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The invariant future mass-shell measure is $d^{3}p/E$, up to fixed normalization factors absorbed in $f$. Its relation to the Lorentz-invariant phase-space measure follows by integrating $2\theta (p^0)\delta (p_{\mu }{p}^{\mu }+m^2)d^{4}p$ over $p^0$ in a local orthonormal tetrad. A particle carries four-momentum $p^{\mu }$ and crosses a local spatial surface with velocity $p^i/p^0$. Integrating the associated momentum flux gives the displayed stress-energy tensor. For photons, $p^{\hat{a}}=E(1,\mathbf{e})$ and $|\mathbf{e}|=1$.

A stationary particle of rest mass $m$ absorbing a photon of energy $E$ acquires total invariant massThis follows from four-momentum conservation, since the incoming photon has momentum magnitude $E/c$. If the system splits into two equal particles, the largest kinematically permitted daughter rest mass is $M/2$, attained when both are at rest in the centre-of-momentum frame. Their common laboratory speed at threshold is $cE/(m{c}^2+E)$. Above this daughter-mass threshold there is insufficient centre-of-momentum energy; below it, relative kinetic energy is available.

The number and energy integrals of a zero-chemical-potential photon gas give $n=2\zeta (3)T^3/{\pi }^2$ and $\rho ={\pi }^2{T}^4/15$ in natural units. Their ratio is the mean energy per photon. It is useful for converting a relic number-density ratio into an energy-density ratio.

Momentum density is momentum per unit physical volume. In a local orthonormal tetrad of the measuring observer it is $q^{\hat{i}}=T^{\hat{0}\hat{i}}$. A perfect fluid in general relativity moving at a small velocity relative to that observer has $q^{\hat{i}}=(\rho +P)v^{\hat{i}}$ to first order. For isotropic background photons, $P=\rho /3$, so the factor is $4\rho /3$ rather than $\rho$. Integrating the kinetic stress-energy tensor gives $q^{\hat{i}}=\int f{p}^{\hat{i}}{d}^{3}p$.

At tree level this QED process has two internal muon exchange diagrams, differing by the order of the two outgoing photon vertices. Their propagator momenta are $p_1-k_1$ and $p_1-k_2$ for incoming muon momentum $p_1$. The amplitude begins at order $e^2$ and obeys the two-photon fermion Ward identity. Identical final photons require the usual factorial correction when integration over the full final-state Lorentz-invariant phase space would otherwise count them twice.

For a temporal photon wave packet $|f,0⟩=\int d^{3}pf(\mathbf{p})a_{\mathbf{p}}^{0†}|0⟩/(2\pi {)}^3$, the oscillator commutator gives $⟨f,0|f,0⟩=-\int d^{3}p|f(\mathbf{p}){|}^2/(2\pi {)}^3<0$ when $f\ne 0$. This is the negative direction of the indefinite Hermitian form, not the divergent normalization of an unsmeared momentum eigenstate. It is excluded by the physical condition in Gupta-Bleuler quantization.

With negligible chemical potential, the Fermi-Dirac distribution has $3/4$ of the boson number density per degree of freedom at equal temperature. A species with $g_{\nu }$ populated states therefore has $n_{\nu }/n_{\gamma }=(3g_{\nu }/8)(T_{\nu }/T_{\gamma }{)}^3$, since photons have two states. This remains the number-density ratio after the neutrino becomes massive if its comoving particle number is conserved.