Schwinger model

{{Short description|Quantum electrodynamics in 1+1 dimensions}}{{Primary sources|date=June 2025}}

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 named after Julian Schwinger who developed it in 1962.{{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}}

The model defines the usual QED Lagrangian density

:$\backslash mathcal\{L\}\; =\; -\; \backslash frac\{1\}\{4g^2\}F\_\{\backslash mu\; \backslash nu\}F^\{\backslash mu\; \backslash nu\}\; +\; \backslash bar\{\backslash psi\}\; (i\; \backslash gamma^\backslash mu\; D\_\backslash mu\; -m)\; \backslash psi$

over a spacetime with one spatial dimension and one temporal dimension. Where $F\_\{\backslash mu\; \backslash nu\}\; =\; \backslash partial\_\backslash mu\; A\_\backslash nu\; -\; \backslash partial\_\backslash nu\; A\_\backslash mu$ is the photon field strength with symmetry group $\backslash mathrm\{U\}(1)$ (unitary group), $D\_\backslash mu\; =\; \backslash partial\_\backslash mu\; -\; iA\_\backslash mu$ is the gauge covariant derivative, $\backslash psi$ is the fermion spinor, $m$ is the fermion mass and $\backslash gamma^0,\; \backslash gamma^1$ 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 $r$, 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.{{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 }}{{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}}

==References==

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{{Quantum field theories}}

Category:Quantum field theory

Category:Quantum electrodynamics

Category:Exactly solvable models

Category:Quantum chromodynamics

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