The Physical Basis of The Direction of Time (The Frontiers ...

100 4 The Quantum Mechanical Arrow of Time
different measurement results (that is, of states φ n Φ n with probabilities |c n | 2 ),
would require the fork of causality to be replaced by a fork of indeterminism.
(The formal ‘plus’ characterizing the superposition would have to become an
‘or’.) This discrepancy represents the quantum measurement problem, that
would be obscured in a phenomenological description by means of reduced
density matrices for the subsystems only. The density matrices resulting from
these two types of forks are identical, since there is no way of distinguishing
these different situations operationally by a local measurement. As emphasized
before, this argument does not explain the fork of indeterminism that lies at
the heart of the probability interpretation.
This measurement problem prevails regardless of the complexity of the
measurement device (that might give rise to thermodynamically irreversible
behavior), and regardless of any perturbations caused by, and in the environment,
since the states Φ may be assumed to describe this complexity completely,
and even to include the whole ‘rest of the Universe’. The popular
argument that quantum mechanical indeterminism might, in analogy to the
classical situation, be caused by thermal fluctuations that occur during a measurement
process (see Sect. 3.3 or Peierls 1985, for example) is incompatible
with universal unitarity. It would instead require the existence of an initial ensemble
of microscopic states which in principle had to determine the outcome.
However, the ensemble entropy of the RHS of (4.32) does not represent an
ensemble that would allow the measurement to be interpreted as in Fig. 3.5.
If both systems in (4.32) are microscopic, the dynamics representing the
fork of causality can even be reversed in practice (the measurement could be
‘undone’) in order to demonstrate that all components still exist. This reversal
leads to observable consequences that may depend on all existing components,
including their relative phases. This excludes the interpretation of (4.32) as
a dynamical fork of indeterminism (a fork between mere possibilities), even
though von Neumann’s fork of causality is defined in terms of wave packets
on a classical configuration space. Therefore, the transition from quantum to
classical (Sect. 4.3) can be understood only if it explains why the fundamental
arena for wave functions often appears as a space of classical configurations.
The interaction (4.32) is an example for the generic case of quantum mechanical
subsystems which are not individually obeying unitary dynamics.
Similar arguments would apply to the ensemble dynamics of systems with
classical correlations (that is, if ρ = ˆP classical ρ). In this case, the effective
subsystem Hamiltonian H φ , say, would depend on the state Φ k of the complementary
system by means of a partial expectation value,
H (k)
φ (t) :=〈Φ k(t)|H|Φ k (t)〉 Op , (4.33)
where H was defined to act on the tensor product of φ and Φ. Therecanbeno
Hamiltonian H φ valid for the whole ensemble any more. This is equivalent to
the induced Hamiltonians of interacting classical mechanical systems, which
are given by