Understanding Quantum Physics: An Advanced Guide for the Perplexed

wavefunction collapse will completely lose its connection with
the problem [92] . Therefore, contrary to Penrose’s expectation, it seems
that the conflict between quantum mechanics and general relativity does not
entail the existence of wavefunction collapse.
    Even though
Penrose’s gravity-induced collapse argument is debatable, the wavefunction
collapse may still exist due to other reasons, and thus Penrose’s concrete
suggestions for the collapse time formula and collapse states also need to be
further examined as some aspects of a phenomenological model. First of all,
let’s analyze Penrose’s collapse time formula Eq. (4.22), according to which
the collapse time of a superposition of two mass distributions is inversely
proportional to the gravitational self- energy of the difference between the
two mass distributions. As we have argued above, the analogy between such a
superposition and an unstable state in quantum mechanics does not exist, and
gravity does not necessarily induce wavefunction collapse either. Thus this
collapse time formula, which is based on a similar application of Heisenberg’s
uncertainty principle to unstable states, will lose its original physical
basis. In particular, the appearance of the gravitational self-energy term in
the formula is in want of a reasonable explanation. In fact, it has already
been shown that this gravitational self-energy term does not represent the
ill-definedness of time-translation operator (or the fuzziness of the
identification between two spacetimes) in the strictly Newtonian regime
(Christian 2001). In this regime, the time-translation operator can be well
defined, but the gravitational self-energy term is not zero. In addition, as
Di´osi (2007) pointed out, the microscopic formulation of the collapse time
formula is unclear and still has some problems (e.g. the cut-off difficulty).
    Next, let’s
examine Penrose’s suggestion for the collapse states. According to Penrose
(1998), the collapse states are the stationary solutions of the
Schrödinger-Newton equation, namely Eq. (2.31) given in Chapter 2. The equation
describes the gravitational self-interaction of a single quantum system, in
which the mass density m|ψ(x, t)| 2 is the source of the classical
gravitational potential. As we have argued in Chapter 2, although a quantum
system has mass density that is measurable by protective measurement, the
density is not real but effective, and it is formed by the ergodic motion of a
localized particle with the total mass of the system. Therefore, there does not
exist a gravitational self-interaction of the mass density. This conclusion can
also be reached by another somewhat different argument. Since charge always
accompanies mass for a charged particle such as an electron [93] , the existence of the gravitational
self-interaction, though which is too weak to be excluded by present
experiments, may further entail the existence of a remarkable electrostatic
self-interaction of the particle [94] , which already contradicts experiments as
we have shown in Chapter 2. This analysis poses a serious objection to the
Schrödinger-Newton equation and Penrose’s suggestion for the collapse states [95] .
    Lastly, we briefly
discuss another two problems of Penrose’s collapse scheme. The first one is the
origin of the randomness of collapse results. Penrose did not consider this
issue in his collapse scheme. If the collapse is indeed spontaneous as implied
by his gravity-induced collapse argument, then the randomness cannot result
from any external influences such as an external noise field, and it can only
come from the studied quantum system and its wave function. The second problem
is energy non-conservation. Although Penrose did not give a concrete model of
wavefunction collapse, his collapse scheme requires the collapse of
superpositions of different positions, while this kind of space collapse
inevitably violates energy conservation [96] . Since the gravitational energy of a
quantum system is much smaller than the energy of the system, Penrose’s
collapse scheme still violates energy conservation even if the gravitational
field is counted [97] . As we have noted earlier, for an isolated system
only the collapse states are energy eigenstates can energy conserve (at the
ensemble level) during the collapse. If the principle of conservation of energy
is indeed universal as widely thought, then the spontaneous collapse models
that violate energy conservation