The Quantum Mechanical Arrow of Time

Abstract

The dynamics of probability distributions on classical phase space, discussed under various aspects in Chap. 3, may be formally translated into quantum mechanics by means of the canonical quantization rules. Many authors of standard textbooks therefore maintain that the foundation of irreversibility in quantum mechanics is the same, in principle, as in classical physics. There could then only be quantitative differences arising from different spectral properties of the `corresponding’ Liouville operators. However, this approach to statistical quantum mechanics completely ignores the fundamentally different interpretation of concepts that formally correspond to one another (such as probability distributions and density operators — see Sect. 4.2). Therefore, it conceals important properties of quantum theory (compare the Introduction), which may be essential for irreversibility in general, viz.:

• 1. The quantum mechanical probability interpretation contains an indeterminism of controversial origin. Most physicists seem to regard it as representing an objective and law-like dynamical indeterminism (cf. Fig. 3.8), and some even as an extra-physical master arrow of time. Others have instead suggested that one may explain the unpredictability of quantum mechanical measurement results (or measurement-like events) in terms of conventional statistical arguments, that is, by means of thermodynamical fluctuations which occur during the required amplification process in the measurement device. If, however, this question is circumvented by interpreting the wave function as representing ‘human knowledge as an intermediate level of reality’ (Heisenberg 1956), Maxwell’s demon, discussed in Sect. 3.3.2, may return through the quantum back door. Therefore, the foundation of irreversibility seems to be intimately related to the interpretation of quantum theory (see Sect. 4.6).

• 2. The quantum theory is kinematically nonlocal. The generic many-particle wave function ψ(r ₁, r ₂,..., r _N ), which represents a ‘pure’ quantum state, describes quantum correlations between subsystems. They are not due to incomplete information (even though they may lead to such statistical correlations in measurements). Similarly, a state of quantum field theory is given by a wave functional of fields which are defined all over space. This ‘entanglement’ is a direct consequence of the superposition principle. In quantum theory, the state of the whole generically does not define states of its parts. This is in fundamental contrast to the completely determined many-particle state of classical mechanics: a point in phase space remains a point when projected onto a subsystem. This kinematical indeterminacy of the parts describes a non-trivial ‘wholeness’ of nature, which cannot, as in classical physics, be interpreted as a mere dynamical interconnectedness (that may lead to statistical correlations in ensembles). Moreover, it has nothing to do with Heisenberg’s uncertainty (or ‘indeterminacy’) relations, which signal the limited validity of classical concepts for describing physical states. The uncertainty relations apply even when individual quantum states are defined for all subsystems. Therefore, the Zwanzig projection of locality (3.37) has in general a nontrivial effect when applied to pure states; it defines non-vanishing local (`physical’) entropy even for a completely determined (‘real’) state of the whole.