The Yang–Mills Functional Integral at Finite Temperature as a Henstock–Kurzweil Path Integral: Matsubara Mode Decomposition and Gribov Copies
Year: 2026
Type: preprint
Abstract: This is the sixth paper of the programme in which the Feynman functional integral is constructed without a Wick rotation, by means of the non-absolute Henstock–Kurzweil (HK) gauge integral. Here the Yang–Mills functional integral is built at finite temperature, on the product T³_L × S¹_β with a thermal circle of circumference β = 1/T, using the same spectral truncation machinery as at the previous stage of the programme. The Matsubara decomposition along S¹_β plays the central role: the retained modes are labelled by a Matsubara index |n| ≤ N and a spatial index |k| ≤ K on a large spatial torus T³_L, and the truncated configuration space V_{N,K,L} has dimension 32(2N+1)(2K+1)³. We prove that the truncated thermal Yang–Mills action is a real polynomial of degree at most four with neither constant nor linear term, construct the truncated thermal gauge group G_{N,K} and show that it is a compact group acting on V_{N,K,L} and leaving the action invariant, establish the Matsubara block decomposition of the Faddeev–Popov operator at the trivial connection together with the factorised product formula for its determinant, and give the Berezin representation of that determinant. The combined integrand I_{N,K}(F,A) = F(A)·Δ_FP^(N,K)(A)·exp(i·S_YM^(N,K)(A)) is HK integrable on every compact box of V_{N,K,L}, and the HK integral is linear in the observable and additive in the box. Gribov copies are present at finite temperature: the Faddeev–Popov determinant depends on the copy, while normalised gauge-invariant averages do not. All of these statements, in their formalised finite-dimensional form, are machine-verified in Lean 4 (Mathlib) and contain no sorry and no new axioms. The scope is stated explicitly: this is Yang–Mills theory truncated to finitely many Matsubara and spatial modes, not the continuum theory at finite temperature; the double limit (N,K,L) → ∞ is formulated as a conjecture and is not proved. Two language versions (English and Russian), the Lean 4 formalisation, and the numerical verification script are provided.
Author: Berdinsky, Yuri N.
Language: English
Subfield: Nuclear and High Energy Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213766657
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