Understanding Quantum Physics: An Advanced Guide for the Perplexed

the
wave function describes is the random discontinuous motion of particles, then
it seems natural to assume that the random motion of particles is the
appropriate noise source to collapse the wave function. This has three merits
at least. First, the noise source and its properties are already known. For
example, the probability of the particles being in certain position, momentum
and energy at each instant is given by the modulus square of their wave
function at the instant. Next, this noise source is not a classical field, and
thus the model can avoid the problems introduced by the field such as the
problem of infinite energy etc (Pearle 2009). Last but not least, the random
discontinuous motion of particles can also manifest itself in the equation of
motion by introducing the collapse evolution of the wave function. In the
following, we will give a more detailed argument for this claim.
    According to the
suggested interpretation of the wave function, the wave function of a quantum
particle is an instantaneous dispositional property of the particle that
determines its random discontinuous motion. However, the wave function is not a
complete description of the instantaneous state of the particle. The
instantaneous state of the particle at a given instant also includes its random
position, momentum and energy at the instant, which may be called the random
part of the instantaneous state of the particle. Although the probability of
the particle being in each random instantaneous state is completely determined
by the wave function, its stay in the state at each instant is a new physical
fact independent of the wave function. Therefore, it seems natural to assume
that the random stays of the particle may have certain physical efficiency that
manifests in the complete equation of motion [62] . Since the motion of the particle is
essentially random, its stay at an instant does not influence its stays at
other instants in any direct way. Then the random stays of the particle can
only manifest themselves in the equation of motion by their influences on the
evolution of the wave function [63] . This forms a feedback in some sense; the wave
function of a particle determines the probabilities of its stays in certain
position, momentum and energy, while its random stay at each instant also
influences the evolution of the wave function in a stochastic way [64] .
    However, the
existence of the stochastic influences on the evolution of the wave function
relies on an important precondition: the discreteness of time. If time is
continuous and instants are durationless, the random stays of a particle can
have no stochastic influence on anything. The reason is as follows. First, the
duration of each random stay of the particle is zero in continuous time. Due to
the randomness of motion, when there are at least two possible instantaneous
states a particle can move between, the particle cannot stay in the same
instantaneous state throughout a finite time. For the joint probability of the
particle being in the same instantaneous state for all infinitely uncountable instants
in the finite time interval is obviously zero, and the total probability of the
particle being in other instantaneous states is not zero at any instant in
between either. In other words, in order that a particle stays in the same
instantaneous state for a finite time, the probability of the particle being in
this instantaneous state must be one all the while during the entire interval.
This is possible only for the banal case where there is only one instantaneous
state the particle can stay and thus there is no motion and its randomness at
all throughout the duration [65] .
    Secondly, the
influence of the random stay of a particle at a durationless instant is zero.
This can be readily understood. If a physical influence is not zero at each
durationless instant, then it may accumulate to infinite during an arbitrarily
short time interval, which should be avoided in physics. Lastly, the
accumulated influence of the random stays during a finite time interval, even
if it can be finite [66] , contains no randomness. For the discontinuity and
randomness of motion exist only at each durationless instant, during which the
influence of the random stay is zero, and they don’t exist during a finite time
interval or even an infinitesimal time interval. For example, the state of
random discontinuous motion in real space, which is defined during an
infinitesimal time interval at