[Paper Review] More About Donsker's Delta Function

Angelika Lascheck, Peter Leukert+2 more|ArXiv.org|Mar 4, 2003

Stochastic processes and financial applications17 references17 citations

TL;DR

This paper extends Donsker's delta function to complex parameters within White Noise Analysis, enabling rigorous treatment of Feynman path integrals and quantum systems on a circle. It establishes convergence for series of delta functions via S-transforms and proves that smeared wave packets yield well-defined Hida distributions, solving the problem of divergent series in quantum mechanical path integrals with winding number contributions.

ABSTRACT

We discuss Donsker's delta function within the framework of White Noise Analysis, in particular its extension to complex arguments. With a view towards applications to quantum physics we also study sums and products of Donsker's delta functions.

Motivation & Objective

• To extend Donsker's delta function to complex arguments within the White Noise Analysis framework.
• To address the convergence issues of infinite series of delta functions in Feynman path integrals, particularly for systems with multiple winding numbers.
• To provide a rigorous mathematical formulation of the propagator for a quantum particle on a circle using Hida distributions.
• To ensure convergence of the S-transform for wave packet states by imposing summability conditions on Fourier coefficients.
• To demonstrate that the resulting path integral satisfies the Schrödinger equation via the T-transform at zero.

Proposed method

• Extends Donsker's delta function to complex parameters using analytic continuation in the sector Re(a²) > 0.
• Applies the S-transform to characterize the generalized function, deriving an explicit integral representation involving Gaussian kernels.
• Uses the S-transform relation SΦ(ξ) = C(ξ)TΦ(−iξ) to extend the transform to complex test functions.
• Constructs a propagator as a sum over winding numbers using Dirac delta functions smeared over the circle.
• Imposes a summability condition ∑|aₗ|exp(½s²l²) < ∞ on Fourier coefficients to ensure convergence of the S-transform series.
• Evaluates the T-transform of the propagator and shows it yields a solution to the Schrödinger equation via TI(0) = ∑aₗexp(−½il²t + ilφ₀).

Experimental results

Research questions

• RQ1 Can Donsker's delta function be rigorously extended to complex parameters within White Noise Analysis?
• RQ2 How can products and infinite series of Donsker's delta functions be defined and handled mathematically?
• RQ3 What conditions ensure convergence of the S-transform for path integral representations involving winding numbers?
• RQ4 Can a quantum particle on a circle be consistently described using Hida distributions and generalized functionals?
• RQ5 Does the resulting path integral formulation yield a solution to the Schrödinger equation?

Key findings

• Donsker's delta function admits an analytic extension to complex parameters in the sector Re(a²) > 0, making it a Hida distribution.
• The S-transform of L(t,a) = ∫₀ᵗ δ(B(s)−a)ds is given by ∫₀ᵗ (1/√(2πs)) exp(−½s⁻¹(∫₀ˢ ξ(τ)dτ − a)²)ds.
• The series ∑ₗ aₗ exp(ilφ) with ∑|aₗ|exp(½s²l²) < ∞ defines a tempered distribution in the White Noise space.
• The T-transform of the propagator is bounded by exp(½(1 + t/s²)|ξ|₀²), ensuring convergence and membership in (S)*.
• The Feynman integral TI(0) = ∑ₗ aₗ exp(−½il²t + ilφ₀) solves the Schrödinger equation for the particle on a circle.
• The formal path integral with δ-functions over winding numbers diverges, but convergence is restored by using wave packet states with summable coefficients.