Understanding Quantum Physics: An Advanced Guide for the Perplexed

the
mass and charge density of an electron, which is measurable by protective measurement
and proportional to the modulus square of its wave function, is not real but
effective; it is formed by the ergodic motion of a localized particle with the
total mass and charge of the electron. If the mass and charge density is real,
i.e., if the mass and charge distributions at different locations exist at the
same time, then there will exist gravitational and electrostatic
self-interactions of the density, the existence of which not only contradicts
experiments but also violates the superposition principle of quantum mechanics.
It is this analysis that reveals the basic existent form of a quantum system
such as an electron in space and time. An electron is a particle [27] . Here the concept of particle is used in
its usual sense. A particle is a small localized object with mass and charge,
and it is only in one position in space at an instant. However, as we have
argued above, the motion of an electron described by its wave function is not
continuous but discontinuous and random in nature. We may say that an electron
is a quantum particle in the sense that its motion is not continuous motion
described by classical mechanics, but random discontinuous motion described by
quantum mechanics.
    Next, let’s
analyze the random discontinuous motion of particles. From a logical point of
view, for the random discontinuous motion of a particle, there should exist a
probabilistic instantaneous condition that determines the probability density
of the particle appearing in every position in space, otherwise it would not "know"
how frequently they should appear in every position in space. This condition
cannot come from otherwhere but must come from the particle itself. In other
words, the particle must have an instantaneous property that determines its
motion in a probabilistic way. This property is usually called indeterministic
disposition or propensity in the literature 29 . In a word, a particle
has a propensity to be in a particular position in space, and the propensity as
a probabilistic instantaneous condition determines the probability density of
the particle appearing in every position in space. This can be regarded as the
physical basis of random discontinuous motion of particles. As a result, the
position of the particle at every instant is random, and its trajectory formed
by the random position series is not continuous at every instant [28] . In short, the motion of the particle is
essentially random and discontinuous [29] .
    Unlike the
deterministic continuous motion, the trajectory function x(t) no longer
provides a useful description for random discontinuous motion. In the
following, we will give a strict description of random discontinuous motion of
particles based on measure theory. For simplicity but without losing
generality, we will mainly analyze the one-dimensional motion that corresponds
to the point set in two-dimensional space and time. The results can be readily
extended to the three-dimensional situation.

    Fig.2 The
description of random discontinuous motion of a single particle
    We first analyze
the random discontinuous motion of a single particle. Consider the state of
motion of the particle in finite intervals ∆t and ∆x near a space-time point (t i ,x j )
as shown in Fig. 2. The positions of the particle form a random, discontinuous
trajectory in this square region. We study the projection of this trajectory in
the t-axis, which is a dense instant set in the time interval ∆t. Let W be the
discontinuous trajectory of the particle and Q be the square region [x j ,
x j + ∆x]× [t i , t i + ∆t]. The dense instant set
can be denoted by π t (W ∩ Q) ∈ R , where π t is the projection on the
t-axis. According to the measure theory, we can define the Lebesgue measure:

    Since the sum of
the measures of all such dense instant sets in the time interval ∆t is equal to
the length of the continuous time interval ∆t, we have:

    Then we can define
the measure density as follows [30] :

    We call it
position measure density or position density in brief. This quantity provides a
strict description of the position distribution of the particle or the relative
frequency of the particle appearing in an infinitesimal space interval dx near
position x during an infinitesimal interval dt near instant t. In other words,
ρ(x, t) provides a strict description of the state of random discontinuous
motion of the