Quantum Mechanical Arrow Of Time Research Articles

Arrow of Time in Rigged Hilbert Space Quantum Mechanics

Arno Bohm and Ilya Prigogine's Brussels–Austin Group have been working on the quantum mechanical arrow of time and irreversibility in rigged Hilbert space quantum mechanics. A crucial notion in Bohm's approach is the so-called preparation/registration arrow. An analysis of this arrow and its role in Bohm's theory of scattering is given. Similarly, the Brussels–Austin Group uses an excitation/de-excitation arrow for ordering events, which is also analyzed. The relationship between the two approaches is initially discussed focusing on their semi-group operators and time arrows. Finally a possible realist interpretation of the rigged Hilbert space formulation of quantum mechanics is considered.

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 ( E R − 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 γ R ⪡ E R .

A quantum mechanical arrow of time and the semigroup time evolution of Gamow vectors

The exponential decay (or growth) of resonances provides an arrow of time which is described as the semigroup time evolution of Gamow vector in a new formulation of quantum mechanics. Another direction of time follows from the fact that a state must first be prepared before observables can be measured in it. Applied to scattering experiments, this produces another quantum mechanical arrow of time. The mathematical statements of these two arrows of time are shown to be equivalent. If the semigroup arrow is interpreted as microphysical irreversibility and if the arrow of time from the prepared in-state to its effect on the detector of a scattering experiment is interpreted as causality, then the equivalence of their mathematical statements implies that causality and irreversibility are interrelated.

Microphysical irreversibility and the time-reversal operation

The division of a scattering experiment into the preparation of a state and the registration of an observable in this state defines an obvious but commonly ignored quantum-mechanical arrow of time. This leads to irreversibility on the microphysical level. The time evolution semigroup for exponentially decaying Gamow states is an example of this microphysical irreversibility. A time-reversal operator exists which is compatible with the quantum-mechanical arrow of time.

The preparation-registration arrow of time in quantum mechanics

A part from the “collapse of the wave function” axiom, there is a quantum mechanical arrow of time expressing the fact that a state needs to be first prepared before observables can be measured in it. We show that this preparation-registration arrow of time is identical to the arrow for exponential decay (or growth) of resonance by Gamow vectors.