Thermal photon bath and partition function

PDF p. 12: S3 opening paragraph

Adopt the article's idealized isotropic photon bath. The algebraic preparation and rotation equations below give a precise meaning to its isotropy for transverse smeared fields. A finite selection of plane-wave modes need not itself have exact rotational symmetry; its Gibbs preparation is specified separately.

Use units with $\mathrm{\hslash }=1$ and let $k_B>0$ be Boltzmann's constant. For a finite regulator with modes $\lambda =(\mathbf{k},ϵ)$ of frequencies ${\omega }_{\lambda }=c\mathrm{\mid }\mathbf{k}\mathrm{\mid }>0$, assume independent bosonic oscillators with free Hamiltonian $H_B={\sum }_{\lambda }{\omega }_{\lambda }{a}_{\lambda }^{†}{a}_{\lambda }$. Here $\mathbf{k}$ is a nonzero wave vector, $ϵ$ labels a transverse polarization, and $a_{\lambda }$ and $a_{\lambda }^{†}$ are the mode annihilation and creation operators. The number basis $\mathrm{\mid }\{n_{\lambda }\}⟩$ is indexed by nonnegative integer occupation numbers, one for each retained mode.

At inverse temperature $\beta >0$, corresponding to temperature $T=1\mathrm{/}(k_B\beta )$, postulate the free bath preparation ${\rho }_B=Z(\beta {)}^{-1}{e}^{-\beta H_B}$, where $Z(\beta )=\mathrm{Tr}(e^{-\beta H_B})={\prod }_{\lambda }(1-e^{-\beta {\omega }_{\lambda }}{)}^{-1}$. The trace and product in this regulated statement are over the finite collection of modes, although each oscillator has infinitely many occupation levels. This state is stationary under the free bath dynamics. The article first treats finite temperature and subsequently considers the vacuum limit.

For the isotropic preparation use the transverse space $\mathfrak{h}=\{g\in L^2({\mathbb{R}}^3,d^{3}k;{\mathbb{C}}^3):\mathbf{k}\cdot g(\mathbf{k})=0\text{ a.e.}\}$, with $⟨g,f⟩=\int \bar{g}\cdot f{d}^{3}k$ linear in $f$. Let $\mathcal{D}=\{g\in C_c^{\mathrm{\infty }}({\mathbb{R}}^3\setminus \{0\};{\mathbb{C}}^3):\mathbf{k}\cdot g(\mathbf{k})=0\}$ and $\omega (\mathbf{k})=c\mathrm{\mid }\mathbf{k}\mathrm{\mid }$. Define $N_{\beta }$ to be multiplication by $(e^{\beta \omega (\mathbf{k})}-1{)}^{-1}$ on its maximal $L^2$ domain. Every test in $\mathcal{D}$ lies in that domain.

The observable algebra $\mathcal{A}$ consists of finite complex linear combinations of formal symbols $W(g)$, $g\in \mathcal{D}$, with unit $W(0)$, product $W(g)W(f)=e^{-i\mathrm{Im}⟨g,f⟩\mathrm{/}2}W(g+f)$ and involution $W(g{)}^{\ast }=W(-g)$. Postulate the preparation given by the complex-linear functional ${\varpi }_{\beta }$ determined by

${\varpi }_{\beta }(W(g))=\exp [-\frac{1}{4}⟨g,(I+2N_{\beta })g⟩]\mathrm{.}$

It is a normalized positive functional: ${\varpi }_{\beta }(W(0))=1$ and ${\varpi }_{\beta }(A^{\ast }A)\ge 0$ for every $A\in \mathcal{A}$. Its physical adoption is the postulate; positivity is supported by the imported representation below. This algebraic state does not require a density operator on the original infinite-volume photon Fock space.

Here rotations are $SO(3)=\{R\in M_3(\mathbb{R}):R^{T}R=I,\det R=1\}$, acting by $U_{R}g(\mathbf{k})=Rg(R^{-1}\mathbf{k})$. Define ${\alpha }_R(W(g))=W(U_{R}g)$ and the free evolution ${\tau }_t(W(g))=W(e^{i\omega t}g)$, extending both linearly. These are algebra automorphisms and ${\alpha }_R{\alpha }_S={\alpha }_{RS}$. The preparation is isotropic and stationary in the explicit sense ${\varpi }_{\beta }\circ {\alpha }_R={\varpi }_{\beta }$ and ${\varpi }_{\beta }\circ {\tau }_t={\varpi }_{\beta }$.

