June  2022, 14(2): 179-272. doi: 10.3934/jgm.2022010

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

Received  December 2020 Published  June 2022 Early access  May 2022

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.

Citation: Alberto S. Cattaneo, Pavel Mnev, Konstantin Wernli. Constrained systems, generalized Hamilton-Jacobi actions, and quantization. Journal of Geometric Mechanics, 2022, 14 (2) : 179-272. doi: 10.3934/jgm.2022010
References:
[1]

A. Alekseev and P. Mnev, One-dimensional Chern–Simons theory, Commun. Math. Phys., 307.1 (2011), 185-227.  doi: 10.1007/s00220-011-1290-1.

[2]

M. AlexandrovM. KontsevichA. Schwarz and O. Zaboronsky, The geometry of the master equation and topological quantum field theory, Int. J. Mod. Phys. A, 12 (1997), 1405-1429.  doi: 10.1142/S0217751X97001031.

[3]

V. I. Arnold, Mathematical Methods of Classical Mechanics, 2$^nd$ edition, Springer, 1989.

[4]

S. Axelrod and I. M. Singer, Chern–Simons perturbation theory, J. Diff. Geom., 39 (1994), 173–213.

[5]

J. F. BarberoG. B. DíazJ. Margalef-Bentabol and E. J. S. Villaseñor, Dirac's algorithm in the presence of boundaries: A practical guide to a geometric approach, Class. Quantum Grav., 36 (2019), 205014.  doi: 10.1088/1361-6382/ab436b.

[6]

I. Batalin and E. Fradkin, A generalized canonical formalism and quantization of reducible gauge theories, Phys. Lett. B, 122 (1983), 157-164.  doi: 10.1016/0370-2693(83)90784-0.

[7]

I. Batalin and G. Vilkovisky, Relativistic S-matrix of dynamical systems with boson and fermion constraints, Phys. Lett. B, 69 (1977), 309-312. 

[8]

M. BershadskyS. CecottiH. Ooguri and C. Vafa, Kodaira–Spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys., 165 (1994), 311-427.  doi: 10.1007/BF02099774.

[9]

S. Bates and A. Weinstein, Lectures on the Geometry of Quantization, Berkeley Mathematics Lecture Notes, 8, American Mathematical Society, Providence, RI, 1997. doi: 10.1016/s0898-1221(97)90217-0.

[10]

F. BonechiA. S. Cattaneo and P. Mnev, The Poisson sigma model on closed surfaces, J. High Energ. Phys., 2012 (2012), 1-27.  doi: 10.1007/JHEP01(2012)099.

[11]

R. Bott and A. S. Cattaneo, Integral invariants of 3-manifolds, J. Diff. Geom., 48 (1998), 91-133. 

[12]

A. S. Cattaneo, From topological field theory to deformation quantization and reduction, Proceedings of the International Congress of Mathematicians, 3 (2006), 338-365. 

[13]

A. S. Cattaneo and I. Contreras, Split canonical relations, Annales Henri Lebesgue, 4 (2021), 155-185.  doi: 10.5802/ahl.69.

[14]

A. S. CattaneoN. Moshayedi and K. Wernli, Globalization for perturbative quantization of nonlinear split AKSZ sigma models on manifolds with boundary, Commun. Math. Phys., 372 (2019), 213-260.  doi: 10.1007/s00220-019-03591-5.

[15]

A. S. Cattaneo and P. Mnev, Remarks on Chern–Simons invariants, Commun. Math. Phys., 293 (2010), 803-836.  doi: 10.1007/s00220-009-0959-1.

[16]

A. S. Cattaneo and P. Mnev, Wave relations, Commun. Math. Phys., 332 (2014), 1083-1111.  doi: 10.1007/s00220-014-2130-x.

[17]

A. S. CattaneoP. Mnev and N. Reshetikhin, Classical BV theories on manifolds with boundary, Commun. Math. Phys., 332 (2014), 535-603.  doi: 10.1007/s00220-014-2145-3.

[18]

A. S. Cattaneo. P. Mnev and N. Reshetikin, Classical and quantum Lagrangian field theories with boundary, Proceedings of the Corfu Summer Institute 2011 School and Workshops on Elementary Particle Physics and Gravity (September 4-18, 2011, Corfu, Greece), SISSA, 2011.

[19]

A. S. CattaneoP. Mnev and N. Reshetikhin, Perturbative quantum gauge theories on manifolds with boundary, Commun. Math. Phys., 357 (2018), 631-730.  doi: 10.1007/s00220-017-3031-6.

[20]

A. S. CattaneoP. Mnev and and N. Reshetikhin, A cellular topological field theory, Commun. Math. Phys., 374 (2020), 1229-1320.  doi: 10.1007/s00220-020-03687-3.

