Momentum Operator - 4-momentum Operator

4-momentum Operator

Inserting the 3d momentum operator above and the energy operator into the 4-momentum (as a 1-form with (+−−−) metric signature):

obtains the 4-momentum operator;

where ∂_μ is the 4-gradient, and the −i ħ becomes +i ħ preceding the 3-momentum operator. This operator occurs in relativistic quantum field theory, such as the Dirac equation and other relativistic wave equations, since energy and momentum combine into the 4-momentum vector above, momentum and energy operators correspond to space and time derivatives, and they need to be first order partial derivatives for Lorentz covariance.

The Dirac operator and Dirac slash of the 4-momentum is given by contracting with the gamma matrices:

If the signature was (−+++), the operator would be

instead.