Evolution equations from an epistemic treatment of time

Abstract

Relativistically, time $t$ is an observable just like position $r$. In quantum theory, $t$ is a parameter, in contrast to the observable $r$. This discrepancy suggests that there exists a more elaborate formalization of time, which encapsulates both perspectives. Such a formalization is proposed in this paper. The evolution is described in terms of sequential time $n\in \mathbf{\mathbb{N}}$, which is updated each time an event occurs. Sequential time $n$ is separated from relational time $t$, which describes distances between events in space-time. There is a space-time associated with each $n$, in which $t$ represents the knowledge at time $n$ about temporal relations. The evolution of the wave function is described in terms of the parameter $\sigma$ that interpolates between sequential times $n$. For a free object we obtain a Stueckelberg equation $\frac{d}{d\sigma}\Psi(r_{4},\sigma)=\frac{ic^{2}\hbar}{2\langle \epsilon\rangle}\Box\Psi(r_{4},\sigma)$, where $r_{4}=(r,ict)$. Here $\sigma$ describes the time $m$ passed between the start of the experiment at time $n$ and the observation at time $n+m$. The parametrization is assumed to be natural, meaning that $\frac{d}{d\sigma}\langle t\rangle=1$, where $\langle t\rangle$ is the expected temporal distance between the events that define $n$ and $n+m$. The squared rest energy $\epsilon_{0}^{2}$ is proportional to the eigenvalue $\tilde{\sigma}$ that describes a 'stationary state' $\Psi(r_{4},\sigma)=\psi(r_{4},\tilde{\sigma})e^{i\tilde{\sigma}\sigma}$. The Dirac equation follows as a `square root' of the stationary state equation from the condition that $\tilde{\sigma}>0$, which follows from the directed nature of $n$. The formalism thus implies that all observable objects have non-zero rest mass, including elementary fermions. The introduction of $n$ releases $t$, so that it can be treated as an observable with uncertainty $\Delta t$.