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Constrained systems, generalized Hamilton-Jacobi actions, and quantization

  • *Corresponding author: Alberto S. Cattaneo

    *Corresponding author: Alberto S. Cattaneo 
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  • 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.

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


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  • Figure 1.  Ghost propagator

    Figure 2.  Ghost vertices

    Figure 3.  $ F $-diagrams

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

    Figure 5.  Physical propagator

    Figure 6.  Linear case

    Figure 7.  Biaffine case

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

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

    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
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