Kuipers, Folkert Stochastic quantization on Lorentzian manifolds. (English) Zbl 1466.83035 J. High Energy Phys. 2021, No. 5, Paper No. 28, 51 p. (2021). Summary: We embed Nelson’s theory of stochastic quantization in the Schwartz-Meyer second order geometry framework. The result is a non-perturbative theory of quantum mechanics on (pseudo-)Riemannian manifolds. Within this approach, we derive stochastic differential equations for massive spin-0 test particles charged under scalar potentials, vector potentials and gravity. Furthermore, we derive the associated Schrödinger equation. The resulting equations show that massive scalar particles must be conformally coupled to gravity in a theory of quantum gravity. We conclude with a discussion of some prospects of the stochastic framework. Cited in 7 Documents MSC: 83C47 Methods of quantum field theory in general relativity and gravitational theory 81T16 Nonperturbative methods of renormalization applied to problems in quantum field theory 53Z05 Applications of differential geometry to physics 62P35 Applications of statistics to physics Keywords:differential and algebraic geometry; models of quantum gravity; stochastic processes PDFBibTeX XMLCite \textit{F. Kuipers}, J. High Energy Phys. 2021, No. 5, Paper No. 28, 51 p. (2021; Zbl 1466.83035) Full Text: DOI arXiv References: [1] K. Schwarzschild, On the gravitational field of a mass point according to Einstein’s theory, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.)1916 (1916) 189 [physics/9905030] [INSPIRE]. · Zbl 1210.83011 [2] Oppenheimer, JR; Snyder, H., On Continued gravitational contraction, Phys. Rev., 56, 455 (1939) · Zbl 0022.28104 [3] Penrose, R., Gravitational collapse and space-time singularities, Phys. Rev. Lett., 14, 57 (1965) · Zbl 0125.21206 [4] Hawking, SW, Singularities in the universe, Phys. Rev. Lett., 17, 444 (1966) [5] Geroch, RP; Traschen, JH, Strings and Other Distributional Sources in General Relativity, Phys. Rev. D, 36, 1017 (1987) [6] G. ’t Hooft and M. J. G. Veltman, One loop divergencies in the theory of gravitation, Ann. Inst. H. Poincare Phys. Theor. A20 (1974) 69 [INSPIRE]. · Zbl 1422.83019 [7] Fényes, I., Eine Wahrscheinlichkeitstheoretische Begründung und Interpretation der Quantenmechanik, Z. Phys., 132, 81 (1952) · Zbl 0048.44201 [8] Kershaw, D., Theory of Hidden Variables, Phys. Rev., 136, B1850 (1964) [9] Nelson, E., Derivation of the Schrödinger equation from Newtonian mechanics, Phys. Rev., 150, 1079 (1966) [10] E. Nelson, Dynamical Theories of Brownian Motion, Princeton University Press (1967). · Zbl 0165.58502 [11] E. Nelson, Quantum Fluctuations, Princeton University Press (1985). · Zbl 0563.60001 [12] A. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer-Verlag (1933). · JFM 59.1154.01 [13] P. Lévy Processus stochastiques et mouvement brownien, Gauthier-Villars (1948). · Zbl 0034.22603 [14] Kac, M., On Distribution of Certain Wiener Functionals, Trans. Am. Math. Soc., 65, 1 (1949) · Zbl 0032.03501 [15] K. Itô, Generalized uniform complex measures in the Hilbertian metric space with their application to the Feynman path integral, in Proc. Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. II, pp. 145-161 (1967). [16] S. A. Albeverio, R. J. Høegh-Krohn and S. Mazzucchi, Mathematical Theory of Feynman Path Integrals, Lecture Notes in Mathematics, vol. 523, Springer-Verlag (2008). · Zbl 1222.46001 [17] Yasue, K., Stochastic calculus of variations, J. Funct. Anal., 41, 327 (1981) · Zbl 0482.60063 [18] Yasue, K., Quantum Mechanics and Stochastic Control Theory, J. Math. Phys., 22, 1010 (1981) [19] Zambrini, JC, Stochastic Dynamics: A Review of Stochastic Calculus, Int. J. Theor. Phys., 24, 277 (1985) · Zbl 0563.60098 [20] Dankel, TG, Mechanics on manifolds and the incorporation of spin into Nelson’s stochastic mechanics, Arch. Rational. Mech. Anal., 37, 192 (1971) · Zbl 0199.28201 [21] Dohrn, D.; Guerra, F., Nelson’s stochastic mechanics on Riemannian manifolds, Lett. Nuovo Cim., 22, 121 (1978) [22] D. Dohrn and F. Guerra, Geodesic correction to stochastic parallel displacement of tensors in Stochastic Behavior in Classical and quantum Hamiltonian Systems, Lect. Notes Phys., vol. 93, pp. 241-249, Springer-Verlag (1979) [DOI]. · Zbl 0431.60099 [23] Dohrn, D.; Guerra, F., Compatibility between the Brownian metric and the kinetic metric in nelson stochastic quantization, Phys. Rev. D, 31, 2521 (1985) [24] Guerra, F.; Morato, LM, Quantization of Dynamical Systems and Stochastic Control Theory, Phys. Rev. D, 27, 1774 (1983) [25] Guerra, F.; Ruggiero, P., New interpretation of the euclidean-markov field in the framework of physical