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**Generalized Feynman integrals using analytic continuation in several complex variables.**
*(English)*
Zbl 0554.60061

Stochastic analysis and applications, Adv. Probab. Relat. Top. 7, 217-267 (1984).

[For the entire collection see Zbl 0541.00008.]

The Feynman integral is similar in spirit to Wiener path integrals, but works with respect to a (non-existent) Wiener measure with an imaginary variance parameter. In an attempt to provide a mathematical foundation of Feynman integral, R. H. Cameron and D. A. Storvick [Analytic functions, Proc. Conf., Kozubnik 1979, Lect. Notes Math. 798, 18-67 (1980; Zbl 0439.28007)] proposed an analytic continuation approach. But, their method does not cover several important cases that arise in quantum physics.

This article reexamines and extends, from an applicatory point of view, the analytic continuation method by considering functions of several complex variables instead of one complex variable as it is in the above case. This is done in the framework of abstract Wiener space and is compared with the Fresnel integral of S. A. Albeverio and R. J. Høegh-Krohn [Mathematical theory of Feynman path integrals. Lect. Notes Math. 523 (1976; Zbl 0337.28009)]. The authors extend their definition so as to cover indefinite bilinear forms in their integral and apply this to solve, in terms of Feynman integral, the Schrödinger equation for the anharmonic oscillator with n-degrees of freedom. Further extension of analytic Feynman integral is needed to discuss the case of infinitely many degrees of freedom. The authors present one such extension and apply it to express the quasi-free states of the harmonic oscillator.

The Feynman integral is similar in spirit to Wiener path integrals, but works with respect to a (non-existent) Wiener measure with an imaginary variance parameter. In an attempt to provide a mathematical foundation of Feynman integral, R. H. Cameron and D. A. Storvick [Analytic functions, Proc. Conf., Kozubnik 1979, Lect. Notes Math. 798, 18-67 (1980; Zbl 0439.28007)] proposed an analytic continuation approach. But, their method does not cover several important cases that arise in quantum physics.

This article reexamines and extends, from an applicatory point of view, the analytic continuation method by considering functions of several complex variables instead of one complex variable as it is in the above case. This is done in the framework of abstract Wiener space and is compared with the Fresnel integral of S. A. Albeverio and R. J. Høegh-Krohn [Mathematical theory of Feynman path integrals. Lect. Notes Math. 523 (1976; Zbl 0337.28009)]. The authors extend their definition so as to cover indefinite bilinear forms in their integral and apply this to solve, in terms of Feynman integral, the Schrödinger equation for the anharmonic oscillator with n-degrees of freedom. Further extension of analytic Feynman integral is needed to discuss the case of infinitely many degrees of freedom. The authors present one such extension and apply it to express the quasi-free states of the harmonic oscillator.

Reviewer: D.Kannan

### MSC:

60H05 | Stochastic integrals |

81Q30 | Feynman integrals and graphs; applications of algebraic topology and algebraic geometry |

81S40 | Path integrals in quantum mechanics |