[21]

A. S. Cattaneo, P. Mnev and K. Wernli, Quantum Chern–Simons theories on cylinders: BV-BFV partition functions, preprint, arXiv: 2012.13983.

[22]

A. S. Cattaneo and M. Schiavina, On time, Lett. Math. Phys., 107 (2017), 375-408.  doi: 10.1007/s11005-016-0907-x.

[23]

I. Contreras, Relational Symplectic Groupoids and Poisson Sigma Models with Boundary, Ph.D Thesis, University of Zurich (2013).

[24]

P. H. Damgaard and M. A. Grigoriev, Superfield BRST charge and the master action, Phys. Lett. B, 474 (2000), 323-330. 

[25]

J. Dereziński, Introduction to Quantization (2020), Available from: https://www.fuw.edu.pl/ derezins/quantize.pdf.

[26]

P. A. M. Dirac, The Lagrangian in quantum mechanics, Physikalische Zeitschrift der Sowjetunion, 3 (1933), 64-72. 

[27]

G. Dito and D. Sternheimer, Deformation quantization: Genesis, developments and metamorphoses, in Deformation Quantization (Strasbourg, 2001), IRMA Lect. Math. Theor. Phys., 1, de Gruyter, Berlin, 2002, 9–54.

[28]

R. P. Feynman, The Principle of Least Action in Quantum Mechanics, Ph.D thesis, Princeton, 1942. doi: 10.1142/9789812567635_0001.

[29]

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965.

[30]

E. Fradkin and G. Vilkovisky, Quantization of relativistic systems with constraints, Phys. Lett. B, 55 (1975), 224-226.  doi: 10.1016/0370-2693(75)90448-7.

[31]

W. Fulton and R. MacPherson, A compactification of configuration spaces, Ann. of Math., 139 (1994), 183-225.  doi: 10.2307/2946631.

[32]

K. Gawedzki, Conformal field theory: A case study, preprint, https://arXiv.org/abs/hep-th/9904145.

[33]

K. Gawedzki, Lectures on conformal field theory, in: Quantum Fields and Strings: A Course for Mathematicians, 2, Amer. Math. Soc., Providence, RI, 1999,727–805. doi: 10.1007/s002200050573.

[34]

A. A. Gerasimov and S. L. Shatashvili, Towards integrability of topological strings I: Three-forms on Calabi–Yau manifolds, J. High Energ. Phys., 2004.11 (2005), 074.  doi: 10.1088/1126-6708/2004/11/074.

[35]

N. Hitchin, The geometry of three-forms in six dimensions, J. Diff. Geom., 55 (2000), 547-576. 

[36]

R. Iraso and P. Mnev, Two-dimensional Yang–Mills theory on surfaces with corners in Batalin–Vilkovisky formalism, Commun. Math. Phys., 370 (2019), 637-702.  doi: 10.1007/s00220-019-03392-w.

[37]

H. M. Khudaverdian, Semidensities on odd symplectic supermanifolds, Commun. Math. Phys., 247 (2004), 353-390.  doi: 10.1007/s00220-004-1083-x.

[38]

J. Kijowski and W. M. Tulczyjew, A Symplectic Framework for Field Theories, Lecture notes in Physics, 107, Springer-Verlag Berlin Heidelberg, 1979.

[39]

K. Kodaira and D. Spencer, On deformations of complex analytic structures. I., Ann. of Math., 67 (1958), 328-401.  doi: 10.2307/1970009.

[40]

P. Libermann and C. Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and Its Applications, 35, D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.

[41]

J.-H. Lu, Multiplicative and Affine Poisson Structures on Lie Groups, Ph.D thesis, UC Berkeley, 1990.

[42] K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, Cambridge University Press, 1987.  doi: 10.1017/CBO9780511661839.
[43]

P. MnevM. Schiavina and K. Wernli, Towards holography in the BV-BFV setting, Ann. Henri Poincaré, 21 (2020), 993-1044.  doi: 10.1007/s00023-019-00862-8.

[44]

N. Reshetikhin, Semiclassical geometry of integrable systems, J. Phys. A, 51 (2018), 164001.  doi: 10.1088/1751-8121/aaaea6.

[45]

P. Ševera, On the origin of the BV operator on odd symplectic supermanifolds, Lett. Math. Phys., 78 (2006), 55-59.  doi: 10.1007/s11005-006-0097-z.

[46]

J. Stasheff, Homological reduction of constrained Poisson algebras, J. Diff. Geom., 45 (1997), 221–240.

[47]

G. Tian, Smoothness of the universal deformation space of compact Calabi–Yau manifolds and its Petersson–Weil metric, in Mathematical Aspects of String Theory (San Diego, Calif., 1986), Adv. Ser. Math. Phys., 1, World Sci. Publishing, Singapore, 1987,629–646.

