Classical Electrodynamic gauge interaction.
Posted by peeterjoot on October 22, 2010
In  chapter 6, we have a statement that in classical mechanics the electromagnetic interaction is due to a transformation of the following form
Let’s verify that this does produce the classical interaction law. Putting a more familiar label on this, we should see that we obtain the Lorentz force law from a transformation of the Hamiltonian.
Recall that the Hamiltonian was defined in terms of conjugate momentum components as
we can take partials to obtain the first of the Hamiltonian system of equations for the motion
With , and taking partials too, we have the system of equations
Starting with the free particle Hamiltonian
we make the transformation required to both the energy and momentum terms
From 2.4b we find
Taking derivatives and employing 2.4a we have
Rearranging and utilizing the convective derivative expansion (ie: chain rule), we have
We guess and expect that the first term of 3.9 is . Let’s verify this
Since we have
Except for a difference in dummy summation variables, this matches what we had in 3.9. Thus we are able to put that into the traditional Lorentz force vector form
It’s good to see that we get the classical interaction from this transformation before moving on to the trickier seeming QM interaction.
 BR Desai. Quantum mechanics with basic field theory. Cambridge University Press, 2009.