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The conventional phase space of classical physics treats space and timedifferently, and this difference carries over to field theories and quantummechanics (QM). In this paper, the phase space is enhanced through two mainextensions. Firstly, we promote the time choice of the Legendre transform to adynamical variable. Secondly, we extend the Poisson brackets of matter fieldsto a spacetime symmetric form. The ensuing "spacetime phase space" is employedto obtain an explicitly covariant version of Hamilton equations forrelativistic field theories. A canonical-like quantization of the formalism isthen presented in which the fields satisfy spacetime commutation relations andthe foliation is quantum. In this approach, the classical action is alsopromoted to an operator and retains explicit covariance through itsnon-separability in the matter-foliation partition. The problem of establishinga correspondence between the new noncausal framework (where fields at differenttimes are independent) and conventional QM is solved through a generalizationof spacelike correlators to spacetime. In this generalization, the Hamiltonianis replaced by the action, and conventional particles by off-shell particles.When the foliation is quantized, the previous map is recovered by conditioningon foliation eigenstates, in analogy with the Page and Wootters mechanism. Wealso provide an interpretation of the correspondence in which the causalstructure of a given theory emerges from the quantum correlations between thesystem and an environment. This idea holds for general quantum systems andallows one to generalize the density matrix to an operator containing theinformation of correlators both in space and time.

Spacetime quantum and classical mechanics with dynamical foliation