Quantum space, quantum time, and relativistic quantum mechanics

We treat space and time as bona fide quantum degrees of freedom on an equal footing in Hilbert space. Motivated by considerations gravity, we focus a paradigm dealing with linear, first-order Hamiltonian momentum constraints that lead to emergent features temporal spatial translations. Unlike the conventional treatment, show Klein-Gordon Dirac equations relativistic mechanics can be unified our applying dispersion relations eigenvalues rather than treating them operator-valued equations. With being treated space, symmetry transformations implemented unitary basis changes giving stronger mechanical footing. Global symmetries, such Lorentz transformations, modify decomposition space; local $U(1)$ gauge are diagonal coordinate do not alter briefly discuss extensions this field theory gravity.