Understanding Quantum Physics: An Advanced Guide for the Perplexed

a given instant, is described by the position
density and position flux density, and they are continuous quantities that
contain no discontinuity and randomness.
    Therefore, if time
is continuous and instants are durationless, then the random stays of a
particle can have no stochastic effects. This also means that the random stays
of a particle can influence the evolution of its wave function in a stochastic
way only when instants are not zero-sized but finite-sized, i.e., when time is
discrete or quantized. Once the duration of each random stay of a particle is
finite, each random stay can have a finite stochastic influence on the
evolution of the wave function. It is worth stressing again that if time is not
discrete but continuous, a particle cannot stay in one of the infinitely many
instantaneous states all through for a finite time; rather, it can only stay
there for one zero-sized instant. But if time is discrete and instants are not
zero-sized but finite-sized, even if a particle stays in an instantaneous state
only for one instant, the duration of its stay is also finite as the instant is
finite-sized. In some sense, the discreteness of time prevents a particle from
jumping from its present instantaneous state to another instantaneous state and
makes the particle stay in the present instantaneous state all through during
each finite-sized instant [67] . Since it has been conjectured that the Planck scale
is the minimum spacetime scale [68] , we will assume that the size of each discrete
instant or the quantum of time is the Planck time in our following analysis [69] .
    To sum up, the
realization of the randomness and discontinuity of motion in the laws of motion
requires that time is discrete. In discrete time, a particle randomly stays in
an instantaneous state with definite position, momentum and energy at each
discrete instant, with a probability determined by the modulus square of its
wave function at the instant. Each random, finite stay of the particle may have
a finite influence on the evolution of its wave function. As we will show in
the next section, the accumulation of such discrete and random influences may
lead to the correct collapse of the wave function, which can then explain the
emergence of definite measurement results. Accordingly, the evolution of the
wave function will be governed by a revised Schrödinger equation, which
includes the normal linear terms and a stochastic nonlinear term that describes
the discrete collapse dynamics. Note that the wave function (as an
instantaneous property of particles) also exists in the discrete time, which
means that the wave function does not change during each discrete instant, and
the evolution of the wave function including the linear Schrödinger evolution
is also discrete.
    4.2.2
Energy conservation and the choices
    Now let’s
investigate the choice problem, namely the problem of determining the states
toward which the collapse tends. The random stay of a particle may have a
stochastic influence on the evolution of its wave function at each discrete
instant. Then when the stochastic influences accumulate and result in the
collapse of the wave function, what are the states toward which collapse tends?
This is the choice problem or preferred basis problem. It may be expected that
the stochastic influences of the motion of a particle on its wave function
should not be arbitrary but be restricted by some fundamental principles. In
particular, it seems reasonable to assume that the resulting dynamical collapse
of the wave function should also satisfy the conservation of energy. As a
result, the collapse states or choices will be the energy eigenstates of the
total Hamiltonian of the system [70] . In the following, we will give a more
detailed analysis of the consequences of this assumption. Its possible physical
basis will be investigated in the next subsection.
    As we have argued
in the last section, for a deterministic evolution of the wave function such as
the linear Schrödinger evolution, the requirement of energy conservation
applies to a single isolated system. However, for a stochastic evolution of the
wave function such as the dynamical collapse process, the requirement of energy
conservation cannot apply to a single system in general but only to an ensemble
of identical systems [71] . It can be proved that only when the collapse states
are energy eigenstates of the total Hamiltonian of a given system, can energy
be conserved for an ensemble