2,207 research outputs found
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
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.Comment: Standard TeX, 36 pages + 3 figures (postscript) Revised abstract and
introduction. To appear in Physical Review
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 of the phase space probability distribution
\la z | \rho | z \ra , where |z \ra are coherent states, and is the
density matrix. The uncertainty principle is expressed in this measure as . For a harmonic oscillator in a thermal state, 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 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 . For the harmonic oscillator, in the
Fokker-Planck regime, we show that increases monotonically. For more
general Hamiltonians, 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 at each moment of time has a lower bound
, 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 .Comment: 36 pages (revised uncorrupted version), Report IC 92-93/2
Classical Dynamics of the Quantum Harmonic Chain
The origin of classical predictability is investigated for the one
dimensional harmonic chain considered as a closed quantum mechanical system. By
comparing the properties of a family of coarse-grained descriptions of the
chain, we conclude that local coarse-grainings in this family are more useful
for prediction than nonlocal ones. A quantum mechanical system exhibits
classical behavior when the probability is high for histories having the
correlations in time implied by classical deterministic laws. But approximate
classical determinism holds only for certain coarse-grainings and then only if
the initial state of the system is suitably restricted. Coarse-grainings by the
values of the hydrodynamic variables (integrals over suitable volumes of
densities of approximately conserved quantities) define the histories usually
used in classical physics. But what distinguishes this coarse-graining from
others? This paper approaches this question by analyzing a family of
coarse-grainings for the linear harmonic chain. At one extreme in the family
the chain is divided into local groups of atoms. At the other extreme the
atoms are distributed nonlocally over the whole chain. Each coarse-graining
follows the average (center of mass) positions of the groups and ignores the
``internal'' coordinates within each group, these constituting a different
environment for each coarse-graining. We conclude that noise, decoherence, and
computational complexity favor locality over nonlocality for deterministic
predictability.Comment: 38 pages RevTeX 3.0 + 4 figures (postscript). Numerous minor
corrections. Submitted to Physical Review
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
moments of time. We use this to construct a probability distribution on the
value of (discrete approximations to) the field equations, , at times. We find that it is peaked around ; thus classical
correlations are exhibited. We show that the width of the peak is
largely independent of the initial state and the uncertainty principle takes
the form , where is
the width of the position samplings, and 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
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
Effective Theories of Coupled Classical and Quantum Variables
We address the issue of coupling variables which are essentially classical to
variables that are quantum. Two approaches are discussed. In the first (based
on collaborative work with L.Di\'osi), continuous quantum measurement theory is
used to construct a phenomenological description of the interaction of a
quasiclassical variable with a quantum variable , where the
quasiclassical nature of is assumed to have come about as a result of
decoherence. The state of the quantum subsystem evolves according to the
stochastic non-linear Schr\"odinger equation of a continuously measured system,
and the classical system couples to a stochastic c-number \x (t) representing
the imprecisely measured value of . The theory gives intuitively sensible
results even when the quantum system starts out in a superposition of
well-separated localized states. The second approach involves a derivation of
an effective theory from the underlying quantum theory of the combined
quasiclassical--quantum system, and uses the decoherent histories approach to
quantum theory.Comment: 25 pages, plain Tex. To appear in proceedings of the conference Open
Systems and Measurement in Relativistic Quantum Theory, Naples, April 3-4,
1998, edited by H.P.Breuer and F.Petruccion
The Feynman propagator for spin foam quantum gravity
We link the notion causality with the orientation of the 2-complex on which
spin foam models are based. We show that all current spin foam models are
orientation-independent, pointing out the mathematical structure behind this
independence. Using the technology of evolution kernels for quantum
fields/particles on Lie groups/homogeneous spaces, we construct a generalised
version of spin foam models, introducing an extra proper time variable and
prove that different ranges of integration for this variable lead to different
classes of spin foam models: the usual ones, interpreted as the quantum gravity
analogue of the Hadamard function of QFT or as a covariant definition of the
inner product between quantum gravity states; and a new class of causal models,
corresponding to the quantum gravity analogue of the Feynman propagator in QFT,
non-trivial function of the orientation data, and implying a notion of
''timeless ordering''.Comment: RevTex, 5 pages, no figures; v2-3:minor typos correcte
The Origin of Time Asymmetry
It is argued that the observed Thermodynamic Arrow of Time must arise from
the boundary conditions of the universe. We analyse the consequences of the no
boundary proposal, the only reasonably complete set of boundary conditions that
has been put forward. We study perturbations of a Friedmann model containing a
massive scalar field but our results should be independent of the details of
the matter content. We find that gravitational wave perturbations have an
amplitude that remains in the linear regime at all times and is roughly time
symmetric about the time of maximum expansion. Thus gravitational wave
perturbations do not give rise to an Arrow of Time. However density
perturbations behave very differently. They are small at one end of the
universe's history, but grow larger and become non linear as the universe gets
larger. Contrary to an earlier claim, the density perturbations do not get
small again at the other end of the universe's history. They therefore give
rise to a Thermodynamic Arrow of Time that points in a constant direction while
the universe expands and contracts again. The Arrow of Time does not reverse at
the point of maximum expansion. One has to appeal to the Weak Anthropic
Principle to explain why we observe the Thermodynamic Arrow to agree with the
Cosmological Arrow, the direction of time in which the universe is expanding.Comment: 41 pages, DAMTP R92/2
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