Quantum mechanical irrebersibility

Microphysical irreversibility is distinguished from the extrinsic irreversibility of open systems. The rigged Hilbert space (RHS) formulation of quantum mechanics is justified based on the foundations of quantum 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 scattering experiment. Within the cintext of a resonance scattering experiment, it is shown how the dichotomy of state and observable leads to a pair of RHSs, one for states and one for observables. Using resonance scattering, it is shown how the Gamow vectors describing decaying states with complex energy eigenvalues (ER − iγ/2) emerge from the first-order resonance poles of the S-matrix. Then, these considerations are extended to S-matrix poles order N and it shown that this leads to Gamow vectors of higher order k = 0, 1, …, N − 1 which are also Jordan vectors of degree k + 1 = 1, 2,…,N. The matrix elements of the self-adjoint Hamiltonian between these vectors from a Jordan block of degree N. The two semigroups of time evolution generated by the Hamiltonian are obtained for Gamow vectors of any order. It is shown how the irreversible time evolution of Gamow vectors enables the derivation 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 γR ⪡ ER.