Scalar Finite-Proper-Time Field Theory as a Complete-History Spectral Calculus
Abstract
We formulate a finite-proper-time (FPT) construction for real scalar λϕ4 theory in which a non-zero lower endpoint s0 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 se≥0, ∑e∈ cse≥ s0 (c∈𝒞(G)). An individual edge or bridge may be arbitrarily short, but no complete closed circulation may collapse below s0. 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 s0 we give an all-order local construction; for λ≥0, 𝒦ϕ̅≥−∂2+m2 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 s02(s2+t2+u2), providing a correlated observable test of the retained scale.
Related
- The Tambara-Yamagami calculus and finite semigroups
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- Nonexistence of complete metrics with uniformly positive scalar curvature
- HDG methods in finite element exterior calculus
BibTeX
@misc{bakr2026scalar,
title = {{Scalar Finite-Proper-Time Field Theory as a Complete-History Spectral Calculus}},
author = {Mustafa Bakr and Tongyu Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.11721},
eprint = {2608.11721},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.11721},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Scalar Finite-Proper-Time Field Theory as a Complete-History Spectral Calculus
AU - Mustafa Bakr
AU - Tongyu Zhang
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2608.11721
UR - https://arxiv.org/abs/2608.11721
AB - 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≥0, ∑_e∈ cs_e≥ s_0 (c∈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≥0, 𝒦_ϕ̅≥−∂^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.
ER -