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When an electron of Lorentz factor $\gamma >1$ annihilates with a stationary equal-mass positron, transverse four-momentum conservation makes the photons' transverse components opposite. Eliminating their positive frequencies gives the displayed relation for noncollinear photon angles to the incoming electron. The exactly collinear forward/backward configuration is physically allowed but makes the quotient $0/0$; its undivided cross-multiplied identity remains valid. The angle relation thus requires nondegeneracy or an appropriate limit, rather than an assertion that every allowed angle pair defines the quotient.

In a local orthonormal tetrad, anisotropic stress is the trace-free spatial part of the stress-energy tensor, conventionally denoted ${\pi }^{ij}$. Some linear cosmological perturbation theory conventions instead use ${\Pi }^{ij}=-{\pi }^{ij}$ and write $T^{ij}=P{\delta }^{ij}-{\Pi }^{ij}$. The sign must be specified before using a photon Euler equation. The photon quadrupole determines the linear photon anisotropic stress.

Bremsstrahlung is electromagnetic radiation emitted when a charged particle accelerates during interaction with another charged particle or field. The microscopic process $e^{-}+Z\to e^{-}+Z+\gamma$ creates a photon, while its inverse absorbs one. Thus it contributes to a photon-number collision source in the photon number current, unlike ordinary Compton scattering. Thermal bremsstrahlung averages this emission over a thermal distribution of charged particles.

The electromagnetic four-potential has a vanishing temporal momentum and a Gauss law constraint in gauge theory. Gauge reduction leaves two transverse canonical pairs. Their canonical commutation relations give the photon creation and annihilation operators. Covariant Gupta-Bleuler quantization instead retains auxiliary polarization modes and selects the same physical state space by a subsidiary condition and null-state quotient.

The free transverse photon field is a Hermitian sum of two transverse-polarization plane-wave modes with creation and annihilation operators. Their standard bosonic commutator gives the transverse equal-time commutator. The normal-ordered Hamiltonian is a sum of positive oscillator energies.

An instantaneous kinetic suppression estimate replaces sound speed by the velocity dispersion of cold dark matter. Nonrelativistic particles in thermal contact with photons have $\sigma \propto a^{-1/2}$; after kinetic decoupling, their momenta redshift and $\sigma \propto a^{-1}$. A background dynamical estimate using total density is then constant during radiation domination and decreases as $a^{-1/2}$ during matter domination. Using only the species' own matter density instead makes the self-gravitating collisionless Jeans length decrease as $a^{-1/2}$ throughout the decoupled nonrelativistic era. Neither instantaneous estimate is the accumulated distance traveled by collisionless free streaming.

The cosmic microwave background last-scattering surface is the shell from which the photons observed in the Cosmic microwave background last scattered around photon decoupling.

A species freezing out while relativistic retains its thermal comoving number. If it is nonrelativistic today, ${\rho }_x\simeq m_x{n}_x$. Comparing with photons gives ${\Omega }_x/{\Omega }_{\gamma }=m_x(n_x/n_{\gamma })/⟨E_{\gamma }⟩$. An upper density fraction therefore bounds its mass inversely with its relic number density. The relation does not apply unchanged to a species whose abundance is depleted by nonrelativistic annihilation.

Detector conversion gain is the number of collected Electrons per analogue-to-digital unit. With independent Poisson distribution photoelectron counts, the shot-noise variance in count units is the mean count divided by the gain. Detected-photon and Electron conventions agree only when each detected photon contributes one Electron.

Double Compton scattering is $e^{-}+\gamma \to e^{-}+\gamma +\gamma$, together with its inverse. The forward reaction has one incoming and two outgoing photons, so it changes their number and can source the divergence of the photon number current. Ordinary Compton scattering has one incoming and one outgoing photon and conserves photon number. Forward and inverse rates can balance without either reaction individually conserving it.

The electromagnetic interaction is the long-range interaction of electrically charged matter with the electromagnetic field. Its massless gauge boson is the photon.

At tree level, the Feynman diagram has an electron-positron current and a muon-antimuon current joined by one virtual photon. In the massless limit, the spin average and final-state spin sum give $\overline{|\mathcal{M}{|}^2}=e^4(1+{\cos }^2\theta )$, and the integrated relativistic scattering cross-section is $4\pi {\alpha }^2/(3s)$.

Thermal electron-positron pairs in chemical equilibrium with photons have opposite chemical potentials when the one-particle energy includes rest energy. For negligible degeneracy and $k_{B}T\ll m_e{c}^2$, the zero-chemical-potential density per charge species is $n_0=2[m_e{k}_{B}T/(2\pi {\hslash }^2){]}^{3/2}{e}^{-m_e{c}^2/(k_{B}T)}$. In that nondegenerate limit, charge neutrality and the classical equilibrium product $n_{-}{n}_{+}=n_0^2$ give $n_{+}=(\sqrt{n_b^2+4n_0^2}-n_b)/2$, with net Electron density $n_b$. Pairs are important when $n_0\gtrsim n_b$. At arbitrary temperature the zero-chemical-potential marker uses the integral $n_0=({\pi }^2{\hslash }^3{)}^{-1}{\int }_0^{\infty }{p}^2[\exp (\sqrt{m_e^2{c}^4+p^2{c}^2}/k_{B}T)+1{]}^{-1}dp$. Its ultrarelativistic limit is $3\zeta (3)(k_{B}T/\hslash c{)}^3/(2{\pi }^2)$. A nonzero net chemical potential shifts the exact abundance boundary; the zero-potential curve is an approximate crossover marker.

For a Higgs doublet with hypercharge $1/2$ and vacuum expectation value $(0,v/\sqrt{2}{)}^T$, its gauge-covariant kinetic term gives the neutral gauge-boson mass matrixin the $(W^3,B)$ basis. Its eigenvalues are zero and $(g^2+g^{\prime 2})v^2/4$, and the Weinberg angle rotation gives the photon and Z boson. The two charged electroweak gauge bosons have $m_W^2=g^2{v}^2/4$. The zero eigenvalue expresses the unbroken electric-charge gauge symmetry.

An exciton is a bound state of an electron and an electron hole. Coupling this material excitation strongly to a photon can form a polariton.

For a blackbody grain with geometric cross-section $\sigma$, mass $m$ and stellar-frame velocity $\mathbf{v}$, define the radiation-pressure coefficient $\beta =L_{\star }\sigma /(4\pi cG{M}_{\star }m)$. To first order in $v/c$, steady absorption and isotropic rest-frame reradiation giveThe relative incoming flux provides $1-\dot{r}/c$; photon reradiation carries stellar-frame momentum proportional to $\mathbf{v}/c$. Subtracting static radiation pressure gives radial drag $-2m\beta G{M}_{\star }\dot{r}/(c{r}^2)$ and transverse drag $-m\beta G{M}_{\star }{v}_{\theta }/(c{r}^2)$. Non-blackbody grains require their actual radiation-pressure efficiency.

In metric signature $(+---)$, a future-pointing timelike four-vector has positive time component larger than its spatial magnitude. Two such vectors have strictly positive Minkowski metric scalar product, since $P\cdot Q=P^0{Q}^0-\mathbf{P}\cdot \mathbf{Q}\ge P^0{Q}^0-|\mathbf{P}||\mathbf{Q}|>0$. In the opposite signature the scalar-product sign reverses, while the causal condition is the same. The future timelike cone is closed under addition, which rules out a null photon momentum equalling a sum of two massive future four-momenta.