Isotropic thermal bath time correlator

PDF p. 12: Eq. (S3.13)

Assume the choice hypotheses of the named suppliers. Work in units $\mathrm{\hslash }=1$, with $c,\beta ,\gamma >0$, and retain the article's definitions $\kappa =d_0\mathcal{E}$ and $B_{\alpha }=-d_0{E}_{\alpha }\mathrm{/}\sqrt{\gamma }$. In continuum notation $\mathrm{\mid }\kappa (\omega ){\mathrm{\mid }}^2{d}^{3}k$ denotes the squared field-coupling measure after the chosen mode-density normalization; it contains no newly inserted $1\mathrm{/}\gamma$. Require $\kappa$ to be Borel and $M={\int }_0^{\mathrm{\infty }}\mathrm{\mid }\kappa (\omega ){\mathrm{\mid }}^2{\omega }^2[2(e^{\beta \omega }-1{)}^{-1}+1]d\omega <\mathrm{\infty }$.

Define the continuum thermal two-point function by this explicit regulator limit. First fix $0<a<b<\mathrm{\infty }$ and the shell $A_{a,b}=\{k\in {\mathbb{R}}^3:a\le c\mathrm{\mid }k\mathrm{\mid }\le b\}$. For each refinement $n$, partition it up to a Lebesgue null set into finitely many positive-volume Borel cells $C$ of diameter at most ${\delta }_n\to 0$, with tags $k_C\in C$, as in the finite-cell lemma. At every tag choose ${\epsilon }_{C1},{\epsilon }_{C2}$ to be a real orthonormal basis of $k_C^{\perp }$. Put ${\omega }_C=c\mathrm{\mid }{k}_C\mathrm{\mid }>0$ and choose finite field coefficients ${\kappa }_C=d_0{\mathcal{E}}_C$ with $\mathrm{\mid }{\kappa }_C{\mathrm{\mid }}^2=w_C={\int }_C\mathrm{\mid }\kappa (c\mathrm{\mid }k\mathrm{\mid }){\mathrm{\mid }}^2{d}^{3}k$, shared by the two polarization modes of that cell. Coefficients are indexed by cells; equal frequencies in different cells need not have equal coefficients.

Use the independent oscillator regulator with modes $r=(C,\lambda )$, $\lambda \in \{1,2\}$, in the finite-mode symmetric Fock realization of the bosonic moment contract. Its occupation basis is $\mathrm{\mid }\mathbf{n}⟩$, its mode operators are $a_r,a_r^{†}$, and its Hamiltonian is the self-adjoint closure of $H_B={\sum }_r{\omega }_C{a}_r^{†}{a}_r$ on the finite occupation span. Adopt its Gibbs preparation ${\rho }_B=Z^{-1}{e}^{-\beta H_B}$, where $Z={\prod }_{C,\lambda }(1-e^{-\beta {\omega }_C}{)}^{-1}$. Specify the coupling-weighted component field $F_i=d_0{E}_i$, $i\in \{1,2,3\}$, as the following particular Hermitian linear coupling permitted by the regularized photon-dipole model:

$F_i={\sum }_{C,\lambda }{\epsilon }_{C\lambda ,i}({\kappa }_C{a}_{C\lambda }+\overline{{\kappa }_C}{a}_{C\lambda }^{†}),B_i=-F_i\mathrm{/}\sqrt{\gamma }\mathrm{.}$

The fields act on the common mode domain $\mathcal{D}={\bigcap }_r(D(a_r)\cap D(a_r^{†}))$. The free dynamics is $U(t)\mathrm{\mid }\mathbf{n}⟩=e^{-it{E}_{\mathbf{n}}}\mathrm{\mid }\mathbf{n}⟩$, where $E_{\mathbf{n}}={\sum }_{C,\lambda }{\omega }_C{n}_{C\lambda }$; put ${\tilde{B}}_i(t)=U(t{)}^{\ast }{B}_{i}U(t)$ on its conjugated domain and ${\mathcal{J}}_{ij}^{(n;a,b)}(t)=\mathrm{Tr}[{\tilde{B}}_i(t)B_j{\rho }_B]$. These products are well defined and trace class, as established below. Refine the cells first, then exhaust $(0,\mathrm{\infty })$ by shells $[1\mathrm{/}m,m]$, $m\ge 2$. This sequential limit exists for every real $t$ and is

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

This is the explicitly normalized, convergent version of the original diagonal expression (S3.13). The original prefactor $8\pi \mathrm{/}(3c^3)$ is retained as attribution in Source notes. It coincides with this formula at numerical $\gamma =1$; for general $\gamma$ and the fixed definitions it is the expression for $\gamma \mathcal{J}$, not for $\mathcal{J}$. No redefinition of $\kappa$, continuum trace-class Gibbs state, ordinary time Fourier integrability, or removal of a nonintegrable ultraviolet cutoff is asserted. A physically fixed ultraviolet form factor can satisfy $M<\mathrm{\infty }$; a bare cubic vacuum spectral weight is outside this absolute-integral statement.

Original article attribution, not an independent proof: PDF p. 12, Eq. (S3.13), states ${\mathcal{J}}_{\alpha \alpha }(t)=\frac{8\pi }{3c^3}{\int }_0^{\mathrm{\infty }}d\omega \mathrm{\mid }\kappa (\omega ){\mathrm{\mid }}^2{\omega }^2[(e^{\beta \omega }-1{)}^{-1}{e}^{i\omega t}+(1+(e^{\beta \omega }-1{)}^{-1})e^{-i\omega t}]$. The original item claim is preserved here in full as attribution; the theorem states the explicit $B$-normalized consequence under the stated convergent regulator contract.

The original source defines $H_{\mathrm{i}\mathrm{n}\mathrm{t}}=\sqrt{\gamma }\sum X_{\alpha }{B}_{\alpha }$ on PDF p. 9, Eq. (S2.2); defines the $B$-operator correlation on p. 10, Eq. (S2.5); and gives both $\kappa =d_0\mathcal{E}$ and $B_{\alpha }=-d_0{E}_{\alpha }\mathrm{/}\sqrt{\gamma }$ on p. 11 around Eq. (S3.3). PDF p. 13 after (S3.18) discusses absorbing density of states into the coupling and later absorbing constants into effective $\gamma ,f(J)$. Those passages motivate checking a normalization convention; they do not supply an explicit $\kappa \mapsto \kappa \mathrm{/}\sqrt{\gamma }$ between the fixed definitions and (S3.13). Here the factor follows directly from the fixed definitions. The source-anchored uncertainty, exact finite diagnostic and complete page-image reading record are retained internally in escalation-support/thermal-correlator/hold-evidence.json as an assessment lead, not an article-defect verdict. Step 8 owns original-article assessment.

The cell construction supplies one specified convergent approximation. It does not prove that every box discretization or every physical bath preparation converges to it. For box coefficients, any identification with this continuum notation must separately match the squared coupling times its mode density, including any $(2\pi {)}^{-3}$ convention. The finite mode field coefficients are $\sqrt{w_C}$ up to phase, and the corresponding $B$ coefficients are $\sqrt{w_C\mathrm{/}\gamma }$. Neither mode density nor a change of the time argument cancels the latter scalar factor. An uncut point-field vacuum weight proportional to ${\omega }^3$ has divergent nonnegative equal-time integral; no oscillatory prescription is invented here.