# Contact Quantization: Quantum Mechanics = Parallel transport

G. Herczeg; E. Latini; Andrew Waldron

Archivum Mathematicum (2018)

- Volume: 054, Issue: 5, page 281-298
- ISSN: 0044-8753

## Access Full Article

top## Abstract

top## How to cite

topHerczeg, G., Latini, E., and Waldron, Andrew. "Contact Quantization: Quantum Mechanics = Parallel transport." Archivum Mathematicum 054.5 (2018): 281-298. <http://eudml.org/doc/294288>.

@article{Herczeg2018,

abstract = {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.},

author = {Herczeg, G., Latini, E., Waldron, Andrew},

journal = {Archivum Mathematicum},

keywords = {quantum mechanics; contact geometry; quantization; contact topology; flat connections; clock ambiguity},

language = {eng},

number = {5},

pages = {281-298},

publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},

title = {Contact Quantization: Quantum Mechanics = Parallel transport},

url = {http://eudml.org/doc/294288},

volume = {054},

year = {2018},

}

TY - JOUR

AU - Herczeg, G.

AU - Latini, E.

AU - Waldron, Andrew

TI - Contact Quantization: Quantum Mechanics = Parallel transport

JO - Archivum Mathematicum

PY - 2018

PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno

VL - 054

IS - 5

SP - 281

EP - 298

AB - 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.

LA - eng

KW - quantum mechanics; contact geometry; quantization; contact topology; flat connections; clock ambiguity

UR - http://eudml.org/doc/294288

ER -

## References

top- Albrecht, A., Iglesias, A., 10.1103/PhysRevD.77.063506, 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]. (2008) MR2996626DOI10.1103/PhysRevD.77.063506
- 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. (1989) MR0984074
- Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D., 10.1007/BF00399745, Lett. Math. Phys. 1 (1977), 521–530. (1977) MR0674337DOI10.1007/BF00399745
- Bieliavsky, P., Cahen, M., Gutt, S., Rawnsley, J., Schwachhöfer, L., 10.1142/S021988780600117X, Int. J. Geom. Methods Mod. Phys. 3 (2006), 375–426, arXiv:math/0511194. (2006) MR2232865DOI10.1142/S021988780600117X
- Bruce, A.J., Contact structures and supersymmetric mechanics, arXiv:1108.5291 [math-ph].
- Čap, A., Slovák, J., [unknown], (2009)
- Cattaneo, A.S., Felder, G., 10.1007/s002200000229, Comm. Math. Phys. 212 (2000), 591–611, arXiv:math/9902090. (2000) Zbl1038.53088MR1779159DOI10.1007/s002200000229
- Dupré, M.J., 10.1016/0022-1236(74)90035-4, J. Funct. Anal. 15 (1974), 244–278. (1974) MR0346541DOI10.1016/0022-1236(74)90035-4
- Fedosov, B.V., 10.4310/jdg/1214455536, J. Differential Geom. 40 (1994), 213–238. (1994) Zbl0812.53034MR1293654DOI10.4310/jdg/1214455536
- Fitzpatrick, S., 10.1016/j.geomphys.2011.07.011, J. Geom. Phys. 61 (2011), 2384–2399. (2011) MR2838515DOI10.1016/j.geomphys.2011.07.011
- Fox, D.J.F., Contact projective structures, Indiana Univ. Math. J. 54 (2005), 1547–1598, arXiv:math/0402332. (2005) Zbl1093.53083MR2189678
- Fradkin, E.S., Vilkovisky, G., 10.1016/0370-2693(75)90448-7, 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. (1975) MR0411451DOI10.1016/0370-2693(75)90448-7
- 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]. (2008) MR3388763
- Grigoriev, M.A., Lyakhovich, S.L., 10.1007/PL00005559, 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. (2001) MR2175993DOI10.1007/PL00005559
- Gukov, S., Witten, E., Branes and quantization, Adv. Theor. Math. Phys. 13 (2009), 1445–1518, arXiv:0809.0305 [hep-th]. (2009) MR2672467
- Herczeg, G., Waldron, A., 10.1016/j.physletb.2018.04.008, Phys.Lett. B 781 (2018), 312–315, arXiv:1709.04557 [hep-th] . (2018) DOI10.1016/j.physletb.2018.04.008
- Kashiwara, M., 10.2977/prims/1195163179, Publ. Res. Inst. Math. Sci. 32 (1) (1996), 1–7. (1996) MR1384750DOI10.2977/prims/1195163179
- Kontsevich, M., 10.1023/B:MATH.0000027508.00421.bf, Lett. Math. Phys. 66 (2003), 157–216, arXiv:q-alg/9709040. (2003) Zbl1058.53065MR2062626DOI10.1023/B:MATH.0000027508.00421.bf
- Krýsl, S., 10.1016/j.difgeo.2013.10.007, Differential Geom. Appl. 33 (2014), 290–297, arXiv:1304.5704 [math.DG]. (2014) MR3159964DOI10.1016/j.difgeo.2013.10.007
- Małkiewicz, P., Miroszewski, A., 10.1103/PhysRevD.96.046003, Phys. Rev. D 96 (2017), 046003, arXiv:1706.00743 [gr-qc]. (2017) MR3852958DOI10.1103/PhysRevD.96.046003
- Manin, Y., Topics in Noncommutative Geometry, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1991. (1991) Zbl0724.17007MR1095783
- Rajeev, S.G., 10.1016/j.aop.2007.05.001, Ann. Physics 323 (2008), 768–782. (2008) MR2404789DOI10.1016/j.aop.2007.05.001
- 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. (1996) MR1409176
- 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. (2002) MR1980911

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.