P-form electrodynamics
In theoretical physics, p-form electrodynamics is a generalization of Maxwell's theory of electromagnetism.
Ordinary (viz. one-form) Abelian electrodynamics
We have a one-form , a gauge symmetry
where is any arbitrary fixed 0-form and is the exterior derivative, and a gauge-invariant vector current with density 1 satisfying the continuity equation
where * is the Hodge dual.
Alternatively, we may express as a ()-closed form, but we do not consider that case here.
is a gauge invariant 2-form defined as the exterior derivative .
satisfies the equation of motion
(this equation obviously implies the continuity equation).
This can be derived from the action
where is the spacetime manifold.
p-form Abelian electrodynamics
We have a p-form , a gauge symmetry
where is any arbitrary fixed (p-1)-form and is the exterior derivative,
and a gauge-invariant p-vector with density 1 satisfying the continuity equation
where * is the Hodge dual.
Alternatively, we may express as a (d-p)-closed form.
is a gauge invariant (p+1)-form defined as the exterior derivative .
satisfies the equation of motion
(this equation obviously implies the continuity equation).
This can be derived from the action
where M is the spacetime manifold.
Other sign conventions do exist.
The Kalb-Ramond field is an example with p=2 in string theory; the Ramond-Ramond fields whose charged sources are D-branes are examples for all values of p. In 11d supergravity or M-theory, we have a 3-form electrodynamics.
Non-abelian generalization
Just as we have non-abelian generalizations of electrodynamics, leading to Yang–Mills theories, we also have nonabelian generalizations of p-form electrodynamics. They typically require the use of gerbes.
References
- Henneaux; Teitelboim (1986), "p-Form electrodynamics", Foundations of Physics 16 (7): 593-617, doi:10.1007/BF01889624
- Bunster, C.; Henneaux, M. (2011). "Action for twisted self-duality". Physical Review D. 83 (12). arXiv:1103.3621. Bibcode:2011PhRvD..83l5015B. doi:10.1103/PhysRevD.83.125015.
- Navarro; Sancho (2012), "Energy and electromagnetism of a differential k-form ", J. Math. Phys. 53, 102501 (2012) doi:10.1063/1.4754817