Quantum Spacetime from Constraints: Wave Equations and Fields

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 Schrödinger, 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 may emerge from entanglement and constraints.