Theorem: Euclidean correction and spectral-state existence

If $H_W\ge 0$, the massless spectral covariance is finite and continuous on compact tests. Its Euclidean kernel is $G_{W,E}=G_{F,E}+\delta G_W$, where

This correction is jointly real-analytic at every pair of regular Euclidean points. The spectral state has the local Hadamard form of Theorem thm:hadamard.

The state remains $t,z$-translation invariant and $(t,z)$-boost invariant. Rotation and dilation invariance hold exactly when the extension has those symmetries.

Proof. Fourier transform the product resolvent in the two Euclidean variables $(\tau ,z)$. The radial angular integral in their momentum plane gives $J_0$, and the normalized transverse modes give the factor $(4{\pi }^2\alpha {)}^{-1}$.

At small $p$, Lemma lem:Cbound and $K_{{\nu }_j}(pr)=O(p^{-{\nu }_j})$, locally uniformly in $r>0$, bound every term by

This is integrable because ${\nu }_j,{\nu }_k<1$. Spatial and angular derivatives preserve this integrability; time and longitudinal derivatives improve it. At large $p$, the bound and the $K$-Bessel asymptotic give exponential decay $e^{-p(r+r^{\prime })}$ times a polynomial.

In particular the equal-time correction is bounded on fixed compact regular spatial pairs. The quadratic form of $B^{-1/2}$, first defined with a positive spectral or mass cutoff, is the Friedrichs form plus this finite correction. Monotone convergence in the spectral integral removes the cutoff. Lemma lem:infrared therefore supplies the remaining Friedrichs part. This proves compact-test finiteness and continuity rather than inferring them from $B\ge 0$.

For analyticity, use the two-dimensional Fourier representation before angular integration. On a small complex neighbourhood of any real regular pair, the imaginary displacement in $(\tau ,z)$ is less than the positive real part of $r+r^{\prime }$. The exponential large-$p$ estimate then persists. The small-$p$ bound is locally uniform in that neighbourhood. Holomorphic parameter integration gives a holomorphic extension, hence real analyticity.

The difference is a Euclidean bisolution. Its continuation is smooth locally, so the same direct Hadamard singularity remains. The spectral construction gives positivity, and the product mass-shell argument proves boosts.