Gaussian equivalent representation of functional integrals in quantum physics

Victor Nikolaevich Popov, Vladimir Sergeevich Yarunin

Stephan A. Baeurle

We report on the specific features of the sign problem in the classical auxiliary field methodology and the strategies employed for its alleviation. In particular, we focus on a new technique based on the method of Gaussian equivalent representation of Efimov and Nogovitsin [Physica (Amsterdam) 234A, 506 (1996)]] with which we could ameliorate the convergence properties significantly. We believe that this technique can also provide an interesting possibility to reduce the sign problem of other methods of computer simulation based on a functional integral approach.

Viktor Nikolayevich Popov

Roman Gielerak, Robert Olkiewicz

It is demonstrated that the thermal structure of the noncritical free Bose Gas is completely described by certain periodic generalized Gaussian stochastic process or equivalently by certain periodic generalized Gaussian random field. Elementary properties of this Gaussian stochastic thermal structure have been established. Gentle perturbations of several types of the free thermal stochastic structure are studied. In particular new models of non-Gaussian thermal structures have been constructed and a new functional integral representation of the corresponding euclidean-time Green functions have been obtained rigorously.

Mingnan Ding, Xiangjun Xing

Path integrals constitute powerful representations for both quantum and stochastic dynamics. Yet despite many decades of intensive studies, there is no consensus on how to formulate them for dynamics in curved space, or how to make them covariant with respect to nonlinear transform of variables (NTV). In this work, we construct a rigorous and covariant formulation of time-slicing path integrals for dynamics in curved space. We first establish a rigorous criterion for equivalence of time-slice Green's function (TSGF) in the continuum limit (Lemma 1). This implies the existence of infinitely many equivalent representations for time-slicing path integral. We then show that, for any dynamics with second order generator, all time-slice actions are equivalent to a Gaussian (Lemma 2). We further construct a continuous family of equivalent path-integral actions parameterized by an interpolation parameter α∈[0,1] (Lemma 3). The action generically contains term linear in Δx , whose concrete form depends on α . Finally we also establish the covariance of our path-integral formalism, by demonstrating how the action transforms under NTV. The α=0 representation of time-slice action is particularly convenient because it is Gaussian and transforms as a scalar, as long as Δx transforms according to Ito's formula .