[Paper Review] Notes on the Schwinger model: regularization and gauge invariance

David Stuart|arXiv (Cornell University)|Jun 5, 2012

Quantum Mechanics and Applications5 references3 citations

TL;DR

This paper presents a rigorous regularization and quantization of the 1+1 dimensional Schwinger model on a circle using point-splitting to construct a gauge-invariant Hamiltonian. It derives a fully regularized Hamiltonian operator in the Schrödinger representation, explicitly resolving anomalies and ensuring gauge invariance through careful treatment of fermionic creation/annihilation operators and Fourier modes of the current, with key results including the absence of anomalous contributions in non-zero Fourier modes of the axial current.

ABSTRACT

The point-splitting computation of the gauge invariant Hamiltonian for the Schwinger model on the circle in a positive energy representation is presented.

Motivation & Objective

• To construct a gauge-invariant Hamiltonian for the Schwinger model on a circle in a positive energy representation.
• To resolve the issue of regularization in the presence of gauge symmetry, particularly concerning bilinear fermionic operators like the axial current and Hamiltonian.
• To ensure consistency of the quantization procedure under large gauge transformations and to handle the zero-mode structure carefully.
• To compute the regularized Fourier modes of the current operators, especially $\jmath_1(m)$ and $\jmath_0(m)$, and verify the absence of anomalies for $m \neq 0$.
• To derive a fully regularized Hamiltonian operator in the Schrödinger representation with explicit dependence on the gauge field coordinate $a$ and fermionic Fock space operators.

Proposed method

• The Schwinger model is formulated in 1+1 dimensions with periodic boundary conditions on a circle of length $L$, using natural units and the radiation gauge of Fermi.
• The classical Hamiltonian is derived via Gauss law, eliminating the longitudinal electric field component, and expressed in terms of the transverse electric field $\dot{a}$ and fermionic bilinears.
• Point-splitting regularization is applied to define bilinear operators such as the axial current and Hamiltonian, ensuring gauge invariance and finiteness of matrix elements.
• The fermionic fields are expanded in Fourier modes, and the creation/annihilation operators $b_m, c_m$ are defined on the zero-charge Fock space $\mathcal{H}_0$.
• The gauge field $a$ is treated as a coordinate operator in the Schrödinger representation over $L^2([0, \frac{2\pi}{L}], da; \mathcal{H}_0)$, enabling explicit operator ordering.
• Fourier modes of the current operators $\jmath_0(m)$ and $\jmath_1(m)$ are computed using anti-commutator structure and delta-function constraints, with careful analysis of matrix elements in the finite-particle subspace.

Experimental results

Research questions

• RQ1 How can a gauge-invariant Hamiltonian be consistently constructed in the Schwinger model on a circle using point-splitting regularization?
• RQ2 What is the role of large gauge transformations in the quantization of the Schwinger model, and how is gauge invariance preserved?
• RQ3 Are there anomalous contributions to the Fourier modes of the axial current $\jmath_1(m)$ for $m \neq 0$ under the Schwinger regularization scheme?
• RQ4 How do the regularized current operators $\jmath_0(m)$ and $\jmath_1(m)$ behave in the Fock space representation, and what is their operator structure?
• RQ5 What is the explicit form of the regularized Hamiltonian in the Schrödinger representation, including all quantum corrections and normal-ordering terms?

Key findings

• The regularized Hamiltonian is derived as a sum of kinetic, free-fermion, and Coulomb interaction terms, with explicit dependence on the gauge field coordinate $a$ and fermionic operators.
• The axial current Fourier mode $\jmath_1(0)$ is found to be anomaly-free and expressed in terms of the regularized axial charge $Q^{5,\text{reg}}$, with no zero-point or divergent contributions.
• For $m \neq 0$, the Fourier modes $\jmath_1(m)$ are free of anomalous contributions due to the vanishing of specific integrals involving the regularization function $\chi_\theta$, ensuring gauge invariance.
• The current operator $\jmath_0(m)$ is also anomaly-free, with its Fourier modes expressed as finite sums of creation and annihilation operators, mapping the finite-particle subspace to itself.
• The Hamiltonian operator is shown to be invariant under large gauge transformations $a \to a + \frac{2\pi}{L}$, confirming gauge invariance of the full construction.
• The final Hamiltonian (1.8) includes a kinetic term in $a$, a free-fermion term with $|k_m|$-weighted occupation numbers, a quadratic term in $a$, and a Coulomb interaction term involving $\sum_{m \neq 0} \frac{1}{k_m^2} \jmath_0(-m)\jmath_0(m)$, all consistently regularized.