Functional Quantum Field Theory in Phase Space

The formulation of Quantum Field Theory (QFT) in phase space offers a unique alternative to operator and path-integral paradigms, providing distinct conceptual advantages for semiclassical expansions. In this work, we present a systematic and self-consistent functional framework that maps stationary Schrodinger functional equations directly onto phase-space star-eigenvalue equations across different spin statistics. Operating within a non-manifestly covariant equal-time formalism, we derive explicit vacuum Wigner functionals for scalar, gauge, and fermionic fields, establishing the rigorous theoretical consistency of the formalism from first principles prior to phenomenological applications. We analyze how ordering prescriptions and continuous symmetries manifest under the functional star-product, including an explicit phase-space formulation of Noether’s theorem and field regularization. Finally, the framework is extended to interacting systems via a functional Rayleigh–Schrodinger perturbative scheme, illustrated explicitly through the non-trivial first-order Wigner functional correction W(1) and the vacuum energy correction for a ϕ4 self-interacting theory, establishing a solid foundation for evaluating real-time quantum field dynamics.