Minimum Virtual Proper Time and Finite Mass--Charge Matching in QED

Abstract

We formulate four-dimensional finite-proper-time QED from a gauge-covariant source functional with a retained spectral lower boundary $s_0=Λ^{-2}>0$ and finite physical duration $\mathcal T$. Physically, the basic $s_0$ operation is a truncated Laplace transform: the arbitrarily small-proper-time part of the spectral integral, which carries the ultraviolet singularity, is excluded without imposing a cutoff on momentum space, while finite $\mathcal T$ expresses that a physical observable is prepared and detected during a finite experiment rather than by an infinite-time measurement. The retained $s_0$ belongs to the defining source construction and controls the correlated finite corrections after mass and charge matching. The charged interval and circle sectors are generated by the same covariant heat semigroup, while the photon covariance carries the same retained lower bound at the source level. At every fixed perturbative order, gauge covariance gives the Ward--Takahashi hierarchy and an assembled graph-level quadratic-form bound establishes ultraviolet finiteness. The physical electron, positron, and transverse-photon poles retain their canonical residues, and the fixed-order cutting relation uses the matched daughter domains on the stated regulated continuation domain. Mass and charge renormalisation are replaced by finite matching, while independent observables retain correlated $s_0$ dependence. At fixed $s_0>0$, the Euclidean source has a nonperturbative definition. Its physical standing-wave response is obtained order by order from the continued FPT amplitudes with the same finite-duration channel data. An exact all-coupling physical boundary realisation requires additional channel-norm control. The limits $s_0\to0^+$, $\mathcal T\to\infty$, and exact all-coupling free-space exhaustion are separate limiting questions and are not used to define the theory.