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A note on quantum Hamiltonian reduction and anomalies

Boris M. Elfimov, Alexey A. Sharapov

TL;DR

This note demonstrates that quantum anomalies can persist in finite-dimensional Hamiltonian systems with first-class constraints, challenging the common view that anomalies require infinite degrees of freedom. Using the BRST-BFV framework and deformation quantization with a Wick-type $\ast$-product, it shows that obstructions in the BRST cohomology, specifically nontrivial $H^1(A)$, can prevent the quantum closure condition from being solvable. An explicit 4D example with a single constraint $T=H-E\approx0$ exhibits such an obstruction, revealing that the algebra of quantum physical observables can be strictly smaller than its classical counterpart due to global geometric features of the reduced phase space. The discussion connects these anomalies to the quantization of contact manifolds and broader geometric structures, offering insight into when and why finite-dimensional reductions fail to admit a consistent quantum theory.

Abstract

Quantization of field-theoretic models with gauge symmetries is often obstructed by quantum anomalies. It is commonly believed that the origin of these anomalies lies in the infinite number of degrees of freedom, which requires completing the model within an appropriate regularization scheme. This paper provides an explicit example of a finite-dimensional Hamiltonian system with first-class constraints whose quantization exhibits anomalies. These anomalies arise from the nontrivial topology of the reduced phase space.

A note on quantum Hamiltonian reduction and anomalies

TL;DR

This note demonstrates that quantum anomalies can persist in finite-dimensional Hamiltonian systems with first-class constraints, challenging the common view that anomalies require infinite degrees of freedom. Using the BRST-BFV framework and deformation quantization with a Wick-type -product, it shows that obstructions in the BRST cohomology, specifically nontrivial , can prevent the quantum closure condition from being solvable. An explicit 4D example with a single constraint exhibits such an obstruction, revealing that the algebra of quantum physical observables can be strictly smaller than its classical counterpart due to global geometric features of the reduced phase space. The discussion connects these anomalies to the quantization of contact manifolds and broader geometric structures, offering insight into when and why finite-dimensional reductions fail to admit a consistent quantum theory.

Abstract

Quantization of field-theoretic models with gauge symmetries is often obstructed by quantum anomalies. It is commonly believed that the origin of these anomalies lies in the infinite number of degrees of freedom, which requires completing the model within an appropriate regularization scheme. This paper provides an explicit example of a finite-dimensional Hamiltonian system with first-class constraints whose quantization exhibits anomalies. These anomalies arise from the nontrivial topology of the reduced phase space.
Paper Structure (5 sections, 37 equations, 1 figure)

This paper contains 5 sections, 37 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Classical Reduction
  3. Quantum Reduction and Anomalies
  4. Example.
  5. Discussion

Figures (1)

  • Figure 1: Left: projection of the isoenergetic surface $\Sigma$ of the Hamiltonian (\ref{['H']}) onto the plane of action variables. Right: the graph depicting the bifurcation of the integral $s_2$ on the isoenergetic surface $\Sigma$.