Generalisation of Maxwell's equations

  • #1
spaghetti3451
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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?
 

Answers and Replies

  • #2
dextercioby
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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.
 
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