Abstract

We present a novel approach to quantum theory construction that involves solving a maximization problem on the Shannon entropy of all possible measurements of a system relative to its initial preparation. By constraining the maximization problem with a phase that vanishes under measurements, we obtain quantum mechanics (vanishing U(1)-valued phase), relativistic quantum mechanics (vanishing Spin^c(3,1)-valued phase) and quantum gravity (also vanishing Spin^c(3,1)-valued phase, but with dilations). The first two cases are equivalent to established theory, even naturally yielding the SU(3)xSU(2)xU(1) gauge symmetries of the Standard Model, whereas the latter case yields the pseudo-Riemannian inner product as its primary observable, effectively building the metric tensor from underlying quantum measurements of spacetime. Finally, the solution is consistent only with 3+1 spacetime dimensions, as it encounters obstructions in all other dimension configurations. This framework integrates quantum mechanics, relativistic quantum mechanics, a quantum metric tensor, spacetime dimensionality, and particle physics gauge symmetries from a simple entropy maximization problem constrained by a vanishing phase.