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quantum mechanics; contact geometry; quantization; contact topology; flat connections; clock ambiguity
Quantization together with quantum dynamics can be simultaneously formulated as the problem of finding an appropriate flat connection on a Hilbert bundle over a contact manifold. Contact geometry treats time, generalized positions and momenta as points on an underlying phase-spacetime and reduces classical mechanics to contact topology. Contact quantization describes quantum dynamics in terms of parallel transport for a flat connection; the ultimate goal being to also handle quantum systems in terms of contact topology. Our main result is a proof of local, formal gauge equivalence for a broad class of quantum dynamical systems—just as classical dynamics depends on choices of clocks, local quantum dynamics can be reduced to a problem of studying gauge transformations. We further show how to write quantum correlators in terms of parallel transport and in turn matrix elements for Hilbert bundle gauge transformations, and give the path integral formulation of these results. Finally, we show how to relate topology of the underlying contact manifold to boundary conditions for quantum wave functions.
[1] Albrecht, A., Iglesias, A.: The clock ambiguity and the emergence of physical laws. Phys. Rev. D 77 (2008), 063506; arXiv:0708.2743 [hep-th]; S. B. Gryb, Jacobi's Principle and the Disappearance of Time Phys. Rev. D 81 (2010), 044035, arXiv:0804.2900 [gr-qc]; S. B. Gryb and K. Thebault, The role of time in relational quantum theories Found. Phys. 42 (2012),1210–1238 arXiv:1110.2429 [gr-qc]. DOI 10.1103/PhysRevD.77.063506 | MR 2996626
[2] Batalin, I., Fradkin, E., Fradkina, T.: Another version for operatorial quantization of dynamical systems with irreducible constraints. Nuclear Phys. B 314 (1989), 158–174, I.A. Batalin and I.V. Tyutin, Existence theorem for the effective gauge algebra in the generalized canonical formalism with abelian conversion of second-class constraints, Internat. J. Modern Phys. A 6 (1991), 3255–3282. MR 0984074
[3] Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Quantum Mechanics as a deformation of classical mechanics. Lett. Math. Phys. 1 (1977), 521–530. DOI 10.1007/BF00399745 | MR 0674337
[4] Bieliavsky, P., Cahen, M., Gutt, S., Rawnsley, J., Schwachhöfer, L.: Symplectic connection. Int. J. Geom. Methods Mod. Phys. 3 (2006), 375–426, arXiv:math/0511194. DOI 10.1142/S021988780600117X | MR 2232865
[5] Bruce, A.J.: Contact structures and supersymmetric mechanics. arXiv:1108.5291 [math-ph].
[6] Čap, A., Slovák, J.:
[7] Cattaneo, A.S., Felder, G.: A path integral approach to the Kontsevich quantization formula. Comm. Math. Phys. 212 (2000), 591–611, arXiv:math/9902090. DOI 10.1007/s002200000229 | MR 1779159 | Zbl 1038.53088
[8] Dupré, M.J.: Classifying Hilbert bundles. J. Funct. Anal. 15 (1974), 244–278. DOI 10.1016/0022-1236(74)90035-4 | MR 0346541
[9] Fedosov, B.V.: A simple geometrical construction of deformation quantization. J. Differential Geom. 40 (1994), 213–238. DOI 10.4310/jdg/1214455536 | MR 1293654 | Zbl 0812.53034
[10] Fitzpatrick, S.: On the geometric quantization of contact manifolds. J. Geom. Phys. 61 (2011), 2384–2399. DOI 10.1016/j.geomphys.2011.07.011 | MR 2838515
