Understanding Quantum Physics: An Advanced Guide for the Perplexed

measurements can be finally expressed in terms of position,
e.g. pointer positions, this amounts to full accordance with all predictions of
quantum mechanics [48] . In this way, it seems that the de Broglie-Bohm
theory can succeed in avoiding the collapse of the wave function.
    However, although
the de Broglie-Bohm theory is mathematically equivalent to quantum mechanics,
there is no clear consensus with regard to its physical interpretation. The
physical contents of the theory contain three parts: the Bohmian particles, the
wave function, and the interaction between them. We first analyze the Bohmian
particles and their physical properties. It is fair to say that what physical
properties a Bohmian particle has is still an unsettled issue, and different
proponents of the theory may have different opinions. For example, it has been
often claimed that a Bohmian particle has mass, as the guiding equation for
each Bohmian particle of a many-body system obviously contains the mass of each
sub-system (Goldstein 2009). Yet it seems unclear whether the mass is inertial
mass or (passive or active) gravitational mass or both or neither. On the other
hand, it has been argued that the mass of a quantum system should be possessed
by its wave function, not by its Bohmian particles (Brown, Dewdney and Horton
1995). It was even claimed (without proof) that a Bohmian particle has no
properties other than its position (Hanson and Thoma 2011). In the last
analysis, in order to know exactly what physical properties a Bohmian particle
has, we need to analyze the guiding equation that defines the laws of motion
for them.
    In the minimum
formulation of the theory, which is usually called Bohmian mechanics (Goldstein
2009) [49] , the guiding equation contains an electromagnetic interaction
term eA(x,t) for the Bohmian particle of a one-particle system with mass m and
charge e in the presence of an external electromagnetic field [50] . According to this equation, the motion of
a Bohmian particle is not only guided by the wave function, but also influenced
by the external vector potential. In particular, the existence of the electromagnetic
interaction term indicates that the Bohmian particle has the charge of the
system, and the charge is localized in its position [51] . Similarly, the appearance of the mass of
the system in the equation indicates that the Bohmian particle also has the
(inertial) mass of the system. Therefore, according to Bohmian mechanics, the
Bohmian particle of a one-particle system such as an electron has the mass and
charge of the system. For example, in the ground state of a hydrogen atom, the
Bohmian particle of the electron in the atom has the mass and charge of the
electron, and it is at rest in a random position relative to the nucleus. That
the Bohmian particle of a one-particle system has the mass and charge of the
system can be seen more clearly from the quantum potential formulation of the
de Broglie-Bohm theory. Its guiding equation contains both an electromagnetic
interaction term and a gravitational interaction term in the presence of
external electromagnetic field and gravitational field, which indicates that
the Bohmian particle has the charge and (passive gravitational) mass of the
system.
    It can be seen
that although a Bohmian particle has mass and charge, the functions of these
properties are not as complete as usual. For example, in Bohmian mechanics, a
charged Bohmian particle responds not to the electric scalar potential, but
only to the magnetic vector potential, and it has no gravitational mass but
only inertial mass. This apparent abnormality is in want of a reasonable
physical explanation. In addition, in the quantum potential formulation,
although the Bohmian particles of a quantum system respond to external
gravitational and electromagnetic potentials, they don't have gravitational and
electromagnetic influences on other charged quantum systems, including their
Bohmian particles. Moreover, the Bohmian particles of a quantum system do not
have gravitational and electromagnetic interactions with each other. Therefore,
the (gravitational) mass and charge of a Bohmian particle are always passive,
i.e., a Bohmian particle is only a receptor of gravitational and
electromagnetic interactions. This characteristic may lead to some problems.
For one, the nonreciprocal interactions will violate the conservation of energy
and momentum (except that the Bohmian particles have no momentum and energy).
At