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arXiv:math/0312294

Global Existence of Bell's Time-Inhomogeneous Jump Process for Lattice Quantum Field Theory

Hans-Otto Georgii, Roderich Tumulka

Published Dec 15, 2003• 10 citations
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Abstract

We consider the time-inhomogeneous Markovian jump process introduced by John S. Bell [Phys.Rep. 137, 49] for a lattice quantum field theory, which runs on the associated configuration space. Its jump rates, tailored to give the process the quantum distribution $|Ψ_t|^2$ at all times $t$, typically exhibit singularities. We establish the existence of a unique such process for all times, under suitable assumptions on the Hamiltonian or the initial state vector $Ψ_0$. The proof of non-explosion takes advantage of the special role of the $|Ψ_t|^2$ distribution.

DOI
10.48550/arxiv.math/0312294

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We consider the time-inhomogeneous Markovian jump process introduced by John S. Bell [Phys.Rep. 137, 49] for a lattice quantum field theory, which runs on the associated configuration space. Its jump rates, tailored to give the process the quantum distribution ∣Ψt∣2|Ψ_t|^2∣Ψt​∣2 at all times ttt, typically exhibit singularities. We establish the existence of a unique such process for all times, under suitable assumptions on the Hamiltonian or the initial state vector Ψ0Ψ_0Ψ0​. The proof of non-explosion takes advantage of the special role of the ∣Ψt∣2|Ψ_t|^2∣Ψt​∣2 distribution.
In his paper (1986 Beables for quantum field theory Phys. Rep. 137 49–54) John S Bell proposed how to associate particle trajectories with a lattice quantum field theory, yielding what can be regarded as a |Ψ| 2 -distributed Markov process on the appropriate configuration space. A similar process can be defined in the continuum, for more or less any regularized quantum field theory; we call such processes Bell-type quantum field theories. We describe methods for explicitly constructing these processes. These concern, in addition to the definition of the Markov processes, the efficient calculation of jump rates, how to obtain the process from the processes corresponding to the free and interaction Hamiltonian alone, and how to obtain the free process from the free Hamiltonian or, alternatively, from the one-particle process by a construction analogous to ‘second quantization’. As an example, we consider the process for a second quantized Dirac field in an external electromagnetic field.
In the paper with the above-noted title, D. T. Gillespie [Phys. Rev. A 49, 1607 (1994)] claims that the theory of Markov stochastic processes cannot provide an adequate mathematical framework for quantum mechanics. In conjunction with the specific quantum dynamics considered there, we give a general analysis of the associated dichotomic jump processes. If we assume that Gillespie's ``measurement probabilities'' are the transition probabilities of a stochastic process, then the process must have an invariant (time independent) probability measure. Alternatively, if we demand the probability measure of the process follow the quantally implemented (via the Born statistical postulate) evolution, then we arrive at the jump process which can be interpreted as a Markov process if restricted to a suitable duration time. However, there is no corresponding Markov process consistent with the Z2{\mathit{Z}}_{2}Z2​ event space assumption, if we require its existence for all times t\ensuremath{\in}R+{\mathbf{R}}_{+}R+​. \textcopyright{} 1996 The American Physical Society.
We discuss in detail how non-Markovian open system dynamics can be described in terms of quantum jumps [J. Piilo et al., Phys. Rev. Lett. 100, 180402 (2008)]. Our results demonstrate that it is possible to have a jump description contained in the physical Hilbert space of the reduced system. The developed non-Markovian quantum jump approach is a generalization of the Markovian Monte Carlo wave function (MCWF) method into the non-Markovian regime. The method conserves both the probabilities in the density matrix and the norms of the state vectors exactly and sheds new light on non-Markovian dynamics. The dynamics of the pure state ensemble illustrates how local-in-time master equation can describe memory effects and how the current state of the system carries information on its earlier state. Our approach solves the problem of negative jump probabilities of the Markovian MCWF method in the non-Markovian regime by defining the corresponding jump process with positive probability. The results demonstrate that in the theoretical description of non-Markovian open systems, there occurs quantum jumps which recreate seemingly lost superpositions due to the memory.
It is shown that non-Markovian master equations for an open system which are local in time can be unravelled through a piecewise deterministic quantum jump process in its Hilbert space. We derive a stochastic Schrödinger equation that reveals how non-Markovian effects are manifested in statistical correlations between different realizations of the process. Moreover, we demonstrate that possible violations of the positivity of approximate master equations are closely connected to singularities of the stochastic Schrödinger equation, which could lead to important insights into the structural characterization of positive non-Markovian equations of motion.