Understanding Quantum Physics: An Advanced Guide for the Perplexed

Di´osi-Penrose criterion. Later, Penrose (1998) further suggested
that the collapse states are the stationary solutions of the Schrödinger-Newton
equation.
    Let’s now analyze
Penrose’s argument. The crux of the matter is whether the conflict between
quantum mechanics and general relativity requires that a quantum superposition
of two space-time geometries must collapse after a finite time. We will argue
in the following that the answer is negative. First of all, although it is
widely acknowledged that there exists a fundamental conflict between the
general covariance principle of general relativity and the superposition
principle of quantum mechanics, it is still a controversial issue what the
exact nature of the conflict is and how to solve it. For example, it is
possible that the conflict may be solved by reformulating quantum mechanics in
a way that does not rely on a definite spacetime background (see, e.g. Rovelli
2011).
    Next, Penrose’s
argument seems too weak to establish a necessary connection between the
conflict and wavefunction collapse. Even though there is an essential
uncertainty in the energy of the superposition of different space-time
geometries, this kind of energy uncertainty is different in nature from the
energy uncertainty of unstable particles or unstable states in usual quantum
mechanics (Gao 2010). The former results from the ill-definedness of the
time-translation operator for the superposed spacetime geometries (though its
nature seems still unclear), while the latter exists in a definite spacetime
background, and there is a well-defined time-translation operator for the
unstable states. Moreover, the decay of these unstable states is a natural
result of the linear Schrödinger evolution, and the process is not random but
deterministic. By contrast, the hypothetical spontaneous decay or collapse of
the superposed space-time geometries is nonlinear and random. In addition, the
decay of an unstable state (e.g. excited state of an atom) is actually not
spontaneous but caused by the background field constantly interacting with it.
In some extreme situations, the state may not decay at all when in a very
special background field with bandgap (Yablonovitch 1987). In short, there
exists no convincing analogy between a superposition of different space-time
geometries and an unstable state in usual quantum mechanics. Accordingly, one
cannot argue for the decay or collapse of the superposition of different
space-time geometries by this analogy. Although an unstable state in quantum
mechanics may decay after a very short time, this does not imply that a
superposition of different space-time geometries should also decay - and,
again, sometimes an unstable state does not decay at all under special
circumstances. To sum up, Penrose’s argument by analogy only has a very limited
force, and especially, it is not strong enough to establish a necessary
connection between the conflict between quantum mechanics and general
relativity and wavefunction collapse.
    Thirdly, it can be
further argued that the conflict does not necessarily lead to the wavefunction
collapse. The key is to realize that the conflict also needs to be solved
before the wavefunction collapse finishes, and when the conflict has been
solved, the wavefunction collapse will lose its basis relating to the conflict.
As argued by Penrose, the quantum superposition of different space-time
geometries and its evolution are both ill-defined due to the fundamental
conflict between the general covariance principle of general relativity and the
superposition principle of quantum mechanics. The ill-definedness seems to
require that the superposition must collapse into one of the definite
space-time geometries, which has no problem of ill-definedness. However, the
wavefunction collapse seems too late to save the superposition from the
"suffering" of the ill-definedness during the collapse. In the final
analysis, the conflict or the problem of ill-definedness needs to be solved
before defining a quantum superposition of different space-time geometries and
its evolution. In particular, the possible collapse evolution of the
superposition also needs to be consistently defined, which again indicates that
the wavefunction collapse does not solve the problem of ill-definedness. On the
other hand, once the problem of ill-definedness is solved and a consistent
description obtained (however this is still an unsolved issue in quantum
gravity), the