Field correlations mean the derivatives of the Weyl expectations: for $g_1,\dots ,g_n\in \mathcal{D}$, $t_1,\dots ,t_n\in \mathbb{R}$, define

$C_{\beta }(g_1,t_1;\dots ;g_n,t_n)=i^{-n}{{\mathrm{\partial }}_{s_1}\cdots {\mathrm{\partial }}_{s_n}{\varpi }_{\beta }(W(s_1{e}^{i\omega t_1}{g}_1)\cdots W(s_n{e}^{i\omega t_n}{g}_n))\mid }_{s_1=\cdots =s_n=0}\mathrm{.}$

All $s_j$ are real. These derivatives exist, and $C_{\beta }(U_R{g}_1,t_1;\dots ;U_R{g}_n,t_n)=C_{\beta }(g_1,t_1;\dots ;g_n,t_n)$. In particular, $C_{\beta }(g,t;f,0)=⟨e^{i\omega t}g,f⟩\mathrm{/}2+\mathrm{Re}⟨e^{i\omega t}g,N_{\beta }f⟩$. To specify vector components at the emitter, take the emitter as spatial origin, fix any nonzero smooth radial smearing $\chi \in C_c^{\mathrm{\infty }}((0,\mathrm{\infty });\mathbb{C})$, and set $g_v(\mathbf{k})=\chi (\mathrm{\mid }\mathbf{k}\mathrm{\mid })(I-\widehat{\mathbf{k}}{\widehat{\mathbf{k}}}^T)v$ for $v\in {\mathbb{R}}^3$, $\widehat{\mathbf{k}}=\mathbf{k}\mathrm{/}\mathrm{\mid }\mathbf{k}\mathrm{\mid }$. Overall field normalization and radial frequency factors belong to the chosen $\chi$. The smeared electric-field component $E_v(t)$ denotes the field whose correlations are $C_{\beta }(g_v,t;\dots )$, equivalently the generator of $s\mapsto W(s{e}^{i\omega t}{g}_v)$ in the imported regular representation. Thus $U_R{g}_v=g_{Rv}$ and its components transform by $E_v(t)\mapsto E_{Rv}(t)$; for the Cartesian basis $e_{\alpha }$ this is $E_{e_{\alpha }}(t)\mapsto {\sum }_{\delta }{R}_{\delta \alpha }{E}_{e_{\delta }}(t)$.

The article also writes its Gibbs expression and partition product in unregulated photon-mode notation; those trace expressions remain formal here. The specified algebraic preparation supplies smeared correlations and their exact isotropy, but does not establish an infinite-volume trace-class Gibbs operator, unsmeared point fields, or a limit of the finite-mode Gibbs regulators. Removing the chosen smearing or relating the two preparations requires a separate convergence contract.

Photon modes and field normalization

Regularized transverse photon dipole bath model

Finite-mode bosonic Gibbs occupations and moments

Transverse one-photon space and helicity projectors

Transverse rotations preserve helicity and generate total angular momentum

Given: The finite positive-frequency regulator, $\beta >0$, $c>0$, the transverse test domain, and A1. The numbered argument establishes the mathematical interfaces of the preparations; their physical adoption remains a postulate.

