\phi^j,\quad R\in\mathrm{O}(N).</math>
The lowest energy state (ground state or vacuum state) of the classical theory is any uniform field {{math|''ϕ''<sub>0</sub>}} satisfying
:<math>\phi_0^i \phi_0^i = \frac{\mu^2}{\lambda}.</math>
Without loss of generality, let the ground state be in the {{math|''N''}}-th direction:
:<math>\phi_0^i = \left(0,\cdots,0,\frac{\mu}{\sqrt{\lambda}}\right).</math>
The original {{math|''N''}} fields can be rewritten as:
:<math>\phi^i(x) = \left(\pi^1(x),\cdots,\pi^{N-1}(x),\frac{\mu}{\sqrt{\lambda}} + \sigma(x)\right),</math>
and the original Lagrangian density as:
:<math>\mathcal{L} = \frac 12 \left(\partial_\mu\pi^k\right)\left(\partial^\mu\pi^k\right) + \frac 12 \left(\partial_\mu\sigma\right)\left(\partial^\mu\sigma\right) - \frac 12 \left(2\mu^2\right)\sigma^2 - \sqrt{\lambda}\mu\sigma^3 - \sqrt{\lambda}\mu\pi^k\pi^k\sigma - \frac{\lambda}{2} \pi^k\pi^k\sigma^2 - \frac{\lambda}{4}\left(\pi^k\pi^k\right)^2,</math>
where {{math|''k'' {{=}} 1, ..., ''N'' − 1}}. The original {{math|O(''N'')}} global symmetry is no longer manifest, leaving only the [[subgroup]] {{math|O(''N'' − 1)}}. The larger symmetry before spontaneous symmetry breaking is said to be "hidden" or spontaneously broken.{{r|peskin|page1=349–350}}

[[Goldstone's theorem]] states that under spontaneous symmetry breaking, every broken continuous global symmetry leads to a massless field called the Goldstone boson. In the above example, {{math|O(''N'')}} has {{math|''N''(''N'' − 1)/2}} continuous symmetries (the dimension of its [[Lie algebra]]), while {{math|O(''N'' − 1)}} has {{math|(''N'' − 1)(''N'' − 2)/2}}. The number of broken symmetries is their difference, {{math|''N'' − 1}}, which corresponds to the {{math|''N'' − 1}} massless fields {{math|''π<sup>k</sup>''}}.{{r|peskin|page1=351}}

On the other hand, when a gauge (as opposed to global) symmetry is spontaneously broken, the resulting Goldstone boson is "eaten" by the corresponding gauge boson by becoming an additional degree of freedom for the gauge boson. The Goldstone boson equivalence theorem states that at high energy, the amplitude for emission or absorption of a longitudinally polarized massive gauge boson becomes equal to the amplitude for emission or absorption of the Goldstone boson that was eaten by the gauge boson.{{r|peskin|page1=743–744}}

In the QFT of [[ferromagnetism]], spontaneous symmetry breaking can explain the alignment of [[magnetic dipole]]s at low temperatures.{{r|zee|page1=199}} In the Standard Model of elementary particles, the [[W and Z bosons]], which would otherwise be massless as a result of gauge symmetry, acquire mass through spontaneous symmetry breaking of the [[Higgs boson]], a process called the [[Higgs mechanism]].{{r|peskin|page1=690}}

====Supersymmetry====
{{Main|Supersymmetry}}

All experimentally known symmetries in nature relate [[boson]]s to bosons and [[fermion]]s to fermions. Theorists have hypothesized the existence of a type of symmetry, called [[supersymmetry]], that relates bosons and fermions.{{r|peskin|zee|page1=795|page2=443}}

The Standard Model obeys [[Poincaré group|Poincaré symmetry]], whose generators are the spacetime [[translation (geometry)|translations]] {{math|''P<sup>μ</sup>''}} and the [[Lorentz transformations]] {{math|''J<sub>μν</sub>''}}.<ref name="WeinbergQFT">{{cite book |last=Weinberg |first=Steven |date=1995 |title=The Quantum Theory of Fields |publisher=Cambridge University Press |isbn=978-0-521-55001-7 |author-link=Steven Weinberg |url-access=registration |url=https://archive.org/details/quantumtheoryoff00stev }}</ref>{{rp|58–60}} In addition to these generators, supersymmetry in (3+1)-dimensions includes additional generators {{math|''Q<sub>α</sub>''}}, called [[supercharge]]s, which themselves transform as [[Weyl fermion]]s.{{r|peskin|zee|page1=795|page2=444}} The symmetry group generated by all these generators is known as the [[super-Poincaré group]]. In general there can be more than one set of supersymmetry generators, {{math|''Q<sub>α</sub><sup>I</sup>'', ''I'' {{=}} 1, ..., ''N''}}, which generate the corresponding {{math|''N'' {{=}} 1}} supersymmetry, {{math|''N'' {{=}} 2}} supersymmetry, and so on.{{r|peskin|zee|page1=795|page2=450}} Supersymmetry can also be constructed in other dimensions,<ref>{{cite arXiv |last1=de Wit |first1=Bernard |last2=Louis |first2=Jan |eprint=hep-th/9801132 |title=Supersymmetry and Dualities in various dimensions |date=1998-02-18 }}</ref> most notably in (1+1) dimensions for its application in [[superstring theory]].<ref>{{cite book |last=Polchinski |first=Joseph |date=2005 |title=String Theory |volume=2 |publisher=Cambridge University Press |isbn=978-0-521-67228-3 |author-link=Joseph Polchinski }}</ref>

