Time's Arrow and Irreversibility in Time‐Asymmetric Quantum Mechanics

Abstract

The aim of this paper is to analyze time‐asymmetric quantum mechanics with respect to the problems of irreversibility and of time’s arrow. We begin with arguing that both problems are conceptually different. Then, we show that, contrary to a common opinion, the theory’s ability to describe irreversible quantum processes is not a consequence of the semigroup evolution laws expressing the non‐time‐reversal invariance of the theory. Finally, we argue that time‐asymmetric quantum mechanics, either in Prigogine’s version or in Bohm’s version, does not solve the problem of the arrow of time because it does not supply a substantial and theoretically founded criterion for distinguishing between the two directions of time.

Acknowledgements

We are very grateful to Arno Bohm for his kind hospitality at the University of Texas at Austin, where this paper was conceived. We are also very grateful to the anonymous referees whose comments greatly improved this final version. This paper was partially supported by grants of the Buenos Aires University, the National Research Council (CONICET), the National Research Agency (FONCYT) of Argentina, the Junta de Castilla y León Project VA013C05, and the FEDER‐Spanish Ministry of Science and Technology Projects DGI BMF 2002–0200 and DGI BMF2002–3773.

Notes

[1] From here, we will not distinguish between mathematical entities (equations and solutions) and physical entities (laws and evolutions).

[2] The fact that we recognize the relevance of Price’s claims about the need of adopting the ‘nowhen’ standpoint does not mean that we agree with his proposal for the solution of the problem of time’s arrow (for a detailed discussion, cf. Castagnino, Lombardi and Lara 2003; Castagnino and Lombardi 2005a, 2005b).

[3] Let us note that we are saying that these arguments are not legitimate in the discussions about the arrow of time: we are not talking about the problem of irreversibility, where such arguments may be acceptable when de facto irreversibility is the question under discussion.

[4] The space Φ has its own topology, which is stronger than the topology that Φ possesses as a subspace of ℋ. The topology in Φ is not given by a norm but usually by a countable infinite family of norms.

[5] The assumptions are: (i) the domain D(A†) of A† includes the space Φ, (ii) for each φ ϵ Φ, A† φ ϵ Φ, and (iii) the A† is continuous on Φ in the own topology of Φ (cf. Schäffer 1970). The duality formula also applies when A is self‐adjoint; in this case, A† = A, and the duality formula becomes:

[6] The possibility of complex generalized eigenvalues of self‐adjoint operators was suggested by Gamow (1928) in the context of the decay of quantum systems. As poles of the resolvent, Gamow vectors were first introduced by Grossmann (1964), independently of RHSs. Later, they were unexpectedly obtained in the RHS formalism as generalized eigenvectors of self‐adjoint operators with complex eigenvalues (Lindblad and Nagel 1970). The association between the poles of the S‐matrix with the vectors in the RHS was established in the 1980s (Bohm 1981, Gadella 1983, 1984): these works showed that RHSs supply the formal representation to the decaying states and resonances heuristically constructed by Gamow.

[7] Second‐and higher‐order poles of the scattering operator S are treated in Bohm et al. (1997) and in Antoniou, Gadella and Pronko (1998).

[8] Therefore, any f(z) analytic in the upper (lower) half‐plane fulfilling condition (iii) uniquely determines its boundary values on the real line, given by a Hardy function on the upper (lower) half‐plane. After the Titchmarsh theorem (Titchmarsh 1937), the reciprocal is also true.

[9] Recall that we want to obtain the evolution operator for the vectors belonging to and, therefore we have to begin with the adjoint of (cf. the duality formula (4))

[10] Bohm recognizes that the origin of the idea of a preparation‐registration arrow can be traced back to the works of Günther Ludwig (1983–1985).