1.1 F1 and F2 supply the finite-mode conventions. On the number basis $H_B$ has eigenvalue ${\sum }_{\lambda }{\omega }_{\lambda }{n}_{\lambda }$. F3 applies because the mode number is finite and all frequencies and $\beta$ are positive; it gives the finite nonzero partition function and normalized Gibbs density. The free evolution and this density are diagonal in the same basis, so conjugation preserves the density, as also established in F2. Choice and Countable Choice are retained for the cited oscillator construction; Dependent Choice is inherited from F2's continuum conventions. [F1, F2, F3, A1, algebra]

1.2 F4 identifies $\mathfrak{h}$ and $\mathcal{D}$. The multiplier ${\gamma }_{\beta }=e^{-\beta \omega }$ is a self-adjoint contraction with $\ker (I-{\gamma }_{\beta })=0$, since $0<e^{-\beta c\mathrm{\mid }\mathbf{k}\mathrm{\mid }}<1$ almost everywhere. Each $g\in \mathcal{D}$ has compact support in an annulus $0<r_0\le \mathrm{\mid }\mathbf{k}\mathrm{\mid }\le r_1$, where $N_{\beta }$ is bounded, so $g\in D(N_{\beta })\subseteq D(N_{\beta }^{1\mathrm{/}2})$. Therefore F6 applies with $N={\gamma }_{\beta }(I-{\gamma }_{\beta }{)}^{-1}=N_{\beta }$. Its representation sends the formal generators to unitaries with the stated product and involution; pulling back the unit-vacuum expectation proves normalization and positivity of ${\varpi }_{\beta }$. The algebra itself is associative because the real-bilinear form $\sigma (g,f)=\mathrm{Im}⟨g,f⟩$ gives the identity $\sigma (g,f)+\sigma (g+f,h)=\sigma (f,h)+\sigma (g,f+h)$ for its multiplication phases; conjugate-linearity and antisymmetry give the involution laws. No finite-dimensional Weyl-algebra theorem or global Gibbs trace is used. [F4, F6, A1, algebra]

2.1 By F5, $U_R$ is unitary, preserves $\mathcal{D}$, and obeys the group law; smoothness and compact support are preserved by the invertible linear substitution. Radiality of $\omega$ gives $N_{\beta }{U}_R=U_R{N}_{\beta }$ on the multiplier domain. Consequently $U_R$ preserves both the multiplication phase and $⟨g,(I+2N_{\beta })g⟩$. The displayed ${\alpha }_R$ thus preserves products and involutions, has inverse ${\alpha }_{R^{-1}}$, and leaves ${\varpi }_{\beta }$ invariant on every generator and hence every finite linear combination. Multiplication by $e^{i\omega t}$ likewise preserves $\mathcal{D}$, the inner product and the covariance, proving the same assertions for ${\tau }_t$ and stationarity. These are local deductions from the explicit preparation, not empirical evidence for it. [F4, F5, step 1.2, algebra]

3.1 Repeated Weyl multiplication expresses the expectation defining $C_{\beta }$ as an exponential of a finite quadratic polynomial in the real $s_j$; all coefficients are finite on $\mathcal{D}$. Thus every displayed derivative exists. Step 2.1 leaves that expectation unchanged under simultaneous $U_R$ on all labels, so differentiation proves invariance of every such correlation. For two labels differentiation gives $⟨e^{i\omega t}g,f⟩\mathrm{/}2+\mathrm{Re}⟨e^{i\omega t}g,N_{\beta }f⟩$, consistently with F6. Finally $P(R\widehat{\mathbf{k}})=RP(\widehat{\mathbf{k}})R^{-1}$ follows directly from $P(n)=I-n{n}^T$ and orthogonality. With radial $\chi$ this proves $U_R{g}_v=g_{Rv}$; real-linearity in $v$ gives the Cartesian transformation stated above. This verifies the intended preparation and correlation symmetry on the specified smearing domain without imposing it on an arbitrary finite mode selection. [F4, F5, F6, step 1.2, step 2.1, algebra] ∎

Use Dereziński's section 9.3, equation (73) and Theorem 40(1),(4), for the left Araki–Woods representation and its vacuum functional. The imported hypotheses are a complex Hilbert space, a self-adjoint $0\le \gamma \le I$ with $\ker (I-\gamma )=0$, and labels in $D([\gamma (I-\gamma {)}^{-1}{]}^{1\mathrm{/}2})$. Here $\gamma ={\gamma }_{\beta }$; step 1.2 checks these hypotheses. The representation's positivity and field-moment formulas are imported; the rotation, stationarity and application to vector smearings are the local calculations above. The article citation supplies attribution of the physical assumption only.