The Lagrangian of a supersymmetric theory must be invariant under the action of the super-Poincaré group.{{r|zee|page1=448}} Examples of such theories include: [[Minimal Supersymmetric Standard Model]] (MSSM), [[N {{=}} 4 supersymmetric Yang–Mills theory|{{math|''N'' {{=}} 4}} supersymmetric Yang–Mills theory]],{{r|zee|page1=450}} and superstring theory. In a supersymmetric theory, every fermion has a bosonic [[superpartner]] and vice versa.{{r|zee|page1=444}}

If supersymmetry is promoted to a local symmetry, then the resultant gauge theory is an extension of general relativity called [[supergravity]].<ref name="NathArnowitt">{{cite journal | last1 = Nath | first1 = P. | last2 = Arnowitt | first2 = R. | year = 1975 | title = Generalized Super-Gauge Symmetry as a New Framework for Unified Gauge Theories | journal = Physics Letters B | volume = 56 | issue = 2| page = 177 | doi=10.1016/0370-2693(75)90297-x| bibcode = 1975PhLB...56..177N }}</ref>

Supersymmetry is a potential solution to many current problems in physics. For example, the [[hierarchy problem]] of the Standard Model—why the mass of the Higgs boson is not radiatively corrected (under renormalization) to a very high scale such as the [[Grand Unified Theory|grand unified scale]] or the [[Planck mass|Planck scale]]—can be resolved by relating the [[Higgs field]] and its super-partner, the [[Higgsino]]. Radiative corrections due to Higgs boson loops in Feynman diagrams are cancelled by corresponding Higgsino loops. Supersymmetry also offers answers to the grand unification of all gauge coupling constants in the Standard Model as well as the nature of [[dark matter]].{{r|peskin|page1=796–797}}<ref>{{Cite journal |last=Munoz |first=Carlos |arxiv=1701.05259 |title=Models of Supersymmetry for Dark Matter |journal=EPJ Web of Conferences |volume=136 |pages=01002 |date=2017-01-18 |bibcode=2017EPJWC.13601002M |doi=10.1051/epjconf/201713601002 |s2cid=55199323 }}</ref>

Nevertheless, experiments have yet to provide evidence for the existence of supersymmetric particles. If supersymmetry were a true symmetry of nature, then it must be a broken symmetry, and the energy of symmetry breaking must be higher than those achievable by present-day experiments.{{r|peskin|zee|page1=797|page2=443}}

====Other spacetimes====
The {{math|''ϕ''<sup>4</sup>}} theory, QED, QCD, as well as the whole Standard Model all assume a (3+1)-dimensional [[Minkowski space]] (3 spatial and 1 time dimensions) as the background on which the quantum fields are defined. However, QFT ''a priori'' imposes no restriction on the number of dimensions nor the geometry of spacetime.

In [[condensed matter physics]], QFT is used to describe [[two-dimensional electron gas|(2+1)-dimensional electron gases]].<ref>{{cite book |last1=Morandi |first1=G. |last2=Sodano |first2=P. |last3=Tagliacozzo |first3=A. |last4=Tognetti |first4=V. |date=2000 |title=Field Theories for Low-Dimensional Condensed Matter Systems |url=https://www.springer.com/us/book/9783540671770 |publisher=Springer |isbn=978-3-662-04273-1 }}</ref> In [[high-energy physics]], [[string theory]] is a type of (1+1)-dimensional QFT,{{r|zee|page1=452}}<ref name="polchinski1" /> while [[Kaluza–Klein theory]] uses gravity in [[extra dimensions]] to produce gauge theories in lower dimensions.{{r|zee|page1=428–429}}

