Off-diagonal correlators vanish in an isotropic photon bath

PDF p. 13: Eq. (S3.15) and following paragraph

Assume the Axiom of Choice, Countable Choice and Dependent Choice. Take the explicitly regulated isotropic thermal photon model, its sequential continuum limit, and all conventions and hypotheses of the completed thermal time-correlator result. In particular $c,\beta ,\gamma >0$, $\kappa$ is Borel, and ${\int }_0^{\mathrm{\infty }}\mathrm{\mid }\kappa (\omega ){\mathrm{\mid }}^2{\omega }^2[2N(\omega )+1]d\omega <\mathrm{\infty }$, where $N(\omega )=(e^{\beta \omega }-1{)}^{-1}$. For Cartesian labels $i,j\in \{1,2,3\}$ and every real $t$, the resulting correlation is ${\mathcal{J}}_{ij}(t)={\delta }_{ij}C(t)$, where

$C(t)=\frac{8\pi }{3\gamma c^3}{\int }_0^{\mathrm{\infty }}\mathrm{\mid }\kappa (\omega ){\mathrm{\mid }}^2{\omega }^2[N(\omega )e^{i\omega t}+(N(\omega )+1)e^{-i\omega t}]d\omega \mathrm{.}$

In particular every off-diagonal correlation vanishes. With the positive-exponential spectral convention, the matrix-valued distributional spectrum is ${\delta }_{ij}\mu$, where $\mu$ is the nonnegative finite measure whose almost-everywhere density $J$ is given by the completed thermal spectral-density result. Thus the spectral density matrix has the representative $J(x)I_3$ almost everywhere. Choose a nonnegative representative at all frequencies, including a value at zero. The positive square root of this representative divided by $2\pi$ is $g(x)=\sqrt{J(x)\mathrm{/}(2\pi )}{I}_3$, as a pointwise matrix identity.

For identification with the cited jump-correlator definition, additionally assume that its bath-trace hypotheses hold and those traces give this correlation, that $C\in L^1(\mathbb{R})$, and that the chosen representative satisfies $J(x)=(2\pi {)}^{-1}{\int }_{\mathbb{R}}C(t)e^{ixt}dt$ at every real $x$. Under these additional hypotheses the frequency-domain jump correlator is the displayed $g(x)$. If its entries are also Lebesgue integrable, the time-domain jump correlator is $g(t)={\int }_{\mathbb{R}}g(x)e^{-ixt}dx$, entrywise. The distributional-spectrum and matrix-square-root conclusions above do not require these additional hypotheses.

For every unitary change of the three channel labels, including a Cartesian-to-spherical change of basis, both scalar matrices remain unchanged under conjugation. Consequently the channels may be expressed in spherical labels without introducing off-diagonal spectral or jump correlations. This is a conditional statement about the model, not an observation.

Isotropic thermal bath time correlator

Thermal bath spectral density and frequency signs

Transverse-polarization completeness and angular average

Jump correlators and bath-memory parameters

Finite support square roots and generalized inverses

Stationary correlation spectra and ULE kernel regularity

Spherical vector components and transition-frequency convention

The Axiom of Choice

The Axiom of Countable Choice (AC_ω)

The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain

The article writes the off-diagonal integrand without the explicit ${\gamma }^{-1}$ factor. In the completed supplier's fixed bath-operator convention, the corresponding displayed article integral represents $\gamma {\mathcal{J}}_{ij}(t)$; both expressions vanish off the diagonal. The source passage is attribution, while the regulated calculation and its normalization are supplied by [F1]. The spectral density is an almost-everywhere representative; assigning values on null sets is a convention and does not by itself prove time or kernel integrability.