In quantum field theory, the Schwinger model is a model describing 1+1D (time + 1 spatial dimension) quantum electrodynamics (QED) which includes electrons, coupled to photons. It is named after Julian Schwinger who developed it in 1962. The model defines the usual QED Lagrangian density
L = − 1 4 g 2 F μ ν F μ ν + ψ ¯ ( i γ μ D μ − m ) ψ {\displaystyle {\mathcal {L}}=-{\frac {1}{4g^{2}}}F_{\mu \nu }F^{\mu \nu }+{\bar {\psi }}(i\gamma ^{\mu }D_{\mu }-m)\psi }
over a spacetime with one spatial dimension and one temporal dimension. Where F μ ν = ∂ μ A ν − ∂ ν A μ {\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }} is the photon field strength with symmetry group U ( 1 ) {\displaystyle \mathrm {U} (1)} (unitary group), D μ = ∂ μ − i A μ {\displaystyle D_{\mu }=\partial _{\mu }-iA_{\mu }} is the gauge covariant derivative, ψ {\displaystyle \psi } is the fermion spinor, m {\displaystyle m} is the fermion mass and γ 0 , γ 1 {\displaystyle \gamma ^{0},\gamma ^{1}} form the two-dimensional representation of the Clifford algebra. This model exhibits confinement of the fermions and as such, is a toy model for quantum chromodynamics. A handwaving argument why this is so is because in two dimensions, classically, the potential between two charged particles goes linearly as r {\displaystyle r} , instead of 1 / r {\displaystyle 1/r} 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 instantons. 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.
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