Ghostbusters: Unitarity and Causality of Non-equilibrium Effective Field Theories

TL;DR

The paper addresses how unitarity and causality are preserved in non-equilibrium EFTs defined on a closed time path by proving the largest time equation (LTE) holds to all loop orders provided propagators are retarded. It introduces a frequency-regulator that preserves the retarded r-a structure and shows ghost contributions vanish, eliminating the need for BRST ghosts. The work connects LTE with dynamical KMS symmetry, demonstrates concrete examples (Brownian motion, Model A, fluctuating hydrodynamics) that exhibit the required pole structure, and discusses additional unitarity constraints and KMS conditions. The results establish a robust foundation for causal, unitary non-equilibrium EFTs and clarify when ghosts are superfluous, with broad implications for fluctuating hydrodynamics and related stochastic EFTs.

Abstract

For a non-equilibrium physical system defined along a closed time path (CTP), a key constraint is the so-called largest time equation, which is a consequence of unitarity and implies causality. In this paper, we present a simple proof that if the propagators of a non-equilibrium effective action have the proper pole structure, the largest time equation is obeyed to all loop orders. Ghost fields and BRST symmetry are not needed. In particular, the arguments for the proof can also be used to show that if ghost fields are introduced, their contributions vanish.

Figures (5)

• Figure 1: Largest time equation. (a) From the perspective of external sources: the path integral is independent of the part where the external sources for upper and lower contours are the same; (b) From the perspective operator insertion: for the operator with the largest time it does not matter whether one inserts it on the upper or lower contour.
• Figure 2: Schematic Feynman rules for \ref{['lamn']}. Dashed lines correspond to $\chi_a$'s, solid lines correspond to $\chi_r$'s. The $r-a$ propagator should be proportional to ${\theta} (t-t')$ while the $a-r$ propagator should be proportional to ${\theta} (t'-t)$. $r-r$ propagator (with some function $f$) is non-vanishing for either orderings of $t$ and $t'$. All the interacting vertices contain at least one $a$-leg.
• Figure 3: Feynman rules for \ref{['Leff-br']}. Dashed lines correspond to $x_a$, solid lines correspond to $x$.
• Figure 4: One-loop contribution to the two-point function of $x_a$.
• Figure 5: Various possibilities: (a) corresponds to the existence of an internal $a$-$r$ loop, which is described by \ref{['ww1']}. In (b), there is an $a$-$r$ loop which starts and ends at the operator with the largest time, which is described by \ref{['ww2']}. In (c) there is an $a$-$r$ line which connects the operator with the largest time with another operator, described by \ref{['ww3']}.

Theorems & Definitions (2)

• Theorem
• proof : Proof: