Constrained systems, generalized Hamilton-Jacobi actions, and quantization

TL;DR

This paper explores constrained classical and quantum systems, detailing the properties of Hamilton-Jacobi actions, providing explicit examples including Chern-Simons theory, and performing perturbative quantization within BV-BFV formalism.

Contribution

It offers a detailed analysis of Hamilton-Jacobi actions in constrained systems, including infinite-dimensional cases, and applies BV-BFV formalism for quantization with explicit examples and results.

Findings

Hamilton-Jacobi action describes constrained systems effectively.

Certain toy models show no quantum corrections in the physical part.

Explicit computations for Chern-Simons and Hitchin models are provided.

Abstract

Mechanical systems (i.e., one-dimensional field theories) with constraints are the focus of this paper. In the classical theory, systems with infinite-dimensional targets are considered as well (this then encompasses also higher-dimensional field theories in the hamiltonian formalism). The properties of the Hamilton-Jacobi (HJ) action are described in details and several examples are explicitly computed (including nonabelian Chern-Simons theory, where the HJ action turns out to be the gauged Wess-Zumino-Witten action). Perturbative quantization, limited in this note to finite-dimensional targets, is performed in the framework of the Batalin-Vilkovisky (BV) formalism in the bulk and of the Batalin-Fradkin-Vilkovisky (BFV) formalism at the endpoints. As a sanity check of the method, it is proved that the semiclassical contribution of the physical part of the evolution operator is still given by the HJ action. Several examples are computed explicitly. In particular, it is shown that the toy model for nonabelian Chern-Simons theory and the toy model for 7D Chern-Simons theory with nonlinear Hitchin polarization do not have quantum corrections in the physical part (the extension of these results to the actual cases is discussed in the companion paper [arXiv:2012.13983]). Background material for both the classical part (symplectic geometry, generalized generating functions, HJ actions, and the extension of these concepts to infinite-dimensional manifolds) and the quantum part (BV-BFV formalism) is provided.