Understanding Quantum Physics: An Advanced Guide for the Perplexed

theory.
    [41] In order to ensure that the nonlinear evolution is unitary and thus the
total probability is conserved in time, the Hamiltonian H(ψ) must be also
Hermitian. Besides, this property is also required to ensure that the energy
eigenvalues (which satisfy the equation H(ψ)ψ(x) = Eψ(x)) are real. When the
Hamiltonian H(ψ) is Hermitian, the Ehrenfest theorem still holds true.
    [42] This will violate the relativistic invariance of momentum eigenstates.
    [43] For more discussions about the arguments for linear quantum dynamics see
Holman (2006) and references therein.
    [44] For example, the collapse to a position eigenstate during an ideal
position measurement is obviously unphysical, as the position eigenstate has
infinite average energy.
    [45] As we have shown in Chapter 2, there are at least three levels of
implications. First, protective measurement can measure the mass and charge
density of a quantum system, which is proportional to the modulus square of the
wave function of the system. This indicates that the mass and charge of a
quantum system are attributes of its wave function. Next, when assuming that
real mass and charge distributions have gravitational and electrostatic
interactions, which has been confirmed not only in the classical domain but
also in the quantum domain for many-body systems, it can be shown that the mass
and charge density of a quantum system is formed by the time average of the
ergodic motion of a localized particle with the total mass and charge of the
system. This indicates that the wave function is a description of the ergodic
motion of particles. Lastly, it can be further argued that the ergodic motion
is not continuous but discontinuous and random. This leads to our suggested
interpretation of the wave function, according to which the wave function in
quantum mechanics is a description of random discontinuous motion of particles.
Most of our critical analysis of the existing solutions to the measurement
problem only depends on the first two implications.
    [46] In other words, the principle of protective measurement and its
implications hold true in any formulation of quantum mechanics that keeps the
linear Schrödinger evolution of the wave function (for microscopic systems) and
the Born rule, such as the de Broglie-Bohm theory and the many-worlds
interpretation. Thus it is legitimate to use them to examine these alternatives
to quantum mechanics. Note that the possible existence of very slow collapse of
the wave function for microscopic systems does not influence the principle of
protective measurement and its implications.
    [47] It has been argued that the wave function living on configuration space
can hardly be considered as a real physical entity due to its
multi-dimensionality (see, e.g. Monton 2002, 2006 and references therein).
However, it seems that this common objection is not conclusive, and one can
still insist on the reality of the wave function living on configuration space
by resorting to some metaphysical arguments. For example, a general strategy is
to show how a many-dimensional world can appear three-dimensional to its
inhabitants, and then argue on that basis that a wavefunction ontology is
adequate to explain our experience (Albert 1996; Lewis 2004). As we argued
earlier, the existence of the effective mass and charge density of a quantum
system, which is measurable by protective measurement, poses a more serious
objection to the wavefunction ontology; even for a single quantum system the
wave function cannot be taken as a field-like entity in three-dimensional space
either. Moreover, the reason is not metaphysical but physical, i.e., the
field-like interpretation contradicts both quantum mechanics and experimental
observations.
    [48] Certainly, as Albert (1992) noted, no theory can have exactly the same
empirical content as quantum mechanics does, as the latter (in the absence of
any satisfactory account of wavefunction collapse) does not have any exact
empirical content.
    [49] For a critical analysis of this minimal formal interpretation see
Belousek (2003).
    [50] Note that for spin 1/2 particles there is also a spin-dependent term
(Holland and Philippidis 2003).
    [51] That a Bohmian particle has no properties other than its position is
possible only when the mass and charge terms disappear in the guiding equation,
but the resulting theory will contradict quantum mechanics and experiments.
    [52] This conclusion relies on the common-sense assumption