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Journal of Mathematical Physics

Stochastic field theory and finite-temperature supersymmetry

Pradip Ghosh, Pratul Bandyopadhyay

Published May 1, 1988• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.527966

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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.
We use the stochastic quantization method to construct a supersymmetric version of the quantum spherical model. This is based on the equivalence between the Brownian motion described by a Langevin equation and the supersymmetric quantum mechanics, which is connected with the existence of the Nicolai map. We investigate the critical behavior of the supersymmetric model at finite as well as at zero temperatures, showing that it exhibits both classical and quantum phase transitions, and determine the critical dimensions.
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.
We consider supersymmetry field theory with supercomponents being the square root of the Bose condensate density, the amplitude of its fluctuations and Grassmannian fields related to the Fermi particles density. The fermion number is demonstrated to be conserved in degenerated Fermi-Bose mixtures with unbroken supersymmetry when the system is invariant with respect to inversion of the time arrow. We show the supersymmetry breaking allows one to derive field equations describing behavior of real Bose-Fermi mixtures. Solution of related field equations reveals the cooling of homogeneously distributed fermions gives first spontaneous rise to strong inhomogeneous fluctuations, while the Bose condensate appears at a lower temperature dependent of the fermion density.
Stochastic field theory for a real scalar field, considering both zero and positive temperatures, is developed from complements to Nelson’s stochastic mechanics. These complements include path integral formulas for the moments of the stochastic process, a functional differential equation for the generating functional, and a virial theorem. Using these and Yasue’s nonstandard analysis formulation of stochastic field theory, a rigorous meaning is given to the path integral formulas for the field moments and to the functional differential equation of the field’s generating functional.
We present a perturbative derivation of the generalized Langevin equation starting from a relativistic quantum field theory at finite temperature. Considering a complex scalar field coupled to a thermal mediator, we use a collective field formulation (Hubbard-Stratonovich transformation) to decouple the non-linear density constraint associated with a localized composite degree of freedom. By performing a saddle-point expansion and explicitly integrating out the collective fluctuations, we obtain the dynamic response kernel for the macroscopic center-of-mass coordinate. We show that localizing the density spontaneously breaks translational invariance, naturally generating a harmonic restoring force that results in an Ornstein-Uhlenbeck process. Furthermore, we resolve the vanishing friction paradox of the free-field limit by employing a resummed Dyson-Schwinger approach. We explicitly demonstrate that the thermal width of the dressed propagator generates finite Ohmic dissipation via Landau damping of the Källén-Lehmann spectral functions. This work provides a transparent, effective field theory bridge between microscopic thermal quantum fields and macroscopic open quantum systems.