Deriving Fundamental Physics via Entropy Maximization Under Linear Constraints

We present a novel derivation of fundamental physical laws by solving a maximization problem on the Shannon entropy of all possible measurements relative to a system's initial state, subject to specific linear constraints. By introducing appropriate linear constraints, we create probability measures that adhere to particular underlying mathematical structures, enabling us to recover various physical theories within a unified framework. Specifically, imposing a U(1) group constraint leads to the emergence of quantum mechanics by incorporating complex probability amplitudes and interference effects. Extending this approach, we apply a Spinc(3,1) group constraint to derive a relativistic quantum theory that naturally includes Lorentz symmetry. Remarkably, in 3+1 dimensions, this method uniquely results in the metric tensor of general relativity through a double-copy mechanism applied to the Dirac current. Furthermore, it inherently incorporates the SU(3)xSU(2)xU(1) gauge symmetries of the Standard Model, providing a unified description of fundamental interactions. These findings highlight the power of entropy maximization under linear constraints to reveal the deep connections between probability theory and the mathematical structures underlying fundamental physics, offering new insights into the emergence of spacetime dimensions and symmetry structures in our universe.