Canonical Quantization of Two Dimensional Gauge Fields

Abstract

$SU(N)$ gauge fields on a cylindrical spacetime are canonically quantized via two routes revealing almost equivalent but different quantizations. After removing all continuous gauge degrees of freedom, the canonical coordinate $A_{\mu }$ (in the Cartan subalgebra $\h$) is quantized. The compact route, as in lattice gauge theory, quantizes the Wilson loop $W$, projecting out gauge invariant wavefunctions on the group manifold $G$. After a Casimir energy related to the curvature of $SU(N)$ is added to the compact spectrum, it is seen to be a subset of the non-compact spectrum. States of the two quantizations with corresponding energy are shifted relative each other, such that the ground state on $G$, ${\chi }_0(W)$, is the first excited state ${\Psi }_1(A_{\mu })$ on $\h$. The ground state ${\Psi }_0(A_{\mu })$ does not appear in the character spectrum as its lift is not globally defined on $G$. Implications for lattice gauge theory and the sum over maps representation of two dimensional QCD are discussed.

32 pages, 3 figures uuencoded, Plain TeX

References in corpus (1)

• Derivation of the Verlinde Formula from Chern-Simons Theory and the G/G model

Cited by in corpus (24)

• Quantum Chromodynamics and Other Field Theories on the Light Cone
• Boundary conditions as constraints
• Dynamical Rearrangement of Gauge Symmetry on the Orbifold S^1/Z_2
• Quantum Mechanics of the Vacuum State in Two-Dimensional QCD with Adjoint Fermions
• Dirac Quantization Gravity-Yang-Mills Systems in 1+1 Dimensions
• Geometry of the physical phase space in quantum gauge models
• Towards a Many-Body Treatment of Hamiltonian Lattice SU(N) Gauge Theory
• Towards Solving QCD - The Transverse Zero Modes in Light-Cone Quantization
• A gauge model for quantum mechanics on a stratified space
• Loop Correlators and Theta States in 2D Yang-Mills Theory
• Lattice QCD-2+1
• On Zero Modes and the Vacuum Problem -- A Study of Scalar Adjoint Matter in Two-Dimensional Yang-Mills Theory via Light-Cone Quantisation
• Matrix string states in pure 2d Yang Mills theories
• On the string interpretation of M(atrix) theory
• Tube Model for Light-Front QCD
• θ-angle monodromy in two dimensions
• Nonperturbative Stochastic Quantization of the Helix Model
• 2d CDT with gauge fields
• Polyakov-Loops and Fermionic Zero Modes in QCD2 on the Torus
• Some Non-Perturbative Aspects of Gauge Fixing in Two Dimensional Yang-Mills Theory
• 2D Yang-Mills Theory as a Matrix String Theory
• Quantum Hamiltonian Reduction of the Schwinger Model
• Topology Change and theta-Vacua in 2D Yang-Mills Theory
• The Path Integral for 1+1-dimensional QCD