Hilbert Space or Gelfand Triplet - Time Symmetric or Time Asymmetric Quantum Mechanics

Abstract

Intrinsic microphysical irreversibility is the time asymmetry observed in exponentially decaying states. It is described by the semigroup generated by the Hamiltonian $\QTR{it}{H}$ of the quantum physical system, not by the semigroup generated by a Liouvillian $\QTR{it}{L}$ which describes the irreversibility due to the influence of an external reservoir or measurement apparatus. The semigroup time evolution generated by $\QTR{it}{H}$ is impossible in the Hilbert Space (HS) theory, which allows only time symmetric boundary conditions and an unitary group time evolution. This leads to problems with decay probabilities in the HS theory. To overcome these and other problems (non-existence of Dirac kets) caused by the Lebesgue integrals of the HS, one extends the HS to a Gel'fand triplet, which contains not only Dirac kets, but also generalized eigenvectors of the self-adjoint $\QTR{it}{H}$ with complex eigenvalues ($E_R-i\Gamma /2$) and a Breit-Wigner energy distribution. These Gamow states $\psi ^G$ have a time asymmetric exponential evolution. One can derive the decay probability of the Gamow state into the decay products described by $\Lambda $ from the basic formula of quantum mechanics $\QTR{cal}{P}(t)=Tr(|\psi ^G> < \psi ^G|\Lambda)$, which in HS quantum mechanics is identically zero. From this result one derives the decay rate $\QTR{group}{\dot c}(t)$ and all the standard relations between $\QTR{group}{\dot c}(0)$, $\Gamma $ and the lifetime $\tau_R$ used in the phenomenology of resonance scattering and decay. In the Born approximation one obtains Dirac's Golden Rule.