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arXiv:hep-th/9503169

Stochastic bosonization in arbitrary dimensions

Luigi Accardi, Yun Gang Lu, I. V. Volovich

Published Mar 24, 1995• 1 citation
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Abstract

A procedure of bosonization of Fermions in an arbitrary dimension is suggested. It is shown that a quadratic expression in the fermionic fields after rescaling time $t\to t/λ^2$ and performing the limit $λ\to0$ (stochastic limit), gives rise to a bosonic operator satisfying the boson canonical commutation relations. This stochastic bosonization of Fermions is considered first for free fields and then for a model with three--linear couplings. The limiting dynamics of the bosonic theory turns out to be described by means of a quantum stochastic differential equations.

DOI
10.48550/arxiv.hep-th/9503169

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A procedure of bosonization of Fermions in an arbitrary dimension is suggested. It is shown that a quadratic expression in the fermionic fields after rescaling time t→t/λ2t\to t/λ^2t→t/λ2 and performing the limit λ→0λ\to0λ→0 (stochastic limit), gives rise to a bosonic operator satisfying the boson canonical commutation relations. This stochastic bosonization of Fermions is considered first for free fields and then for a model with three--linear couplings. The limiting dynamics of the bosonic theory turns out to be described by means of a quantum stochastic differential equations.
We consider two strategies for stochastic quantization. With the first, one posits an additional time dimension (fictitious time) and describes the evolution of classical fields by means of the Langevin equation. One then evaluates stochastic averages of the field functions. In the limit that the fictitious time goes to infinity, these approach the time ordered correlation functions of canonically quantized field theory. We conclude that, while this strategy successfully describes QED and other quantum field theories, it is contrived and probably lacks deep physical significance. With the second strategy for field quantization, one begins with a classical action in either one or two extra dimensions coupled to an external random source. We review a method of quantizing bosonic fields which uses this strategy. Furthermore, we present an analogous method for quantizing fermion fields and a possible new way of quantizing interacting fermion and boson fields. Finally, we discuss applications to quantum mechanics and stochastically quantize the simple harmonic oscillator.
Dynamics of quantum systems which are stochastically perturbed by linear coupling to the reservoir can be studied in terms of quantum stochastic differential equations (for example, quantum stochastic Liouville equation and quantum Langevin equation).In order to work it out one needs to define the quantum Brownian motion.As far as only its boson version has been known until recently, in the present paper we present the definition which makes it possible to consider the fermion Brownian motion as well.
The time evolution of quantum fields is shown to be equivalent to a time-symmetric Fokker-Planck equation.Results are obtained using a Q-function representation, including fermion-fermion, boson-boson, and fermion-boson interactions with linear, quadratic, cubic, and quartic Hamiltonians, typical of QED and many other cases.For local boson-boson coupling, the resulting probability distribution is proved to have a positive, time-symmetric path integral and action principle, leading to a forward-backward stochastic process in both time directions.The solution corresponds to a c-number field equilibrating in an additional dimension.Paths are stochastic trajectories of fields in space-time, which are samples of a statistical mechanical steady state in a higher-dimensional space.We derive numerical methods and examples of solutions to the resulting stochastic partial differential equations in a higher time dimension, giving agreement with examples of simple bosonic quantum dynamics.This approach may lead to useful computational techniques for quantum field theory, as well as to new ontological models of physical reality.
We show that in a model of a particle interacting with a quantum field, the field operators rescaled according to the prescriptions of the stochastic limit, obey q -commutational relations with q depending on time. After the stochastic limit, due to the nonlinearity, the particle and field degrees of freedom become entangled even at a kinematical level in the sense that the field and the atomic variables no longer commute but give rise to a new algebra with new commutation relations replacing the boson ones. This allows to give a simple proof of the fact that the non-crossing half-planar diagrams give the dominating contribution in a weak-coupling regime and to calculate explicitly the correlations associated with the new algebra.
Abstract In this chapter, Langevin equations (or Ito stochastic differential equations, SDEs) are derived that are equivalent to Fokker–Planck equations for bosons and fermions. The approach involves replacing modal phase space variables by stochastic phase space variables satisfying Ito SDEs containing c-number Wiener stochastic variables, as well as functions of stochastic phase variables. It is shown that if the Ito SDE quantities are related to drift and diffusion terms in the FPE, then the time dependences of phase space averages of arbitrary functions equal those for stochastic averages of these functions. Using Takagi factorisation, Ito SDE quantities satisfying the required relationship are found, for both bosons and fermions. The Wiener variable numbers differ—2n for bosons, 2 n2 for fermions. For fermion unnormalised B distributions, both the drift and the diffusion Ito SDE terms are linear in the stochastic phase variables. SDEs are applied to fluctuations and quantum correlation functions.