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International Journal of Modern Physics A

STOCHASTIC FIELD THEORY AND RENORMALIZATION GROUP APPROACH TO CRITICAL BEHAVIORS IN THERMO FIELD DYNAMICS

B. BHATTACHARYA

Published May 20, 1989• 0 citations
Publisher

Abstract

We have studied here the critical behaviors in a simple model from the point of view of the renormalization group at finite temperature utilizing the Stochastic field theoretic approach towards a finite temperature field theory. To this end, thermofield dynamics has been formulated in terms of Stochastic fields in the external and internal space and the thermal average of the two-point correlation function of the internal field functions is related with the order parameter. The thermodynamical functions and the critical phenomena are then studied constructing the generating functionals involving Stochastic fields.

DOI
10.1142/s0217751x8900087x

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We have studied here the critical behaviors in a simple model from the point of view of the renormalization group at finite temperature utilizing the Stochastic field theoretic approach towards a finite temperature field theory. To this end, thermofield dynamics has been formulated in terms of Stochastic fields in the external and internal space and the thermal average of the two-point correlation function of the internal field functions is related with the order parameter. The thermodynamical functions and the critical phenomena are then studied constructing the generating functionals involving Stochastic fields.
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Stochastic dynamics of a nonconserved scalar order parameter near its critical point, subject to random stirring and mixing, is studied using the field theoretic renormalization group. The stirring and mixing are modelled by a random external Gaussian noise with the correlation function ∝δ(t−t′)k4−d−y\proptoδ(t-t') k^{4-d-y}∝δ(t−t′)k4−d−y and the divergence-free (due to incompressibility) velocity field, governed by the stochastic Navier--Stokes equation with a random Gaussian force with the correlation function ∝δ(t−t′)k4−d−y′\proptoδ(t-t') k^{4-d-y'}∝δ(t−t′)k4−d−y′. Depending on the relations between the exponents yyy and y′y'y′ and the space dimensionality ddd, the model reveals several types of scaling regimes. Some of them are well known (model A of equilibrium critical dynamics and linear passive scalar field advected by a random turbulent flow), but there are three new nonequilibrium regimes (universality classes) associated with new nontrivial fixed points of the renormalization group equations. The corresponding critical dimensions are calculated in the two-loop approximation (second order of the triple expansion in yyy, y′y'y′ and ε=4−dε=4-dε=4−d).
The finite-temperature behavior of supersymmetry is considered from the viewpoint of stochastic field theory. To this end, it is considered that Nelson’s stochastic mechanics may be generalized to the quantization of a Fermi field when the classical analog of such a field is taken to be a scalar nonlocal field where the internal space is anisotropic in nature such that when quantized this gives rise to two internal helicities corresponding to fermion and antifermion. Stochastic field theory at finite temperature is then formulated from stochastic mechanics which incorporates Brownian motion in the external space as well as in the internal space of a particle. It is shown that when the anisotropy of the internal space is suppressed so that the internal time ξ0 vanishes and the internal space variables are integrated out one has supersymmetry at finite temperature. This result is true for T=0, also. However, at this phase equilibrium will be destroyed. Thus for a random process van Hove’s result involving quantum mechanical operators, i.e., that when supersymmetry remains unbroken at T=0 it will also remain unbroken at T≠0, occurs. However, this formalism indicates that when at T=0 broken supersymmetry results, supersymmetry may be restored at a critical temperature Tc.
Stochastic field theory at finite temperature, when it is formulated from stochastic mechanics, which incorporates Brownian motion in both the external, as well as the internal, space of a particle, and the external and internal fields, and which are written as thermal doublets, gives rise to finite temperature quantum field theory in Minkowski space. The Haag–Hugenholtz–Winnik, (HHW) formalism of modular conjugation and the Tomita–Takesaki theory of modular Hilbert algebra then follow from this finite temperature quantum field theoretical formalism. It is shown that there is an inherent discrete Z2 symmetry among the fields of the doublet, and the equilibrium condition is ensured by this symmetry. When this symmetry is spontaneously broken, thermodynamical equilibrium is destroyed.