Approximate Decoherence of Histories and 't Hooft's Deterministic Quantum Theory

This paper explores the possibility that an exactly decoherent set of histories may be constructed from an approximately decoherent set by small distortions of the operators characterizing the histories. In particular, for the case of histories of positions and momenta, this is achieved by doubling the set of operators and then finding, amongst this enlarged set, new position and momentum operators which commute, so decohere exactly, and which are ``close'' to the original operators. The enlarged, exactly decoherent, theory has the same classical dynamics as the original one, and coincides with the so-called deterministic quantum theories of the type recently studied by 't Hooft. These results suggest that the comparison of standard and deterministic quantum theories may provide an alternative method of characterizing emergent classicality. A side-product is the surprising result that histories of momenta in the quantum Brownian motion model (for the free particle in the high-temperature limit) are exactly decoherent.Comment: 41 pages, plain Te

The Isaacson expansion in quantum cosmology

This paper is an application of the ideas of the Born-Oppenheimer (or slow/fast) approximation in molecular physics and of the Isaacson (or short-wave) approximation in classical gravity to the canonical quantization of a perturbed minisuperspace model of the kind examined by Halliwell and Hawking. Its aim is the clarification of the role of the semiclassical approximation and the backreaction in such a model. Approximate solutions of the quantum model are constructed which are not semiclassical, and semiclassical solutions in which the quantum perturbations are highly excited.Comment: Revtex, 11 journal or 24 preprint pages. REPLACEMENT: A comment on previous work by Dowker and Laflamme is corrected. Utah preprint UU-REL-93/3/1

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 $\Delta F$ is largely independent of the initial state and the uncertainty principle takes the form $2 \sigma^2 \ (\Delta F)^2 \ge { \hbar^2 / t^2 }$, where $\sigma$ 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.Comment: 40 pages (plain Tex), Imperial College Preprin

An Information-Theoretic Measure of Uncertainty due to Quantum and Thermal Fluctuations

We study an information-theoretic measure of uncertainty for quantum systems. It is the Shannon information $I$ of the phase space probability distribution \la z | \rho | z \ra , where |z \ra are coherent states, and $\rho$ is the density matrix. The uncertainty principle is expressed in this measure as $I \ge 1$. For a harmonic oscillator in a thermal state, $I$ coincides with von Neumann entropy, - \Tr(\rho \ln \rho), in the high-temperature regime, but unlike entropy, it is non-zero at zero temperature. It therefore supplies a non-trivial measure of uncertainty due to both quantum and thermal fluctuations. We study $I$ as a function of time for a class of non-equilibrium quantum systems consisting of a distinguished system coupled to a heat bath. We derive an evolution equation for $I$. For the harmonic oscillator, in the Fokker-Planck regime, we show that $I$ increases monotonically. For more general Hamiltonians, $I$ settles down to monotonic increase in the long run, but may suffer an initial decrease for certain initial states that undergo ``reassembly'' (the opposite of quantum spreading). Our main result is to prove, for linear systems, that $I$ at each moment of time has a lower bound $I_t^{min}$, over all possible initial states. This bound is a generalization of the uncertainty principle to include thermal fluctuations in non-equilibrium systems, and represents the least amount of uncertainty the system must suffer after evolution in the presence of an environment for time $t$.Comment: 36 pages (revised uncorrupted version), Report IC 92-93/2

Somewhere in the Universe: Where is the Information Stored When Histories Decohere?

We investigate the idea that decoherence is connected with the storage of information about the decohering system somewhere in the universe. The known connection between decoherence of histories and the existence of records is extended from the case of pure initial states to mixed states. Records may still exist but are necessarily imperfect. We formulate an information-theoretic conjecture about decoherence due to an environment: the number of bits required to describe a set of decoherent histories is approximately equal to the number of bits of information thrown away to the environment in the coarse-graining process. This idea is verified in a simple model consisting of a particle coupled to an environment that can store only one bit of information. We explore the decoherence and information storage in the quantum Brownian motion model. It is shown that the variables that the environment naturally measures and stores information about are the Fourier components of the function $x(t)$ (describing the particle trajectory). The records storing the information about the Fourier modes are the positions and momenta of the environmental oscillators at the final time. Decoherence is possible even if there is only one oscillator in the environment. The information count of the histories and records in the environment add up according to our conjecture. These results give quantitative content to the idea that decoherence is related to ``information lost''.Comment: 48 pages, plain Tex. Second revisio

Effective Theories of Coupled Classical and Quantum Variables from Decoherent Histories: A New Approach to the Backreaction Problem

We use the decoherent histories approach to quantum theory to derive the form of an effective theory describing the coupling of classical and quantum variables. The derivation is carried out for a system consisting of a large particle coupled to a small particle with the important additional feature that the large particle is also coupled to a thermal environment producing the decoherence necessary for classicality. The effective theory is obtained by tracing out both the environment and the small particle variables. It consists of a formula for the probabilities of a set of histories of the large particle, and depends on the dynamics and initial quantum state of the small particle. It has the form of an almost classical particle coupled to a stochastic variable whose probabilities are determined by a formula very similar to that given by quantum measurement theory for continuous measurements of the small particle's position. The effective theory gives intuitively sensible answers when the small particle is in a superposition of localized states.Comment: 27 pages, plain Te

Decoherent histories analysis of the relativistic particle

The Klein-Gordon equation is a useful test arena for quantum cosmological models described by the Wheeler-DeWitt equation. We use the decoherent histories approach to quantum theory to obtain the probability that a free relativistic particle crosses a section of spacelike surface. The decoherence functional is constructed using path integral methods with initial states attached using the (positive definite) ``induced'' inner product between solutions to the constraint equation. The notion of crossing a spacelike surface requires some attention, given that the paths in the path integral may cross such a surface many times, but we show that first and last crossings are in essence the only useful possibilities. Different possible results for the probabilities are obtained, depending on how the relativistic particle is quantized (using the Klein-Gordon equation, or its square root, with the associated Newton-Wigner states). In the Klein-Gordon quantization, the decoherence is only approximate, due to the fact that the paths in the path integral may go backwards and forwards in time. We compare with the results obtained using operators which commute with the constraint (the ``evolving constants'' method).Comment: 51 pages, plain Te

Hilbert space of wormholes

Wormhole boundary conditions for the Wheeler--DeWitt equation can be derived from the path integral formulation. It is proposed that the wormhole wave function must be square integrable in the maximal analytic extension of minisuperspace. Quantum wormholes can be invested with a Hilbert space structure, the inner product being naturally induced by the minisuperspace metric, in which the Wheeler--DeWitt operator is essentially self--adjoint. This provides us with a kind of probabilistic interpretation. In particular, giant wormholes will give extremely small contributions to any wormhole state. We also study the whole spectrum of the Wheeler--DeWitt operator and its role in the calculation of Green's functions and effective low energy interactions.Comment: 23 pages, 2 figures available upon request, REVTE

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, $U = (\Delta p)^2 (\Delta q)^2$, 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 $1-\sigma$ 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.Comment: 35 pages (plain Tex, revised to avoid corruption during file transmission), Imperial College preprint 92-93/25 (1994

Cosmological perturbations and classical change of signature

Cosmological perturbations on a manifold admitting signature change are studied. The background solution consists in a Friedmann-Lemaitre-Robertson- Walker (FLRW) Universe filled by a constant scalar field playing the role of a cosmological constant. It is shown that no regular solution exist satisfying the junction conditions at the surface of change. The comparison with similar studies in quantum cosmology is made.Comment: 35 pages, latex, 2 figures available at [email protected], to appear in Physical Review