Scalar Finite-Proper-Time Field Theory as Spectral Operator Calculus

Abstract

We formulate a finite-proper-time construction for Euclidean scalar $\lambda\phi^4$ theory in which a retained endpoint $s_0$ is assigned to complete internal histories rather than independently to individual Schwinger segments. The theory is defined through paired open and closed spectral functions and their Fr\'echet/Duhamel hierarchy. Functional differentiation inserts operators along an existing proper-time history and partitions its total length, while interaction vertices sew separately complete histories. We introduce a corresponding complete-history diagrammatic calculus and derive the resulting one- and two-loop structures. We distinguish the retained endpoint from an auxiliary regulator or renormalisation-group scale and test whether its effects survive ordinary parameter matching and admissible field redefinitions. For the one-loop four-point function, fixing the renormalised mass, field normalisation, and quartic coupling leaves a finite momentum-dependent remainder. We establish a perturbative non-redundancy description of the retained endpoint against finite renormalisable-parameter redefinitions and local $S$-matrix-preserving field redefinitions at this order. The low-energy theory can be represented as an effective field theory, but its higher-derivative coefficients are not independent. This provides a concrete distinction between a retained finite proper-time scale and an arbitrary cutoff prescription.