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&#39;&#39; (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&#39;,...,x_N&#39;)= ψ(x_1,...,x_N)ψ*(x_1&#39;,...,x_N&#39;) by a multiplicative factor, which, if the positions {x_1,...,x_N; x_1&#39;,...,x_N&#39;} 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<J) of ((|x_K-x_K&#39;|/R)(|x_I&#39;-x_J||x_I-x_J&#39;|/|x_I-x_J||x_I&#39;-x_J&#39;|))^{-24M^2}. Here we show that if each uniform-mass ball is replaced by a grainy ball or more general-shaped lump of similar size consisting of a number, n, of well-spaced small balls of mass m and radius r and, in the above formula, R is replaced by r, M by m and the products are taken over all Nn positions of all the small balls, then the result is well-approximated by replacing R in the original formula by a new value R_eff. This suggests that the original formula will apply in general to physically realistic lumps -- be they macroscopic lumps of ordinary matter with the grains atomic nuclei etc. or be they atomic nuclei themselves with their own (quantum) grainy substructure -- provided R is chosen suitably. In the case of a cubical lump consisting of n=(2L+1)^3 small balls (L > 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.