Constrained systems, generalized Hamilton-Jacobi actions, and quantization

Alberto S. Cattaneo*, Pavel Mnev, Konstantin Wernli

*Kontaktforfatter

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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 [21]). 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.

OriginalsprogEngelsk
TidsskriftJournal of Geometric Mechanics
Vol/bind14
Udgave nummer2
Sider (fra-til)179-272
Antal sider94
ISSN1941-4889
DOI
StatusUdgivet - jun. 2022

Bibliografisk note

Funding Information:
This research was (partly) supported by the NCCR SwissMAP, funded by the Swiss National Science Foundation. A.S.C. and K.W. acknowledge partial support of SNF Grant No. 200020 192080. K. W. also acknowledges support from a BMS Dirichlet postdoctoral fellowship and the SNF Postdoc.Mobility grant

Funding Information:
We thank Samson Shatashvili for suggesting the study of 7D abelian Chern–Simons theory in the quantum BV-BFV formalism, now in [21, Section 6.3], which was the original motivation out of which this paper grew. We also thank Francesco Bonechi, Alejandro Cabrera, Ivan Contreras, Philippe Mathieu, Nicolai Reshetikhin, Pavel Safronov, Michele Schiavina, Stephan Stolz, Alan Weinstein, Ping Xu, and Donald Youmans for useful discussions. This research was (partly) supported by the NCCR SwissMAP, funded by the Swiss National Science Foundation. A.S.C. and K.W. acknowledge partial support of SNF Grant No. 200020 192080. K. W. also acknowledges support from a BMS Dirichlet postdoctoral fellowship and the SNF Postdoc.Mobility grant P2ZHP2 184083, and would like to thank the Humboldt-Universität Berlin, in particular the group of Dirk Kreimer, and the university of Notre Dame for their hospitality.

Publisher Copyright:
© 2022 American Institute of Mathematical Sciences. All rights reserved.

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