Schwinger Model

Prerequisites: Quantum Electrodynamics. Full path (33 pages)

Intuitive
Concrete
Abstract
Why is it interesting?

Intuitive

The Schwinger model is quantum electrodynamics in a world with one space dimension and one time dimension, usually with massless electrons. It is a toy model: nobody thinks the real world looks like this. It is studied because it can be solved exactly, and because the exact solution shows several effects that are believed to happen in quantum chromodynamics, where they are very hard to prove.

In one space dimension the electric field between two charges does not spread out. Field lines have nowhere to go, so the field between a positive and a negative charge is constant and the energy grows in proportion to their separation. Pulling the pair apart costs more and more energy, which is the same behaviour that keeps quarks inside protons. In the full quantum theory, electron-positron pairs from the vacuum can screen such charges, and whether external charges stay confined depends on the electron mass and on the size of the charges.

The exact solution shows that electrons and positrons never appear as free particles. They bind into a single kind of electrically neutral particle with a mass set by the charge, even though the electrons themselves are massless and the photon in one space dimension has no propagating waves of its own. Schwinger found in 1962 that the photon in this model acquires a mass in this way without any Higgs field.

The model also has a quantum anomaly: a symmetry of the classical equations, the separate conservation of left-moving and right-moving electrons, is destroyed by quantum effects in the presence of an electric field. The anomaly is responsible for the mass of the neutral particle, and it parallels the way the eta prime meson in QCD gets a large mass.

Concrete

We use natural units $\hbar = c = 1$ and metric signature $(+,-)$ . The Lagrangian of the Schwinger model is

$\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \bar\psi\, i\gamma^\mu\left(\partial_\mu + i e A_\mu\right)\psi - m\bar\psi\psi ,$

with $\mu,\nu \in \{0,1\}$ , a two-component Dirac spinor $\psi$ , and $m = 0$ for the original massless model. In $1+1$ dimensions the coupling $e$ has dimensions of mass, so the theory is super-renormalisable and $e$ sets the only scale when $m = 0$ .

Gauss law and confinement. In one space dimension the field strength has a single component, the electric field $E = F_{01}$ , and Gauss's law reads $\partial_1 E = e j^0$ . For static external charges $\pm q$ at separation $L$ , and ignoring the dynamical fermions, the field between them is constant, $|E| = q$ , and the energy stored in it is $\frac{1}{2}q^2 L$ : a linear potential. With dynamical massless fermions external charges are screened; with $m \neq 0$ charges that are not integer multiples of $e$ remain confined. A photon has no transverse polarisations in one space dimension, so $A_\mu$ has no propagating degrees of freedom of its own.

The anomaly. In two dimensions $\gamma^\mu\gamma^5 = \pm\epsilon^{\mu\nu}\gamma_\nu$ (the sign depends on conventions), so the axial current is dual to the vector current, $j_5^\mu = \pm\epsilon^{\mu\nu}j_\nu$ . Classically both are conserved when $m = 0$ . Quantum mechanically, a gauge invariant regularisation gives

$\partial_\mu j_5^\mu = \frac{e}{2\pi}\,\epsilon^{\mu\nu}F_{\mu\nu} = \frac{e}{\pi}\, E$

with $\epsilon^{01} = 1$ , up to an overall sign convention. Physically, an electric field accelerates right movers and left movers in opposite directions in momentum space, pulling states out of the Dirac sea and changing the number of right movers minus left movers.

