Hilbert Space or Gel'fand Triplet. Time-Symmetric or Time-Asymmetric Quantum Mechanics

Abstract

Intrinsic microphysical irreversibility is thetime asymmetry observed in exponentially decayingstates. It is described by the semigroup generated bythe Hamiltonian H of the quantum physical system, not by the semigroup generated by a Liouvillian Lwhich describes the irreversibility due to the influenceof an external reservoir or measurement apparatus. Thesemigroup time evolution generated by H is impossible in the Hilbert space (HS) theory, which allowsonly time-symmetric boundary conditions and a unitarygroup time evolution. This leads to problems with decayprobabilities in the HS theory. To overcome these and other problems (nonexistence of Dirac kets)caused by the Lebesgue integrals of the HS, one extendsthe HS to a Gel'fand triplet, which contains not onlyDirac kets, but also generalized eigenvectors of the self-adjoint H with complex eigenvalues(ER – iΓ/2) and a Breit-Wignerenergy distribution. These Gamow statesψG have a time-asymmetric exponentialevolution. One can derive the decay probability of the Gamow state into the decay productsdescribed by Λ from the basic formula of quantummechanics ℘(t) = Tr(|ψG ›‹ψG|Λ), which in HS quantum mechanicsis identically zero. From this result one derives the decay rate ℘(t) and all the standard relations between℘(0), Γ, and the lifetimeτR used in the phenomenology of resonancescattering and decay. In the Born approximation oneobtains Dirac's Golden Rule.