Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Evaluating Feynman path integrals via the polynomial path family.

Gabriel Bouch, Philip L. Bowers

Published Jan 1, 2007• 39 citations
OpenAlex

Abstract

We present a difficult and detailed calculation of propagators for quadratic potentials using a filtration of the space of variational paths around the classical path by the dense set of polynomial paths filtered by degree. This presents evidence for a cohesive explanation of path integrals as limits of finite-dimensional integrations using a sequence of filtration measures determined for specific dense families of paths by the requirement that the limit yield the free particle propagator.

Related papers

A NUMERICAL METHOD FOR EVALUATING FEYNMAN PATH INTEGRALS

Richard J. Creswick

A method for numerically evaluating Feynman path integrals based on the density of actions is presented. The method is illustrated by calculating the propagator for the free particle and the simple harmonic oscillator.

Modern Physics Letters B•May 20, 1995•4 citations

More accurate discrete-time approximation of the Feynman path integral

Daniel R. Stump

A discrete-time approximation of the Feynman path integral is derived, for which the potential is approximated at each step by a quadratic polynomial; this is more accurate than the usual approximation for which the potential is approximated by a constant. This approximation is studied for several simple examples, by Monte Carlo calculations of the path integral.

Physical Review A•Jul 1, 1987•7 citations

On approximations of Feynman path integrals

Tapani Hyttinen

We study approximations of Feynman path integrals in finite dimensional spaces and how the approximations determine the propagator.

arXiv:2208.06206•Aug 12, 2022•0 citations

The Feynman integral as a limit of complex measures

José L. Martínez-Morales

The fundamental solution of the Schrödinger equation for a free particle is a distribution. This distribution can be approximated by a sequence of smooth functions. It is defined for each one of these functions, a complex measure on the space of paths. For certain test functions, the limit of the integrals of a test function with respect to the complex measures, exists. We define the Feynman integral of one such function by this limit.

arXiv:0906.1230•Jun 5, 2009•0 citations

Heuristic Variational Analysis of the Single-Slice Feynman Path Integral: Conceptual Insights into Short-Time Propagation

Mandemaker, Menno

AbstractThis work investigates a variational perspective on the Feynman path integral, aiming to explore connections between stationary action, probability amplitude, integrating over path evolutions, and the semiclassical limit. The variational analysis employed should be viewed as conceptually explorative and heuristic rather than strictly formal. It explores how variations of path evolutions in the path integral at the level of a single infinitesimal slice can yield meaningful insight into the behavior of quantum trajectories. A local variational analysis of the free particle and a particle near a constant potential provides insight into how the stationary-action principle underlies both classical and quantum behavior, through a local relationship between the classical slice-action and a functional over the phase-weighting of quantum path evolutions. In particular, this functional is proportional to the slice-action itself up to additive and multiplicative constants. This proportionality implies that the stationarity of this functional is equivalent to the stationarity of the slice-action, which in turn generates the semiclassical stationary path evolution. The resulting formulation produces results consistent with known short-time propagators for the free particle and a particle near a constant potential, including tunneling behavior.

•Nov 17, 2025•0 citations

An elementary derivation of the harmonic oscillator propagator

L. Moriconi

The harmonic oscillator propagator is found straightforwardly from the free particle propagator within the imaginary-time Feynman path integral formalism. The derivation is simple, and requires only elementary mathematical manipulations and no clever use of Hermite polynomials, annihilation and creation operators, cumbersome determinant evaluations, or involved algebra.

arXiv:physics/0402069•American Journal of Physics•Aug 13, 2004•16 citations
We present a difficult and detailed calculation of propagators for quadratic potentials using a filtration of the space of variational paths around the classical path by the dense set of polynomial paths filtered by degree. This presents evidence for a cohesive explanation of path integrals as limits of finite-dimensional integrations using a sequence of filtration measures determined for specific dense families of paths by the requirement that the limit yield the free particle propagator.
AbstractThis work investigates a variational perspective on the Feynman path integral, aiming to explore connections between stationary action, probability amplitude, integrating over path evolutions, and the semiclassical limit. The variational analysis employed should be viewed as conceptually explorative and heuristic rather than strictly formal. It explores how variations of path evolutions in the path integral at the level of a single infinitesimal slice can yield meaningful insight into the behavior of quantum trajectories. A local variational analysis of the free particle and a particle near a constant potential provides insight into how the stationary-action principle underlies both classical and quantum behavior, through a local relationship between the classical slice-action and a functional over the phase-weighting of quantum path evolutions. In particular, this functional is proportional to the slice-action itself up to additive and multiplicative constants. This proportionality implies that the stationarity of this functional is equivalent to the stationarity of the slice-action, which in turn generates the semiclassical stationary path evolution. The resulting formulation produces results consistent with known short-time propagators for the free particle and a particle near a constant potential, including tunneling behavior.