Evolution equations from an epistemic treatment of time

TL;DR

This paper proposes a formalism where time is treated as an observable like position, introducing sequential and relational time, leading to evolution equations that unify quantum and relativistic perspectives and imply all objects have non-zero rest mass.

Contribution

It introduces a novel formalization of time using sequential and relational concepts, deriving evolution equations that unify quantum mechanics and relativity.

Findings

Derives a Stueckelberg equation for wave function evolution.

Shows Dirac equation as a square root of stationary state equation.

Implements a formalism where all observable objects have non-zero rest mass.

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 $\Delta t$.