Bosonization and the mass. A massless Dirac fermion in $1+1$ dimensions is equivalent to a massless real scalar $\phi$ , with the dictionary

$\bar\psi i\gamma^\mu\partial_\mu\psi \leftrightarrow \frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi, \qquad j^\mu = \bar\psi\gamma^\mu\psi \leftrightarrow \frac{1}{\sqrt\pi}\,\epsilon^{\mu\nu}\partial_\nu\phi .$

The charge density is then $j^0 = \frac{1}{\sqrt\pi}\partial_1\phi$ , and Gauss's law integrates to $E = \frac{e}{\sqrt\pi}\phi$ (plus a constant background field, which we set to zero here). The electric field energy $\frac{1}{2}E^2$ becomes a mass term for $\phi$ , and the Hamiltonian density is

$\mathcal{H} = \frac{1}{2}\Pi^2 + \frac{1}{2}(\partial_1\phi)^2 + \frac{e^2}{2\pi}\phi^2 .$

The massless Schwinger model is therefore a free boson of mass

$m_\gamma = \frac{e}{\sqrt{\pi}} .$

This neutral boson is the only particle in the spectrum. With $m \neq 0$ the mass term bosonizes to an interaction proportional to $m\cos(2\sqrt\pi\,\phi - \theta)$ , where $\theta$ is the constant background field in units of $e/2\pi$ , and the model is no longer free.

Abstract

Index theorem and topological sectors. Work in Euclidean signature on a compact surface such as the torus $T^2$ . A $U(1)$ gauge field is a connection on a line bundle, and the flux is quantised, $\frac{e}{2\pi}\int F = k \in \mathbb{Z}$ , the first Chern number. The Atiyah-Singer index theorem gives, up to an orientation convention,

$n_+ - n_- = \frac{e}{2\pi}\int_{T^2} F = k$

for the numbers $n_\pm$ of zero modes of the Dirac operator of positive and negative chirality. The anomaly is the statement that the fermion measure is not invariant under a chiral rotation $\psi \to e^{i\alpha\gamma^5}\psi$ , $\bar\psi \to \bar\psi e^{i\alpha\gamma^5}$ : the Jacobian is $e^{-2i\alpha k}$ in the sector of Chern number $k$ (the sign again depends on conventions). The path integral splits into a sum over $k$ , and the sectors may be weighted by $e^{i\theta k}$ . A chiral rotation shifts $\theta \to \theta - 2\alpha$ when $m = 0$ , so in the massless model $\theta$ can be rotated away, while for $m \neq 0$ it is a physical parameter. In the Hamiltonian picture on a spatial circle, $\theta$ labels the eigenvalue $e^{i\theta}$ of the operator implementing a large gauge transformation of winding number one (gauge transformations $S^1 \to U(1)$ are classified by $\pi_1(U(1)) = \mathbb{Z}$ ) on physical states. These are the analogues of the instantons and $\theta$ -vacua of the QCD vacuum, with the Chern number replacing the instanton number.

Chiral symmetry. The axial $U(1)$ is broken explicitly by the anomaly. In the $k = \pm 1$ sectors the Dirac operator has exactly one zero mode, of a single chirality fixed by the sign of $k$ , which soaks up the Grassmann integration in $\langle\bar\psi\psi\rangle$ ; these sectors give a non-zero chiral condensate $\langle\bar\psi\psi\rangle \propto e\cos\theta$ . Coleman's theorem forbids spontaneous breaking of a continuous global symmetry in two dimensions; there is no conflict, because the axial symmetry is not a symmetry of the quantum theory, and correspondingly no massless Goldstone boson appears. This is the two-dimensional counterpart of the $U(1)_A$ problem of QCD.

Exact solution and generalisations. Through bosonization the massless model maps to the free massive scalar theory above, so all correlation functions of gauge invariant operators can be computed exactly. Cluster decomposition requires working in a definite $\theta$ vacuum, which is a superposition of states of different winding number. Generalisations include several fermion flavours, where the bosonized theory has one massive and several massless or light modes, the massive model, which is not exactly solvable, and two-dimensional QCD in the limit of many colours.

Why is it interesting?

Massless (1+1) QED has many properties that resemble QCD: confinement, theta-vacuum, axial anomaly, … and is exactly soluble

http://physik.uni-graz.at/schladming2015/LectureNotes/Kharzeev4.pdf

Schwinger Model

Intuitive

Concrete

Abstract

Why is it interesting?

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