Understanding Quantum Physics: An Advanced Guide for the Perplexed

discontinuous motion of particles as
protective measurement suggests, then this analysis may be helpful for solving
the problem of the incompatibility of quantum mechanics with special relativity [102] .
    5.1 Distorted picture of motion
    Let’s first see
how the picture of random discontinuous motion is distorted by the Lorentz transformation
that leads to the relativity of simultaneity.
    5.1.1
Single particle picture
    For the random
discontinuous motion of a particle, the particle has a propensity to be in any
possible position at a given instant, and the probability density of the particle
appearing in each position x at a given instant t is determined by the modulus
square of its wave function, namely ρ(x, t) = |ψ(x, t)| 2 . The
physical picture of the motion of the particle is as follows. At a discrete
instant the particle randomly stays in a position, and at the next instant it
will still stay there or randomly appear in another position, which is probably
not in the neighborhood of the previous position. In this way, during a time
interval much larger than the duration of one instant, the particle will move discontinuously
throughout the whole space with position probability density ρ(x, t). Since the
distance between the locations occupied by the particle at two neighboring
instants may be very large, this jumping process is obviously nonlocal. In the
non-relativistic domain where time is absolute, the nonlocal jumping process is
the same in every inertial frame. But in the relativistic domain, the jumping
process will look different in different inertial frames due to the Lorentz
transformation. Let’s give a concrete analysis.
    Suppose a particle
is in position x 1 at instant t 1 and in position x 2 at instant t 2 in an inertial frame S. In another inertial frame S
with velocity v relative to S, the Lorentz transformation leads to:

    Since the jumping
process of the particle is nonlocal, the two events (t 1 , x 1 )
and (t 2 , x 2 ) may readily satisfy the spacelike separation
condition |x 2 − x 1 | > c|t 2 − t 1 |.
Then we can always select a possible velocity (v < c) that leads to t 2 = t 1 :

    But obviously the
two positions of the particle in frame S , namely x 1 and x 2 ,
are not equal. This means that in frame S the particle will be in two different
positions x 1 and x 2 at the same time at instant t 1 .
In other words, it seems that there are two identical particles at instant t 1 in frame S . Note that the velocity of S relative to S may be much smaller than
the speed of light, and thus the appearance of the two-particle picture is
irrelevant to the high-energy processes described by relativistic quantum field
theory, e.g. the creation and annihilation of particles.
    The above result
shows that for any pair of events in frame S that satisfies the spacelike
separation condition, there always exists an inertial frame in which the
two-particle picture will appear. Since the jumping process of the particle in
frame S is essentially random, it can be expected that the two-particle picture
will appear in the infinitely many inertial frames in an even way. Then during
an arbitrary finite time interval, in each inertial frame the measure of the
instants at which there are two particles in appearance, which is equal to the
finite time interval divided by the total number of the frames that is
infinite, will be zero. Moreover, there may also exist situations where the
particle is at arbitrarily many positions at the same time at an instant in an
inertial frame, though the measure of these situations is also zero. Certainly,
at nearly all instants which measure is one, the particle is still in one
position at an instant in all inertial frames. Therefore, the many-particle
appearance of the random discontinuous motion of a particle cannot be measured
in principle.
    However, for the
random discontinuous motion of a particle, in any inertial frame different from
S, the Lorentz transformation will inevitably make the time order of the random
stays of the particle in S reversal and disorder, as the discontinuous motion
of the particle is nonlocal and most neighboring random stays are spacelike
separated events. In other words, the time order is not Lorentz invariant.
Moreover, the set of the instants at which the time order of the random stays
of the particle is reversed has finite measure, which may be close to one. As
we will see below, this reversal and disorder of time order will lead to more
distorted