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This theoretical study demonstrates that standard wave equations, including Schrödinger, Klein-Gordon, and Dirac, can emerge naturally from a relational quantum framework where spacetime coordinates are absent. By deriving solutions directly from global constraints on energy and momentum in 1+1 dimensions, the work reinforces the hypothesis that quantum dynamics arise from entanglement rather than background geometry. While significant for foundational physics, these results remain at the theoretical level and do not yet indicate immediate applicability to current quantum computing or engineering architectures.
Quantum spacetime from constraints: wave equations and fields
arXiv:2508.12698v4 Announce Type: replace-cross 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.
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 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 in spacetime may emerge from entanglement and constraints.
arXiv:2508.12698 [gr-qc]
(or arXiv:2508.12698v4 [gr-qc] for this version)
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