Schwinger model

Quantum electrodynamics in 1+1 dimensions

In physics, the Schwinger model, named after Julian Schwinger, is the model^[1] describing 1+1D (1 spatial dimension + time) Lorentzian quantum electrodynamics which includes electrons, coupled to photons.

The model defines the usual QED Lagrangian

{\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 $F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }$ is the $U(1)$ photon field strength, $D_{\mu }=\partial _{\mu }-iA_{\mu }$ is the gauge covariant derivative, $\psi$ is the fermion spinor, $m$ is the fermion mass and $\gamma ^{0},\gamma ^{1}$ form the two-dimensional representation of the Clifford algebra.

This model exhibits confinement of the fermions and as such, is a toy model for QCD. A handwaving argument why this is so is because in two dimensions, classically, the potential between two charged particles goes linearly as $r$ , instead of $1/r$ 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.^[2]^[3]

References

^ Schwinger, Julian (1962). "Gauge Invariance and Mass. II". Physical Review. 128 (5). Physical Review, Volume 128: 2425–2429. Bibcode:1962PhRv..128.2425S. doi:10.1103/PhysRev.128.2425.
^ Schwinger, Julian (1951). "The Theory of Quantized Fields I". Physical Review. 82 (6). Physical Review, Volume 82: 914–927. Bibcode:1951PhRv...82..914S. doi:10.1103/PhysRev.82.914. S2CID 121971249.
^ Schwinger, Julian (1953). "The Theory of Quantized Fields II". Physical Review. 91 (3). Physical Review, Volume 91: 713–728. Bibcode:1953PhRv...91..713S. doi:10.1103/PhysRev.91.713.

Quantum field theories

Theories

Models

Regular	Born–Infeld Euler–Heisenberg Ginzburg–Landau Non-linear sigma Proca Quantum electrodynamics Quantum chromodynamics Quartic interaction Scalar electrodynamics Scalar chromodynamics Soler Yang–Mills Yang–Mills–Higgs Yukawa
Low dimensional	2D Yang–Mills Bullough–Dodd Gross–Neveu Schwinger Sine-Gordon Thirring Thirring–Wess Toda
Conformal	2D free massless scalar Liouville Minimal Polyakov Wess–Zumino–Witten
Supersymmetric	4D N = 1 N = 1 super Yang–Mills Seiberg–Witten Super QCD Wess–Zumino
Superconformal	6D (2,0) ABJM N = 4 super Yang–Mills
Supergravity	4D N = 1 Higher dimensional N = 8 Pure 4D N = 1
Topological	BF Chern–Simons
Particle theory	Chiral Fermi MSSM Nambu–Jona-Lasinio NMSSM Standard Model Stueckelberg