Path integrals for functionals are studied from the point of view of the continuum calculus proposed earlier [J. Math. Phys. 17, 1988 (1976)]. The weak distributions of Skorohod in an infinite-dimensional Hilbert space and the p-integral method of continuum calculus are employed to derive a formula for the functional integral, which is in turn evaluated through a natural extension of the weak distribution expression. Generalizations are made to measures with density functions in the function space. As a demonstration, the formula is tested against the polynomial functionals studied by Friedrichs, and valid results are obtained for the general case.
Path integrals for functionals are studied from the point of view of the continuum calculus proposed earlier [J. Math. Phys. 17, 1988 (1976)]. The weak distributions of Skorohod in an infinite-dimensional Hilbert space and the p-integral method of continuum calculus are employed to derive a formula for the functional integral, which is in turn evaluated through a natural extension of the weak distribution expression. Generalizations are made to measures with density functions in the function space. As a demonstration, the formula is tested against the polynomial functionals studied by Friedrichs, and valid results are obtained for the general case.
The continuum calculus, proposed earlier [J. Math. Phys. 17, 1988 (1976)], is here applied to the development of a functional version of Laplace transform and a method for the evaluation of path integrals containing a Gaussian-like measure. Two methods in functional integration are proposed. First, the Gaussian integral for polynomial functionals and consequently functionals that can be expanded in Taylor series are examined. Formula (2.12) is derived. Next, we define the Laplace transform in the function space through the weak distribution formulation of Skorohod. Comparison of both approaches enables us to determine the expression of the functional integeral through a series of Laplace transformations. The second formula is given in Eq. (4.5). The latter formula is applicable to all Laplace transformable functionals. As an illustration of the utility of the formulas derived, we evaluate the integral of a cosine functional by methods 1 and 2, and obtain consistent results. Further applications to quantum mechanics are also presented. We examine the cases of a free particle,the quantum harmonic oscillator, the forced oscillator, and charged particle in a magnetic field. In all cases, we obtain correct results in comparison with known expressions. A numerical procedure is employed in the calculation of infinite products. The usefulness of the p-integral method is stressed.
(and its generalisations that include magnetic fields) can be roughly divided into two classes. In the aproaches of the first class that we shall not discuss here in detail the Feynman integral is not supposed to be a genuine integral, but is specified as some generalised functional on an appropriate space of functions, which can be defined, for example, as the limit of certain discrete approximations (see e.g. [16], [44] and more recent papers [24], [25], [34]), by means of analytical continuation (see e.g. [26], [23] and references therein), by extensions of Parceval’s identity and by related axiomatic definitions (see [2], [5], [30], [11], [42] and references therein) or by means of the white noise analysis (see [22]), see also [46] for the discussion of path integral applied to the Dirac equation. These approaches still cover only a very restrictive class of potentials, for example, singular potentials were considered only by white noise analysis approach but only in one-dimensional case (see e.g. [6] and references therein). In the approaches of the second class, one tries to define the infinite-dimensional Feynmann integral as a genuine integral over a bona fide σ-additive measure on an appropriate space of trajectories. The first attempt made in [19] to construct such a measure was erronious and led to understanding that there is no direct generalisation of Wiener measure that can give an analog of FeynmanKac formula for the case of Schrodinger operators. A correct construction of the Feynman integral in terms of the Wiener measure was proposed in [15] and was based on the idea of the rotation of the classical trajectories in complex
Techniques for integration in function spaces which are not necessarily vector spaces are presented in the light of DeWitt-Morette’s recent redefinition of path integrals, which does not involve the usual limiting process, and is therefore free from the ambiguities inherent in this approach. General translated Gaussian measures defined by their Fourier transforms are introduced on various path spaces and used to derive generalized moments formulas, and what is essentially the Feynman–Kac formula for the expression of solutions of the Schrödinger equation as functional integrals.
"Feynman path integrals are ubiquitous in quantum physics, even if a large part of the scientific community still considers them as a heuristic tool that lacks a sound mathematical definition. Our book aims to refute this prejudice, 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. This second edition presents a detailed discussion of the general theory of complex integration on infinite dimensional spaces, providing on one hand a unified view of the various existing approaches to the mathematical construction of Feynman path integrals and on the other hand a connection with the classical theory of stochastic processes. Moreover, new chapters containing recent applications to several dynamical systems have been added. This book bridges between the realms of stochastic analysis and the theory of Feynman path integration. It is accessible to both mathematicians and physicists"--
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)).
In order to preserve the leading role of the action principle in formulating all field theories one needs quantum field theory, with the associated BRST symmetry, and Feynman–DeWitt–Faddeev–Popov ghost fields. Such fields result from the fiber-bundle structure of the space of histories, but the physics-oriented literature used them formally because a rigorous theory of measure and integration was lacking. Motivated by this framework, this paper exploits the previous work of Gill and Zachary, where the use of Banach spaces for the Feynman integral was proposed. The Henstock–Kurzweil integral is first introduced, because it makes it possible to integrate functions like [Formula: see text]. The Lebesgue measure on [Formula: see text] is then built and used to define the measure on every separable Hilbert space. The subsequent step is the construction of a new Hilbert space [Formula: see text], which contains [Formula: see text] as a continuous dense embedding, and contains both the test functions [Formula: see text] and their dual [Formula: see text], the Schwartz space of distributions, as continuous embeddings. This space allows us to construct the Feynman path integral in a manner that maintains its intuitive and computational advantages. We also extend this space to [Formula: see text], where [Formula: see text] is any separable Hilbert space. Last, the existence of a unique universal definition of time, [Formula: see text], that we call historical time, is proven. We use [Formula: see text] as the order parameter for our construction of Feynman’s time ordered operator calculus, which in turn is used to extend the path integral in order to include all time-dependent groups and semigroups with a reproducing kernel representation.