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North-Holland mathematical library

An Example of a Stochastic Quantum Process: Interaction of a Quantum Particle with a Boson Field

Bernard Gaveau

Published Jan 1, 1984• 1 citation
Publisher
DOI
10.1016/s0924-6509(08)70391-x

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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.
Abstract. In the context of Markov processes, we show a new scheme to derive a dual process and duality function based on a boson representation. This scheme is applicable to a case in which a generator is expressed by boson creation and annihilation operators. For some stochastic processes, duality relations have been known, which connect continuous time Markov processes with discrete state space and those with continuous state space. We clarify that using a generating function approach and the Doi-Peliti method, a birth-death process (or discrete random walk model) is naturally connected to a differential equation with continuous variables, which would be interpreted as a dual Markov process. The key point in the derivation is to use bosonic coherent states for a bra state, instead of a usual projection state. As examples, we apply the scheme to a simple birth-coagulation process and a Brownian momentum process. The generator of the Brownian momentum process is written by operators in SU(1, 1), and using a boson realization for operators in SU(1, 1) we show that the same scheme is available. PACS numbers: 05.40.-a, 02.50.-r,82.20.-wDuality in interacting particle systems and boson representation 2 1.