[48]

F. Valach and D. Youmans, Schwarzian quantum mechanics as a Drinfeld–Sokolov reduction of $BF$ theory, J. High Energ. Phys., 2020 (2020). doi: 10.1007/jhep12(2020)189.

show all references

References:
[1]

A. Alekseev and P. Mnev, One-dimensional Chern–Simons theory, Commun. Math. Phys., 307.1 (2011), 185-227.  doi: 10.1007/s00220-011-1290-1.

[2]

M. AlexandrovM. KontsevichA. Schwarz and O. Zaboronsky, The geometry of the master equation and topological quantum field theory, Int. J. Mod. Phys. A, 12 (1997), 1405-1429.  doi: 10.1142/S0217751X97001031.

[3]

V. I. Arnold, Mathematical Methods of Classical Mechanics, 2$^nd$ edition, Springer, 1989.

[4]

S. Axelrod and I. M. Singer, Chern–Simons perturbation theory, J. Diff. Geom., 39 (1994), 173–213.

[5]

J. F. BarberoG. B. DíazJ. Margalef-Bentabol and E. J. S. Villaseñor, Dirac's algorithm in the presence of boundaries: A practical guide to a geometric approach, Class. Quantum Grav., 36 (2019), 205014.  doi: 10.1088/1361-6382/ab436b.

[6]

I. Batalin and E. Fradkin, A generalized canonical formalism and quantization of reducible gauge theories, Phys. Lett. B, 122 (1983), 157-164.  doi: 10.1016/0370-2693(83)90784-0.

[7]

I. Batalin and G. Vilkovisky, Relativistic S-matrix of dynamical systems with boson and fermion constraints, Phys. Lett. B, 69 (1977), 309-312. 

[8]

M. BershadskyS. CecottiH. Ooguri and C. Vafa, Kodaira–Spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys., 165 (1994), 311-427.  doi: 10.1007/BF02099774.

[9]

S. Bates and A. Weinstein, Lectures on the Geometry of Quantization, Berkeley Mathematics Lecture Notes, 8, American Mathematical Society, Providence, RI, 1997. doi: 10.1016/s0898-1221(97)90217-0.

[10]

F. BonechiA. S. Cattaneo and P. Mnev, The Poisson sigma model on closed surfaces, J. High Energ. Phys., 2012 (2012), 1-27.  doi: 10.1007/JHEP01(2012)099.

[11]

R. Bott and A. S. Cattaneo, Integral invariants of 3-manifolds, J. Diff. Geom., 48 (1998), 91-133. 

[12]

A. S. Cattaneo, From topological field theory to deformation quantization and reduction, Proceedings of the International Congress of Mathematicians, 3 (2006), 338-365. 

[13]

A. S. Cattaneo and I. Contreras, Split canonical relations, Annales Henri Lebesgue, 4 (2021), 155-185.  doi: 10.5802/ahl.69.

[14]

A. S. CattaneoN. Moshayedi and K. Wernli, Globalization for perturbative quantization of nonlinear split AKSZ sigma models on manifolds with boundary, Commun. Math. Phys., 372 (2019), 213-260.  doi: 10.1007/s00220-019-03591-5.

[15]

A. S. Cattaneo and P. Mnev, Remarks on Chern–Simons invariants, Commun. Math. Phys., 293 (2010), 803-836.  doi: 10.1007/s00220-009-0959-1.

[16]

A. S. Cattaneo and P. Mnev, Wave relations, Commun. Math. Phys., 332 (2014), 1083-1111.  doi: 10.1007/s00220-014-2130-x.

[17]

A. S. CattaneoP. Mnev and N. Reshetikhin, Classical BV theories on manifolds with boundary, Commun. Math. Phys., 332 (2014), 535-603.  doi: 10.1007/s00220-014-2145-3.

[18]

A. S. Cattaneo. P. Mnev and N. Reshetikin, Classical and quantum Lagrangian field theories with boundary, Proceedings of the Corfu Summer Institute 2011 School and Workshops on Elementary Particle Physics and Gravity (September 4-18, 2011, Corfu, Greece), SISSA, 2011.

[19]

A. S. CattaneoP. Mnev and N. Reshetikhin, Perturbative quantum gauge theories on manifolds with boundary, Commun. Math. Phys., 357 (2018), 631-730.  doi: 10.1007/s00220-017-3031-6.

[20]

A. S. CattaneoP. Mnev and and N. Reshetikhin, A cellular topological field theory, Commun. Math. Phys., 374 (2020), 1229-1320.  doi: 10.1007/s00220-020-03687-3.

[21]

A. S. Cattaneo, P. Mnev and K. Wernli, Quantum Chern–Simons theories on cylinders: BV-BFV partition functions, preprint, arXiv: 2012.13983.

