Lagrangian of a Force Law, with magnetic monopole

[praharmitra]

We know the covariant Lagrangian of the Lorentz force law. However, in presence of a magnetic monopole, one must add another term to the force law. This term, of course is the Dual Field Tensor, along with the magnetic charge 'g' as follows -


$$m \frac{d^2 x^{\mu}}{d {\tau}^2} = \frac{e}{c}F^{\mu \nu} \frac{d x_{\nu}}{d \tau} + \frac{g}{c}G^{\mu \nu} \frac{d x_{\nu}}{d \tau}$$

Where G is the dual field tensor

Could anyone help me with writing the Lagrangian that gives rise to this equation?
Here is what I have tried till now.

1. Solve the original Lagrangian, get the equation of motion. Add the additional term, of G, and work backwards. However, I have realized that the required relation between A (potential) and a (dual potential, if I can call it that) is non-linear and non-local.

$$G^{ \mu \nu } = \partial_{\mu} a_{\nu} - \partial_{\nu} a_{\mu}$$


2. So i tried hit and trial, wrote down, using covariance, symmetry, and dimensions as my guides, several possible Lagrangians, and got the corresponding equation of motion, hoping to give the one that i want.

However, I have been unlucky so far. Have been working on it for several hours.

Can such a Lagrangian be written down explicitly in terms of A and a??

 

[diazona]

Have you tried adding additional terms to the regular Lagrangian which take the same form as the known terms, but using magnetic charge/current/potential instead of their electric counterparts? e.g. there's a term $$J^{\mu}A_{\mu}$$, so try adding $$j^{\mu}a_{\mu}$$ where j is magnetic 4-current. That's the first thing I would try.

 

[praharmitra]

no u misunderstood my question. The lagrangian you are talking about will give the Maxwell's Equations. I am not looking for that. I solved that.

What I am looking for is the Lagrangian which will tell me how a magnetic monopole interacts with electric and magnetic fields. I know from texts that the force law for a magnetic monopole is given by the equation as I have written. I want to arrive at a Lagrangian given the force law.

The non-magnetic monopole Lagrangian ( Action) which I am looking to modify is

Integral of $$-mc ds - e/c A_{\mu} d x^{\mu}$$

I cannot add terms like magnetic 4-current, because that tells me how charge produces fields.

Also, $$a_{\mu}$$ hasn't been defined yet. I am yet to calculate a in terms of A, which as I mentioned is non-linear, and non-local. This is what is causing issues

And yes, I am trying to modify th eregular Lagrangian only, but nothing yet :(