Scalar Finite-Proper-Time Field Theory as a Complete-History Spectral Calculus

Abstract

We formulate a finite-proper-time (FPT) construction for real scalar $λϕ^4$ theory in which a non-zero lower endpoint $s_0$ is retained as physical spectral data for complete virtual histories. Functional differentiation partitions a pre-existing history, while interaction sewing creates closed momentum circulations. Local momentum conservation identifies the primitive closed circulations with graph circuits; applying the retained endpoint condition gives \[ s_e\ge0, \qquad \sum_{e\in c}s_e\ge s_0 \quad(c\in\mathcal{C}(G)). \] An individual edge or bridge may be arbitrarily short, but no complete closed circulation may collapse below $s_0$. This routing-independent prescription differs from damping every propagator. We prove that it gives a positive loop quadratic form and ultraviolet-finite massive amplitudes, including overlapping short-distance regions. Under a physical cut, precisely the circuit conditions crossed by the cut disappear, leaving the independently constructed daughter domains together with the ordinary pole residues and positive scalar phase space. At fixed $s_0$ we give an all-order local construction; for $λ\ge0$, $\mathcal{K}_{\barϕ}\ge-\partial^2+m^2$ gives a background-uniform heat-kernel bound. The open Euclidean history kernel is virtual spectral machinery rather than the physical propagator, and physical positivity is imposed on complete boundary amplitudes. After local matching, the first non-constant one-loop on-shell correction is proportional to $s_0^2(s^2+t^2+u^2)$, providing a correlated observable test of the retained scale.