[22]

A. S. Cattaneo and M. Schiavina, On time, Lett. Math. Phys., 107 (2017), 375-408.  doi: 10.1007/s11005-016-0907-x.

[23]

I. Contreras, Relational Symplectic Groupoids and Poisson Sigma Models with Boundary, Ph.D Thesis, University of Zurich (2013).

[24]

P. H. Damgaard and M. A. Grigoriev, Superfield BRST charge and the master action, Phys. Lett. B, 474 (2000), 323-330. 

[25]

J. Dereziński, Introduction to Quantization (2020), Available from: https://www.fuw.edu.pl/ derezins/quantize.pdf.

[26]

P. A. M. Dirac, The Lagrangian in quantum mechanics, Physikalische Zeitschrift der Sowjetunion, 3 (1933), 64-72. 

[27]

G. Dito and D. Sternheimer, Deformation quantization: Genesis, developments and metamorphoses, in Deformation Quantization (Strasbourg, 2001), IRMA Lect. Math. Theor. Phys., 1, de Gruyter, Berlin, 2002, 9–54.

[28]

R. P. Feynman, The Principle of Least Action in Quantum Mechanics, Ph.D thesis, Princeton, 1942. doi: 10.1142/9789812567635_0001.

[29]

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965.

[30]

E. Fradkin and G. Vilkovisky, Quantization of relativistic systems with constraints, Phys. Lett. B, 55 (1975), 224-226.  doi: 10.1016/0370-2693(75)90448-7.

[31]

W. Fulton and R. MacPherson, A compactification of configuration spaces, Ann. of Math., 139 (1994), 183-225.  doi: 10.2307/2946631.

[32]

K. Gawedzki, Conformal field theory: A case study, preprint, https://arXiv.org/abs/hep-th/9904145.

[33]

K. Gawedzki, Lectures on conformal field theory, in: Quantum Fields and Strings: A Course for Mathematicians, 2, Amer. Math. Soc., Providence, RI, 1999,727–805. doi: 10.1007/s002200050573.

[34]

A. A. Gerasimov and S. L. Shatashvili, Towards integrability of topological strings I: Three-forms on Calabi–Yau manifolds, J. High Energ. Phys., 2004.11 (2005), 074.  doi: 10.1088/1126-6708/2004/11/074.

[35]

N. Hitchin, The geometry of three-forms in six dimensions, J. Diff. Geom., 55 (2000), 547-576. 

[36]

R. Iraso and P. Mnev, Two-dimensional Yang–Mills theory on surfaces with corners in Batalin–Vilkovisky formalism, Commun. Math. Phys., 370 (2019), 637-702.  doi: 10.1007/s00220-019-03392-w.

[37]

H. M. Khudaverdian, Semidensities on odd symplectic supermanifolds, Commun. Math. Phys., 247 (2004), 353-390.  doi: 10.1007/s00220-004-1083-x.

[38]

J. Kijowski and W. M. Tulczyjew, A Symplectic Framework for Field Theories, Lecture notes in Physics, 107, Springer-Verlag Berlin Heidelberg, 1979.

[39]

K. Kodaira and D. Spencer, On deformations of complex analytic structures. I., Ann. of Math., 67 (1958), 328-401.  doi: 10.2307/1970009.

[40]

P. Libermann and C. Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and Its Applications, 35, D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.

[41]

J.-H. Lu, Multiplicative and Affine Poisson Structures on Lie Groups, Ph.D thesis, UC Berkeley, 1990.

[42] K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, Cambridge University Press, 1987.  doi: 10.1017/CBO9780511661839.
[43]

P. MnevM. Schiavina and K. Wernli, Towards holography in the BV-BFV setting, Ann. Henri Poincaré, 21 (2020), 993-1044.  doi: 10.1007/s00023-019-00862-8.

[44]

N. Reshetikhin, Semiclassical geometry of integrable systems, J. Phys. A, 51 (2018), 164001.  doi: 10.1088/1751-8121/aaaea6.

[45]

P. Ševera, On the origin of the BV operator on odd symplectic supermanifolds, Lett. Math. Phys., 78 (2006), 55-59.  doi: 10.1007/s11005-006-0097-z.

[46]

J. Stasheff, Homological reduction of constrained Poisson algebras, J. Diff. Geom., 45 (1997), 221–240.

[47]

G. Tian, Smoothness of the universal deformation space of compact Calabi–Yau manifolds and its Petersson–Weil metric, in Mathematical Aspects of String Theory (San Diego, Calif., 1986), Adv. Ser. Math. Phys., 1, World Sci. Publishing, Singapore, 1987,629–646.

[48]

F. Valach and D. Youmans, Schwarzian quantum mechanics as a Drinfeld–Sokolov reduction of $BF$ theory, J. High Energ. Phys., 2020 (2020). doi: 10.1007/jhep12(2020)189.

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}} $
Table1 
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
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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