A quantum theory that applies to the (closed) universe needs to be time asymmetric because of the cosmological arrow of time. The preparation $\ensuremath{\Rightarrow}$ registration arrow of time (a state must be prepared before an observable can be detected in it) of the quantum mechanics of measured systems provides a phenomenological reason for an asymmetric semigroup time evolution. The standard theory in the Hilbert space (HS) is inadequate for either since the mathematics of the HS allows only reversible unitary group evolution and time symmetric boundary conditions. The mathematical theory that describes time-asymmetric quantum physics in addition to providing the mathematics for the Dirac kets is the rigged Hilbert space (RHS) theory. It uses a pair of RHS's of Hardy class with complementary analyticity property, one for the prepared states (``in states'') and the other for the registered observables (``out states''). The RHS's contain Gamow kets which have all the properties needed to represent decaying states and resonances. Gamow kets have only asymmetric time evolution. The neutral kaon system is used to show that quasistationary microphysical systems can be experimentally isolated if their time of preparation can be accurately identified. The theoretical predictions for a Gamow ket have the same features as the observed decay probabilities, including the time ordering. This time ordering is the same as the time ordering in the probabilities of histories for the quantum universe. The fundamental quantum mechanical arrow of time represented by the semigroup in the RHS is therefore the same as the cosmological arrow of time, assuming that the universe can be considered a closed quantum system.
The meaning of time asymmetry in quantum physics is discussed.On the basis of a mathematical theorem, the Stone-von Neumann theorem, the solutions of the dynamical equations, the Schrödinger equation (1) for states or the Heisenberg equation (6a) for observables are given by a unitary group.Dirac kets require the concept of a RHS (rigged Hilbert space) of Schwartz functions; for this kind of RHS a mathematical theorem also leads to time symmetric group evolution.Scattering theory suggests to distinguish mathematically between states (defined by a preparation apparatus) and observables (defined by a registration apparatus (detector)).If one requires that scattering resonances of width Γ and exponentially decaying states of lifetime τ = Γ should be the same physical entities (for which there is sufficient evidence) one is led to a pair of RHS's of Hardy functions and connected with it, to a semigroup time evolution t 0 ≤ t < ∞, with the puzzling result that there is a quantum mechanical beginning of time, just like the big bang time for the universe, when it was a quantum system.The decay of quasi-stable particles is used to illustrate this quantum mechanical time asymmetry.From the analysis of these processes, we show that the properties of rigged Hilbert spaces of Hardy functions are suitable for a formulation of time asymmetry in 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.
Experiments on quantum systems are usually divided into preparation of states and the registration of observables. Using the traditional mathematical methods (the Hilbert space and Schwartz space of distribution theory), it is not possible to distinguish mathematically between observables and states. The Hilbert space as well as Schwartz space boundary conditions for the dynamical equations lead by mathematic theorems (Stone-von Neumann) to unitary group with — ∞ < t < ∞ . But in the experimental set-up, one clearly distinguishes between the preparation of a state and the registration of an observable in that state. Furthermore, a state must be prepared first before an observable can be measured in this state (causality). This suggests time asymmetric boundary conditions for the dynamical equations of quantum theory. Such boundary conditions have been provided by Hardy space in the Lax-Phillips theory for electromagnetic and acoustic scattering phenomena. The Paley-Wiener theorem for Hardy space then leads to semi-group and time asymmetry in quantum physics. It introduces a finite "beginning of time" t 0 for a time asymmetric quantum theory, which have been observed as an ensemble of finite times t ( i ) 0 , the onset times of dark periods in the quantum jump experiments on a single ion.
The Rigged Hilbert Space (RHS) theory of resonance scattering and decay is reviewed and contrasted with the standard Hilbert space (HS) theory of quantum mechanics. The main difference is in the choice of boundary conditions. Whereas the conventional theory allows for the in-states
ϕ+ and the out-states (observables)
ψ− of the S-matrix elements
(ψ−,ϕ+)=(ψout,Sϕin) any elements of the HS $\H$,
{ψ−}={ϕ+}(=)˝, the RHS theory chooses the boundary conditions~:
ϕ+∈Φ−⊂⊂˝Φ−×,
ψ−∈Φ+⊂⊂˝Φ+×, where
Φ− (
Φ+) are Hardy class spaces associated to the lower (upper) half-plane of the second sheet of the analytically continued S-matrix. This can be phenomenologically justified by causality. The two RHS's for states
ϕ+ and observables
ψ− provide new vectors which are not in $\H$, e.g. the Dirac-Lippmann-Schwinger kets
∣E±∈Φ∓× (solutions of the Lippmann-Schwinger equation with
±iε respectively) and the Gamow vectors
∣ER−iΓ/2±∈Φ∓×. The Gamow vectors
∣ER−iΓ/2− have all the properties that one heuristically needs for quasistable states. In addition, they give rise to asymmetric time evolution expressing irreversibility on the microphysical level.
Quantum mechanics in the Rigged Hilbert Space formulation describes quasistationary phenomena mathematically rigorously in terms of Gamow vectors. We show that these vectors exhibit microphysical irreversibility, related to an intrinsic quantum mechanical arrow of time, which states that preparation of a state has to precede the registration of an observable in this state. Moreover, the Rigged Hilbert Space formalism allows the derivation of an exact golden rule describing the transition of a pure Gamow state into a mixture of interaction-free decay products.