Four-momentum

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In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional space-time. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in space-time. The covariant four-momentum of a particle with three-momentum \vec p = (p_x, p_y, p_z) and energy E is

\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.

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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^{\mu\nu} p_\mu p_\nu = {E^2 \over c^2} - |\vec p|^2 = m^2c^2

where we use the SI units convention that

\eta^{\alpha\beta} = \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.

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_{\mu\nu} 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.

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

  1. The total energy E = - p0 is conserved.
  2. 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 pA and pB 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 qμ = pAμ + pBμ, 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 Aμ 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^{\mu\nu} p_{\nu}}{m} = \frac{1}{2m} \frac{d}{dt} |p|^2 = \frac{1}{2m} \frac{d}{dt} (-m^2c^2) = 0 .

For applications in relativistic quantum mechanics, it is useful to define a "canonical" momentum four-vector, Pμ, 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.

  • Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd). Oxford: Oxford University Press. ISBN 0-19-853952-5. 
Retrieved from "http://en.wikipedia.org/wiki/Four-momentum"
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