Parisi-Wu Stochastic Quantization Method Research Articles

Parisi–Wu stochastic quantization for relativistic particles in a weak gravitational plane wave

The problem of relativistic particles moving in the background of a weak gravitational plane wave is solved by the use of Parisi–Wu stochastic quantization method. After having solved the corresponding Langevin equations, we have calculated exactly the correlation product of two fields known as a Green's function at the thermal-equilibrium limit for the fictitious time. By addition of an infinitesimal imaginary part of the mass, the existence of the limit of the correlation function at the equilibrium is assured. Analytical and exact expressions of the wave functions are obtained both for Klein–Gordon (KG) and Dirac particles.

The fractional oscillator process with two indices

We introduce a new fractional oscillator process which can be obtained as a solution of a stochastic differential equation with two fractional orders. Basic properties such as fractal dimension and short-range dependence of the process are studied by considering the asymptotic properties of its covariance function. By considering the fractional oscillator process as the velocity of a diffusion process, we derive the corresponding diffusion constant, fluctuation–dissipation relation and mean-square displacement. The fractional oscillator process can also be regarded as a one-dimensional fractional Euclidean Klein–Gordon field, which can be obtained by applying the Parisi–Wu stochastic quantization method to a nonlocal Euclidean action. The Casimir energy associated with the fractional field at positive temperature is calculated by using the zeta function regularization technique.

Green’s function for spinless particle via Parisi-Wu stochastic quantization method

An exact and analytic Green function for a spinless particle in interaction with an electromagnetic plane wave field, expressed in the coordinate gauge is given by Parisi-Wu stochastic quantization method. We separate the classical calculations from those related to the quantum fluctuation term. We have used a perturbative treatment relying on phase and configuration spaces formulation.

Stochastic quantization of nonlocal fields

Quantization of fractional Klein–Gordon field and fractional electromagnetic potential in the Coulomb gauge and the temporal gauge can be carried out using a modification of the quantization method based on stochastic mechanics. The usual oscillator process is replaced by a fractional oscillator process which satisfied a fractional Langevin equation. Properties of these fractional fields are discussed. We also show that the Parisi–Wu stochastic quantization method can be applied to fractional fields.

Basic Ideas of Stochastic Quantization

We describe basic ideas of the stochastic quantization which was originally proposed by Parisi and W u. We start from a brief survey of stochastic-dynamical approaches to quantum mechanics, as a historical background, in which one can observe important charac­ teristics of the Parisi-Wu stochastic quantization method that are different from others. Next we give an outline of the stochastic quantization, in which a neutral scalar field is quantized as a simple example. We show that this method enables us to quantize gauge fields without resorting to the conventional gauge-fixing procedure and the Faddeev-Popov trick. Furthermore, we introduce a generalized (kerneled) Langevin equation to extend the mathe­ matical formulation of the stochastic quantization: Its illustrative application is given by a quantization of dynamical systems with bottomless actions. Finally, we develop a general formulation of stochastic quantization within the framework of a ( 4 +I)-dimensional field theory.

BRST STOCHASTIC QUANTIZATION

After a brief review of the BRST formalism and of the Parisi-Wu stochastic-quantization method, the BRST-stochastic-quantization scheme is introduced. This scheme allows the second quantization of constrained Hamiltonian systems in a manifestly gauge-symmetry-preserving way. The examples of the relativistic particle, the spinning particle and the bosonic string are worked out in detail. The paper is closed with a discussion on the interacting field theory associated with the relativistic-point-particle system.

Stochastic Quantization of System under Constraints through Phase Space Formulation

The Parisi-Wu stochastic quantization method is extended to a new method in the phase space. We first show that this method applied to a regular dynamical system yields the equilibrium distribution, in the stochastic process with respect to a fictitious time, which is the same formula as given by the conventional path-integral method. Next we present an extension of this phase space formulation applicable to non-Abelian gauge systems. The generalized Fokker-Planck equation gives naturally the equilibrium distribution which coincides with the Faddeev-Popov formula. The relation between this method and stochastic gauge fixing schemes is briefly discussed.

Possible Derivation of Energy Gaps or Hadron Masses from Fictitious-Time Correlations in Stochastic Quantization Method

The Parisi-Wu stochastic quantization method offers us one of the powerful tools for nonperturbative approaches to quantum field theory, especially, for numerical simulations of long-range correlations to give energy gaps or hadron masses. I) The method is so designed as to produce quantum theory as a thermal equilibrium limit of a hypothetical stochastic process with respect to a fictitious time other than the ordinary time. Usually we calculate physical quantities after erasing fictitious time dependences. In fact, in many papers of numerical simulations by means of the stochastic quantization, energy gaps or hadron masses have so far been derived from ordinary space-time correlations but not from fictitious-time ones.l),2) However, the stationary correlation function given by the stochastic quantization method must be subject to a dispersion formula, implicitly existing inside its Fourier transform, which gives a certain relation between the ordinary momentum variables and the fictitious-time frequency. So it would be naturally expected that we can extract information about energy gaps or hadron masses from the fictitioustime dependence of the correlation function. In this paper we discuss such a problem in the case of free field, with some perspective to interacting fields, and then carry out numerical simulations of fictitious-time correlations and energy gaps in the case of a nonrelativistic particle in a fixed potential. Consider a neutral scalar field ¢(x, t) with particle mass m, depending on D-dimensional Euclidean space-time coordinate x and fictitious time t. In the case of free field, stochastic quantization of ¢ (x, t) starts with the Langevin equation

Self-regularized field theory in Parisi-Wu stochastic quantization method : M. Namiki and Y. Yamanaka, ∗Waseda University, Tokyo, Japan

Self-regularized field theory in Parisi-Wu stochastic quantization method : M. Namiki and Y. Yamanaka, ∗Waseda University, Tokyo, Japan

Fermions and stochastic quantization.

We investigate the conditions under which fermions can be described by the Parisi-Wu stochastic quantization method. For free fermions we find that no restrictions are placed on the time approach to equilibrium. On the other hand, for interacting theories a time direction is picked out.

Stochastic quantization of supersymmetric field theory

The Parisi-Wu stochastic quantization method is applied to supersymmetric field theory. The Langevin equation, which reproduces the Green functions of euclidean field theory, is written in terms of superfields. Supersymmetric U(1) theory under gauge fixing and the large N reduction in chiral SU( N) theory are discussed. Regularization based on the stochastic method is studied also.

The new functional approach to field theory by De Alfaro, Fubini and Furlan and its connection to the Parisi-Wu stochastic quantization

We show how to derive the De Alfaro-Fubini-Furlan functional approach to field theory, starting directly from the Parisi-Wu stochastic quantization method. The main ingredient is the use of the Nicolai transformation here applied to the vacuum-vacuum generating functional of the Fokker-Planck dynamics.