HOW CLOSE TO IDEAL IS THE PHOTON GAS - CORRECTIONS TO PLANCKS LAWS AT KT LESS-THAN ME

At temperatures well below the electron rest mass, the electron-positron concentrations in black-body radiation (BBR) are negligible, and deviations from Planck's laws are due to the photon-photon coupling described (in natural units) by the classic Euler-Heisenberg local interaction Lagrangean density ( α2 360π2me4){(E2 - B2)2 + 7(E · B)2}. Though unobservably small, these corrections answer the question in the title. They are best expressed in terms of the (frequency-independent) shift in the refractive index κ = (1 + Δκ) of BBR, where Δκ = α2( kT me)4 44π2 2025 ≈ 7.5 × 10-35( T 300)4. There are fractional changes of 3 Δκ 2 in the free-energy density and the pressure; 7 Δκ 2 in the energy density; and 2 Δκ in the mean-square electric field in any frequency range, whence only the intensity of the Planck spectrum is shifted but not its shape. The dielectric constant ε = (1 + Δε) and magnetic susceptibility μ = (1 + Δμ) of BBR are equal, with Δε = Δμ = Δκ, whence we compare the BBR shifts with those in an ordinary linear nondispersive medium having ε=μ=κ εμ. © 1991.