Evolution equations from an epistemic treatment of time

Abstract

Relativistically, time t may be seen as 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 ∈ 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 σ that interpolates between sequential times n. For a free object we obtain a Stueckelberg equation d dσ Ψ(r 4 , σ ) = ic 2 h 2 ε ✷Ψ(r 4 , σ ), where r 4 = (r, ict). Here σ describes the time m passed from the start of the experiment at time n and the observation at time n + m. The parametrization is assumed to be natural in the sense that d dσ t = 1, where t is the expected temporal distance between the events that define n and n + m. The squared rest energy ε 2 0 is proportional to the eigenvalue σ that describes a stationary state Ψ(r 4 , σ ) = ψ(r 4 , σ )e i σ σ . The Dirac equation follows as a square root of the stationary state equation from the condition σ > 0, which is a consequence of 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 ∆t.