P-form electrodynamics

Template:Short description In theoretical physics, Template:Math-form electrodynamics is a generalization of Maxwell's theory of electromagnetism.

Ordinary (via. one-form) Abelian electrodynamics

We have a two-form $𝐀$, a gauge symmetry

$𝐀\to 𝐀+d\alpha ,$

where $\alpha$ 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

$d\star 𝐉=0,$

where $\star$ is the Hodge star operator.

Alternatively, we may express $𝐉$ as a closed Template:Math-form, but we do not consider that case here.

$𝐅$ is a gauge-invariant 2-form defined as the exterior derivative $𝐅=d𝐀$.

$𝐅$ satisfies the equation of motion

$d\star 𝐅=\star 𝐉$

(this equation obviously implies the continuity equation).

This can be derived from the action

$S=\underset{M}{\int }[\frac{1}{2}𝐅\wedge \star 𝐅-𝐀\wedge \star 𝐉],$

where $M$ is the spacetime manifold.

p-form Abelian electrodynamics

We have a Template:Math-form $𝐁$, a gauge symmetry

$𝐁\to 𝐁+d\mathit{\alpha },$

where $\alpha$ is any arbitrary fixed Template:Math-form and $d$ is the exterior derivative, and a gauge-invariant [[p-vector|Template:Math-vector]] $𝐉$ with density 1 satisfying the continuity equation

$d\star 𝐉=0,$

where $\star$ is the Hodge star operator.

Alternatively, we may express $𝐉$ as a closed Template:Math-form.

$𝐂$ is a gauge-invariant Template:Math-form defined as the exterior derivative $𝐂=d𝐁$.

$𝐁$ satisfies the equation of motion

$d\star 𝐂=\star 𝐉$

(this equation obviously implies the continuity equation).

This can be derived from the action

$S=\underset{M}{\int }[\frac{1}{2}𝐂\wedge \star 𝐂+(-1{)}^p𝐁\wedge \star 𝐉]$

where Template:Math is the spacetime manifold.

Other sign conventions do exist.

The Kalb–Ramond field is an example with Template:Math in string theory; the Ramond–Ramond fields whose charged sources are D-branes are examples for all values of Template:Math. In eleven-dimensional 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 Template:Math-form electrodynamics. They typically require the use of gerbes.

References

• Henneaux; Teitelboim (1986), "Template:Math-Form electrodynamics", Foundations of Physics 16 (7): 593-617, Template:Doi
• Template:Cite journal
• Navarro; Sancho (2012), "Energy and electromagnetism of a differential Template:Math-form ", J. Math. Phys. 53, 102501 (2012) Template:Doi