Understanding Quantum Physics: An Advanced Guide for the Perplexed

stipulate that these distributions
are real, because they may be effective and formed by the ergodic motion of a
localized particle with the total mass and charge of the system, and
especially, the effective mass and charge distributions have no gravitational
and electrostatic self-interactions, which is consistent with the superposition
principle. Thus this view begs the question and leaves the origin of mass and
charge density as a mystery. On the other hand, the assumption that real mass
and charge distributions have gravitational and electrostatic self-interactions
has been confirmed not only in the classical domain but also in the quantum
domain for many-body systems. For example, two charged quantum systems such as
two electrons, which represent two real charge distributions, do have
electrostatic interactions. Thus it is reasonable to expect that this
assumption also holds true for individual quantum systems. Our following
analysis will show that this assumption, when combining with the superposition
principle, can help to reveal the physical origin of the mass and charge
density of a quantum system.
    [18] It has been argued that the existence of a gravitational
self-interaction term in the Schrödinger-Newton equation does not have a
consistent Born rule interpretation (Adler 2007). The reason is that the
probability of simultaneously finding a particle in different positions is
zero.
    [19] By contrast, the potential strength of the gravitational
self-interaction for a free electron is about 4 × 10 −89 .
    [20] Note that even if there are only two masses and charges in space at a
given instant, the densities formed by their motion also have gravitational and
electrostatic interactions. Therefore, the mass and charge density of a quantum
system can only be formed by the ergodic motion of one localized particle with
the total mass and charge of the system.
    [21] At a particular time the charge density is either zero (if the electron
is not there) or singular (if the electron is inside the infinitesimally small
region including the space point in question).
    [22] Note that in Nelson’s stochastic mechanics, the electron, which is
assumed to undergo a Brownian motion, moves only within a region bounded by the
nodes (Nelson 1966). This ensures that the theory can be equivalent to quantum
mechanics in a limited sense. Obviously this sort of motion is not ergodic and
cannot generate the required charge density distribution. Likewise, some
variants of stochastic mechanics (Bell 1986b; Vink 1993; Barrett, Leifer and
Tumulka 2005), which assume that the motion of particles is discrete random
jump but still nonergodic, cannot be consistent with protective measurement
either. In addition, it has been argued that stochastic mechanics is
inconsistent with quantum mechanics (Glabert, H¨anggi and Talkner 1979;
Wallstrom 1994). Glabert, H¨anggi and Talkner (1979) argued that the
Schrödinger equation is not equivalent to a Markovian process, and the various
correlation functions used in quantum mechanics do not have the properties of
the correlations of a classical stochastic process. Wallstrom (1994) further
showed that one must add by hand a quantization condition, as in the old
quantum theory, in order to recover the Schrödinger equation, and thus the
Schrödinger equation and the Madelung hydrodynamic equations are not
equivalent. In fact, Nelson (2005) also showed that there is an empirical
difference between the predictions of quantum mechanics and his stochastic
mechanics when considering quantum entanglement and nonlocality. For example,
for two widely-separated but entangled harmonic oscillators, the two theories
predict totally different statistics; stochastic mechanics predicts that measurements
of the position of the first one at time T (oscillation period) and the
position of the second one at time 0 do not interfere with each other, while
quantum mechanics predicts that there exists a strong correlation between them.
    [23] The word "cause" used here only denotes a certain
instantaneous condition determining the change of position, which may appear in
the laws of motion. Our analysis is irrelevant to whether the condition has
causal power or not.
    [24] This deterministic instantaneous condition has been often called
intrinsic velocity (Tooley 1988).
    [25] In discrete space and time, the motion will be a discrete jump across
space along a fixed direction at each time unit, and thus it will become
continuous