# A Generalisation of Maxwell's equations

1. Oct 30, 2016

### spaghetti3451

The electromagnetic action in the language of differential geometry is given by

$\displaystyle{S \sim \int F \wedge \star F},$

where $A$ is the one-form potential and $F={\rm d}A$ is the two-form field strength.

At the extremum of the action $S$, $F$ is constrained by ${\rm d}F=0$ and ${\rm d}\star F=0$.

Now, generalise the above action to

$\displaystyle{S \sim \int H \wedge \star H}$

where $B$ is the two-form potential and $H={\rm d}B$ is the three-form field strength.

At the extremum of the action $S$, $H$ is constrained by ${\rm d}H=0$ and ${\rm d}\star H=0$.

Are there any qualitative differences between the two sets of equations in $d+1$-dimensions?

2. Oct 31, 2016

### dextercioby

Well, the em field is an abelian gauge 1-form field. The n-form abelian field has no obvious particularities in the Lagrangian formalism. Only when you try to quantize it with a path integral, you get complications due to the fact that the Hamiltonian constraints are n-1 fold reducible, hence the ghost+anti-ghost spectrum is much wider.