Parisi-Wu Stochastic Quantization Research Articles

Global Solutions to Elliptic and Parabolic $${\Phi^4}$$ Φ 4 Models in Euclidean Space

We prove the existence of global solutions to singular SPDEs on $${\mathbb{R}^{\rm d}}$$ with cubic nonlinearities and additive white noise perturbation, both in the elliptic setting in dimensions d = 4, 5 and in the parabolic setting for d = 2, 3. We prove uniqueness and coming down from infinity for the parabolic equations. A motivation for considering these equations is the construction of scalar interacting Euclidean quantum field theories. The parabolic equations are related to the $${\Phi^{4}_d}$$ Euclidean quantum field theory via Parisi–Wu stochastic quantization, while the elliptic equations are linked to the $${\Phi^{4}_{d-2}}$$ Euclidean quantum field theory via the Parisi–Sourlas dimensional reduction mechanism.

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.

Stochastic quantization for a system of N identical interacting Bose particles

We apply stochastic quantization to a system of N interacting identical bosons in an external potential Φ, by means of a general stationary-action principle. The collective motion is described in terms of a Markovian diffusion on , with joint density and entangled current velocity field , in principle of non-gradient form, related to one another by the continuity equation. Dynamical equations relax to those of canonical quantization, in some analogy with Parisi–Wu stochastic quantization. Thanks to the identity of particles, the one-particle marginal densities ρ, in the physical space , are all the same and it is possible to give, under mild conditions, a natural definition of the single-particle current velocity, which is related to ρ by the continuity equation in . The motion of single particles in the physical space comes to be described in terms of a non-Markovian three-dimensional diffusion with common density ρ and, at least at dynamical equilibrium, common current velocity v. The three-dimensional drift is perturbed by zero-mean terms depending on the whole configuration of the N-boson interacting system. Finally, we discuss in detail under which conditions the one-particle dynamical equations, which in their general form allow rotational perturbations, can be particularized, up to a change of variables, to the Gross–Pitaevskii equations.

Fractional derivative quantum fields at positive temperature

This paper considers fractional generalization of finite temperature Klein–Gordon (KG) field and vector potential in covariant gauge and static temporal gauge. Fractional derivative quantum field at positive temperature can be regarded as a collection of infinite number of fractional thermal oscillators. Generalized Riemann zeta function regularization and heat kernel techniques are used to obtain the high temperature expansion of free energy associated with the fractional KG field. We also show that quantization of the fractional derivative fields can be carried out by using the Parisi–Wu stochastic quantization.

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.

Quantizing Yang-Mills theory: From Parisi-Wu stochastic quantization to a global path integral

Based on a generalization of the stochastic quantization scheme we recently proposed a generalized, globally defined Faddeev-Popov path integral density for the quantization of Yang-Mills theory. In this talk first our approach on the whole space of gauge potentials is shortly reviewed; in the following we discuss the corresponding global path integral on the gauge orbit space relating it to the original Parisi-Wu stochastic quantization scheme.

Global path integral quantization of Yang–Mills theory

Based on a generalization of the stochastic quantization scheme recently a modified Faddeev–Popov path integral density for the quantization of Yang–Mills theory was derived, the modification consisting in the presence of specific finite contributions of the pure gauge degrees of freedom. Due to the Gribov problem the gauge fixing can be defined only locally and the whole space of gauge potentials has to be partitioned into patches. We propose a global path integral density for the Yang–Mills theory by summing over all patches, which can be proven to be manifestly independent of the specific local choices of patches and gauge fixing conditions, respectively. In addition to the formulation on the whole space of gauge potentials we discuss the corresponding global path integral on the gauge orbit space relating it to the original Parisi–Wu stochastic quantization scheme and to a proposal of Stora, respectively.

Stochastic Quantum Gravity inDDimensions

The generalizations of the Parisi-Wu stochastic quantization scheme that are necessary for its application to D-dimensional General Relativity are reviewed and some open problems pointed out.

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.

Quantum Field Theory with Fictitious Time in Stochastic Quantization

A model with spontaneous breakdown of symmetry in quantum field theory is studied in the framework of the Parisi-Wu stochastic quantization (SQ). Looking at the conventional treatment of the model in the canonical and the path-integral quantizations including the problems of many inequivalent Hilbert spaces and measures, we explicitly construct the SQ formulation for the model in the conventional treatment of quantum field theory. The real power of SQ is found in extending quantum field theory beyond the existing frameworks; SQ easily avoids the problems imposed by choices of Hilbert space or measure. We demonstrate it by introducing c-numbers depending on the fictitious time into the starting formal Langevin equations for the model. This new formulation in SQ thus obtained is mathematically well-defined, leading to a definite calculational scheme. The result from this shows that the masses and number of excitation modes change from those in the conventional treatment.

1/N expansion of the nonlinear sigma model and its renormalization through stochastic quantization.

The 1/{ital N} expansion of the nonlinear {sigma} model is considered in the framework of Parisi-Wu stochastic quantization. The expansion is implemented both in the Langevin approach and also in the functional-integral representation of stochastic processes. Whereas in the first method quantization is possible only by means of an artifice, the functional-integral approach is straightforward and, thanks to a graphical Ward identity and the Becchi-Rouet-Stora-Tyutin symmetry characteristic of the formulation, we show the renormalizability of the model in two and three dimensions.

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 THE KINK SOLUTION OF ϕ4 FIELD THEORY

The method of Parisi-Wu Stochastic quantization in quantum field theory is compared to earlier work in classical field equations. The method is applied to solve for the propagator of ϕ4 field theory by perturbing the Kink solution.

Chiral anomaly for parafermions from stochastic quantization.

We obtain a generalization of Fujikawa's result on the chiral anomaly for spinor fields obeying para-Fermi statistics of order p. This is done within the context of Parisi-Wu stochastic quantization extended to parafermion spinor fields. For p=1, we recover the standard result for an ordinary Fermi field.

Nonlinear parabolic stochastic differential equations with additive colored noise onR d �R +: A regulated stochastic quantization

We prove the existence of solutions to the nonlinear parabolic stochastic differential equation $$({\partial \mathord{\left/ {\vphantom {\partial {\partial t}}} \right. \kern-\nulldelimiterspace} {\partial t}} - \Delta )\varphi = - V'(\varphi ) + \eta _c $$ for polynomialsV of even degree with positive leading coefficient and ν c a gaussian colored noise process onR d ×R +. When ν c is colored enough that the gaussian solution to the linear problem has Holder continuous covariance, the nongaussian processes are almost surely realized by continuous functions. Uniqueness, regularity properties, asymptotic perturbation expansions and nonperturbative fluctuation bounds are obtained for the infinite volume processes. These equations are a cutoff version of the Parisi-Wu stochastic quantization procedure forP(ϕ) d models, and the results of this paper rigorously establish the nonperturbative nature of regularization via modification of the noise process. In the limit ν c → gaussian white noise we find that the asymptotic expansion and the rigorous bounds agree for processes corresponding to the (regulated) stochastic quantization of super-renormalizable and small coupling, strictly renormalizable scalar field theories and disagree for nonrenormalizable models.

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.

Regularization schemes for stochastic quantization.

We show that some regularization schemes in conventional quantum field theory can be extended to the Parisi-Wu stochastic-quantization scheme. In particular, we discuss the Pauli-Villars and the dimensional-regularization methods.

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