The transition was done from the 4-vector of the generalized velocity to a new 4-vector of the Hamiltonian, specifying the energy and the momentum of substance in fundamental fields.
We shall introduce 4-vector of the generalized velocity with the covariant index:
delta]] is the rotor of 4-vector of generalized velocities [S.
mu]] according to , and the 4-vector of generalized velocity (66):
or -]] particle is described by a 4-vector function [W.
is not related to a Hamiltonian because (13) is a Lorentz scalar whereas the Hamiltonian is a 0-component of a 4-vector.
The first term on the right side of (2) is the relativistic damping-force 4-vector
that leads to (1) in the nonrelativistic limit.
Now, density is a 0-component of a 4-vector and the same is true for energy.
3]] and be a 0-component of a 4-vector satisfying the continuity equation
Notice that with the 4-vector
formulation Maxwell equation are written in terms of two sets of equations.
mu]] is the 4-vector potential of the electromagnetic field and [j.
k] a magnetic interaction field that is separate from the 4-vector [A.