Generally covariant evolution equations from a cognitive treatment of time

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 $\mathrm{\Delta }t$ 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.