On time. (English) Zbl 1364.81221
Summary: This note describes the restoration of time in one-dimensional parameterization-invariant (hence timeless) models, namely, the classically equivalent Jacobi action and gravity coupled to matter. It also serves as a timely introduction by examples to the classical and quantum BV-BFV formalism as well as to the AKSZ method.
MSC:
| 81T70 | Quantization in field theory; cohomological methods |
| 70S05 | Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems |
| 83C47 | Methods of quantum field theory in general relativity and gravitational theory |
| 81T45 | Topological field theories in quantum mechanics |
| 81V17 | Gravitational interaction in quantum theory |
| 81T60 | Supersymmetric field theories in quantum mechanics |
Keywords:
parametrization invariant Lagrangian; Jacobi action; one-dimensional gravity; BV; BFV; AKSZ; spinning particle; supersymmetryReferences:
| [1] | Alexandrov, M., Kontsevich, M., Schwarz, A., Zaboronsky, O.: “The geometry of the master equation and topological quantum field theory. Int. J. Mod. Phys. <Emphasis Type=”Bold”>A12, 1405-1430 (1997) · Zbl 1073.81655 · doi:10.1142/S0217751X97001031 |
| [2] | Barbour, J.: The nature of time. arXiv:0903.3489 · Zbl 1082.83036 |
| [3] | Batalin, I.A., Fradkin, E.S.: A generalized canonical formalism and quantization of reducible gauge theories. Phys. Lett. B 122, 157-164 (1983) · Zbl 0967.81508 · doi:10.1016/0370-2693(83)90784-0 |
| [4] | Batalin, I.A., Vilkovisky, G.A.: Gauge algebra and quantization. Phys. Lett. B 102, 27-31 (1981) · doi:10.1016/0370-2693(81)90205-7 |
| [5] | Batalin, I.A., Vilkovisky, G.A.: Relativistic S-matrix of dynamical systems with boson and fermion costraints. Phys. Lett. B 69, 309-312 (1977) · doi:10.1016/0370-2693(77)90553-6 |
| [6] | Becchi, C., Rouet, A., Stora, R.: Renormalization of the abelian Higgs-Kibble model. Commun. Math. Phys. 42, 127-162 (1975) · doi:10.1007/BF01614158 |
| [7] | Bonechi, F., Cattaneo, A.S., Mnëv, P.: The Poisson sigma model on closed surfaces. JHEP 2012(99), 1-27 (2012) · Zbl 1306.81290 |
| [8] | Brink, L., Deser, S., Zumino, B., Di Vecchia, P., Howe, P.: Local supersymmetry for spinning particles. Phys. Lett. B 64, 435-438 (1976) · doi:10.1016/0370-2693(76)90115-5 |
| [9] | Cattaneo, A.S., Mnëv, P., Reshetikhin, N.: Classical BV theories on manifolds with boundaries. Commun. Math. Phys. 332, 535-603 (2014) · Zbl 1302.81141 · doi:10.1007/s00220-014-2145-3 |
| [10] | Cattaneo, A.S., Mnëv, P., Reshetikhin, N.: Classical and quantum Lagrangian field theories with boundary. In: Proceedings of the Corfu Summer Institute 2011 School and Workshops on Elementary Particle Physics and Gravity. arXiv:1207.0239 · Zbl 1302.81141 |
| [11] | Cattaneo, A.S.: Mnëv, P., Reshetikhin, N.: Perturbative quantum gauge theories on manifolds with boundary. Commun. Math. Phys. arXiv:1507.01221 · Zbl 1390.81381 |
| [12] | Cattaneo, A.S., Schiavina, M.: BV-BFV approach to general relativity: Einstein-Hilbert action. J. Math. Phys. 57, 023515, 17 (2016) · Zbl 1336.83007 |
| [13] | Cattaneo, A.S., Schiavina, M.: BV-BFV approach to general relativity II: Palatini-Holst action (in preparation) · Zbl 1336.83007 |
| [14] | Geztler, E.: The Batalin-Vilkovisky formalism of the spinning particle. arXiv:1511.02135 |
| [15] | Geztler, E.: The spinning particle with curved target. arXiv:1605.04762 · Zbl 1302.81141 |
| [16] | Hartle, J.B., Hawking, S.W.: Wave function of the Universe. Phys. Rev. D 28, 2960 (1983) · Zbl 1370.83118 · doi:10.1103/PhysRevD.28.2960 |
| [17] | Jacobi, C.G.J.: Das Princip der kleinsten Wirkung. Ch. 6. In: Clebsch, A. (ed.) Vorlesungen über Dynamik. Available at Gallica-Math. Reimer, Berlin (1866) · Zbl 1125.81040 |
| [18] | Kijowski, J., Tulczyjew, M.: A Symplectic Framework for Field Teories, Lecture Notes in Physics, vol. 107. Springer, New york (1979) · Zbl 0439.58002 |
| [19] | Kostant, B., Sternberg, S.: Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras. Ann. Phys. 176, 49-113 (1987) · Zbl 0642.17003 · doi:10.1016/0003-4916(87)90178-3 |
| [20] | Roytenberg, D.: AKSZ-BV formalism and courant algebroid-induced topological field theories. Lett. Math. Phys. 79, 143-159 (2007) · Zbl 1125.81040 · doi:10.1007/s11005-006-0134-y |
| [21] | Schiavina, M.: BV-BFV Approach to general relativity, PhD thesis, Zurich (2015) · Zbl 1336.83007 |
| [22] | Tyutin, I.V.: Gauge invariance in field theory and statistical physics in operator formalism. Lebedev Institute preprint No. 39 (1975). arXiv:0812.0580 · Zbl 1125.81040 |
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