Free Understanding Quantum Physics: An Advanced Guide for the Perplexed by Shan Gao Page B

state
is linear or nonlinear, they should have a fundamental status in any theory
formulated in space and time such as quantum mechanics. The second point is
that the above argument implicitly assumes that the nonlinear evolution H(ψ) is
universal, i.e., that it applies to all possible states. If the nonlinear
evolution only applies to some special states, then the evolution may still
conserve energy and momentum. For example, suppose the nonlinear evolution H(ψ)
= P 2 /2m + α|ψ| 2 applies only to the momentum eigenstates
e i(px−Et) and the linear evolution H(ψ) = P 2 /2m applies
to the superpositions of momentum eigenstates, then energy and momentum are
still conserved during the evolution. On the other hand, it has been argued
that the universal nonlinear quantum dynamics has a serious drawback, namely
that the description of composite systems depends on a particular basis in a
Hilbert space (Czachor 1996). If a nonlinear quantum evolution only applies to
certain privileged bases due to some reason, then such nonlinear quantum
dynamics may be logically consistent and also conserve energy and momentum (Gao
2004).
    The second
implication of the above derivation of the conservation laws is that spacetime
translation invariance implies the conservation of energy and momentum for
individual states, not for an ensemble of identical systems. As in the
derivation of the free Schrödinger equation, we only refer to an isolated
system and never refer to any ensemble of identical systems in the derivation
of the conservation laws. Moreover, the transformations of spacetime
translation also apply to a single isolated system. Therefore, what the
derivation tells us is that spacetime translation invariance implies the
conservation of energy and momentum for the linear evolution of the states of
an isolated system. The conservation of energy and momentum for a single system
means that the objective probability distributions of energy eigenvalues and
momentum eigenvalues are constant during the evolution of the state of the
system. As argued before, the objective probability can be well understood
according to the suggested interpretation of the wave function in terms of
random discontinuous motion. Similarly, our analysis of nonlinear evolutions
also shows that a universal nonlinear evolution violates the conservation of
energy and momentum for individual systems.
    This implication
raises a further issue. It is well known that the conservation of energy and
momentum in quantum mechanics refers to an ensemble of identical systems, not
to individual systems, and its precise statement is that the probability
distributions of the measurement results of energy and momentum for an ensemble
of identical isolated systems are the same at every instant during the
evolution of the systems in the ensemble. But as we have argued above, the
derivation of the conservation laws based on spacetime translation invariance
is for individual isolated systems, not for an ensemble of these systems. The
derivation never refers to the measurements of these systems either. Therefore,
there is still a gap (which maybe very large) between the derivation and the
conservation laws in quantum mechanics. Undoubtedly we must analyze the
measurement process in order to fill the gap. We will postpone the detailed
analysis of the measurement problem to the next section. Here we only want to
answer a more general question. If the conservation laws in quantum mechanics
are indeed valid as widely thought, then what are their implications for the
evolution of individual states?
    First of all, the
evolution of the state of an isolated system cannot contain a universal
deterministic nonlinear evolution, which applies to all possible states;
otherwise the evolution will violate the conservation of energy and momentum
not only at the individual level but also at the ensemble level. Next, the
evolution may contain linear evolutions as well as special deterministic
nonlinear