Gauging Open EFTs from the top down

TL;DR

The paper provides explicit top-down constructions of Open EFTs for gauged degrees of freedom within the Schwinger–Keldysh framework, carefully accounting for gauge fixing via BRST and the diagonal BRST symmetry that survives the in-in boundary conditions. It demonstrates how the Feynman–Vernon influence functional remains gauge-invariant when dressed with Wilson lines or Stückelberg fields, across both unbroken and spontaneously broken phases, and across several explicit Abelian models. Through a sequence of concrete calculations—gauged Caldeira–Leggett, spinor/scalar QED in thermal states, and the Abelian Higgs–Kibble model—the work clarifies how environmental integration induces nonlocal memory effects, dissipation, and noise while preserving the two-copy gauge structure. Finally, it develops a bottom-up open EFT program for electromagnetism at the level of the in-in action, outlining how diagonal BRST invariance and medium symmetries constrain the most general nonlocal quadratic influence functionals, with clear prescriptions for locality limits and decoupling. The results have broad implications for real-time gauge theories in cosmology, plasma physics, and beyond, including potential extensions to non-Abelian theories and gravity.

Abstract

We present explicit top-down calculations of Open EFTs for gauged degrees of freedom with a focus on the effects of gauge fixing. Starting from the in-in contour with two copies of the action, we integrate out the charged matter in various $U(1)$ gauge theories to obtain the Feynman-Vernon influence functional for the photon, or, in the case of symmetry breaking, for the photon and Stückelberg fields. The influence functional is defined through a quantum path integral, which -- as is always the case when quantizing gauge degrees of freedom -- contains redundancies that must be eliminated via a gauge-fixing procedure. We implement the BRST formalism in this setting. The in-in boundary conditions break the two copies of BRST symmetry down to a single diagonal copy. Nevertheless the single diagonal BRST is sufficient to ensure that the influence functional is itself gauge invariant under two copies of gauge symmetries, retarded and advanced, regardless of the choice of state or symmetry-breaking phase. We clarify how this is consistent with the decoupling limit where the global advanced symmetry is generically broken by the state. We illustrate our results with several examples: a gauge field theory analogue of the Caldeira-Leggett model, spinor QED with fermions integrated out, scalar QED in a thermal state, the Abelian Higgs-Kibble model in the spontaneously broken state with the Higgs integrated out, and Abelian Higgs-Kibble model coupled to a charged bath in a symmetry-broken phase. The latter serves as an example of an open system for Stückelberg/Goldstone fields.

Figures (1)

• Figure 1: The closed-time-path interpretation of the late-time density matrix components in Eq. (\ref{['rho_inin']}) (left) and the in-in generating functional $Z_{\text{in-in}}$ in Eq. (\ref{['Zinin']}) (right). In both cases there is a sum over the components of the initial state $\rho_{\mathrm{in}}$ together with unitary time-evolution operators to and from the initial conditions. In the case of $Z_{\text{in-in}}$ there is also a sum over final states, producing a closed time path.