arXiv subjects

J. J. Halliwell

Publications and source records attributed to J. J. Halliwell.

65 records · Page 4Linked to original sources

arXiv1997

Coupling Classical and Quantum Variables using Continuous Quantum Measurement Theory

We propose a system of equations to describe the interaction of a quasiclassical variable $X$ with a set of quantum variables $x$ that goes beyond the usual mean field approximation. The idea is to regard the quantum system as continuously and imprecisely measured by the classical system. The effective equations of motion for the classical system therefore consist of treating the quantum variable $x$ as a stochastic c-number $\x (t) $ the probability distibution for which is given by the theory of continuous quantum measurements. The resulting theory is similar to the usual mean field equations (in which $x$ is replaced by its quantum expectation value) but with two differences: a noise term, and more importantly, the state of the quantum subsystem evolves according to the stochastic non-linear Schrodinger equation of a continuously measured system. In the case in which the quantum system starts out in a superposition of well-separated localized states, the classical system goes into a statistical mixture of trajectories, one trajectory for each individual localized state.

quant-ph↗

arXiv1996

Decoherence of Hydrodynamic Histories: A Simple Spin Model

In the context of the decoherent histories approach to the quantum mechanics of closed systems, Gell-Mann and Hartle have argued that the variables typically characterizing the quasiclassical domain of a large complex system are the integrals over small volumes of locally conserved densities -- hydrodynamic variables. The aim of this paper is to exhibit some simple models in which approximate decoherence arises as a result of local conservation. We derive a formula which shows the explicit connection between local conservation and approximate decoherence. We then consider a class of models consisting of a large number of weakly interacting components, in which the projections onto local densities may be decomposed into projections onto one of two alternatives of the individual components. The main example we consider is a one-dimensional chain of locally coupled spins, and the projections are onto the total spin in a subsection of the chain. We compute the decoherence functional for histories of local densities, in the limit when the number of components is very large. We find that decoherence requires two things: the smearing volumes must be sufficiently large to ensure approximate conservation, and the local densities must be partitioned into sufficiently large ranges to ensure protection against quantum fluctuations.

quant-ph↗

arXiv1995

Alternative Derivation of the Hu-Paz-Zhang Master Equation for Quantum Brownian Motion

Hu, Paz and Zhang [ B.L. Hu, J.P. Paz and Y. Zhang, Phys. Rev. D {\bf 45} (1992) 2843] have derived an exact master equation for quantum Brownian motion in a general environment via path integral techniques. Their master equation provides a very useful tool to study the decoherence of a quantum system due to the interaction with its environment. In this paper, we give an alternative and elementary derivation of the Hu-Paz-Zhang master equation, which involves tracing the evolution equation for the Wigner function. We also discuss the master equation in some special cases.

quant-ph↗

arXiv1995

An Operator Derivation of the Path Decomposition Expansion

The path decomposition expansion is a path integral technique for decomposing sums over paths in configuration space into sums over paths in different spatial regions. It leads to a decomposition of the configuration space propagator across arbitrary surfaces in configuration space. It may be used, for example, in calculations of the distribution of first crossing times. The original proof relied heavily on the position representation and in particular on the properties of path integrals. In this paper, an elementary proof of the path decomposition expansion is given using projection operators. This leads to a version of the path decomposition expansion more general than the configuration space form previously given. The path decomposition expansion in momentum space is given as an example.

quant-ph↗

arXiv1994

Generalized Uncertainty Relations and Long Time Limits for Quantum Brownian Motion Models

We study the time evolution of the reduced Wigner function for a class of quantum Brownian motion models. We derive two generalized uncertainty relations. The first consists of a sharp lower bound on the uncertainty function, $U = (Δp)^2 (Δq)^2 $, after evolution for time $t$ in the presence of an environment. The second, a stronger and simpler result, consists of a lower bound at time $t$ on a modified uncertainty function, essentially the area enclosed by the $1-σ$ contour of the Wigner function. In both cases the minimizing initial state is a non-minimal Gaussian pure state. These generalized uncertainty relations supply a measure of the comparative size of quantum and thermal fluctuations. We prove two simple inequalites, relating uncertainty to von Neumann entropy, and the von Neumann entropy to linear entropy. We also prove some results on the long-time limit of the Wigner function for arbitrary initial states. For the harmonic oscillator the Wigner function for all initial states becomes a Gaussian at large times (often, but not always, a thermal state). We derive the explicit forms of the long-time limit for the free particle (which does not in general go to a Gaussian), and also for more general potentials in the approximation of high temperature.

gr-qc↗

arXiv1994

A Review of the Decoherent Histories Approach to Quantum Mechanics

I review the decoherent (or consistent) histories approach to quantum mechanics, due to Griffiths, to Gell-Mann and Hartle, and to Omnes. This is an approach to standard quantum theory specifically designed to apply to genuinely closed systems, up to and including the entire universe. It does not depend on an assumed separation of classical and quantum domains, on notions of measurement, or on collapse of the wave function. Its primary aim is to find sets of histories for closed systems exhibiting negligble interference, and therefore, to which probabilities may be assigned. Such sets of histories are called consistent or decoherent, and may be manipulated according to the rules of ordinary (Boolean) logic. The approach provides a framework from which one may predict the emergence of an approximately classical domain for macroscopic systems, together with the conventional Copenhagen quantum mechanics for microscropic subsystems. In the special case in which the total closed system naturally separates into a distinguished subsystem coupled to an environment, the decoherent histories approach is closed related to the quantum state diffusion approach of Gisin and Percival.

gr-qc↗

arXiv1994

Decoherent Histories and Quantum State Diffusion

We demonstrate a close connection between the decoherent histories (DH) approach to quantum mechanics and the quantum state diffusion (QSD) picture, for open quantum systems described by a master equation of Lindblad form. The (physically unique) set of variables that localize in the QSD picture also define an approximately decoherent set of histories in the DH approach. The degree of localization is related to the degree of decoherence, and the probabilities for histories prescribed by each approach are essentially the same.

gr-qc↗

arXiv1993

Quantum mechanical composition laws in reparameterization invariant systems

This paper gives a brief description of the derivation of a composition law for the propagator of a relativistic particle, in a sum over histories quantization. The extension of this derivation to the problem of finding a composition law for quantum cosmology is also discussed. (For the proceedings of Journees Relativistes 93)

gr-qc↗

arXiv1993

Aspects of the Decoherent Histories Approach to Quantum Mechanics

I give an informal overview of the decoherent histories approach to quantum mechanics, due to Griffiths, to Omnès, and to Gell-Mann and Hartle is given. Results on the connections between decoherence, records, correlation and entropy are described. The emphasis of the presentation is on understanding the broader meaning of the conditions of consistency and decoherence, and in particular, the extent to which they permit one to assign definite properties to the system. The quantum Brownian motion model is briefly discussed. (To appear in proceedings of the workshop, "Stochastic Evolution of Quantum States in Open Systems and Measurement Processes", Budapest, March, 1993, edited by L.Diosi).

gr-qc↗

arXiv1993

Quantum-Mechanical Histories and the Uncertainty Principle. II. Fluctuations About Classical Predictability

This paper is concerned with two questions in the decoherent histories approach to quantum mechanics: the emergence of approximate classical predictability, and the fluctuations about it necessitated by the uncertainty principle. We consider histories characterized by position samplings at $n$ moments of time. We use this to construct a probability distribution on the value of (discrete approximations to) the field equations, $F = m \ddot x + V'(x) $, at $n-2$ times. We find that it is peaked around $F=0$; thus classical correlations are exhibited. We show that the width of the peak $ ΔF$ is largely independent of the initial state and the uncertainty principle takes the form $2 σ^2 \ (ΔF)^2 \ge { \hbar^2 / t^2 } $, where $σ$ is the width of the position samplings, and $t$ is the timescale between projections. We determine the modifications to this result when the system is coupled to a thermal environment. We show that the thermal fluctuations become comparable with the quantum fluctuations under the same conditions that decoherence effects come into play. We also study an alternative measure of classical correlations, namely the conditional probability of finding a sequence of position samplings, given that particular initial phase space data have occurred. We use these results to address the issue of the formal interpretation of the probabilities for sequences of position samplings in the decoherent histories approach to quantum mechanics. The decoherence of the histories is also briefly discussed.

gr-qc↗

arXiv1993

Quantum-Mechanical Histories and the Uncertainty Principle: I.Information-Theoretic Inequalities

This paper is generally concerned with understanding how the uncertainty principle arises in formulations of quantum mechanics, such as the decoherent histories approach, whose central goal is the assignment of probabilities to histories. We first consider histories characterized by position or momentum projections at two moments of time. Both exact and approximate (Gaussian) projections are studied. Shannon information is used as a measure of the uncertainty expressed in the probabilities for these histories. We derive a number of inequalities in which the uncertainty principle is expressed as a lower bound on the information of phase space distributions derived from the probabilities for two-time histories. We go on to consider histories characterized by position samplings at $n$ moments of time. We derive a lower bound on the information of the joint probability for $n$ position samplings. Similar bounds are derived for histories characterized by samplings of other variables. All lower bounds on the information of histories have the general form $\ln \left( V_H / V_S \right) $, where $V_H$ is a volume element of history space, which we define, and $V_S$ is the volume of that space probed by the projections. We thus obtain a concise and general form of the uncertainty principle referring directly to the histories description of the system, and making no reference to notions of phase space.

gr-qc↗