[11] Fox, D.J.F.: Contact projective structures. Indiana Univ. Math. J. 54 (2005), 1547–1598, arXiv:math/0402332. MR 2189678 | Zbl 1093.53083
[12] Fradkin, E.S., Vilkovisky, G.: Quantization of relativistic systems with constraints. Phys. Lett. B 55 (1975), 224–226, I.A. Batalin and G.A. Vilkovisky, Relativistic s-matrix of dynamical systems with boson and fermion constraints, Phys. Lett. B 69 (1977), 309–312; E.S. Fradkin and T. Fradkina, Phys. Lett. B 72 (1978), 343–348; I. Batalin and E.S. Fradkin, La Rivista del Nuovo Cimento 9 (1986), 1–48. DOI 10.1016/0370-2693(75)90448-7 | MR 0411451
[13] Geiges, H.: An Introduction to Contact Topology. Cambridge University Pres, 2008, and P. Ševera, Contact geometry in lagrangian mechanics, J. Geom. Phys. 29 (1999), 235–242; A. Bravetti, C.S. Lopez-Monsalvo and F. Nettel, Contact symmetries and Hamiltonian thermodynamics, Ann. Phys. 361 (2015), 377-400, arXiv:1409.7340; A. Bravetti, H. Cruz and D. Tapias, Contact Hamiltonian dynamics, arXiv:1604.08266[math-ph]. MR 3388763
[14] Grigoriev, M.A., Lyakhovich, S.L.: Fedosov Deformation Quantization as a BRST Theory. Comm. Math. Phys. 218 (2001), 437–457, hep-th/0003114. See also G. Barnich and M. Grigoriev, A. Semikhatov and I. Tipunin, Parent Field Theory and Unfolding in BRST First-Quantized Terms, 260, (2005), 147–181, hep-th/0406192. DOI 10.1007/PL00005559 | MR 2175993
[15] Gukov, S., Witten, E.: Branes and quantization. Adv. Theor. Math. Phys. 13 (2009), 1445–1518, arXiv:0809.0305 [hep-th]. MR 2672467
[16] Herczeg, G., Waldron, A.: Contact geometry and quantum mechanics. Phys.Lett. B 781 (2018), 312–315, arXiv:1709.04557 [hep-th] . DOI 10.1016/j.physletb.2018.04.008
[17] Kashiwara, M.: Quantization of contact manifolds. Publ. Res. Inst. Math. Sci. 32 (1) (1996), 1–7. DOI 10.2977/prims/1195163179 | MR 1384750
[18] Kontsevich, M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66 (2003), 157–216, arXiv:q-alg/9709040. DOI 10.1023/ | MR 2062626 | Zbl 1058.53065
[19] Krýsl, S.: Cohomology of the de Rham complex twisted by the oscillatory representation. Differential Geom. Appl. 33 (2014), 290–297, arXiv:1304.5704 [math.DG]. DOI 10.1016/j.difgeo.2013.10.007 | MR 3159964
[20] Małkiewicz, P., Miroszewski, A.: Internal clock formulation of quantum mechanics. Phys. Rev. D 96 (2017), 046003, arXiv:1706.00743 [gr-qc]. DOI 10.1103/PhysRevD.96.046003 | MR 3852958
[21] Manin, Y.: Topics in Noncommutative Geometry. M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1991. MR 1095783 | Zbl 0724.17007
[22] Rajeev, S.G.: Quantization of contact manifolds and thermodynamics. Ann. Physics 323 (2008), 768–782. DOI 10.1016/j.aop.2007.05.001 | MR 2404789
[23] Schwarz, A.S.: Superanalogs of symplectic and contact geometry and their applications to quantum field theory. Topics in statistical and theoretical physics, vol. 177, Amer. Math. Soc. Transl. Ser. 2, 1996, Adv. Math. Sci., 32, arXiv:hep-th/9406120, pp. 203–218. MR 1409176
[24] Yoshioka, A.: Contact Weyl manifold over a symplectic manifold. Lie groups, geometric structures and differential equations – one hundred years after Sophus Lie. Adv. Stud. Pure Math. 37 (2002), 459–493, A. Yoshioka, Il Nuov. Cim. 38C (2015), 173. MR 1980911
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