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For future-directed particles with total four-momentum $P$ and metric signature $(-+++)$, the positive square root of $-P_{\mu }{P}^{\mu }$ is their energy in the frame where total spatial momentum vanishes. Its invariant definition remains valid at a collision in spacetime.

In an isolated relativistic collision, total four-momentum is conserved; its time component states conservation of total relativistic energy.

The inclusive product of hadronic-current matrix elements, summed over final states with a Lorentz-invariant phase-space measure. Electromagnetic current conservation makes both indices transverse to the exchanged four-momentum.

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 $|\mathbf{e}|=1$.

For the positive-sign covariant gauge-fixing inverse, $p^a{G}_{ab}=\xi p_b/p^2$. It is not identically zero for fixed $\xi \ne 0$ and is a nonzero vector at a nonzero non-null four-momentum. Null momenta require the chosen pole prescription, and individual zero components are possible. The Landau gauge limit suppresses this longitudinal part. The Ward identity removes longitudinal contributions from physical conserved-current amplitudes.

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.

For four-momentum $p^{\mu }$ and four-velocity $u^{\mu }$, the covariant Lorentz-force law isIts spatial and temporal components are

The stabilizer of a timelike four-momentum in $D$-dimensional Minkowski spacetime has connected group $SO(D-1)$. Massive particle polarizations form representations of this group.

A massive spin-one particle transforms as a vector under its massive particle little group $SO(D-1)$. Its $D-1$ polarizations are orthogonal to its timelike four-momentum. Under transverse $SO(D-2)$ rotations they split into a vector and a scalar.

The stabilizer of a nonzero null four-momentum has connected group $ISO(D-2)$. For ordinary finite-polarization particles its translation part acts trivially, leaving a representation of $SO(D-2)$.

A massless supermultiplet is a positive-energy unitary irreducible representation of the Super-Poincaré algebra with null four-momentum. For finite helicity states in four dimensions, half the supercharges act trivially and the others form a fermionic Fock space.

A particle polarization specifies a physical spin state at fixed four-momentum. Massive spin states transform under the rotation little group $SO(D-1)$; finite-spin massless states transform under its transverse rotation part $SO(D-2)$. A massive two-form field has $\left(\genfrac{}{}{0ex}{}{D-1}{2}\right)$ such states, while a two-form gauge field has $\left(\genfrac{}{}{0ex}{}{D-2}{2}\right)$. Polarization components must be counted after field equations and any gauge equivalence are imposed.