On the functional formalism in quantum field theory

Giuseppe Mussardo

Abstract Chapter 7 covers the main reasons for adopting the methods of quantum field theory (QFT) to study the critical phenomena. It presents both the canonical quantization and the path integral formulation of the field theories as well as the analysis of the perturbation theory. The chapter also covers transfer matrix formalism and the Euclidean aspects of QFT, the field theory of the Ising model, Feynman diagrams, correlation functions in coordinate space, the Minkowski space and the Legendre transformation and vertex functions. Everything in this chapter will be needed sooner or later, since it highlights most of the relevant aspects of quantum field theory.

Nima Moshayedi

José A. R. Cembranos, Marcos Skowronek

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.

Khavtgain Namsrai

Luc Jean-Marie Darmé, Joerg Jaeckel, Marek Lewicki

We present a formalism based on the functional Schr\"odinger equation to analyze time-dependent tunneling in quantum field theory at the semiclassical level. The full problem is reduced step by step to a finite dimensional quantum mechanical setup and solved using the Wentzel-Kramers-Brillouin approximation. As an example, we consider tunneling from a homogeneous oscillating initial state in scalar quantum field theory.

Carlos Farina de Souza