Path integrals on finite sets. (English) Zbl 0853.58026

The author uses the formalism of the Feynman path integral to “solve” \[ -{1 \over i} {\partial \over \partial t} \phi_t = H_0 \phi_t \] where \(H_0\) is a self-adjoint operator on \(L^2(M) = \mathbb{C}^M\), \(M\) being a finite set. The paths are functions of \(\mathbb{R}\) with values in \(M\). The path integral here is a family of measures parametrized by \(t'\) and \(t\) with values in the space of operators on \(L^2(M)\). The relationship of these measures to the measures of a Markov process is studied and it is shown that these measures are concentrated on a certain set of step measures. Some applications to certain hypothetical systems are also considered.


58D30 Applications of manifolds of mappings to the sciences
81S40 Path integrals in quantum mechanics
46G10 Vector-valued measures and integration
Full Text: DOI


[1] Albeverio S. and Høegh-Krohn R.: Mathematical Theory of Feynman Path Integrals, Lecture Notes in Math., 523, Springer, New York, 1976. · Zbl 0337.28009
[2] Billingsley P.: Convergence of Probability Measures, Wiley, New York, 1968. · Zbl 0172.21201
[3] Bourbaki N.: Intégration, Ch. IX, Hermann, Paris, 1969.
[4] Cameron R. R. and Storvick D. A.: A simple definition of the Feynman integral with applications, Mem. Amer. Math. Soc. 46(288) (1983), 1-46. · Zbl 0527.28015
[5] Dollard J. D. and Friedman Ch. N.: Product integration, Encyclopedia Math. Appl. 10, Addison Wesley, New York, 1979. · Zbl 0454.28002
[6] de Witt Morette C.: Feynman’s path integrals. Definition without limiting procedure, Comm. Math. Phys. 28 (1972), 47-67. · Zbl 0239.46041
[7] Doob J. L.: Stochastic Processes, Wiley, New York, 1953.
[8] Feynman R. P.: Space-time approach to nonrelativistic quantum mechanics, Rev. Modern Phys. 20 (1948), 367-387. · Zbl 1371.81126
[9] Feynman R. P. and Hibbs A. R.: Quantum Mechanics and Path Integrals, McGraw-Hill, Englewood Cliffs, 1965. · Zbl 0176.54902
[10] Gill, R. D.: Personal communication.
[11] Gikhman I. I. and Skorokhod A. V.: Introduction to the Theory of Random Processes, W.B. Saunders, New York, 1969. · Zbl 0132.37902
[12] Grothendieck A.: Produits tensoriels topologiques et espaces nucléaries, Mem. Amer. Math. Soc. 16, (1966).
[13] Hida T., Kuo H.-H., Potthoff J., and Streit L.: White Noise. An Infinite Dimensional Calculus, Kluwer Acad. Publ., Dordrecht, 1993. · Zbl 0771.60048
[14] Jofriet, R. H.: A generalization of the Trotter product formula (unpublished).
[15] Meyer P. A.: Probabilités et potentiel, Hermann, Paris, 1966.
[16] Muldowney P.: A General Theory of Integration in Fuction Spaces, Pitman Research Notes, Pitman, London, 1987. · Zbl 0623.28008
[17] Nelson E.: Feynman integrals and the Schrödinger equation, J. Math. Phys. 5 (1964), 332-343. · Zbl 0133.22905
[18] Nelson E.: Regular probability measures on function space, Ann. of Math. 69 (1959), 630-643. · Zbl 0087.13102
[19] Schwartz, L.: Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures, Oxford Univ. Press, 1973. · Zbl 0298.28001
[20] Simon B.: Functional Integration and Quantum Physics, Academic Press, New York, 1979. · Zbl 0434.28013
[21] Thomas, E. G. F.: Integral representations in conuclear cones, to appear in J. Convex Anal. · Zbl 0837.46009
[22] Trotter H. F.: On the product of semigroups of operators, Proc. Amer. Math. Soc. 10 (1959), 545-551. · Zbl 0099.10401
[23] Truman A.: The polygonal path formulation of the Feynman path integrals, Lecture Notes in Phys. 106, Springer, New York, 1979, pp. 73-102.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.