Schwinger model: Difference between revisions
Jump to navigation
Jump to search
imported>LaundryPizza03 added Category:Quantum chromodynamics using HotCat |
imported>ReyHahn primary |
||
| Line 1: | Line 1: | ||
{{Short description|Quantum electrodynamics in 1+1 dimensions}} | {{Short description|Quantum electrodynamics in 1+1 dimensions}}{{Primary sources|date=June 2025}} | ||
The model defines the usual [[ | In [[quantum field theory]], the '''Schwinger model''' is a model describing 1+1D (time + 1 spatial dimension) [[quantum electrodynamics]] (QED) which includes [[Dirac equation|electrons]], coupled to [[Maxwell's equations|photons]]. It is named after named after [[Julian Schwinger]] who developed it in 1962.<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> | ||
The model defines the usual QED [[Lagrangian (field theory)|Lagrangian density]] | |||
:<math> \mathcal{L} = - \frac{1}{4g^2}F_{\mu \nu}F^{\mu \nu} + \bar{\psi} (i \gamma^\mu D_\mu -m) \psi</math> | :<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> | 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 photon [[Electromagnetic tensor|field strength]] with symmetry group <math> \mathrm{U}(1) </math> ([[unitary group]]), <math> D_\mu = \partial_\mu - iA_\mu </math> is the [[gauge covariant derivative]], <math> \psi </math> is the [[Dirac spinor|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 [[ | This model exhibits [[colour confinement|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 <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}} | ||
</ref> | </ref> | ||
Latest revision as of 08:21, 16 June 2025
Template:Short descriptionScript error: No such module "Unsubst".
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.[1]
The model defines the usual QED Lagrangian density
over a spacetime with one spatial dimension and one temporal dimension. Where is the photon field strength with symmetry group (unitary group), is the gauge covariant derivative, is the fermion spinor, is the fermion mass and 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 , instead of 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]