The paper is devoted to the construction of an “integral” on an infinite-dimensional space, combining the approaches proposed previously and at the same time the simplest. A new definition of the construction and study its properties on a special class of functionals is given. An introduction of a quasi-scalar product, an orthonormal system, and applications in physics (path integral, loop space, functional derivative) are proposed. In addition, the paper contains a discussion of generalized functionals.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. J. Daniell, “Integrals in An Infinite Number of Dimensions,” Ann. Math. 20, No. 4, 281–288 (1919).
L. D. Faddeev and A. A. Slavnov, Gauge Fields: An Introduction To Quantum Theory, The Benjamin-Cummings Publishing Company (1980).
V. N. Popov, Functional Integrals in Quantum Field Theory and Statistical Physics, Springer Netherlands (1983).
L. A. Takhtajan, Quantum Mechanics for Mathematicians, American Mathematical Society, Graduate Series in Mathematics, 95 (2008).
J. M. Rabin, “Introduction to quantum field theory for mathematicians,” in: Geometry and quantum field theory, IAS/Park City, Math. Ser. 1 (1991), pp. 185–269.
Yu. L. Daletskii and S. V. Fomin, Measures and Differential Equations in Infinite-Dimensional Spaces, Nauka, Moscow (1983).
R. P. Feynman, “Space-Time Approach to Non-Relativistic Quantum Mechanics,” Rev. Modern Phys., 20, 367–387 (1948).
N. Wiener, “The average of an analytical functional,” Proc. Nat. Acad. Sci. USA 7, 253–260 (1921).
M. Kac, “On Distributions of Certain Wiener Functionals,” Trans. Amer. Math. Soc., 65, 1–13 (1949).
E. T. Shavgulidze and O. G. Smolyanov, Continual Integrals, Moscow State University Press, Moscow (1990).
G. M. Fichtenholz, Course of Differential and Integral Calculus, Vol. 2, Science, Moscow (1970).
V. S. Vladimirov, Generalized Functions in Mathematical Physics, Mir, Moscow (1979).
A. V. Ivanov and N. V. Kharuk, “Heat kernel: proper time method, Fock-Schwinger gauge, path integral representation, and Wilson line,” arXiv:1906.04019 [hep-th] (2019).
Ya. A. Butko, “Feynman Formulas and Functional Integrals for Diffusion with Drift in a Domain on a Manifold,” Math. Notes, 83, 301–316 (2008).
A. K. Kravtseva, “Infinite-Dimensional Schr¨odinger Equations with Polynomial Potentials and Representation of Their Solutions via Feynman Integrals,” Math. Notes, 94, 824–828 (2013).
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 487, 2019, pp. 140–150.
Rights and permissions
About this article
Cite this article
Ivanov, A.V. Notes on Functional Integration. J Math Sci 257, 518–525 (2021). https://doi.org/10.1007/s10958-021-05499-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-021-05499-9