Lemma: Horizon pullback and singularity subtraction

Let $j_U$ have covariance (eq:vacuumcurrent) in the affine coordinate $U$. On $U<0$, set

This defines a positive quasifree current state with the canonical commutator (eq:currentCCR). Its covariance is (eq:thermalcurrent). Moreover,

Proof. For $g\in {C^{\infty }}_c(\mathbb{R})$, change variables from $u$ to $U$:

The transformed test is smooth and compactly supported in $U<0$; its extension by zero is a valid test on the affine line. Positivity therefore follows from (eq:vacuumpositive). The antisymmetric covariance is preserved by this change of variable, which proves the current commutator.

The covariance is the pullback distribution

The boundary-value prescription is well defined because $U$ is a smooth strictly increasing coordinate.

Away from $u=u^{\prime }$, elementary substitution gives

At the diagonal, the pullback keeps the same positive-frequency boundary orientation. Equivalently, factor $U(u)-U(u^{\prime })=(u-u^{\prime })D(u,u^{\prime })$ with $D>0$, and use the boundary-value distribution in the variable $u-u^{\prime }$. This gives exactly (eq:thermalcurrent), including its distributional extension.

Finally,

at $z=0$. The only real singularity of $C_{\kappa }$ is its diagonal singularity, which coincides with that of $C_0$. Their difference therefore extends smoothly, with the value in (eq:subtraction).