# The Lorentz force derived from the Klein Gordon equation

1. Nov 20, 2008

### Hans de Vries

The Lorentz Force and Maxwell's equations derived from Klein Gordon's equation

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http://www.physics-quest.org/Book_Lorentz_force_from_Klein_Gordon.pdf

I posted several new chapters of my book lately, mostly involving the Klein Gordon equation.

This chapter shows how the Lorentz Force has a quantum mechanical base and is the
result of the U(1) group represention of the complex Klein Gordon field with electric charge.
( The interaction between $A^\mu$ and $\psi$ goes via the phase of the field )

After some introductions on the Aharonv-Bohm effect the Wilson loops are discussed in
combination with the U(1) symmetry. Thereafter we go to the derivation.

First the acceleration operator is obtained by applying the Hamiltonian twice on the position
operator of the Klein Gordon field.

Applying the acceleration operator on the Klein Gordon field with interaction gives the
the Lorentz Force, the acceleration is a result of the Lorentz Force.

There are lots of illustrations, for instance figures 11.3 through 11.6 show how the wave
nature leads to each of the individual components of the Lorentz force.

Regards, Hans

Code (Text):
[B]The Lorentz force derived from Klein Gordon's equation[/B]

11.1   Klein Gordon equation with EM interaction. . . . . . . . 2
11.2   Aharonov Bohm effect and experiments . . . . . . . . . . 4
11.3   The scalar phase and Wilson Loops. . . . . . . . . . . . 7
11.4   Lorentz force from the acceleration operator . . . . . . 10
11.5   Parallel electric Lorentz force term . . . . . . . . . . 14
11.6   Orthogonal electric Lorentz force term . . . . . . . . . 15
11.7   Parallel magnetic Lorentz force term . . . . . . . . . . 16
11.8   Orthogonal magnetic Lorentz force term . . . . . . . . . 17
11.9   The total four-vector Lorentz force. . . . . . . . . . . 18
11.10  Maxwell's equations. . . . . . . . . . . . . . . . . . . 20
11.11  Covariant derivative and gauge invariance. . . . . . . . 23

Last edited: Nov 20, 2008
2. Nov 20, 2008

### Naty1

Very interesting to see some illustrations and explanation of physical interactions underlying QFT...thanks!!! I had never seen an experimental set ups for QFT..

Why derive Lorentz Force and Maxwells equations from Klein Gorden? In other words, what is the motivation and are there new consequences of being able to do so?

3. Nov 20, 2008

### Hans de Vries

Thank you, Naty1

The complex Klein Gordon field is the simplest field incorporating the electric charge
and it is important to understand the time evolution of the field, which can be interpreted
as a charge-current density, under influence of an arbitrary four potential field $A^\mu$

The charge-current density of the Klein Gordon field is also one of the two components
of the charge-current density of the Dirac field, the other component is the charge-current
density from spin.

The next step is to repeat this for the time evolution of the Dirac field and to recover
equations like the Bargmann-Michel-Telegdi equation for the motion of spin directly
from the Dirac equation itself.

Regards, Hans.