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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Decoherence in the density matrix describing quantum three-geometries and the emergence of classical spacetime

Τ. Padmanabhan

Published May 15, 1989• 101 citations
Publisher

Abstract

We construct the quantum gravitational density matrix \ensuremath{\rho}(${g}_{\ensuremath{\alpha}\ensuremath{\beta}}$,${g}_{\ensuremath{\alpha}\ensuremath{\beta}}^{\mathcal{'}}$) for compact three-geometries by integrating out a set of unobserved matter degrees of freedom from a solution to the Wheeler-DeWitt equation \ensuremath{\Psi}[${g}_{\ensuremath{\alpha}\ensuremath{\beta}}$,${q}_{k(\mathrm{matter})]}$. In the adiabatic approximation, \ensuremath{\rho} can be expressed as exp(-${l}^{2}$) where ${l}^{2}$(${g}_{\ensuremath{\alpha}\ensuremath{\beta}}$,${g}_{\ensuremath{\alpha}\ensuremath{\beta}}^{\mathcal{'}}$) is a specific ``distance'' measure in the space of three-geometries. This measure depends on the volumes of the three-geometries and the eigenvalues of the Laplacian constructed from the three-metrics. The three-geometries which are ``close together'' (${l}^{2}$\ensuremath{\ll}1) interfere quantum mechanically; those which are ``far apart'' (${l}^{2}$\ensuremath{\gg}1) are suppressed exponentially and hence contribute decoherently to \ensuremath{\rho}. Such a suppression of ``off-diagonal'' elements in the density matrix signals classical behavior of the system. In particular, three-geometries which have the same intrinsic metric but differ in size contribute decoherently to the density matrix. This analysis provides a possible interpretation for the semiclassical limit of the wave function of the Universe.

