Backward wave oscillator

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, a gauge symmetry

$\mathbf {A} \rightarrow \mathbf {A} +d\alpha$

where α is any arbitrary fixed 0-form and d is the exterior derivative, and a gauge-invariant vector current J with density 1 satisfying the continuity equation

$d*\mathbf {J} =0$

where * is the Hodge dual.

Alternatively, we may express J as a (d − 1)-closed form.

F is a gauge invariant 2-form defined as the exterior derivative $\mathbf {F} =d\mathbf {A}$.

A satisfies the equation of motion

$d*\mathbf {F} =*\mathbf {J}$

(this equation obviously implies the continuity equation).

This can be derived from the action

$S=\int _{M}\left[{\frac {1}{2}}\mathbf {F} \wedge *\mathbf {F} -\mathbf {A} \wedge *\mathbf {J} \right]$

where M is the spacetime manifold.

p-form Abelian electrodynamics

We have a p-form B, a gauge symmetry

$\mathbf {B} \rightarrow \mathbf {B} +d\mathbf {\alpha }$

where α is any arbitrary fixed (p-1)-form and d is the exterior derivative,

and a gauge-invariant p-vector J with density 1 satisfying the continuity equation

$d*\mathbf {J} =0$

where * is the Hodge dual.

Alternatively, we may express J as a (d-p)-closed form.

C is a gauge invariant (p+1)-form defined as the exterior derivative $\mathbf {C} =d\mathbf {B}$.

B satisfies the equation of motion

$d*\mathbf {C} =*\mathbf {J}$

(this equation obviously implies the continuity equation).

This can be derived from the action

$S=\int _{M}\left[{\frac {1}{2}}\mathbf {C} \wedge *\mathbf {C} +(-1)^{p}\mathbf {B} \wedge *\mathbf {J} \right]$

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

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• Navarro; Sancho (2012), "Energy and electromagnetism of a differential k-form ", J. Math. Phys. 53, 102501 (2012) doi:10.1063/1.4754817

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