Minimal Proper Time and Deterministic Microstates: Emergent Quantum Fields and Relativistic Spacetime

We develop a top-down counterpart of the minimal proper-time formulation of quantum field theory previously introduced as an effective bottom-up framework.

Starting from a deterministic pre-geometric substrate of causally ordered events, we show how coarse-graining over microscopic histories leads, at low energies, to an effective Nambu-like quantum dynamics.

The elementary deterministic update is identified with the minimal proper-time step, while the growth of coarse-grained equivalence classes controls both the ultraviolet dissipative correction and the scale dependence of the effective quantization strength, encoded in a running Planck constant.

In this way, the proper-time cutoff kernel of the bottom-up formulation acquires a microscopic interpretation as the inverse growth of unresolved deterministic histories.

In the infrared limit, the dissipative term vanishes and standard unitary quantum field theory is recovered.

The same coarse-grained structure also provides a natural setting for an emergent relativistic spacetime geometry, compatible in the macroscopic limit with Einstein gravity.

The resulting picture suggests a common deterministic origin for minimal-scale structure, quantum behavior, and relativistic spacetime.