Understanding Quantum Physics: An Advanced Guide for the Perplexed

the worst, it may already suggest that the hypothetical Bohmian particles
are redundant entities in the theory (and their role in solving the measurement
problem is ad hoc), since they have no any influence on other entities in the
theory such as the wave function. Note that these problems do not exist for the
wave function; the evolution of the wave function of a charged quantum system
is influenced by both electric scalar potential and magnetic vector potential,
as well as by gravitational potential, and the wave functions of two charged
quantum systems also have gravitational and electromagnetic interactions with
each other.
    Another suggestion
of the non-existence of Bohmian particles concerns the mass and charge
distributions of a one-particle system such as an electron. As we have shown
above, the guiding equation in the de Broglie-Bohm theory requires that the
Bohmian particle of a one-particle system has the mass and charge of the
system, and the mass and charge are localized in a position where the Bohmian
particle is. On the other hand, as noted before, protective measurement shows
that the mass and charge of a one-particle system such as an electron are not
localized in one position but distributed throughout space, and the mass and
charge density in each position is proportional to the modulus square of its
wave function there. Therefore, the de Broglie-Bohm theory is inconsistent with
the results of protective measurement concerning the mass and charge
distributions of a quantum system [52] . This poses a serious objection to the de
Broglie-Bohm theory.
    Now let's turn to
the wave function in the de Broglie-Bohm theory. Admittedly, the interpretation
of the wave function in the theory has been debated by its proponents. For
example, the wave function has been regarded as a field similar to
electromagnetic field (Bohm 1952), an active information field (Bohm and Hiley
1993), a field carrying energy and momentum (Holland 1993), a causal agent more
abstract than ordinary fields (Valentini 1997), a component of physical law (Durr,
Goldstein and Zangh`i 1997), and a dispositional property of Bohmian particles
(Belot 2011) etc. Notwithstanding the differences between these existing
interpretations, they are inconsistent with the meaning of the wave function as
implied by the results of protective measurement. To say the least, they fail
to explain the existence of the mass and charge density for a charged quantum
system, which is measurable by protective measurement and proportional to the
modulus square of the wave function of the system. Our previous analysis shows
that the mass and charge density of a quantum system is formed by the ergodic
motion of a localized particle with the total mass and charge of the system,
which is discontinuous and random in nature. Thus the wave function describes
the state of random discontinuous motion of particles, and at a deeper level,
it represents the property of the particles that determines their random
discontinuous motion. Since the principle of protective measurement is based on
the linear Schrödinger evolution of the wave function and the Born rule, which
also hold true in the de Broglie-Bohm theory, its implications, especially the
resulting interpretation of the wave function, are still valid in the theory.
    The realistic
interpretation of the wave function poses another serious threat against the
Bohmian-particles explanation of the guiding equation imposed by the de
Broglie-Bohm theory. The guiding equation is only a mathematical transformation
of the relation between the density ρ and the flux density j for the wave
function; the relation is j = ρv, while the guiding equation is v=j/ρ. Since
the wave function of a quantum system is not merely a probability amplitude for
the predictions of measurement results, but also a realistic description of the
physical state of the system as implied by protective measurement [53] , the guiding equation already has a
physical explanation relating only to the realistic wave function. Inasmuch as
a fundamental mathematical equation in a physical theory has a unique physical
explanation, the additional explanation of the guiding equation relating to the
hypothetical Bohmian particles will be improper [54] . In addition, the positions of the Bohmian
particles as added (hidden) variables seem redundant too [55] . In some sense, there are already
additional variables besides the wave function for the random