Minkowski space-time, Phys. Rev. Lett., 31, 1022 (1973) [26] Guerra, F.; Ruggiero, P., A note on relativistic Markov processes, Lett. Nuovo Cim., 23, 528 (1978) [27] Guerra, F.; Loffredo, MI, Stochastic equations for the Maxwell field, Lett. Nuovo Cim., 27, 41 (1980) [28] Guerra, F., Structural Aspects of Stochastic Mechanics and Stochastic Field Theory, Phys. Rept., 77, 263 (1981) [29] T. Kodama and T. Koide, Variational Principle of Hydrodynamics and Quantization by Stochastic Process, arXiv:1412.6472 [INSPIRE]. · Zbl 1348.81282 [30] Marra, R.; Serva, M., Variational principles for a relativistic stochastic mechanics, Ann. Inst. H. Poincare Phys. Theor., 53, 97 (1990) · Zbl 0701.60106 [31] L. M. Morato and L. Viola, Markov diffusions in comoving coordinates and stochastic quantization of the free relativistic spinless particle, J. Math. Phys.36 (1995) 4691 [Erratum ibid.37 (1996) 4769] [quant-ph/9505007] [INSPIRE]. · Zbl 0841.60090 [32] Garbaczewski, P.; Klauder, JR; Olkiewicz, R., The Schrödinger problem, Levy processes and all that noise in relativistic quantum mechanics, Phys. Rev. E, 51, 4114 (1995) [33] Pavon, M., On the stochastic mechanics of the free relativistic particle, J. Math. Phys., 42, 4846 (2001) · Zbl 1016.81010 [34] L. Fritsche and M. Haugk, Stochastic Foundation of Quantum Mechanics and the Origin of Particle Spin, arXiv:0912.3442 [INSPIRE]. · Zbl 1037.81058 [35] Parisi, G.; Wu, YS, Perturbation Theory Without Gauge Fixing, Sci. Sin., 24, 483 (1981) · Zbl 1480.81051 [36] Damgaard, PH; Huffel, H., Stochastic Quantization, Phys. Rept., 152, 227 (1987) [37] Damgaard, PH; Tsokos, K., Stochastic Quantization With Fermions, Nucl. Phys. B, 235, 75 (1984) [38] Markopoulou, F.; Smolin, L., Quantum theory from quantum gravity, Phys. Rev. D, 70, 124029 (2004) [39] Erlich, J., Stochastic Emergent Quantum Gravity, Class. Quant. Grav., 35, 245005 (2018) · Zbl 1431.83049 [40] P. de la Pena, A. M. Cetto and A. Valdes Hernandez, The Emerging Quantum, Springer International Publishing (2015). · Zbl 1300.81001 [41] Olavo, LSF; Lapas, LC; Figueiredo, A., Foundations of quantum mechanics: The Langevin equations for QM, Annals Phys., 327, 1391 (2012) · Zbl 1242.81092 [42] Petroni, NC; Morato, LM, Entangled states in stochastic mechanics, J. Phys. A, 33, 5833 (2000) · Zbl 1010.81006 [43] Gaeta, G., EPR and stochastic mechanics, Phys. Lett. A, 175, 267 (1993) [44] Nelson, E., Review of stochastic mechanics, J. Phys. Conf. Ser., 361, 012011 (2012) [45] Wallstrom, TC, On the derivation of the Schrödinger equation from Stochastic Mechanics, Found. Phys. Lett., 2, 113 (1988) [46] Wallstrom, TC, Inequivalence between the Schrödinger equation and the Madelung hydrodynamic equations, Phys. Rev. A, 49, 1613 (1994) [47] M. Derakhshani, A Suggested Answer To Wallstrom’s Criticism: Zitterbewegung Stochastic Mechanics I, arXiv:1510.06391 [INSPIRE]. [48] M. Derakhshani, A Suggested Answer To Wallstrom’s Criticism: Zitterbewegung Stochastic Mechanics II, arXiv:1607.08838 [INSPIRE]. [49] I. Schmelzer, An answer to the Wallstrom objection against Nelsonian stochastics, arXiv:1101.5774. [50] Wick, GC, Properties of Bethe-Salpeter Wave Functions, Phys. Rev., 96, 1124 (1954) · Zbl 0057.21202 [51] J. Schwinger, Four-dimensional Euclidean formulation of quantum field theory, in Proc. 8th Annual International Conference on High Energy Physics, pp. 134-140 (1958) [INSPIRE]. [52] Symanzik, K., Euclidean Quantum Field Theory. I. Equations for a Scalar Model, J. Math. Phys., 7, 510 (1966) [53] Nelson, E., Construction of quantum fields from Markoff fields, J. Funct. Anal., 12, 97 (1973) · Zbl 0252.60053 [54] L. Schwartz, Semi-Martingales and their Stochastic Calculus on Manifolds, Presses de l’Université de Montréal (1984). [55] P. A. Meyer, A differential geometric formalism for the Itô calculus, in Stochastic Integrals, Lecture Notes in Mathematics, vol. 851, Springer (1981) [DOI]. · Zbl 0457.60031 [56] M. Emery, Stochastic Calculus in Manifolds, Springer-Verlag (1989) [DOI]. · Zbl 0697.60060 [57] Campos, CM; de Leon, M.; de Diego, DM; Vankerschaver, J., Unambiguous Formalism for Higher-Order Lagrangian Field Theories, J. Phys. A, 42, 475207 (2009) · Zbl 1231.58005 [58] Campos, CM; de Leon, M.; de Diego, DM, Constrained Variational Calculus for Higher Order Classical Field Theories, J. Phys. A, 43, 455206 (2010) · Zbl 1248.58010 [59] Shucker, DS, Stochastic Mechanics of Systems with Zero Potential, J. Funct. Anal., 38, 146 (1980) · Zbl 0447.60042 [60] Pavon, M., A new formulation of stochastic mechanics, Phys. Lett. A, 209, 143 (1995) · Zbl 1020.81546 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.