p-form electrodynamics

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

Ordinary (via. one-form) Abelian electrodynamics

We have a 1-form $\mathbf{𝐀}$, a gauge symmetry

$\mathbf{𝐀}\to \mathbf{𝐀}+d\alpha ,$

where $\alpha$ is any arbitrary fixed 0-form and $d$ is the exterior derivative, and a gauge-invariant vector current $\mathbf{𝐉}$ with density 1 satisfying the continuity equation

$d\star \mathbf{𝐉}=0,$

where $\star$ is the Hodge star operator.

Alternatively, we may express $\mathbf{𝐉}$ as a closed (n − 1)-form, but we do not consider that case here.

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

$\mathbf{𝐅}$ satisfies the equation of motion

$d\star \mathbf{𝐅}=\star \mathbf{𝐉}$

(this equation obviously implies the continuity equation).

This can be derived from the action

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

where $M$ is the spacetime manifold.

p-form Abelian electrodynamics

We have a p-form $\mathbf{𝐁}$, a gauge symmetry

$\mathbf{𝐁}\to \mathbf{𝐁}+d\mathit{\alpha },$

where $\alpha$ is any arbitrary fixed (p − 1)-form and $d$ is the exterior derivative, and a gauge-invariant p-vector $\mathbf{𝐉}$ with density 1 satisfying the continuity equation

$d\star \mathbf{𝐉}=0,$

where $\star$ is the Hodge star operator.

Alternatively, we may express $\mathbf{𝐉}$ as a closed (n − p)-form.

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

$\mathbf{𝐁}$ satisfies the equation of motion

$d\star \mathbf{𝐂}=\star \mathbf{𝐉}$

(this equation obviously implies the continuity equation).

This can be derived from the action

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

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 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 p-form electrodynamics. They typically require the use of gerbes.

References

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