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On using physical properties to make mathematical choices in quantum mechanical scattering

Joshua Russell Hebert

Published Jan 1, 2015• 0 citations

Abstract

Traditional Quantum Scattering theory is built upon a Hilbert space formalism. The nature of the Hilbert space has led to a number of ideas about the nature of scattering systems, chief among them being the understanding that the decay of unstable quan- tum systems cannot be purely exponential and that there can only be an approximate relationship between the width Γ of a scattering resonance and the lifetime τ of the resonant state whose formation gives rise to it. That such physical conclusions are de- rived from purely mathematical properties of the underlying topological vector space inform the notion that, in choosing a particular space, one is choosing the mathemat- ical representation of the physical universe for the system. With this notion in mind, purely physical characteristics of quantum systems are used to answer the question: “What is the proper mathematical space for quantum mechanical calculations?” In the particular case of quantum scattering, it is shown that this approach casts the choice of the Hilbert space into doubt and motivates instead the use of rigged Hilbert spaces employing Hardy spaces; in these Hardy rigged Hilbert spaces, exponential decay is permissible and the time evolution of decaying states is governed by semigroups. The original theories concerning deviation from exponential decay in the Hilbert space are revisited to reveal how their conclusions flow from a mathematical property whose physical interpretation is problematic. Finally, an application of a Hardy rigged Hilbert space formalism is presented: Time Asymmetric Quantum Mechanics, as developed by A. Bohm, M. Gadella, and S. Wickramasekara.

DOI: 10.15781/t2930w

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Traditional Quantum Scattering theory is built upon a Hilbert space formalism. The nature of the Hilbert space has led to a number of ideas about the nature of scattering systems, chief among them being the understanding that the decay of unstable quan- tum systems cannot be purely exponential and that there can only be an approximate relationship between the width Γ of a scattering resonance and the lifetime τ of the resonant state whose formation gives rise to it. That such physical conclusions are de- rived from purely mathematical properties of the underlying topological vector space inform the notion that, in choosing a particular space, one is choosing the mathemat- ical representation of the physical universe for the system. With this notion in mind, purely physical characteristics of quantum systems are used to answer the question: “What is the proper mathematical space for quantum mechanical calculations?” In the particular case of quantum scattering, it is shown that this approach casts the choice of the Hilbert space into doubt and motivates instead the use of rigged Hilbert spaces employing Hardy spaces; in these Hardy rigged Hilbert spaces, exponential decay is permissible and the time evolution of decaying states is governed by semigroups. The original theories concerning deviation from exponential decay in the Hilbert space are revisited to reveal how their conclusions flow from a mathematical property whose physical interpretation is problematic. Finally, an application of a Hardy rigged Hilbert space formalism is presented: Time Asymmetric Quantum Mechanics, as developed by A. Bohm, M. Gadella, and S. Wickramasekara.

The states (Schrödinger picture) and observables (Heisenberg picture) in the standard quantum theory evolve symmetrically in time, given by the unitary group with time extending over − ∞ < t < + ∞ . This time evolution is a mathematical consequence of the Hilbert space boundary condition for the dynamical differential equations. However, this unitary group evolution violates causality. Moreover, it does not solve an old puzzle of Wigner: How does one describe excited states of atoms which decay exponentially, and how is their lifetime τ related to the Lorentzian width Γ? These question can be answered if one replaces the Hilbert space boundary condition by new, Hardy space boundary conditions. These Hardy space boundary conditions allow for a distinction between states (prepared by a preparation apparatus) and observables (detected by a registration apparatus). The new Hardy space quantum theory is time asymmetric, i.e, the time evolution is given by the semigroup with t 0 ≤ t < + ∞ , which predicts a finite "beginning of time" t 0 , where t 0 is the ensemble of time at which each individual system has been prepared. The Hardy space axiom also leads to the new prediction: the width Γ and the lifetime τ are exactly related by τ = ℏ/Γ .

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^+)=(ψ^{out},S ϕ^{in})

any elements of the HS $\H$,

\{ψ^-\}=\{ϕ^+\}(=\H)

, the RHS theory chooses the boundary conditions~:

ϕ^+\inΦ_-\subset\H\subsetΦ_-^\times

,

ψ^-\inΦ_+\subset \H\subset Φ_+^\times

, 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^{\pm}\inΦ_{\mp}^{\times}

(solutions of the Lippmann-Schwinger equation with

\pm iε

respectively) and the Gamow vectors

|E_R-iΓ/2^\pm\inΦ_{\mp}^\times

. The Gamow vectors

|E_R-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.

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.

Many useful concepts for a quantum theory of scattering and decay (like Lippmann-Schwinger kets, purely outgoing boundary conditions, exponentially decaying Gamow vectors, causality) are not well defined in the mathematical frame set by the conventional (Hilbert space) axioms of quantum mechanics. Using the Lippmann-Schwinger equations as the takeoff point and aiming for a theory that unites resonances and decay, we conjecture a new axiom for quantum mechanics that distinguishes mathematically between prepared states and detected observables. Suggested by the two signs

\\pm i\\epsilon

of the Lippmann-Schwinger equations, this axiom replaces the one Hilbert space of conventional quantum mechanics by two Hardy spaces. The new Hardy space theory automatically provides Gamow kets with exponential time evolution derived from the complex poles of the

S

-matrix. It solves the causality problem since it results in a semigroup evolution. But this semigroup brings into quantum physics a new concept of the semigroup time

t=0

, a beginning of time. Its interpretation and observations are discussed in the last section.

For a quantum theory that includes exponentially decaying states and Breit–Wigner resonances, which are related to each other by the lifetime–width relation , where τ is the lifetime of the decaying state and Γ is the width of the resonance, one has to go beyond the Hilbert space and beyond the Schwartz–Rigged Hilbert Space of the Dirac formalism. One has to distinguish between prepared states, using a space , and detected observables, using a space , where −(+) refers to analyticity of the energy wavefunction in the lower (upper) complex energy semiplane. This differentiation is also justified by causality: a state needs to be prepared first, before an observable can be measured in it. The axiom that will lead to the lifetime–width relation is that Φ + and Φ − are Hardy spaces of the upper and lower semiplane, respectively. Applying this axiom to the relativistic case for the variable leads to semigroup transformations into the forward light cone (Einstein causality) and a precise definition of resonance mass and width.