Time Asymmetric Quantum Mechanics

TL;DR

This paper explores the concept of time asymmetry in quantum mechanics, proposing a mathematical framework based on rigged Hilbert spaces of Hardy functions that introduces a quantum mechanical beginning of time, contrasting with traditional time-symmetric theories.

Contribution

It introduces a novel formulation of quantum time asymmetry using Hardy function rigged Hilbert spaces, leading to a semigroup time evolution and a quantum beginning of time.

Findings

Time asymmetry can be modeled with Hardy space rigged Hilbert spaces.

A semigroup time evolution replaces the traditional unitary group.

Quantum systems exhibit a beginning of time similar to the universe's Big Bang.

Abstract

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 Schr\"odinger 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 $\Gamma$ and exponentially decaying states of lifetime $\tau=\frac{\hbar}{\Gamma}$ 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 $t_{0}\leq t<\infty$, 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.