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Modern Physics Letters A

QUANTUM FIELD THEORY OF A DISSIPATIVE SYSTEM

B. BHATTACHARYA, K. Hajra

Published Feb 28, 1995• 0 citations
Publisher

Abstract

Quantum dissipative scalar and fermionic fields are studied here from classical stochastic fields. These classical stochastic fields are the outcome of the relativistic generalization of Nelson's stochastic mechanics based on a new microlocal geometry. Results show that the dissipation is the external classical phenomena whereas quantum nature comes from within (microlocal structure).

DOI
10.1142/s0217732395000508

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Quantum dissipative scalar and fermionic fields are studied here from classical stochastic fields. These classical stochastic fields are the outcome of the relativistic generalization of Nelson's stochastic mechanics based on a new microlocal geometry. Results show that the dissipation is the external classical phenomena whereas quantum nature comes from within (microlocal structure).
The dissipative scalar field theory by means of the deformation quantization formalism is studied. Following the ideas presented by G. Dito and F. J. Turrubiates [Phys. Lett. A352, 309 (2006)] for quantum mechanics, a star product which contains the dissipative effect for the damped oscillation modes of the field is constructed. Employing this approach the expectation values of some observables in the quantum mechanical case as well as certain correlation functions for the field case are obtained under a particular dissipative process.
A model independent generalization of quantum mechanics, including the usual as well as the dissipative quantum systems, is proposed. The theory is developed deductively from the basic principles of the standard quantum theory, the only new qualitative assumption being that we allow the wave operator at time t of a quantum system to be non-differentiable (in t) in the usual sense, but only in an appropriately defined (Sec. 5) stochastic sense. The resulting theory is shown to lead to a natural generalization of the usual quantum equations of motion, both in the form of the Schrödinger equation in interaction representation (Sec. 6) and of the Heisenberg equation (Sec. 8). The former equation leads in particular to a quantum fluctuation-dissipation relation of Einstein’s type. The latter equation is a generalized Langevin equation, from which the known form of the generalized master equation can be deduced via the quantum Feynmann-Kac technique (Secs. 9 and 10). For quantum noises with increments commuting with the past the quantum Langevin equation defines a closed system of (usually nonlinear) stochastic differential equations for the observables defining the coefficients of the noises. Such systems are parametrized by certain Lie algebras of observables of the system (Sec. 10). With appropriate choices of these Lie algebras one can deduce generalizations and corrections of several phenomenological equations previously introduced at different times to explain different phenomena. Two examples are considered: the Lie algebra [q, p]=i (Sec. 12), which is shown to lead to the equations of the damped harmonic oscillator; and the Lie algebra of SO(3) (Sec. 13) which is shown to lead to the Bloch equations. In both cases the equations obtained are independent of the model of noise. Moreover, in the former case, it is proved that the only possible noises which preserve the commutation relations of p, q are the quantum Brownian motions, commonly used in laser theory and solid state physics.
The relation between quantum electrodynamics and (classical) stochastic electrodynamics is elucidated by means of a general construction which associates with every Gaussian quantum field (for example, vacuum-free fields or coherent states at zero or nonzero temperature) a classical random field, which in the case of quantum electrodynamics yields stochastic electrodynamics. The associated classical and quantum theories yield the same results for all nondisturbing (causally separated) field measurements. This is an additional classical characteristic of coherent quantum states which implies that Bell's inequality is satisfied by quantum field measurements. Furthermore, stochastic electrodynamics is identified as the Wigner distribution of quantum electrodynamics. We point out that in these Gaussian states stochastic electrodynamics is more suitable as a classical model of the quantum field than is the Euclidean field, which results from applying Nelson's stochastic mechanics to each field mode.
Arguments are provided for the plausibility that quantum mechanics is a stochastic theory and that many quantum phenomena derive from the existence of a real noise consisting of vacuum fluctuations of the fields existing in nature. I revisit stochastic electrodynamics, a theory that studies classical systems of electrically charged particles immersed in a real electromagnetic zeropoint field with spectral density proportional to the cube of the frequency, Planck\'{}s constant appearing as the parameter fixing the scale. Asides from briefly reviewing known results, I introduce a calculational quantumlike method resting upon the commutator of stochastic processes. An intuitive explanation is offered for some typical quantum features, like the uncertainty principle, the entanglement and the fact that quantum noise looks as not dissipative.
We consider a quantization of relativistic wave equations which allows to treat quantum fields together with interacting particles at a finite time. We discuss also a dissipative interaction with the environment. We introduce a stochastic wave function whose dynamics is determined by a non-linear Schrödinger-type evolution equation in an additional time parameter. The correct classical limit requires the proper time interpretation of the time parameter. An average over the proper time leads to the conventional quantum field theory of particles which are free at an infinite space separation. We consider models with scalar and vector fields on a pseudoriemannian manifold. A quantization of the Einstein gravity in this approach is briefly discussed.
Tracking a real trajectory of a quantum particle still has been treated as the interpretation problem. It shall be expressed by a Brownian (stochastic) motion suggested by E. Nelson, however, the well-defined mechanism of field generation from a stochastic particle hasn't been proposed yet. For the improvement of this, I propose the extension of Nelson's quantum dynamics, for describing a relativistic scalar electron with its radiation equivalent to the Klein-Gordon particle and field system.