Abelian Maxwell Theory as a Continuum Henstock-Kurzweil Integral: Gauge Invariance, Spectral Truncation, and Stability
Abstract
This is the eighth paper of a programme constructing the Feynman functional integral without a Wick rotation, based on the non-absolutely convergent Henstock–Kurzweil (HK) integral and on Kuelbs–Steadman spaces. Here the bosonic sector of an abelian gauge theory — Maxwell theory — is constructed as a continuum HK integral on the four-torus T^4: with the oscillating cylindrical premeasure of the previous stage, an explicit spectral truncation |n_mu| <= N (dim V_N = 4(2N+1)^4), Landau gauge fixing, and the continuum limit N -> infinity understood as convergence in the Kuelbs–Steadman space KS_2. Machine-verified: the quadraticity and the gauge invariance of the continuum Maxwell action; the diagonalisation of the truncated action in the Fourier modes; the independence of the abelian Faddeev–Popov operator of the connection and the equality Delta_FP = 1 of the normalised determinant; the unique solvability of the Landau condition on every non-constant mode (the abelian case has no Gribov copies); the HK integrability of F e^{iS_M^{(N)}} on compact boxes; the closed form of the gauge-fixed Fresnel value Z_N = (2 pi i)^{M_N} prod_{0<|n|<=N} |2 pi n|^{-2}, M_N = (2N+1)^4 - 1, and its independence of the truncation scheme; the projective consistency of the normalised premeasure, the Cauchy property of the truncated integrals in KS_2 and the uniqueness of the limit; the nilpotency and the invariance of the abelian BRS transformation together with the Slavnov–Taylor identities. The scope is stated honestly: the zeta-regularised value Delta_M = exp(zeta'(0)) remains a paper-level statement; the interacting non-abelian theory is not constructed here, and the Gribov problem is not solved — the abelian case simply has no Gribov copies, and the present paper serves as a test case for the continuum-limit machinery. A deterministic numerical script confirms all eight checks. The paper also contains preparatory material for the non-abelian case: the explicit linear deformation of the Faddeev–Popov operator with the bound ||M_1^{(N,K)}(A)|| <= C||A|| and positive definiteness for small fields; the cut-out violation functional together with its elementary bound; and a precisely stated singular-direction conjecture, which remains a conjecture and is nowhere used as an established statement. An explicit summability criterion for the increments in KS_2 is added as well.
Related
- Gribov Copies and Gauge Invariance: A Henstock–Kurzweil Continuum Integral on the SU(3) Lattice
- Normalized CutOut in the Non-Abelian Continuum Henstock–Kurzweil Integral: The Cartan Class of Configurations and zeta-Regularization of the Faddeev–Popov Determinant
- Normalized CutOut for the quasi-abelian two-mode class in the non-abelian continuum Henstock–Kurzweil integral: banded Faddeev–Popov operator and numerical comparison with lattice data
- Algebraic Completeness of Feynman Premeasures: The Henstock-Kurzweil Path Integral and Operator Wick Calculus in Lean 4
- Henstock--Kurzweil Gauge Integral in the Non--Gaussian Regime: A Machine--Verified Construction
BibTeX
@misc{berdinsky2026abelian,
title = {{Abelian Maxwell Theory as a Continuum Henstock-Kurzweil Integral: Gauge Invariance, Spectral Truncation, and Stability}},
author = {Yu. N. Berdinsky},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23440893},
url = {https://zenodo.org/records/23440893},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Abelian Maxwell Theory as a Continuum Henstock-Kurzweil Integral: Gauge Invariance, Spectral Truncation, and Stability
AU - Yu. N. Berdinsky
PY - 2026
DA - 2026/10/17
PB - Zenodo
DO - 10.5281/zenodo.23440893
UR - https://zenodo.org/records/23440893
AB - This is the eighth paper of a programme constructing the Feynman functional integral without a Wick rotation, based on the non-absolutely convergent Henstock–Kurzweil (HK) integral and on Kuelbs–Steadman spaces. Here the bosonic sector of an abelian gauge theory — Maxwell theory — is constructed as a continuum HK integral on the four-torus T^4: with the oscillating cylindrical premeasure of the previous stage, an explicit spectral truncation |n_mu| <= N (dim V_N = 4(2N+1)^4), Landau gauge fixing, and the continuum limit N -> infinity understood as convergence in the Kuelbs–Steadman space KS_2. Machine-verified: the quadraticity and the gauge invariance of the continuum Maxwell action; the diagonalisation of the truncated action in the Fourier modes; the independence of the abelian Faddeev–Popov operator of the connection and the equality Delta_FP = 1 of the normalised determinant; the unique solvability of the Landau condition on every non-constant mode (the abelian case has no Gribov copies); the HK integrability of F e^{iS_M^{(N)}} on compact boxes; the closed form of the gauge-fixed Fresnel value Z_N = (2 pi i)^{M_N} prod_{0<|n|<=N} |2 pi n|^{-2}, M_N = (2N+1)^4 - 1, and its independence of the truncation scheme; the projective consistency of the normalised premeasure, the Cauchy property of the truncated integrals in KS_2 and the uniqueness of the limit; the nilpotency and the invariance of the abelian BRS transformation together with the Slavnov–Taylor identities. The scope is stated honestly: the zeta-regularised value Delta_M = exp(zeta'(0)) remains a paper-level statement; the interacting non-abelian theory is not constructed here, and the Gribov problem is not solved — the abelian case simply has no Gribov copies, and the present paper serves as a test case for the continuum-limit machinery. A deterministic numerical script confirms all eight checks. The paper also contains preparatory material for the non-abelian case: the explicit linear deformation of the Faddeev–Popov operator with the bound ||M_1^{(N,K)}(A)|| <= C||A|| and positive definiteness for small fields; the cut-out violation functional together with its elementary bound; and a precisely stated singular-direction conjecture, which remains a conjecture and is nowhere used as an established statement. An explicit summability criterion for the increments in KS_2 is added as well.
ER -