Saturday, June 26, 2010

The "N" Field

I have recently been exposed to a terrific idea, proposed by my friend Bill Polson, while enjoying a nice breakfast overlooking the coast.

He expressed a few ideas about Lagrangians that I think are rather revolutionary.

We usually think of an action integral is the integral of a scalar Lagrangian function along a parametrized path, with time as the parameter. The path that minimizes the result of this integral is the path that a particle takes.

Bill Polson introduced the idea of a vector valued Lagrangian. This is something I've done as well, but Bill does something rather special with his vector Lagrangian. He forms the action integral out of a line integral on a closed loop. The first branch of the loop represents the path the particle might take, while the returning branch contains a slight variation. He is then able to make a correlation, via Stokes Law, between the abstract idea of minimizing action, and the geometric properties of the vector Lagrangian field.

Bill introduces a vector field he calls N, which should contain vector and pseudo-vector portions. The action is stationary when integrated over the "area" contained inside of the closed path. He defined N as the "curl" of the vector Lagrangian field, but he meant the generalization of the curl to 4 dimensions, of course.

Now, I think this idea is rockin' as is. However, Bill went further and figured out what N needed to be in order to reproduce the Lagrangian of a charged particle in an EM field.

Again, the curl here is a generalization of the 3D curl of a vector field into 4 dimensions. Also, there are several factors such as c, charge and mass that Bill neglected for convenience.

Using the Clifford Algebra notation, this is how I would define N

Here, I have inserted the mass, charge, and speed of light constants where they are appropriate. If I replace F with its definition in terms of the four-vector potential A, this becomes

The quantity under the conjugation bar is the canonical momentum p of a particle in an EM field. Using this we can build an analogy. As p is to A, N is to F.

Now we have

Or, just

The Action integral SBP expressed in terms of Clifford Algebra notation is now

Now, let's compare this with my version of the action integral, which is an integral of a scalar Lagrangian density function over a volume.

Thus, according to this definition of action, my vector valued Lagrangian field is

I don't define an N field, but my vector Lagrangian is the same as Bill's apart from a constant factor.