Editing Schwinger model

{{Short description|Quantum electrodynamics in 1+1 dimensions}} 
In physics, the '''Schwinger model''', named after [[Julian Schwinger]],  is the model<ref>{{Cite journal  | last = Schwinger  | first = Julian  | title = Gauge Invariance and Mass. II | journal = Physical Review | publisher = Physical Review, Volume 128  | date = 1962  | volume = 128 | issue = 5 | pages = 2425–2429  | doi = 10.1103/PhysRev.128.2425  | bibcode =1962PhRv..128.2425S}}</ref> describing 1+1D (1 spatial dimension + time) ''[[Minkowski Space|Lorentzian]]'' [[quantum electrodynamics]] which includes [[Dirac equation|electrons]], coupled to [[Maxwell's equations|photons]]. 

The model defines the usual [[quantum electrodynamics|QED]] Lagrangian

:<math> \mathcal{L} = - \frac{1}{4g^2}F_{\mu \nu}F^{\mu \nu} + \bar{\psi} (i \gamma^\mu D_\mu -m) \psi</math>

over a [[spacetime]] with one spatial dimension and one temporal dimension.  Where <math> F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu</math> is the <math> U(1) </math> photon field strength, <math> D_\mu = \partial_\mu - iA_\mu </math> is the gauge covariant derivative, <math> \psi </math> is the fermion spinor, <math> m </math> is the fermion mass and <math> \gamma^0, \gamma^1 </math> form the two-dimensional representation of the Clifford algebra.

This model exhibits [[colour confinement|confinement]] of the fermions and as such, is a toy model for [[Quantum chromodynamics|QCD]]. A handwaving argument why this is so is because in two dimensions, classically, the potential between two charged particles goes linearly as <math>r</math>, instead of <math>1/r</math> in 4 dimensions, 3 spatial, 1 time. This model also exhibits a [[spontaneous symmetry breaking]] of the U(1) symmetry due to a [[chiral condensate]] due to a pool of [[instanton]]s. The [[photon]] in this model becomes a massive particle at low temperatures. This model can be solved exactly and is used as a [[toy model]] for other more complex theories.<ref>{{Cite journal  | last = Schwinger  | first = Julian  | title =The Theory of Quantized Fields I  | journal = Physical Review | publisher = Physical Review, Volume 82  | date = 1951  | volume = 82 | issue = 6 | pages = 914–927  | doi = 10.1103/PhysRev.82.914  | bibcode =1951PhRv...82..914S| s2cid = 121971249 }}</ref><ref>{{Cite journal  | last = Schwinger  | first = Julian  | title =The Theory of Quantized Fields II | journal = Physical Review | publisher = Physical Review, Volume 91  | date = 1953  | volume = 91 | issue = 3 | pages = 713–728  | url =   https://digital.library.unt.edu/ark:/67531/metadc1021287/| doi = 10.1103/PhysRev.91.713  | bibcode =1953PhRv...91..713S}}
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==References== 
{{reflist}}

{{Quantum field theories}}

[[Category:Quantum field theory]]
[[Category:Quantum electrodynamics]]
[[Category:Exactly solvable models]]
[[Category:Quantum chromodynamics]]


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