What is it about?
In this article, we report the evaluation of the functional integral with conditional Wiener measure for the an-harmonic oscillator, where the an-harmonicity is represented by symmetric, x^4 term. Moreover, we choose all model parameters (mass, frequency, and an-harmonicity) as time-dependent functions. The result is the equation for the propagator of the an-harmonic oscillator. We showed the principal possibility, to sum up, the perturbative expansion of the an-harmonic part of the propagator.
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Why is it important?
The Gaussian functional integral methods offer the elegant method for solving the important but simple physical models. Beyond the Gaussian integrals, the functional integral methods will play a more significant role in contemporary physics. The non-perturbative results could help understand the phase transitions description, the non-abelian field theory problems, and the challenge in quantum gravity.
Perspectives
I see the perspectives in two ways. First, we must expand the method of evaluation of the functional integral by the addition of another nonperturbative term to the action and try to simplify the summation of the perturbative expansion of the an-harmonic part of the propagator. Second, let us try to solve the physical problems connected to phase transitions and conformally reduced gravity, where, as authorities claim, the Einstein-Hilbert action can be converted to a kind of scalar phi^4 theory.
Juraj Boháčik
Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia
Read the Original
This page is a summary of: Time-dependent propagator for an-harmonic oscillator with quartic term in potential, Journal of Mathematical Physics, February 2021, American Institute of Physics,
DOI: 10.1063/5.0018545.
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