# How many dimensions for EM?

• PatrickPowers
In summary, Maxwell's equations use four dimensions, but it depends on the specific situation and definition of the EM force. For the old Lorenz force, 4D is sufficient, but for more general forces, such as in Stuckelberg theory, a higher number of dimensions is necessary.

#### PatrickPowers

Maxwell's equations use four dimensions. But wouldn't we need more dimensions to completely describe the EM force. I'm thinking of polarization and twist. So how many mathematical dimensions are necessary for a complete description? That is, how big do the matrices need to be and how many degrees of freedom do they have.

The electromagnetic field is a massless vector field, which means that the free field has two independent physically relevant field degrees of freedom. Of course, spacetime has four dimensions. One could think about fields in higher spacetime dimensions, i.e., look for the unitary representations of the Poincare group in more than four dimensions.

I'm thinking of polarization and twist.
... are not twist and polarization included in Maxwell's equations? Look how they come about!

Of course, you can represent anything you like in as many dimensions as you like.
Depends what you want to do with it.

@vanhees71: I was kinda interpreting the question to be asking about the minimum number of dimensions needed to completely describe all of E-Mag. I still think that's 4 - though it is possible to do it in any number like you say.

We can also do it in less for specific situations with a lot of symmetry.

PatrickPowers said:
Maxwell's equations use four dimensions. But wouldn't we need more dimensions to completely describe the EM force. I'm thinking of polarization and twist. So how many mathematical dimensions are necessary for a complete description? That is, how big do the matrices need to be and how many degrees of freedom do they have.

It depends what you mean by «EM force«. If you mean the old Lorenz force, then 4D are enough. If you mean more general forces (e.g. the internal force in a two-body covariant system) then you must go beyond Maxwell and special relativity. For instance the EM forces in Stuckelberg theory are defined in a 8N dimensional phase space. For a two-body system, the force is a function in 16 dimensions.