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Proceedings of the Steklov Institute of Mathematics

Quantum Anomalies via Differential Properties of Lebesgue–Feynman Generalized Measures

John Edward Gough, Tudor Stefan Ratiu, Oleg Georgievich Smolyanov

Published Sep 1, 2020• 3 citations
Publisher

Abstract

Abstract We address the problem concerning the origin of quantum anomalies, which has been the source of disagreement in the literature. Our approach is novel as it is based on the differentiability properties of families of generalized measures. To this end, we introduce a space of test functions over a locally convex topological vector space, and define the concept of logarithmic derivatives of the corresponding generalized measures. In particular, we show that quantum anomalies are readily understood in terms of the differential properties of the Lebesgue–Feynman generalized measures (equivalently, of the Feynman path integrals). We formulate a precise definition for quantum anomalies in these terms.

DOI
10.1134/s0081543820050077

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Abstract We address the problem concerning the origin of quantum anomalies, which has been the source of disagreement in the literature. Our approach is novel as it is based on the differentiability properties of families of generalized measures. To this end, we introduce a space of test functions over a locally convex topological vector space, and define the concept of logarithmic derivatives of the corresponding generalized measures. In particular, we show that quantum anomalies are readily understood in terms of the differential properties of the Lebesgue–Feynman generalized measures (equivalently, of the Feynman path integrals). We formulate a precise definition for quantum anomalies in these terms.
The definition of Feynman path integrals (Feynman functional integrals) as integrals with respect to a generalized measure, called the Lebesgue–Feynman measure in the paper and being an infinite-dimensional analogue of the classical Lebesgue measure on finite-dimensional Euclidean space, is discussed. This definition, which is a formalization of Feynman’s original definition, is different from those used previously in the mathematical literature. It makes it possible to give a description of the origin of quantum anomaly which is a mathematically correct version of the description given in the book Path Integrals and Quantum Anomalies by K. Fujikawa and H. Suzuki (Oxford, 2004) (and erroneously qualified as wrong in the book Functional Integration: Action and Symmetries by P. Cartier and C. DeWitt-Morette (Cambridge Univ. Press, Cambridge, 2006)).
Feynman-Path Integral in Banach Space: In 1940, R.P. Feynman attempted to find a mathematical representation to express quantum dynamics of the general form for a double-slit experiment. His intuition on several slits with several walls in terms of Lagrangian instead of Hamiltonian resulted in a magnificent work. It was known as Feynman Path Integrals in quantum physics, and a large part of the scientific community still considers them a heuristic tool that lacks a sound mathematical definition. This paper aims to refute this prejudice, by providing an extensive and self-contained description of the mathematical theory of Feynman Path Integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. About a hundred years after the beginning of modern physics, it was realized that light could in fact show behavioral characteristics of both waves and particles. In 1927, Davisson and Germer demonstrated that electrons show the same dual behavior, which was later extended to atoms and molecules. We shall follow the method of integration with some modifications to construct a generalized Lebesgue-Bochner-Stieltjes (LBS) integral of the form , where u is a bilinear operator acting in the product of Banach spaces, f is a Bochner summable function, and μ is a vector-valued measure. We will demonstrate that the Feynman Path Integral is consistent and can be justified mathematically with LBS integration approach.
It has been stated in the physics literature that the probability amplitude for a quantum system after a continuous position measurement can be expressed by the restricted or the weighted Feynman path integral containing information about the result and the resolution of its measurement. In the present paper it is proved that the restricted Feynman path integrals can be defined in a rigorous sense. More specifically, we consider a quantum system consisting of some particles and simultaneously perform a continuous measurement of the positions of all particles in the time interval [0,T][0,T][0,T]. Then, it is shown rigorously that the restricted Feynman path integral after the measurement can be defined in the L2L^2L2 space and satisfies the corresponding non-self-adjoint Schroeodinger equation. In addition, our results are generalized to a quantum spin system where a continuous position measurement for all spin components of all particles is simultaneously performed, though the results obtained are inadequate for general use.
In this paper, we define a new concept of analytic Feynman integral on theWiener space, which is called the generalized analytic Feynman integral, to explain various physical circumstances. Furthermore, we evaluate the generalized analytic Feynman integrals for several important classes of functionals.We also establish various properties of these generalized analytic Feynman integrals. We conclude the paper by giving several applications involving the Cameron-Storvick theorem and quantum mechanics.
Evolution semigroups generated by pseudo-differential operators are considered. These operators are obtained by different (parameterized by a number τ) procedures of quantization from a certain class of functions (or symbols) defined on the phase space. This class contains Hamilton functions of particles with variable mass in magnetic and potential fields and more general symbols given by the Lévy-Khintchine formula. The considered semigroups are represented as limits of n-fold iterated integrals when n tends to infinity. Such representations are called Feynman formulae. Some of these representations are constructed with the help of another pseudo-differential operator, obtained by the same procedure of quantization; such representations are called Hamiltonian Feynman formulae. Some representations are based on integral operators with elementary kernels; these are called Lagrangian Feynman formulae. Langrangian Feynman formulae provide approximations of evolution semigroups, suitable for direct computations and numerical modeling of the corresponding dynamics. Hamiltonian Feynman formulae allow to represent the considered semigroups by means of Feynman path integrals. In the article, a family of phase space Feynman pseudomeasures corresponding to different procedures of quantization is introduced. The considered evolution semigroups are represented as phase space Feynman path integrals with respect to these Feynman pseudomeasures, i.e., different quantizations correspond to Feynman path integrals with the same integrand but with respect to different pseudomeasures. This answers Berezin’s problem of distinguishing a procedure of quantization on the language of Feynman path integrals. Moreover, the obtained Lagrangian Feynman formulae allow also to calculate these phase space Feynman path integrals and to connect them with some functional integrals with respect to probability measures.