Understanding Quantum Physics: An Advanced Guide for the Perplexed

is usually in the order
of 10 7 , the collapse time of the quantum superposition of two
different conscious perceptions will be
    τ c ≈ hE P /(∆E) 2 ≈ ( 2.8MeV/100GeV ) 2 ≈ 10 -9 s, (4.21)
    Since the normal
conscious time of a human being is in the order of several hundred
milliseconds, the collapse time is much shorter than the normal conscious time.
Therefore, our conscious perceptions are always definite according to the
energy-conserved collapse model.
    4.5 Critical comments on other dynamical
collapse models
    In this section,
we will give a critical analysis of other dynamical collapse models. These
models can be sorted into two categories. The first one may be called spontaneous
collapse models, in which the dynamical collapse of the wave function is
assumed to happen even for an isolated system. They include the gravity-induced
wavefunction collapse model (Di´osi 1989; Penrose 1996), the GRW model
(Ghirardi, Rimini and Weber 1986) [89] etc. The second category may be called
interaction-induced collapse models, which assume that the dynamical collapse
of the wave function of a given system results from its particular interaction
with a noise field. One typical example is the CSL model (Pearle 1989;
Ghirardi, Pearle and Rimini 1990) [90] . In the following, we will primarily
analyze Penrose’s gravity-induced wavefunction collapse model and the CSL
model, which are generally regarded as two of the most promising models of
wavefunction collapse.
    4.5.1
Penrose’s gravity-induced wavefunction collapse model
    It seems very
natural to guess that the collapse of the wave function is induced by gravity.
The reasons include: (1) gravity is the only universal force being present in
all physical interactions; (2) gravitational effects grow with the size of the
objects concerned, and it is in the context of macroscopic objects that linear
superpositions may be violated. The gravity-induced collapse conjecture can be
traced back to Feynman (1995) [91] . In his Lectures on Gravitation, he considers the
philosophical problems in quantizing macroscopic objects and contemplates on a
possible breakdown of quantum theory. He said, "I would like to suggest
that it is possible that quantum mechanics fails at large distances and for
large objects, it is not inconsistent with what we do know. If this failure of
quantum mechanics is connected with gravity, we might speculatively expect this
to happen for masses such that GM 2 / c = 1, of M near 10 −5 grams."
    Penrose (1996)
further proposed a concrete gravity-induced collapse argument. The argument is
based on a profound and fundamental conflict between the general covariance
principle of general relativity and the superposition principle of quantum mechanics.
The conflict can be clearly seen by considering the superposition state of a
static mass distribution in two different locations, say position A and
position B. On the one hand, according to quantum mechanics, the valid
definition of such a superposition requires the existence of a definite
space-time background, in which position A and position B can be distinguished.
On the other hand, according to general relativity, the space-time geometry,
including the distinguishability of position A and position B, cannot be
predetermined, and must be dynamically determined by the position superposition
state. Since the different position states in the superposition determine
different space-time geometries, the space-time geometry determined by the whole
superposition state is indefinite, and as a result, the superposition and its
evolution cannot be consistently defined. In particular, the definition of the
time-translation operator for the superposed space-time geometries involves an
inherent ill-definedness, and this leads to an essential uncertainty in the
energy of the superposed state. Then by analogy Penrose argued that this
superposition, like an unstable particle in usual quantum mechanics, is also
unstable, and it will decay or collapse into one of the two states in the
superposition after a finite lifetime. Furthermore, Penrose suggested that the
essential energy uncertainty in the Newtonian limit is proportional to the
gravitational self-energy E ∆ of the difference between the two mass
distributions, and the collapse time, analogous to the half-life of an unstable
particle, is
    T ≈ h/E ∆ .
(4.22)
    This criterion is
very close to that put forward by Di´ osi (1989) earlier, and it is usually
called the