Constrained systems, generalized HamiltonJacobi actions, and quantization
Abstract
Mechanical systems (i.e., onedimensional field theories) with constraints are the focus of this paper. In the classical theory, systems with infinitedimensional targets are considered as well (this then encompasses also higherdimensional field theories in the hamiltonian formalism). The properties of the HamiltonJacobi (HJ) action are described in details and several examples are explicitly computed (including nonabelian ChernSimons theory, where the HJ action turns out to be the gauged WessZuminoWitten action). Perturbative quantization, limited in this note to finitedimensional targets, is performed in the framework of the BatalinVilkovisky (BV) formalism in the bulk and of the BatalinFradkinVilkovisky (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 ChernSimons theory and the toy model for 7D ChernSimons 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 infinitedimensional manifolds) and the quantum part (BVBFV formalism) is provided.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 DOI:
 10.48550/arXiv.2012.13270
 arXiv:
 arXiv:2012.13270
 Bibcode:
 2020arXiv201213270C
 Keywords:

 Mathematical Physics;
 High Energy Physics  Theory;
 Mathematics  Symplectic Geometry;
 81T70;
 53D22;
 70H20;
 53D55;
 53D50 (Primary);
 81T13;
 81S10;
 70H15;
 57R56;
 81T45 (Secondary)
 EPrint:
 96 pages, 9 figures. Updated references