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arXiv:gr-qc/0506039

The robustness of a many-body decoherence formula of Kay under changes in graininess and shape of the bodies

Varqa Abyaneh, Bernard S. Kay

Published Jun 6, 2005• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.gr-qc/0506039

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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 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}[${g}_{\ensuremath{\alpha}\ensuremath{\beta}}$,qk(matter)]{q}_{k(\mathrm{matter})]}qk(matter)]​. In the adiabatic approximation, \ensuremath{\rho} can be expressed as exp(-l2{l}^{2}l2) where l2{l}^{2}l2(${g}_{\ensuremath{\alpha}\ensuremath{\beta}}$,${g}_{\ensuremath{\alpha}\ensuremath{\beta}}^{\mathcal{'}}$) 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'' (l2{l}^{2}l2\ensuremath{\ll}1) interfere quantum mechanically; those which are ``far apart'' (l2{l}^{2}l2\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.
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.
Consider a quantum particle of mass M in R3, described at time 0 by a wave function ϕ(x) with dispersion Δ, interacting independently with a collection of N particles of mass m. Using only Schroedinger's Quantum Mechanics we prove that when N becomes large and m/M becomes small, and if the information at time t>0 about the N particles of small mass in negleted, the system admits a "classical" description, i.e. a description in which the coherence of the wave function over distances of the order of mM-1N-1Δ have disappeared. We consider this a first step towards proving that most "sufficiently large" quantum systems interacting with an uncontrolled environment admit a classical description at least for position measurements.
We restate Kay's 1998 hypothesis which simultaneously offers an objective definition for the entropy of a closed system, a microscopic foundation for the Second Law, a resolution of the Information Loss (and other) Black-Hole Puzzle(s) and an objective mechanism for decoherence. Presupposing a conventional unitary theory of low-energy quantum gravity, it offers all this by taking the physical density operator of a closed system to be the partial trace of its total density operator (assumed pure) over gravity and by defining its physical entropy to be its `matter-gravity entanglement entropy'. We also recall Kay's 1998 modified non-relativistic (many-body) quantum mechanics based on Kay's hypothesis with a Newtonian approximation to quantum gravity. In this modification, we find formal expectation values for certain `observables' such as momentum-squared and Parity are altered but those for functions of positions are unaltered. However, by arguing that every real measurement can ultimately be taken to be a position measurement, we prove that, in practice, it is impossible to detect any alteration at all and, in particular, we predict no alteration for Roger Penrose's experiment. Nevertheless, Kay's modification contains no Schrödinger Cat-like states, and also allows an `events' interpretation which we tentatively propose and begin to explore. We also obtain a Second-Law type result for a non-relativistic toy-model closed system and argue that similar results will apply for a wide class of model Newtonian and post-Newtonian closed systems although we argue that ordinary actual lab-sized systems can never be treated as closed for the purpose of calculating their entropy. Compared with `collapse models' such as GRW, Kay's Newtonian theory does a similar job while being free from ad hoc assumptions.
The recent results of [J. Dubail, J.-M. Stéphan, J. Viti, P. Calabrese, Scipost Phys. 2, 002 (2017)], which aim at providing access to large scale correlation functions of inhomogeneous critical one-dimensional quantum systems —e.g. a gas of hard core bosons in a trapping potential— are extended to a dynamical situation: a breathing gas in a time-dependent harmonic trap. Hard core bosons in a time-dependent harmonic potential are well known to be exactly solvable, and can thus be used as a benchmark for the approach. An extensive discussion of the approach and of its relation with classical and quantum hydrodynamics in one dimension is given, and new formulas for correlation functions, not easily obtainable by other methods, are derived. In particular, a remarkable formula for the large scale asymptotics of the bosonic \boldsymbol{n} 𝐧 -particle function \boldsymbol{\left< \Psi^\dagger (x_1,t_1) \dots \Psi^\dagger (x_n,t_n) \Psi(x_1',t_1') \dots \Psi(x_n',t_n') \right>} is obtained. Numerical checks of the approach are carried out for the fermionic two-point function —easier to access numerically in the microscopic model than the bosonic one— with perfect agreement.