Abstract
We investigate the implication of the nonlinear and non-local multi-particle Schrödinger–Newton equation for the motion of the mass centre of an extended multi-particle object, giving self-contained and comprehensible derivations. In particular, we discuss two opposite limiting cases. In the first case, the width of the centre-of-mass wave packet is assumed much larger than the actual extent of the object, in the second case it is assumed much smaller. Both cases result in nonlinear deviations from ordinary free Schrödinger evolution for the centre of mass. On a general conceptual level we include some discussion in order to clarify the physical basis and intention for studying the Schrödinger–Newton equation.
Highlights
How does a quantum system in a non-classical state gravitate? There is no unanimously accepted answer to this seemingly obvious question
In this paper we consider the (N + 1)-particle Schrödinger–Newton equation for a function Ψ : 1+3(N+1) →, where 3 (N + 1) arguments correspond to the three coordinates each of (N + 1) particles of masses m0, m1, ⋯, mN, and one argument is given by the (Newtonian absolute) time t
Hamiltonian would be a double integral over Ψ †(x) Ψ (x) Ψ †(x′) Ψ (x′) / ∥ x − x′∥, where Ψ is the field operator. (See, e.g., chapter 11 of [17] for a text-book account of nonrelativistic QFT.) This term will lead to divergent self-energies, which one renormalizes through normal ordering, and pointwise Coulomb interactions of pairs. This is just the known and accepted strategy followed in deriving the multi-particle Schrödinger equation for charged point-particles from QED
Summary
How does a quantum system in a non-classical state gravitate? There is no unanimously accepted answer to this seemingly obvious question. From that perspective it would make little sense to use one-particle expectation values on the right-hand side of Einsteins equation, for their associated classical gravitational field according to (4) will not be any reasonable approximation of the (strongly fluctuating) fundamentally quantum gravitational field This has been rightfully stressed recently [5, 6]. If we consider the possibility that gravity stays fundamentally classical, as we wish to do so here, we are led to contemplate the strict (and not just approximate) sourcing of gravitational fields by expectation values rather than operators In this case we do get nonlinear self-interactions due to gravity in the equations, even for the one-particle amplitudes. For this we start afresh from a multi-particle version of the Schrödinger–Newton equation
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