In Minkowski space, the flat [[metric tensor (general relativity)|metric]] {{math|''η<sub>μν</sub>''}} is used to [[raising and lowering indices|raise and lower]] spacetime indices in the Lagrangian, ''e.g.''
:<math>A_\mu A^\mu = \eta_{\mu\nu} A^\mu A^\nu,\quad \partial_\mu\phi \partial^\mu\phi = \eta^{\mu\nu}\partial_\mu\phi \partial_\nu\phi,</math>
where {{math|''η<sup>μν</sup>''}} is the inverse of {{math|''η<sub>μν</sub>''}} satisfying {{math|''η<sup>μρ</sup>η<sub>ρν</sub>'' {{=}} ''δ<sup>μ</sup><sub>ν</sub>''}}. 
For [[quantum field theory in curved spacetime|QFTs in curved spacetime]] on the other hand, a general metric (such as the [[Schwarzschild metric]] describing a [[black hole]]) is used:
:<math>A_\mu A^\mu = g_{\mu\nu} A^\mu A^\nu,\quad \partial_\mu\phi \partial^\mu\phi = g^{\mu\nu}\partial_\mu\phi \partial_\nu\phi,</math>
where {{math|''g<sup>μν</sup>''}} is the inverse of {{math|''g<sub>μν</sub>''}}. 
For a real scalar field, the Lagrangian density in a general spacetime background is
:<math>\mathcal{L} = \sqrt{|g|}\left(\frac 12 g^{\mu\nu} \nabla_\mu\phi \nabla_\nu\phi - \frac 12 m^2\phi^2\right),</math>
where {{math|''g'' {{=}} det(''g<sub>μν</sub>'')}}, and {{math|∇<sub>''μ''</sub>}} denotes the [[covariant derivative]].<ref>{{cite book |last1=Parker |first1=Leonard E. |last2=Toms |first2=David J. |date=2009 |title=Quantum Field Theory in Curved Spacetime |url=https://archive.org/details/quantumfieldtheo00park |url-access=limited |publisher=Cambridge University Press |page=[https://archive.org/details/quantumfieldtheo00park/page/n58 43] |isbn=978-0-521-87787-9 }}</ref> The Lagrangian of a QFT, hence its calculational results and physical predictions, depends on the geometry of the spacetime background.

====Topological quantum field theory====
{{Main|Topological quantum field theory}}

The correlation functions and physical predictions of a QFT depend on the spacetime metric {{math|''g<sub>μν</sub>''}}. For a special class of QFTs called [[topological quantum field theories]] (TQFTs), all correlation functions are independent of continuous changes in the spacetime metric.<ref>{{cite arXiv |last1=Ivancevic |first1=Vladimir G. |last2=Ivancevic |first2=Tijana T. |eprint=0810.0344v5 |title=Undergraduate Lecture Notes in Topological Quantum Field Theory |class=math-th |date=2008-12-11 }}</ref>{{rp|36}} QFTs in curved spacetime generally change according to the ''geometry'' (local structure) of the spacetime background, while TQFTs are invariant under spacetime [[diffeomorphism]]s but are sensitive to the ''[[topology]]'' (global structure) of spacetime. This means that all calculational results of TQFTs are [[topological invariant]]s of the underlying spacetime. [[Chern–Simons theory]] is an example of TQFT and has been used to construct models of quantum gravity.<ref>{{cite book |last=Carlip |first=Steven |author-link=Steve Carlip |date=1998 |title=Quantum Gravity in 2+1 Dimensions |url=https://www.cambridge.org/core/books/quantum-gravity-in-21-dimensions/D2F727B6822014270F423D82501E674A |publisher=Cambridge University Press |pages=27–29 |isbn=9780511564192 |doi=10.1017/CBO9780511564192 |arxiv=2312.12596 }}</ref> Applications of TQFT include the [[fractional quantum Hall effect]] and [[topological quantum computer]]s.<ref>{{cite journal |last1=Carqueville |first1=Nils |last2=Runkel |first2=Ingo |arxiv=1705.05734 |title=Introductory lectures on topological quantum field theory |journal=Banach Center Publications |year=2018 |volume=114 |pages=9–47 |doi=10.4064/bc114-1 |s2cid=119166976 }}</ref>{{rp|1–5}} The world line trajectory of fractionalized particles (known as [[anyons]]) can form a link configuration in the spacetime,<ref>{{Cite journal |author-link=Edward Witten |first=Edward |last=Witten |title=Quantum Field Theory and the Jones Polynomial |journal=[[Communications in Mathematical Physics]] |volume=121 |issue=3 |pages=351–399 |year=1989 |mr=0990772 |bibcode = 1989CMaPh.121..351W |doi = 10.1007/BF01217730 |s2cid=14951363 |url=http://projecteuclid.org/euclid.cmp/1104178138 }}</ref> which relates the braiding statistics of anyons in physics to the
link invariants in mathematics. Topological quantum field theories (TQFTs) applicable to the frontier research of topological quantum matters include Chern-Simons-Witten gauge theories in 2+1 spacetime dimensions, other new exotic TQFTs in 3+1 spacetime dimensions and beyond.<ref>{{Cite journal |first1=Pavel|last1=Putrov |first2=Juven |last2=Wang | first3=Shing-Tung | last3=Yau    |title=Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions |journal=[[Annals of Physics]] |volume=384 |issue=C |pages=254–287 |year=2017|doi =10.1016/j.aop.2017.06.019|arxiv=1612.09298 |bibcode=2017AnPhy.384..254P |s2cid=119578849 }}</ref>