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Four-momentum

In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime. The covariant four-momentum of a particle with three-momentum

vec p = (p_x, p_y, p_z)

and energy

E

begin{pmatrix} p_0 p_1 p_2 p_3 end{pmatrix} = begin{pmatrix} -E p_x p_y p_z end{pmatrix}

The four-momentum is useful in relativistic calculations because it is a Lorentz vector. This means that it is easy to keep track of how it transforms under Lorentz transformations.

Minkowski norm: p²

Calculating the Minkowski norm of the four-momentum gives a Lorentz invariant quantity equal (up to factors of the speed of light c) to the square of the particle's proper mass:

- |p|^2 = - eta^{munu} p_mu p_nu = {E^2 over c^2} - |vec p|^2 = m^2c^2

where we use the SI units convention that

eta^{alphabeta} = begin{pmatrix}

-1/c^2 & 0 & 0 & 0 0 & 1 & 0 & 0 0 & 0 & 1 & 0 0 & 0 & 0 & 1 end{pmatrix}

is the reciprocal of the metric tensor of special relativity. Because $|p|^2!$ is Lorentz invariant, its value is not changed by Lorentz transformations, i.e. boosts into different frames of reference.

Relation to four-velocity

For a massive particle, the four-momentum is given by the particle's invariant mass times the particle's four-velocity:

p_mu = m , eta_{munu} U^nu!

where the four-velocity is

begin{pmatrix} U^0 U^1 U^2 U^3 end{pmatrix} = begin{pmatrix} gamma gamma v_x gamma v_y gamma v_z end{pmatrix}

and $gamma = frac{1}{sqrt{1-(frac{v}{c})^2}}$ is the Lorentz factor and c is the speed of light.

Conservation of four-momentum

The conservation of the four-momentum yields two conservation laws for "classical" quantities:

The total energy E = - p₀ is conserved.
The classical three-momentum $vec p$ is conserved.

Note that the mass of a system of particles may be more than the sum of the particles' rest masses, since kinetic energy in the system center-of-mass frame counts as system mass. As an example, two particles with the four-momentums {-5 Gev, 4 Gev/c, 0, 0} and {-5 Gev, -4 Gev/c, 0, 0} each have (rest) mass 3 Gev/c² separately, but their total mass (the system mass) is 10 Gev/c². If these particles were to collide and stick, the mass of the composite object would be 10 Gev/c².

One practical application from particle physics of the conservation of the invariant mass involves combining the four-momenta p^A and p^B of two daughter particles produced in the decay of a heavier particle with four-momentum q to find the mass of the heavier particle. Conservation of four-momentum gives q_μ = p^A_μ + p^B_μ, while the mass M of the heavier particle is given by -|q|² = M²c². By measuring the energies and three-momenta of the daughter particles, one can reconstruct the invariant mass of the two-particle system, which must be equal to M. This technique is used, e.g., in experimental searches for Z' bosons at high-energy particle colliders, where the Z' boson would show up as a bump in the invariant mass spectrum of electron-positron or muon-antimuon pairs.

If an object's mass does not change, the Minkowski inner product of its four-momentum and corresponding four-acceleration A^μ is zero. The acceleration is proportional to the time derivative of the momentum divided by the particle's mass, so

p_{mu} A^{mu} = p_{mu} frac{d}{dt} frac{eta^{munu} p_{nu}}{m} = frac{1}{2m} frac{d}{dt} |p|^2 = frac{1}{2m} frac{d}{dt} (-m^2c^2) = 0 .

Canonical momentum in the presence of an electromagnetic potential

For applications in relativistic quantum mechanics, it is useful to define a "canonical" momentum four-vector, $P_mu$ , which is the sum of the four-momentum and the product of the electric charge with the four-vector potential:

P_{mu} = p_{mu} + q A_{mu} !

where the four-vector potential is a result of combining the scalar potential and the vector potential:

begin{pmatrix} A_0 A_1 A_2 A_3 end{pmatrix} = begin{pmatrix} -phi A_x A_y A_z end{pmatrix}

This allows the potential energy from the charged particle in an electrostatic potential and the Lorentz force on the charged particle moving in a magnetic field to be incorporated in a compact way into the Schroedinger equation.