  The conventional phase space of classical physics treats space and time
differently, and this difference carries over to field theories and quantum
mechanics (QM). In this paper, the phase space is enhanced through two main
extensions. First, we promote the time choice of the Legendre transform to a
dynamical variable. Second, we extend the Poisson brackets of matter fields to
a spacetime symmetric form. The ensuing "spacetime phase space" is employed to
obtain an explicitly covariant version of Hamilton equations for relativistic
field theories. A canonical-like quantization of the formalism is then
presented in which the fields satisfy spacetime commutation relations and the
foliation is quantum. In this approach, the classical action is also promoted
to an operator and retains explicit covariance through its non-separability in
the matter-foliation partition. The problem of establishing a correspondence
between the new noncausal framework (where fields at different times are
independent) and conventional QM is solved through a generalization of
spacelike correlators to spacetime. In this generalization, the Hamiltonian is
replaced by the action, and conventional particles by off-shell particles. When
the foliation is quantized, the previous map is recovered by conditioning on
foliation eigenstates, in analogy with the Page and Wootters mechanism. We also
provide an interpretation of the correspondence in which the causal structure
of a given theory emerges from the quantum correlations between the system and
an environment. This idea holds for general quantum systems and allows one to
generalize the density matrix to an operator containing the information of
correlators both in space and time.
