Understanding Quantum Physics: An Advanced Guide for the Perplexed

motion with constant velocity in the continuous limit.
    [26] In the next chapter, we will derive this equation of free motion from
fundamental physical principles. This will make the argument given here more
complete. Besides, the derivation itself may also provide an argument for
discontinuous motion that does not resort to direct experience, as the equation
of free motion does not permit the persisting existence of the local state of
continuous motion. For details see Section 3.4.
    [27] However, the analysis cannot tell us the precise size and possible structure
of an electron.
    [28] Recall that a function x(t) is continuous if and only if for every t and
every real number ε > 0, there exists a real number δ > 0 such that
whenever a point t 0 has distance less than δ to t, the point x(t 0 )
has distance less than ε to x(t).
    [29] However, there is an exception. When the probability density function is
a special δ-function such as δ(x − x(t)), where x(t) is a continuous function
of t, the motion of the particle is deterministic and continuous. In addition,
even for a general probability density function it is still possible that the
random position series forms a continuous trajectory, though the happening
probability is zero.
    [30] The existence of this limit relies on the continuity of the evolution of
the probabilistic instantaneous condition or propensity of a particle that
determines its random discontinuous motion.
    [31] Note that the relation between j(x, t) and ψ(x, t) depends on the
concrete evolution under an external potential such as electromagnetic vector
potential. By contrast, the relation ρ(x, t) = |ψ(x, t)| 2 holds true
universally, independent of the concrete evolution.
    [32] For a many-particle system in an entangled state, the propensity
property is possessed by the whole system. See Chapter 5 for a detailed
analysis of the physical picture of quantum entanglement.
    [33] Note that for random discontinuous motion the properties (e.g. position)
of a quantum system in a superposed state are indeterminate in the sense of
usual hidden variables, though they do have definite values at each instant.
This makes the theorems that restrict hidden variables such as the
Kochen-Specker theorem (Kochen and Specker 1967) irrelevant.
    [34] But if the spin state of a particle is entangled with its spatial state
and the branches of the entangled state are well separated in space, the
particle in different branches will have different spin, and it will also
undergo random discontinuous motion between these different spin states. This
is the situation that usually happens during a spin measurement.
    [35] This is an important presupposition in our derivation. We will consider
the possible case of nonlinearity of H in the next section.
    [36] Different from the derivation given below, most existing “derivations”
of the energy-momentum relation are based on the somewhat complex analysis of
an elastic collision process. Moreover, they resort to either some Newtonian
limit (e.g. p = mv) or some less fundamental relation (e.g. p = Eu/c 2 )
or even some mathematical intuition (e.g. four-vectors) (see Sonego and Pin
2005 and references therein).
    [37] Alternatively we can obtain the transformations of momentum and energy
by directly requiring the relativistic invariance of momentum eigenstate e i(px−Et) ,
which leads to the relation px − Et = p 0 x 0 − E 0 t 0 .
Note that any superposition of momentum eigenstates is also invariant under the
coordinates transformation. The reason is that it is a scalar that describes
the physical state of a quantum system, and when observed in different
reference frames it should be the same (except an absolute phase). This also
means that the state evolution equation must be relativistically invariant in
nature. However, if the relativistic invariant equation is replaced by the
nonrelativistic approximation such as the Schrödinger equation, the state will
no longer satisfy the relativistic invariance.
    [38] According to the analysis here, it seems that we can in principle avoid
talking about mass in modern physics from a more fundamental point of view (cf.
Okun 2009).
    [39] This also means that the Klein-Gordon equation can be derived in the
relativistic domain when assuming that the wave function is a number function.
    [40] In order to derive the complete Schrödinger equation in a fundamental
way, we need a fundamental theory of interactions such as quantum field