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The possibility of the non-perturbative an-harmonic correction to Mehler's formula for propagator of the harmonic oscillator

J. Boháčik, P. Prešnajder, P. Augustín

Abstract

We find the possibility of the non-perturbative an-harmonic correction to Mehler's formula for propagator of the harmonic oscillator. We evaluate the conditional Wiener measure functional integral with a term of the fourth order in the exponent by an alternative method as in the conventional perturbative approach. In contrast to the conventional perturbation theory, we expand into power series the term linear in the integration variable in the exponent. We discuss the case, when the starting point of the propagator is zero. We present the results in analytical form for positive and negative frequency.

The possibility of the non-perturbative an-harmonic correction to Mehler's formula for propagator of the harmonic oscillator

Abstract

We find the possibility of the non-perturbative an-harmonic correction to Mehler's formula for propagator of the harmonic oscillator. We evaluate the conditional Wiener measure functional integral with a term of the fourth order in the exponent by an alternative method as in the conventional perturbative approach. In contrast to the conventional perturbation theory, we expand into power series the term linear in the integration variable in the exponent. We discuss the case, when the starting point of the propagator is zero. We present the results in analytical form for positive and negative frequency.

Paper Structure

This paper contains 12 sections, 280 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Evaluation of the path integral
  3. Path integral as the continuum limit of the $N-1$ dimensional integral.
  4. The analysis of the an-harmonic correction.
  5. Conclusions.
  6. Evaluation of the $N-$ dimensional integral
  7. Evaluation of the $\mathcal{W}_{N}$ by the recurrent summation over $k_i$ indices
  8. The continuum limit of the square-root factor in the Eq.( \ref{['recur104']}).
  9. The continuum limit of the exponential factor in the Eq.(\ref{['recur104']}).
  10. The evaluation of the an-harmonicity correction in the Eq.(\ref{['recur104']}).
  11. Evaluation of the continuum limit of the an-harmonic correction
  12. Algebra of integrals.

Figures (1)

  • Figure 1: $b$ dependence of the exponential part of the Mehler's formula for an-harmonic oscillator (\ref{['con1']}) when the model parameters were fixed as $\beta = c = 1,\ a = 2.$