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Four-momentum
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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 and energy is
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.
he reciprocal of the metric tensor of special relativity.
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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 and energy is
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:
where we use the SI units convention that
is the reciprocal of the metric tensor of special relativity. Because 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:
where the four-velocity is
and 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 is conserved.
Note that the invariant 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 and potential energy from forces between the particles contribute to the invariant mass. As an example, two particles with four-momenta (-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
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, , which is the sum of the four-momentum and the product of the electric charge with the four-vector potential:
where the four-vector potential is a result of combining the scalar potential and the vector potential:
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.
See also
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