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

4.6 The Time Arrow in Various Interpretations of Quantum Theory 127
quantum theory in a deterministic way. However, this is not surprising, since
it leaves Schrödinger’s wave function entirely unchanged, while the assumed
trajectories for classical states, which would determine all observed quantities
according to this model, have to remain unobservable and in drastic
conflict with classical intuition (‘surrealistic’) in order to reproduce the empirically
confirmed quantum probabilities by means of their postulated statistical
distribution. Because of the ‘phenomenological’ wave–particle dualism
(see Sect. 4.3.2), it also remains controversial in this theory whether the classical
configurations must contain photon positions or electromagnetic fields
(Holland 1993).
Although wave functions and trajectories in configuration space are equally
assumed to be real in this theory, 8 they are treated quite differently. While
the former are usually regarded as ‘given’, the latter are always represented
by an ensemble (without thereby contributing to the entropy). Their initial
probability distribution in this ensemble, which has to be regarded as incomplete
information, is postulated to comply with the Born rule. Since the
Bohm trajectories themselves remain unobservable, they can be said to serve
as no more than artificial and empirically unfounded selectors for the ‘active’
branch of the global wave function, to which the actual trajectory would be
confined according to its dynamics. For example, entropy is calculated, in
the form S[ ˆP |ψ〉〈ψ|] with an appropriate Zwanzig projection ˆP ,fromsucha
component ψ – as though the wave function had been reduced by a real collapse
(see Sect. 5 of Dürr, Goldstein and Zanghi 1993). While this description
requires the same fact-like time asymmetry of the global wave function as decoherence,
the selection of subsets of trajectories defines an external arrow of
time. A justification of this different treatment of wave functions and Bohm
trajectories is not at all obvious (see Zeh 1999b).
Similarly to Bohm’s theory, collapse theories (Pearle 1976, Ghirardi, Rimini,
and Weber 1986) and the Everett interpretation (Everett, 1957) also
assume the wave function to represent a real physical object. This is in contrast
to genuine hidden variables theories, which intend to derive or explain
the wave function from some (hoped-for) more fundamental level of description.
These latter theories are affected by various no-go theorems (such as
Bell’s theorem) if they are assumed to be local. Otherwise, however, it is hard
to see what could be gained from them in comparison to the global wave
function itself as a nonlocal object.
While collapse theories propose stochastic modifications of the Schrödinger
equation, the Everett interpretation is based on the concept of ‘splitting ob-
8 “No one can understand this theory until he is willing to think of ψ as a real
objective field rather than just a ‘probability amplitude’ ” (Bell 1981). This
statement may apply to quantum theory in general – quite in contrast to an
analogous statement about Bohm’s trajectories, which are empirically unfounded
and thus have no more than ‘religious’ status. A stochastic theory can always be
deterministically completed by means of unobservable variables: any by definition
unobservable (pseudo-)random number generator will do.