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 four-momentum of a particle with three-momentum \vec p = (p^x, p^y, p^z) and energy E is

p^{\mu} = \left( \frac{E}{c}, p^x, p^y, p^z \right)

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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[edit] 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 mass:

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

where we have chosen to use the convention gμν = diag(1,-1,-1,-1) for the metric. Because this quantity is Lorentz invariant, its value is not changed by Lorentz transformations, i.e., boosts into different frames of reference.

[edit] 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 = mU^\mu = m\left( \gamma c , \gamma u^x , \gamma u^y ,\gamma u^z \right) = \left( \gamma m c^2 /c , \gamma m u^x , \gamma m u^y ,\gamma m u^z \right) = \left( {E \over c} , p^x , p^y , p^z \right)

where \gamma m c^2 = E \,\! is the energy of the moving body, and c is the speed of light.

[edit] Conservation of four-momentum

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

  1. The total energy E = p0c is conserved.
  2. The classical three-momentum \vec p is conserved.
  3. The norm of the four-momentum m2c2 is conserved, so the particle's mass is conserved. This is also called the invariant mass.

In reactions between an isolated handful of particles, the total four-momentum is conserved, and therefore, so is the system mass (this is mass in the ordinary sense of rest mass, as used for m\,\! above). 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, 4, 0, 0} and {5, -4, 0, 0} both have (rest) mass 3, but their total mass (the system mass) is 10. If these particles were to collide and stick, the mass of the composite object would be 10.

One practical application from particle physics of the conservation of the invariant mass involves combining the four-momenta p1 and p2 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 = p1 + p2, while the mass M of the heavier particle is given by q2 = 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.

The Minkowski inner product pμaμ of a four-momentum pμ and the corresponding four-acceleration aμ is always zero. This can be understood as follows. The acceleration is equal to the time derivative of the momentum divided by the particle's mass, so

p^{\mu} a_{\mu} = p^{\mu} \frac{d}{dt} \frac{p_{\mu}}{m} = \frac{1}{2m} \frac{d}{dt} p^2 = \frac{1}{2m} \frac{d}{dt} m^2 = 0

because the mass does not vary with time.

[edit] 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, Pμ, which is the sum of the four-momentum and the product of the charge with the four-vector potential:

P^\mu = p^\mu +\frac{q}{c}\phi ^\mu

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.

[edit] See also

[edit] References

  • Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd). Oxford: Oxford University Press. ISBN 0-19-853952-5.
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