Generally covariant evolution equations from a cognitive treatment of time

TL;DR

This paper introduces a formalization of time based on cognition, leading to quantum evolution equations that incorporate both sequential and relational time, aiming to unify perspectives in relativity and quantum theory.

Contribution

It proposes a new conceptual framework for time in physics, deriving evolution equations that integrate cognitive insights with quantum and relativistic principles.

Findings

Derivation of quantum evolution equations with a continuous parameter  interpolating discrete times.

Time becomes an observable with uncertainty in measurement contexts.

The evolution equations resemble those suggested by Stueckelberg and relate to quantum gravity.

Abstract

The treatment of time in relativity does not conform to that in quantum theory. To resolve the discrepancy, a formalization of time is introduced in an accompanying paper, starting from the assumption that the treatment of time in physics must agree with our cognition. The formalization has two components: sequential time $n$ and relational time $t$. The evolution of physical states is described in terms of $n$. The role of $t$ is to quantify 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 distances between present and past events. This approach leads to quantum evolution equations expressed in terms of a continuous evolution parameter $\sigma$, which interpolates between discrete sequential times $n$. Rather than describing the evolution of the world at large, these evolution equations provide probabilites of a set of predefined outcomes in well-defined experimental contexts. When the context is designed to measure spatio-temporal position $(x,t)$, time $t$ becomes an observable with Heisenberg uncertainty $\Delta t$ on the same footing as $x$. The corresponding evolution equation attains the same symmetric form as that suggested by Stueckelberg in 1941. When the context is such that the metric of space-time is measured, the corresponding evolution equation may be seen as an expression of quantum gravity. In short, the aim of this paper is to propose a coherent conceptual basis for the treatment of time in evolution equations, in so doing clarifying their meaning and domain of validity.