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

In quantum physics, disturbance due to a measurement is not negligible. This requires the time parameter $t$ in the Schrödinger 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 Schrödinger (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.