Understanding Quantum Physics: An Advanced Guide for the Perplexed

wave
function may be such a stochastic nonlinear evolution.
    To summarize, we
have analyzed the relationships between the conservation of energy and
momentum, spacetime translation invariance and the linearity of quantum
dynamics. It has been often claimed that the conservation of energy and
momentum is a conservation law resulting from the requirement of spacetime
translation invariance. However, this common-sense view is not wholly right.
Only when assuming the linearity of quantum dynamics, can spacetime translation
invariance lead to the conservation of energy and momentum. Moreover, the
connection between invariance of natural laws and conservation laws is for
individual states, not for an ensemble of identical states. Although a
nonlinear evolution of the wave function can readily satisfy spacetime
translation invariance, the invariance can no longer lead to the conservation
of energy and momentum, let alone determining the form of the nonlinear evolution.
Rather, a universal nonlinear evolution that applies to all possible states
will inevitably violate the conservation of energy and momentum.
    Since the
conservation of energy and momentum is required by spacetime translation
invariance only for the linear evolution of the wave function of an isolated
system, the principle cannot exclude the existence of a possible nonlinear
evolution that may violate it. In other words, spacetime translation invariance
is no longer a reason to require that the evolution of the wave function of an
isolated system must conserve energy and momentum. On the other hand, the
conservation of energy and momentum may still hold true for an ensemble of
identical isolated systems as claimed by the standard quantum mechanics. Therefore,
a (universal) stochastic nonlinear evolution of the wave function may exist.
Although such evolutions cannot conserve energy and momentum for individual
states, it may conserve energy and momentum at the ensemble level. However,
unlike the linear evolution, which is natural in the sense that its form can be
uniquely determined by the invariance requirements, the stochastic nonlinear
evolution must have a physical origin, and its form can only be determined by
the underlying mechanism. In the next chapter, we will investigate the possible
stochastic nonlinear evolution of the wave function.

 
    Chapter 4
    The Solution to the Measurement Problem
    Was the wavefunction of the world
waiting to jump for thousands of millions of years until a single-celled living
creature appeared? Or did it have to wait a little longer, for some better
qualified system ... with a Ph.D.? ... Do we not have jumping then all the
time?
    —John
Bell
    In standard
quantum mechanics, it is postulated that when a wave function is measured by a
macroscopic device, it will no longer follow the linear Schrödinger equation,
but instantaneously collapse to one of the wave functions that correspond to
definite measurement results. However, this collapse postulate is ad hoc [44] , and the theory does not tell us why and
how a definite measurement result appears (Bell 1990).
    There are in
general two ways to solve the measurement problem. The first one is to
integrate the collapse evolution with the normal Schrödinger evolution into a
unified dynamics, e.g. in the dynamical collapse theories (Ghirardi 2008). The
second way is to reject the collapse postulate and assume that the Schrödinger
equation completely describes the evolution of the wave function. There are two
main alternative theories for avoiding collapse. The first one is the de
Broglie-Bohm theory (de Broglie 1928; Bohm 1952), which takes the wave function
as an incomplete description and adds some hidden variables to explain the
emergence of definite measurement results. The second is the many-worlds
interpretation (Everett 1957; DeWitt and Graham 1973), which assumes the
existence of many equally real worlds corresponding to different possible
results of quantum experiments and still regards the unitarily evolving wave
function as a complete description of the total worlds.
    It has been in hot
debate which solution to the measurement problem is the right one or in the
right direction. One of the main reasons is that the physical meaning of the
wave function is not well understood. The failure of making sense of the wave
function is partly because the problem is only investigated in the context of
conventional impulse measurements. As we have seen in the previous