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At a nonzero Euclidean momentum $k$, the longitudinal projector of a vector field is $P_L^{ab}=k^a{k}^b/k^2$. Its Lorentzian mixed-index form is $(P_L{)}^a{}_b=p^a{p}_b/p^2$ for non-null four-momentum. Together with the transverse projector of a vector field, it obeys $P_T+P_L=I$, $P_T{P}_L=0$, $P_T^2=P_T$, and $P_L^2=P_L$. This formula is not a pointwise decomposition at null four-momentum.

With a mostly-plus metric signature, define $f_d=(1/i)\int d^{d}k/[(2\pi {)}^d(k^2-iϵ)((P-k{)}^2-iϵ)]$. Wick rotation and a Feynman parameter give $f_d=\Gamma (2-d/2)(4\pi {)}^{-d/2}{\int }_0^1[x(1-x)P^2{]}^{d/2-2}dx$. At nonzero Euclidean $P^2$, the Gamma function supplies the pole. Exceptional massless momentum can introduce infrared ambiguity.

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.

For the normalized isotropic cutoff model, collisions redistribute the direction of relative velocity uniformly while keeping its length. With center $C=(v+v_{\ast })/2$ and relative speed $r=|v-v_{\ast }|$, outgoing velocities are $C\pm r\sigma /2$. Momentum and kinetic energy are preserved, and the collision rate has no relative-speed factor. The displayed operator uses unit total angular rate. General Maxwell molecule kernels can have nonconstant angular dependence.

The mass-normalized first velocity moment of a distribution on phase space. For a velocity-only distribution, omit the position integral. Nonzero finite mass and a finite first moment are required. For a probability distribution this is its expected value; its unnormalized counterpart is the momentum.

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 $q^{\hat{i}}=\int f{p}^{\hat{i}}{d}^{3}p$.

For a triangular-profile line plume with centreline density $\rho$, speed $W$, half-width $b$, ambient density ${\rho }_0$, and inward relative edge speed $e$, volume, mass and momentum conservation giveSubtracting mass conservation from ${\rho }_0$ times volume conservation also gives buoyancy conservation. Batchelor entrainment supplies a closure for $e$.

For equal-mass elastic collisions, one reflection parametrization is $v^{\prime }=v-((v-v_{\ast })\cdot n)n$ and $v_{\ast }^{\prime }=v_{\ast }+((v-v_{\ast })\cdot n)n$. The fixed-$n$ transformation preserves pairwise momentum, kinetic energy and velocity-pair measure. A nonnegative kernel invariant under particle interchange and collision reversal yields a gain-loss operator. Its weak Boltzmann collision identity proves conservation and its entropy identity proves the Boltzmann H theorem. Unlike the Maxwell molecule collision operator, this general kernel may depend on relative speed.

The interaction vertex $-i\lambda$ and scalar propagator $-i/(p^2-iϵ)$ give one four-point bubble diagram for each of three momentum channels, each with Feynman-diagram symmetry factor $1/2$. The two-point tadpole diagram is zero as a scaleless integral in dimensional regularization. If full proper vertices include the free quadratic kernel, ${\hat{\tau }}_2^{(0)}=-p^2$; the interaction self-energy convention instead starts at zero.

An operator-valued distribution assigns a linear operator on a common dense domain to each smooth test function of compact support $f$, continuously and linearly in the appropriate weak sense. A quantized field is generally of this type: $\varphi (x)$ is formal point notation, while the smeared $\varphi (f)$ has operator meaning. Equal-time canonical commutation relations and momentum oscillators are interpreted after smearing, which avoids treating a Dirac delta distribution or a coincident field as a finite ordinary operator. Products at coincident points can need further regularization even when individual smeared fields are defined.

Integration by parts against $1$, $v_i$ and $|v{|}^2/2$ gives these equations for mass, unnormalized momentum and kinetic energy. With $M_0\ne 0$, the mean is $u(t)=e^{-t}{u}_0$, and $E(t)=d{M}_0/2+(E_0-d{M}_0/2)e^{-2t}$. Normalizing by mass is invalid if that mass is zero.