Causality in quantum physics, the ensemble of beginnings of time, and the dispersion relations of wave function

TL;DR

This paper explores the distinction between physical and experimental time in quantum mechanics, proposing a time asymmetric framework where quantum state evolution is described by semigroup solutions constrained by dispersion relations.

Contribution

It introduces a new perspective on quantum time evolution, emphasizing the ensemble of beginnings of time and the necessity of Hardy functions satisfying dispersion relations.

Findings

Quantum time parameter differs from experimental clock time.

Standard Hilbert space evolution extends over all real times, conflicting with physical constraints.

Semigroup evolution with Hardy functions aligns with the ensemble of beginnings of time.

Abstract

In quantum physics, disturbance due to a measurement is not negligible. This requires the time parameter $t$ in the Schr\"odinger or Heisenberg equation to be considered differently from a time continuum of experimenter's clock $T$ on which physical events are recorded. It will be shown that $t$ represents an ensemble of time intervals on $T$ during which a microsystem travels undisturbed. In particular $t=0$ represents the ensemble of preparation events that we refer to as the ensemble of beginnings of time. This restricts $t$ to be $0\leq t<\infty$. But such a time evolution of quantum states cannot be achieved in the Hilbert space $L^2$ functions because due to the Stone-von Neumann theorem this time evolution is given by the unitary group with $t$ extending $-\infty<t<\infty$. Hence one needs solutions of the Schr\"odinger (and Heisenberg) equation under time asymmetric boundary condition in which only the semigroup time evolution is allowed. This boundary condition is fulfilled by the energy wave functions for quantum states (and as well as for observables) which are smooth Hardy function satisfying the Hilbert transform, called the dispersion relation in physics.