Understanding Quantum Physics: An Advanced Guide for the Perplexed

that gravitational fields must be also
quantized.
    [94] If there is a gravitational self-interaction but no electrostatic
self-interaction for a charged particle, e.g. an electron, then the charge and
mass of an electron will be located in different positions and have different
density distributions in space, though they are described by the same wave
function. Concretely speaking, the mass density of an electron is m e |ψ(x,
t)| 2 as in the Schrödinger-Newton equation, whereas its charge
density is not e|ψ(x, t)| 2 but only localized in a single position
(which permits no electrostatic self-interaction). This result seems very
unnatural and has no experimental support either.
    [95] Since the Schrödinger-Newton equation is the non-relativistic
realization of the typical model of semiclassical gravity, in which the source
term in the classical Einstein equation is taken as the expectation of the
energy momentum operator in the quantum state (Rosenfeld 1963), our analysis
also presents a serious objection to the approach of semiclassical gravity.
Although the existing arguments against the semiclassical gravity models seem
so strong, they are still not conclusive (Carlip 2008; Boughn 2009). This new
analysis of the Schrödinger-Newton equation may shed some new light on the
solution of the issue.
    [96] Di´osi (2007) explicitly pointed out that the von-Neumann Newton
equation, which may be regarded as one realization of Penrose’s collapse
scheme, obviously violates conservation of energy. Another way to understand
this conclusion is to realize that the energy-conserved wavefunction collapse
cannot result from the spacetime geometry difference between the branches in a
superposition as suggested by Penrose’s collapse scheme. The reason is that
there is no difference of spacetime geometries for two different momentum
eigenstates. A momentum eigenstate does not influence its background spacetime
geometry, as its energy density is zero throughout the whole space. Thus if a
superposition of two momentum eigenstates does collapse into one of them, the
collapse cannot result from the difference of spacetime geometries in the
superposition. As a result, Penrose’s gravity-induced collapse argument does
not lead to the energy-conserved wavefunction collapse, and if it does lead to
some sort of wavefunction collapse, the collapse cannot conserve energy.
    [97] This is contrary to Penrose’s own expectation. According to Penrose
(2004), "There is the advantage with the gravitational OR scheme put
forward above that the energy uncertainty in E G would appear to
cover such a potential non-conservation, leading to no actual violation of
energy conservation. This is a matter that needs further study, however. It would
seem that there is some kind of trade-off between the apparent energy
difficulties in the OR process and the decidedly non-local (and curiously
slippery) nature of gravitational energy...".
    [98] As admitted by Pearle (2009), "When, over 35 years ago, ... I had
the idea of introducing a randomly fluctuating quantity to cause wave function
collapse, I thought, because there are so many things in nature which fluctuate
randomly, that when the theory is better developed, it would become clear what
thing in nature to identify with that randomly fluctuating quantity. Perhaps
ironically, this problem of legitimizing the phenomenological CSL collapse
description by tying it in a natural way to established physics remains almost
untouched." Related to this legitimization problem is that the two
parameters which specify the model are ad hoc (Pearle 2007). These two
parameters, which were originally introduced by Ghirardi, Rimini and Weber
(1986), are a distance scale, a ≈ 10 5 cm, characterising the distance
beyond which the collapse becomes effective, and a time scale, λ −1 ≈
10 16 sec, giving the rate of collapse for a microscopic system. If
wavefunction collapse is a fundamental physical process related to other
fundamental processes, the parameters should be able to be written in terms of
other physical constants.
    [99] Pearle (2009) further argued that compatibility with general relativity
requires a gravitational force exerted upon matter by the w-field. However, as
Pearle (2009) admitted, no convincing connection (for example, identification
of metric fluctuations, dark matter or dark energy with w(x, t)) has yet
emerged, and the legitimization problem (i.e. the problem of endowing physical
reality to the