Time Evolution of Quantum Effects

Abstract

For quantum effects $a$ and $b$ we define the $a$-evolution of $b$ at time $t$ denoted by $b(t\mid a)$. We interpret $b(t\mid a)$ as the influence that $a$ has on $b$ at time $t$ when $a$ occurs, but is not measured at time $t=0$. Using $b(t\mid a)$ we define the time-dependent sequential product $a[t]b$. This is interpreted as an effect that results from first measuring $a$ and then measuring $b$ after a time delay $t$. Various properties of $a[t]b$ are derived and it is shown that $a[t]b$ is constant in time if and only if $a$ and $b$ commute or $a$ is a multiple of a projection. These concepts are extended to observables for a quantum system. The ideas are illustrated with some examples.