Manuscript Em03-G@1 · Em03-G-v1.tex · theorem · 0bb3a8a3

Theorem: Finite signed excess radiative energy

Causality, interference energy, and switching work › The exact additional occupation kernel · lines 1904–1972 · proof included or linked

Em03-G@1#thm:energy, Em03-G@1#eq:occupationchange, Em03-G@1#eq:totalenergy

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The diagonal change of the outgoing occupation kernel is

δ n_{α} (ω) = ∣ d_{1, α} (ω) ∣^{2} + ∣ d_{2, α} (ω) ∣^{2} - 2 q u_{y} Im (e_{1, α} (ω) d_{2, α} (ω)) .

Its frequency-weighted integral is absolutely convergent:

δ E_{α} = \int_{0}^{\infty} ω δ n_{α} (ω) d ω .

The total excess is

δ E_{tot} = ∥ ω^{1/2} a_{1} ∥_{0}^{2} + ∥ ω^{1/2} a_{2} ∥_{0}^{2} - 2 q u_{y} Im ⟨ ω^{1/2} C a_{1}, ω^{1/2} a_{2} ⟩ .

It equals $δ E_{\infty} + δ E_{hor}$ .

The change is not asserted nonnegative. The interference term can change the background energy, and is not a rank-one mean-amplitude addition of the Em03-F type.

Proof. Let $N_{in} = diag (n_{H}, 0)$ . Gaussian differentiation of the Weyl characteristic function gives

ω_{U} (a (v)^{†} sin (2 X (a_{1}))) = 2 q ⟨ a_{1}, N_{in} v ⟩,

and the cosine contraction vanishes. Also

R = X cos (2 X (a_{1})) - Y sin (2 X (a_{1})), ⟨ Z R ⟩ = i q u_{y} .

Expand the occupation expectation using (eq:annconjugation). The squared displacement terms give $∣ d_{1} ∣^{2} + ∣ d_{2} ∣^{2}$ . Their source cross term is $- 2 q u_{y} Im (d_{1} d_{2})$ . The contractions with the background add $- 4 q u_{y} Im ((S N_{in} a_{1}) d_{2})$ . Since $C = I + 2 N_{in}$ , this is (eq:occupationchange). The computation can first be made on compact-band test kernels and then extended by continuity.

Lemma lem:compactregularity and pointwise scattering unitarity give

∥ ω^{1/2} d_{j} ∥_{0} < \infty, ∥ ω^{1/2} e_{1} ∥_{0} < \infty.

Cauchy–Schwarz proves absolute integrability of the interference term. Summing the two ends and using scattering unitarity proves (eq:totalenergy).

Displayed formulas in this passage

Em03-G@1#eq:occupationchange · line 1907
$δ n_{α} (ω) = ∣ d_{1, α} (ω) ∣^{2} + ∣ d_{2, α} (ω) ∣^{2} - 2 q u_{y} Im (e_{1, α} (ω) d_{2, α} (ω)) .$
Em03-G@1#eq:totalenergy · line 1923
$δ E_{tot} = ∥ ω^{1/2} a_{1} ∥_{0}^{2} + ∥ ω^{1/2} a_{2} ∥_{0}^{2} - 2 q u_{y} Im ⟨ ω^{1/2} C a_{1}, ω^{1/2} a_{2} ⟩ .$
unlabeled · line 1918
$δ E_{α} = \int_{0}^{\infty} ω δ n_{α} (ω) d ω .$
unlabeled · line 1942
$ω_{U} (a (v)^{†} sin (2 X (a_{1}))) = 2 q ⟨ a_{1}, N_{in} v ⟩,$
unlabeled · line 1948
$R = X cos (2 X (a_{1})) - Y sin (2 X (a_{1})), ⟨ Z R ⟩ = i q u_{y} .$
unlabeled · line 1965
$∥ ω^{1/2} d_{j} ∥_{0} < \infty, ∥ ω^{1/2} e_{1} ∥_{0} < \infty.$

Theorem: Finite signed excess radiative energy

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