Journal of Mathematical Physics

Stochastic representations of Feynman integration

William Boos

Published Dec 1, 2007• 1 citation

Abstract

For polynomially bounded potentials V such that H=H0+V is essentially self-adjoint on S(Rd)⊆D(H0)∩D(V), this essay offers two reconstructions of Feynman’s sum over histories as the unitary image of a genuine integral with respect to Wiener measure μ of a functional σtx(ω) defined on the space W of Brownian paths ω into momentum space Rd. The first representation, based on Feynman’s original argument, “lifts” σtx(ω) from a “convolutional Trotter product formula” for the Fourier-transformed image φ̂t(p) of the time-evolved wave function φt(x)=exp(−itH)φ(x) in L2(Rd). The second—which varies and extends a construction introduced in a slightly different context by Albeverio and Høegh-Krohn [Mathematical Theory of Feynman Integrals, Springer Lecture Notes in Mathematics Vol. 523 (Springer, New York, 1976)]—lifts the functional σtx(ω) from a “convolutional Dyson expansion” of the time-evolved momentum-space function φ̂t(p).

DOI: 10.1063/1.2812416

Related papers

Feynman path integrals and quantum mechanics as h/→0

Aubrey Truman

When the space of paths is a certain Hilbert space H, we show how to extend the Feynman path integral ℱ of DeWitt and Albeverio and Hoegh-Krohn. Our extension enables us to integrate a wider class of functionals on H. We establish a new representation for the wavefunction in nonrelativistic quantum mechanics—the quasiclassical representation. Using our extension of ℱ and the quasiclassical representation, we discuss the problem of obtaining classical mechanics as the limiting case of quantum mechanics when h/→0.

Journal of Mathematical PhysicsOct 1, 197661 citations

Mathematics of the Feynman path integral

Vassili Nikitich Kolokoltsov

(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

Jan 1, 20012 citations

Stochastic interpretation of Feynman path integral

Zbigniew Haba

A class of functions ℋnF was chosen, which are boundary values of holomorphic functions. It was shown that if the potential V∈ℋnF and the initial wave function ψ∈ℋnF then the solution of the Schrödinger equation is ψt∈ℋnF. The unitary evolution ψ → ψt can be expressed by an integral with respect to the Wiener measure. To each ψt∈ℋnF is associated a complex Markov process qt fulfilling stochastic Hamilton equations. The time evolution of ψ as well as multitime correlation functions of arbitrary observables in quantum mechanics take the form of expectation values with respect to qt. Some statistical and ergodic aspects of quantum mechanics resulting from the stochastic interpretation of Feynman’s sum over trajectories are discussed.

Journal of Mathematical PhysicsDec 1, 199413 citations

Notes on the Feynman integral. III. The Schroedinger equation

Gerald Johnson, David Skoug

In the setting of Cameron and Storvick's recent theory we show that the solution of an integral equation formally equivalent to the Schroedinger equation is expressible as the analytic Feynman integral of a function on ^-dimensional Wiener space of the form F(X) = exp{/ o '0(ίs, X(s) + I) ds}ψ(X(t\+ ?).Here Xis an Revalued continuous function on [0, /] such that Z(0) = 0, ?G R\ and ψ and θ(s, ) are Fourier-Stieltjes transforms.REMARK 3.2.Corollary 3.2 and Lemma 3.2 will yield the measurability properties that we will need as we continue.For example, let θ be in % and for λ > 0 and fin W let G: [0, ί 0 ] X C[0, t 0 ] -> C be defined by The Pacific

Pacific Journal of MathematicsApr 1, 198345 citations

Semiclassical stochastic representation of the Feynman integral

Zbigniew Haba

We show that for a wide class of analytic potentials the solution of the Schrodinger equation can be expressed by the Wiener integral. We express the Feynman kernel by the Brownian bridge. We prove an asymptotic semiclassical expansion of the solution of the Schrodinger equation as well as of the Feynman kernel. We show that for a small time the van Vleck formula holds true as the limit h(cross) to 0 of the Feynman kernel.

Journal of Physics A Mathematical and GeneralOct 7, 199414 citations

On the physical and mathematical foundations of quantum physics via functional integrals

Giampiero V. M. Esposito, Tepper L. Gill

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.

arXiv:2408.16404International Journal of Modern Physics AJun 12, 20251 citation