QUANTUM-MECHANICAL IRREVERSIBILITY

Microphysical irreversibility is distinguished from the extrinsic irre versibility of open systems. The rigged Hilbert space (RHS) formulatio n of quantum mechanics is justified based on the foundations of quantu m mechanics. Unlike the Hilbert space formulation of quantum mechanics , the rigged Hilbert space formulation of quantum mechanics allows for the description of decay and other irreversible processes because it allows for a preferred direction of time for time evolution generated by a semi-bounded, essentially self-adjoint Hamiltonian. This quantum mechanical arrow of time is obtained and applied to a resonance scatte ring experiment. Within the context of a resonance scattering experime nt, it is shown how the dichotomy of state and observable leads to a p air of RHSs, one for states and one for observables. Using resonance s cattering, it is shown how the Gamow vectors describing decaying state s with complex energy eigenvalues (E(R) - i Gamma/2) emerge from the f irst-order resonance poles of the S-matrix. Then, these considerations are extended to S-matrix poles of order N and it is shown that this l eads to Gamow vectors of higher order k = 0, 1,..., N - 1 which are al so Jordan vectors of degree k + 1 = 1,2,...,N. The matrix elements of the self-adjoint Hamiltonian between these vectors form a Jordan block of degree N. The two semigroups of time evolution generated by the Ha miltonian are obtained for Gamow vectors of any order. It is shown how the irreversible time evolution of Gamow vectors enables the derivati on of an exact Golden Rule for the calculation of decay probabilities, from which the standard (approximate) Golden Rule is obtained as the Born approximation in the limit Gamma(R) much less than E(R).