How many dimensions are needed to fully describe the electromagnetic force?

[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.

 

[vanhees71]

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.

 

[Simon Bridge]

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.

 

[juanrga]

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.

 

[PJC01]

I can say that the number of dimensions required to fully describe the electromagnetic force is a topic of ongoing research and debate. While Maxwell's equations do use four dimensions (three spatial dimensions and one time dimension) to describe the behavior of electromagnetic fields, it is understood that there may be additional dimensions necessary to fully capture all aspects of the electromagnetic force.

For example, as you mentioned, polarization and twist are important properties of electromagnetic waves that can be described using additional mathematical dimensions. In fact, theories such as string theory and extra dimensions propose the existence of additional dimensions beyond the four we are familiar with, which could potentially play a role in describing the electromagnetic force.

In terms of the size of matrices and degrees of freedom needed for a complete description, it is difficult to give a specific answer as it would depend on the specific theory or model being used. However, it is safe to say that a complete description of the electromagnetic force would require a significant number of mathematical dimensions and degrees of freedom.

In conclusion, while four dimensions are currently used to describe the electromagnetic force, it is likely that more dimensions will be needed for a complete and comprehensive understanding of this fundamental force. Continued research and advancements in theoretical physics will help us better understand the true nature of the electromagnetic force and the dimensions needed to fully describe it.