Understanding Quantum Physics: An Advanced Guide for the Perplexed

will have been excluded. By contrast, although
the interaction-induced collapse models such as the CSL model also violate
energy conservation in their present formulations, there is still hope that
when counting the energy of external noise field the total energy may be
conserved in these models (Pearle 2000; Bassi, Ippoliti and Vacchini 2005).
Let’s turn to the CSL model now.
    4.5.2
The CSL model
    In the CSL model,
the collapse of the wave function of a quantum system is assumed to be caused
by its interaction with a classical scalar field, w(x, t). The collapse states
are the eigenstates of the smeared mass density operator, and the mechanism
leading to the suppression of the superpositions of macroscopically different
states is fundamentally governed by the integral of the squared differences of
the mass densities associated to the superposed states. It may be expected that
the introduction of the noise field can help to solve the problems plagued by
the spontaneous collapse models, e.g. the problems of energy non-conservation
and the origin of randomness etc. However, one must first answer what field the
noise field is and especially why it can collapse the wave functions of all
quantum systems. The validity of the CSL model strongly depends on the
existence of this hypothetical noise field. In this subsection, we will mainly
analyze this important legitimization problem of the CSL model [98] .
    Whatever the
nature of the noise field w(x, t) is, it cannot be quantum in the usual sense
since its coupling to a quantum system is not a standard coupling between two
quantum systems. The coupling is anti-Hermitian (Bassi 2007), and the equation
of the resulting dynamical collapse is not the standard Schrödinger equation
with a stochastic potential either. According to our current understandings,
the gravitational field is the only universal field that might be not
quantized, though this possibility seems extremely small in the view of most
researchers. Therefore, it seems natural to identify this noise field with the
gravitational field. In fact, it has been argued that in the CSL model the
w-field energy density must have a gravitational interaction with ordinary
matter (Pearle and Squires 1996; Pearle 2009). The argument of Pearle and
Squires (1996) can be summarized as follows [99] .
    There are two
equations which characterize the CSL model. The first equation is a modified
Schrödinger equation, which expresses the influence of an arbitrary field w(x,
t) on the quantum system. The second equation is a probability rule which gives
the probability that nature actually chooses a particular w(x, t). This
probability rule can also be interpreted as expressing the influence of the
quantum system on the field. As a result, w(x, t) can be written as follows:
    w(x, t) = w 0 (x,
t)+ < A(x, t) >, (4.23)
    where A(x, t) is
the mass density operator smeared over the GRW scale a, < A(x, t) > is
its quantum expectation value, and w 0 (x, t) is a Gaussian randomly
fluctuating field with zero drift, temporally white noise in character and with
a particular spatial correlation function. Then the scalar field w(x, t) that
causes collapse can be interpreted as the gravitational curvature scalar with
two sources, the expectation value of the smeared mass density operator and an
independent white noise fluctuating source. This indicates that the CSL model
is based on the semi-classical gravity, and the smeared mass density is the
source of the gravitational potential. Note that the reality of the field w(x,
t) requires that the smeared mass density of a quantum system is real 58 .
    According to our
previous analysis, however, a quantum system does not have a real mass density
distribution in space, no matter it is smeared or not. Moreover, although the
approach of semi-classical gravity may be consistent in the context of
dynamical collapse models (Pearle and Squires 1996; Ghirardi 2008), it may have
been excluded as implied by the analysis. Besides, as we have pointed out in
Section 2, protective measurement shows that a quantum system has an effective
mass density proportional to the modulus square of its wave function. Thus the
assumed existence of the smeared mass density in the CSL model, even if it is
effective, also contradicts protective measurement. Note that it is crucial
that the mass density be smeared over the GRW scale a in the CSL model; without
such a smearing the energy excitation of particles undergoing collapse