Constrained systems, generalized Hamilton-Jacobi actions, and quantization

Alberto S. Cattaneo; Pavel Mnev; Konstantin Wernli

doi:10.3934/jgm.2022010

Article Contents

2022, Volume 14, Issue 2: 179-272. Doi: 10.3934/jgm.2022010

This issue Previous Article Preface to special issue in honor of Kirill C. H. Mackenzie: Part Ⅱ Next Article Modular class of Lie

$ \infty $

-algebroids and adjoint representations

Constrained systems, generalized Hamilton-Jacobi actions, and quantization

1.
Insitute of Mathematics, University of Zurich, Switzerland
2.
University of Notre Dame, USA, St. Petersburg Department, V. A. Steklov Institute of Mathematics of the Russian Academy of Sciences, Russia
3.
Centre for Quantum Mathematics, University of Southern Denmark, Denmark

^*Corresponding author: Alberto S. Cattaneo
^*Corresponding author: Alberto S. Cattaneo

Received: December 2020

Early access: May 23, 2022

Published online: May 23, 2022

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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.

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Mathematics Subject Classification: Primary: 81T70, 53D22, 70H20, 53D55, 53D50; Secondary: 81T13, 81S10, 70H15, 57R56, 81T45.

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Figure 1. Ghost propagator

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Figure 2. Ghost vertices

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Figure 3. $ F $-diagrams

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Figure 4. $ \mathbb{W} $-diagrams

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Figure 5. Physical propagator

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Figure 6. Linear case

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Figure 7. Biaffine case

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Figure 8. Composition of the partition function $ Z_{\rm{II}}^{\rm{new}} $

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Figure 9. Composition of the partition function $ Z_{\rm{INL}} $

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Contents
1. Introduction	181
1.1. Structure of the paper	183
1.2. How to read this paper	183
1.3. Notations	183
1.4. Teaser	184
2. Hamilton–Jacobi for nondegenerate actions	186
3. Systems with one constraint	193
3.1. Nondegenerate Lagrange multiplier	193
3.2. The general case	194
4. Systems with several constraints	197
4.1. The strictly involutive case	199
4.2. The Lie algebra case	201
4.3. The general case	205
5. Systems with nontrivial evolution and constraints	205
5.1. Classical mechanics	205
5.2. The free relativistic particle	209
6. Generalized generating functions for “bad” endpoint conditions	211
6.1. No evolution and no constraints	211
6.2. The partial Legendre transform	213
7. Infinite-dimensional targets	214
7.1. Three-dimensional abelian Chern–Simons theory	214
7.2. Nonabelian Chern–Simons theory	215
7.3. Nonabelian BF theory	216
7.4. More examples: 2D Yang–Mills theory and electrodynamics in general dimension	216
7.5. Higher-dimensional Chern-Simons theory	219
7.6. A nonlinear polarization in 7D Chern–Simons theory and the Kodaira–Spencer action	220
8. BFV, AKSZ and BV	224
9. An outline of elements of BV-BFV quantization	224
9.1. The classical BV-BFV setting	224
9.2. The quantum BV-BFV setting	224
10. BV-BFV boundary structures for linear polarizations	233
10.1. Three cases	233
10.2. The boundary structure in Case Ⅰ	234
10.3. The boundary structure in Case Ⅱ	234
10.4. The boundary structure in Case Ⅲ	234
11. BV-BFV quantization with linear polarizations	236
11.1. Quantization in Case Ⅱ	236
11.2. Quantization in Case Ⅲ	238
11.3. Quantization in Case Ⅰ	239
11.4. Gluing	245
11.5. Quantum mechanics	246
11.6. The quantum relativistic particle	248
12. BV-BFV quantization with nonlinear polarizations	248
12.1. Boundary structure	248
12.2. Gluing	248
12.3. The Case ⅢNL	248
12.4. The computation of ZINL	252
Appendix A. Symplectic geometry and generating functions	252
A.1. Symplectic spaces	253
A.2. Symplectic manifolds	258
A.3. Generalized Hamilton–Jacobi actions with infinite-dimensional targets	264
Appendix B. The modified differential quantum master equation	266
B.1. Assumptions	266
B.2. The propagator and the mdQME	268
B.3. Proof of the mdQME	268
B.4. Historical remarks	270
Acknowledgments	270
REFERENCES	271

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