DOI
10.1103/physrevd.39.2924

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In ``Decoherence of macroscopic closed systems within Newtonian quantum gravity'' (Kay B S 1998 Class. Quantum Grav. 15 L89-L98) it was argued that, given a many-body Schroedinger wave function ψ(x_1,...,x_N) for the centre-of-mass degrees of freedom of a closed system of N identical uniform-mass balls of mass M and radius R, taking account of quantum gravitational effects and then tracing over the gravitational field amounts to multiplying the position-space density matrix ρ(x_1,...,x_N; x_1',...,x_N')= ψ(x_1,...,x_N)ψ*(x_1',...,x_N') by a multiplicative factor, which, if the positions {x_1,...,x_N; x_1',...,x_N'} are all much further away from one another than R, is well-approximated by the product from 1 to N over I, J, K (I 0) of radius r with centres at the vertices of a cubic lattice of spacing a (assumed to be very much bigger than 2r) and side 2La we establish the bound e^{-1/3}(r/a)^{1/n}La < R_eff < 2\sqrt 3(r/a)^{1/n} La.
We discuss the manner in which the gravitational field becomes classical in quantum cosmology. This involves two steps. First, one must show that the quantum state of the gravitational field becomes strongly peaked about a set of classical configurations. Second, one must show that the system is in one of a number of states of a relatively permanent nature that have negligible interference with each other. This second step involves decoherence---destruction of the off-diagonal terms in the density matrix, representing interference. To introduce the notion of decoherence, we discuss it in the context of the quantum theory of measurement, following the environment-induced superselection approach of Zurek. We then go on to discuss the application of these ideas to quantum cosmology. We show, in a simple homogeneous isotropic model, that the density matrix of the Universe will decohere if the long-wavelength modes of an inhomogeneous massless scalar field are traced out. These modes effectively act as an environment which continuously ``monitors'' the scale factor. The coherence width is very small except in the neighborhood of a classical bounce. This means that one cannot really say that a classical solution bounces because the notion of classical spacetime does not apply. The coherence width decreases as the scale factor increases, which has implications for the arrow of time. We also show, using decoherence arguments, that the WKB component of the wave function of the Universe which represents expanding universes has negligible interference with the collapsing component. This justifies the usual assumption that they may be treated separately.
We propose a modification of the coupling between classical spacetime geometry and quantum matter in which the spacetime metric at each event is sourced by the stress-energy tensor evaluated on a locally decohered density matrix constructed from the causal past of that event. This replaces the usual semiclassical expectation value with a coarse-grained stress tensor derived from a decoherence channel acting on the quantum state restricted to the causal past. In this framework, decoherence determines which sector of the quantum state can influence classical spacetime geometry: coherent phase correlations remain present in the quantum state but do not source classical spacetime geometry. The resulting effective stress tensor is formulated to be covariant and compatible with the Bianchi identities under branchwise conservation. The framework avoids macroscopic-superposition paradoxes of the traditional semiclassical approach and predicts the absence of gravitationally mediated entanglement in the Bose–Marletto–Vedral protocol. Observation of such entanglement would therefore falsify the proposal.
A theory recently proposed by the author aims to explain decoherence and the thermodynamical behaviour of closed systems within a conservative, unitary framework for quantum gravity by assuming that the operators tied to the gravitational degrees of freedom are unobservable and equating physical entropy with matter - gravity entanglement entropy. Here we obtain preliminary results on the extent of decoherence this theory predicts. We treat first a static state which, if one were to ignore quantum gravitational effects, would be a quantum superposition of two spatially displaced states of a single classically well describable ball of uniform mass density in empty space. Estimating the quantum gravitational effects on this system within a simple Newtonian approximation, we obtain formulae which predict, for example, that as long as the mass of the ball is considerably larger than the Planck mass, such a would-be-coherent static superposition will actually be decohered whenever the separation of the centres of mass of the two ball-states exceeds a small fraction (which decreases as the mass of the ball increases) of the ball radius. We then obtain a formula for the quantum-gravitational correction to the would-be-pure density matrix of a non-relativistic many-body Schrödinger wavefunction and argue that this formula predicts decoherence between configurations which differ (at least) in the `relocation' of a cluster of particles of Planck mass. We estimate the entropy of some simple model closed systems, finding a tendency for it to increase with `matter-clumping' suggestive of a link with existing phenomenological discussions of cosmological entropy increase.
This paper exploits an analogy with a time-independent Schrodinger equation to propose a concrete interpretation of a 'wavefunction of the universe' psi which solves the Wheeler-DeWitt equation for a minisuperspace quantum cosmology, and then uses this interpretation to define information theoretic notions of entropy. Given a wavefunction psi depending, in a configuration space representation, on some set of variables q 0 , . . ., q N , one is instructed to pick one of the q A , say q 0 identical to a, as a 'time' and then view psi (Q; a) as an unnormalised conditional probability amplitude for finding the values Q identical to (q 1 , . . ., q N ) at 'time' a. By identifying the 'natural' measure dQ on an a=constant hypersurface in minisuperspace, rho (Q, Q'; a)dQ varies as psi *(Q; a) psi (Q'; a)dQ is interpreted as representing a density matrix in the Q-representation; and, by viewing the Q as abstract operators for fixed a, rho is interpreted as an abstract density matrix. This interpretation admits no a priori notion of unitary evolution with respect to a 'translation' da, unitarity being viewed instead as a derived concept arising only in a semiclassical limit. The object then is to use rho to define reasonable measures of 'coarse-grained entropy' in accord with physical intuition which could serve as a diagnostic for an 'arrow of time'. Thus, e.g., by interpreting the diagonal components rho ( lambda i , lambda i ; a) in some O a -representation as conditional probabilities p i for measuring the eigenvalues lambda i of some operator O a , one is led to consider an observable dependent entropy S 0 (a)=- Sigma i p i log p i . And similarly, at least when the a=constant hypersurfaces are flat, one can map psi into a 'Wigner function' f w and then use that f w to define a 'Wigner function entropy' S w .