# On the effective potential of Duru-Kleinert path integrals

## TL;DR

This paper introduces a new method for evaluating the effective potential in Duru-Kleinert path integrals, allowing for arbitrary ordering prescriptions and analyzing their dependence, with applications to radial oscillator and Coulomb systems.

## Contribution

A novel approach that avoids restriction to specific ordering prescriptions, enabling analysis of ordering dependence in path integrals for quantum systems.

## Key findings

- Path integrals are independent of the ordering parameter.
- Explicit dependence on the splitting parameter remains.
- Application to radial oscillator and Coulomb systems confirms the method's validity.

## Abstract

We propose a new method to evaluate the effective potential in the path integral for the fixed-energy amplitude as well as for the pseudotime evolution kernel in the formalism by Duru and Kleinert. Restriction to the postpoint or the prepoint prescriptions in formulating time sliced path integrals is avoided by leaving off the use of expectation values for correction terms. This enables us to consider an arbitrary ordering prescription and to examine the ordering dependence of the effective potential. To investigate parameter dependences, we introduce the ordering parameter $\alpha$ in addition to the splitting parameter $\lambda$ in the formulation of the time sliced path integral. The resulting path integrals are found to be independent of the ordering parameter although the explicit dependence, given by a contribution proportional to $(1-2\lambda)^{2}$, on the splitting parameter remains. As an application, we check the relationship between path integrals for the radial oscillator and the radial Coulomb system in arbitrary dimensions.