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

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.