Quantum spacetime from constraints: wave equations and fields

TL;DR

This paper demonstrates that standard quantum wave equations naturally emerge from a relational, constraint-based quantum spacetime model in 1+1 dimensions, supporting the idea that spacetime and dynamics arise from entanglement.

Contribution

It extends the relational quantum spacetime framework to derive wave equations directly from constraints without assuming external spacetime.

Findings

Wave equations like Schrödinger, Klein-Gordon, and Dirac emerge from the model.

Solutions are derived directly from constraints, not from external spacetime assumptions.

Second quantization formalism is discussed within this framework.

Abstract

In previous works, we showed that both time and space can emerge from entanglement within a globally constrained quantum Universe, with no background coordinates. By extending the Page and Wootters quantum time formalism to include both quantum clocks and rods, and imposing global constraints on total energy and momentum, we constructed a fully relational model of quantum spacetime. Here we take a further step: working in 1+1 dimensions, we show that the standard wave equations governing quantum particles (the Schr\"odinger, Klein-Gordon and Dirac equations) emerge naturally from this framework. The solutions of the equations are derived directly from the constraints, without assuming any external spacetime structure. The second quantization formalism is also implemented and discussed. Our results provide further support for the idea that quantum dynamics in spacetime may emerge from entanglement and constraints.