Understanding Quantum Physics: An Advanced Guide for the Perplexed

In addition, although the quantum observer has a dispositional
property relating to his superposition state, the property is still a definite
property of the unique observer and thus cannot correspond to the existence of
many minds.
    [61] Moreover, it can be expected that the conscious perception of the
observer is none of the perceptions corresponding to the brain states in the
superposition because these states have the same status.
    [62] This is distinct from the case of continuous motion. For the latter, the
position of a particle at each instant is completely determined by the
deterministic instantaneous condition at the instant, and thus the position of
the particle has no influence on the deterministic instantaneous condition.
    [63] In fact, since the random stays of a particle as one part of its
instantaneous state are completely random, the complete evolution equation of
the instantaneous state of the particle is only about the evolution of the wave
function. Therefore, the random stays of the particle can only manifest
themselves in the complete equation of motion by their stochastic influences on
the evolution of the wave function.
    [64] In other words, the wave function of a particle determines its random
discontinuous motion, while the motion also influences the evolution of the
wave function reciprocally.
    [65] Unfortunately, this banal case does not exist. Due to the uncertainty
relation between position and momentum in quantum mechanics, there are always
infinitely many different instantaneous states (with definite position and
momentum) where a particle can stay at any time.
    [66] Our analysis of a concrete model in the next section will show that
under some reasonable assumptions the accumulated influence of the random stays
during a finite time interval is still zero when time is continuous.
    [67] This means that the minimum duration of the random stay of a particle in
a definite position or momentum or energy is always a discrete instant. It can
be imagined that the duration of the random stay of a particle in an eigenvalue
of energy is a discrete instant, but the duration of its random stay in each
position is still zero as in continuous space and time. In this case, however,
the position probability distribution of the particle cannot be uniquely
determined during its stay in the definite energy for a general state of motion
where the energy branches are not wholly separated in space. Moreover, it seems
that only the duration of the random stay of a particle in the eigenvalue of
every property is the same can the (objective) probability distributions of all
these properties be consistent with those given by the modulus square of the
wave function in quantum mechanics.
    [68] Note that the existing arguments, which are based on some sort of
combination of quantum theory and general relativity (see, e.g. Garay 1995 for
a review), do not imply but only suggest that space and time are discrete.
Moreover, the meanings and realization of discrete spacetime are also different
in the existing models of quantum gravity.
    [69] It has been conjectured that a fundamental theory of physics may be
formulated by three natural constants: the Planck time (t P ), the
Planck length (l P ) and the Planck constant (h ), and all other
physical constants are expressed by the combinations of them (Gao 2006b). For
example, the speed of light is c = l P /t P , and the
Einstein gravitational constant is κ = 8πl P t P /h. In this
sense, the quantum motion in discrete space and time, represented by the above
three constants, is more fundamental than the phenomena described by the
special and general theory of relativity, represented by the speed of light and
the gravitational constant, respectively. However, even if this conjecture
turns out to be right, it is still a big challenge how to work out the details
(see Gao 2011c for an initial attempt).
    [70] For the superpositions of degenerate energy eigenstates of a
many-particle system, a further collapse rule is needed. We will discuss this
issue later on.
    [71] As we will see later, the conservation of energy may also hold true at
the individual level for the collapse evolution of some special wave functions.
    [72] If the phase of an energy eigenstate also changes with time, then the
probability distribution of energy eigenvalues will in general be changed for
each identical system in the ensemble, and as a result, energy will be not
conserved even at the ensemble