fourmomenta

Four-momenta are four-vectors used in special relativity to describe the energy and momentum of particles in a relativistically invariant way. A particle’s four-momentum is p^μ = (E/c, p_x, p_y, p_z), where E is the total energy and p is the three-momentum. In natural units where c = 1, this becomes p^μ = (E, p).

The Minkowski norm of the four-momentum is p^μ p_μ = (E/c)^2 − p^2, an invariant under Lorentz transformations. This relation leads to the mass-shell condition E^2 = p^2 c^2 + m^2 c^4, where m is the particle’s rest mass. For massless particles, m = 0 and E = pc.

Four-momentum transforms as a four-vector under Lorentz transformations, so the results of dot products with other four-vectors are frame-independent. In any physical process, the total four-momentum is conserved: ∑ p^μ_in = ∑ p^μ_out. This conservation holds in decays and scattering, and is a central constraint in collider physics.

The time component of p^μ corresponds to energy (divided by c if using units with c explicit), while the spatial components give the momentum. In the center-of-mass frame, the total spatial momentum vanishes, and the total energy relates to the invariant mass of the system.

The concept extends to systems of particles, where the sum of their four-momenta characterizes the overall energy-momentum content. Four-momenta provide a compact, frame-independent language for relativistic kinematics and conservation laws.