Lemma: Thermal spectral density

The distribution (eq:thermalcurrent) has Fourier transform (eq:thermalfourier). It has the spectral representation

understood in ${\mathcal{S}}^{\prime }(\mathbb{R})$. In particular,

Proof. Let

It is analytic in the strip $-{\beta }_{\mathrm{H}}<\mathrm{Im}z<0$. Its poles are $z=\mathrm{i}n{\beta }_{\mathrm{H}}$, $n\in \mathbb{Z}$, and its principal part at every pole is

For $0<\eta <{\beta }_{\mathrm{H}}$, consider

The integrand decays exponentially as $|\mathrm{Re}z|\to \infty$.

For $\nu >0$, close the contour upwards. The double-pole residue at $z=\mathrm{i}n{\beta }_{\mathrm{H}}$ is

The poles enclosed are $n\ge 0$. Taking rectangles whose horizontal edges stay halfway between consecutive poles, and then taking their limits, gives

For $\nu <0$, close downwards with clockwise orientation. The enclosed poles are $n\le -1$, so

The horizontal edges vanish because of the sign of $\nu$, and the vertical edges vanish by the real-direction exponential decay.

Putting $z=s-\mathrm{i}\eta$ shows that

For fixed $\eta$, the left-hand Fourier transform is continuous, so the value at $\nu =0$ is the continuous limit $1/(2{\beta }_{\mathrm{H}})$. As $\eta \downarrow 0$, the right-hand side converges in ${\mathcal{S}}^{\prime }(\mathbb{R})$: on positive frequencies it has at most linear growth; on negative frequencies it decays exponentially, uniformly for $0<\eta \le {\beta }_{\mathrm{H}}/2$. This proves the regulator removal and (eq:thermalfourier).

Separating positive and negative frequencies in the inverse Fourier transform yields (eq:thermalspectral). Formula (eq:detailedbalance) follows directly.