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On quantization of systems with actions unbounded from below. (English. Russian original) Zbl 0962.81522
Theor. Math. Phys. 109, No. 2, 1379-1387 (1996); translation from Teor. Mat. Fiz. 109, No. 2, 175-186 (1996).
Summary: We consider two possible approaches to the problem of the quantization of systems with actions unbounded from below: the Borel summation method applied to the perturbation expansion in the coupling constant and the method based on the kerneled Langevin equation for stochastic quantization. In the simplest case of an anharmonic oscillator, the first method produces Schwinger functions, even though the corresponding path integral diverges. The solutions of the kerneled Langevin equation are studied both analytically and numerically. The fictitious time averages are shown to have limits that can be considered as the Schwinger functions. The examples demonstrate that both methods may give the same result.

MSC:
81S20 Stochastic quantization
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[1] S. Graffi, V. Grecchi, and B. Simon,Phys. Lett. B,32, 631–634 (1970). · doi:10.1016/0370-2693(70)90564-2
[2] J.-P. Eckmann, J. Magnen, and R. Seneor,Commun. Math. Phys.,39, 251–271 (1975). · doi:10.1007/BF01705374
[3] J. Avron, I. Herbst, and B. Simon, ”Schrödinger operators with magnetic fields,” Preprint, Princeton Univ. (1977). · Zbl 0399.35029
[4] A. Erdélyi et al. (eds.),Higher Transcendental Functions (Based on notes left by H. Bateman), McGraw-Hill, New York (1953).
[5] R. Z. Has’minskii,Stochastic Stability of Differential Equations, Sijthoff & Nordhoff, Alphen Van den Rijn, Rockville (1980).
[6] M. Namiki,Stochastic Quantization, Springer (1992).
[7] J. D. Breit, S. Gupta, and A. Zaks,Nucl. Phys. B,233, 61 (1984). · doi:10.1016/0550-3213(84)90170-6
[8] S. Tanaka, M. Namiki, I. Ohba, M. Mizutani, N. Komoike, and M. Kanenaga,Phys. Lett. B,288, 129 (1992). · doi:10.1016/0370-2693(92)91966-D
[9] M. Kanenaga, M. Mizutani, M. Namiki, I. Ohba, and S. Tanaka,Prog. Theor. Phys.,91, 599–610 (1994). · doi:10.1143/PTP.91.599
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