P-form electrodynamics

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

Ordinary (via. one-form) Abelian electrodynamics

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

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

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

${\displaystyle d{\star }\mathbf {J} =0,}$

where ${\displaystyle {\star }}$ is the Hodge star operator.

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

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

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

${\displaystyle d{\star }\mathbf {F} ={\star }\mathbf {J} }$

(this equation obviously implies the continuity equation).

This can be derived from the action

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

where ${\displaystyle M}$ is the spacetime manifold.

p-form Abelian electrodynamics

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

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

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

${\displaystyle d{\star }\mathbf {J} =0,}$

where ${\displaystyle {\star }}$ is the Hodge star operator.

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

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

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

${\displaystyle d{\star }\mathbf {C} ={\star }\mathbf {J} }$

(this equation obviously implies the continuity equation).

This can be derived from the action

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

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 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, doi:10.1007/BF01889624
• Bunster, C.; Henneaux, M. (2011). "Action for twisted self-duality". Physical Review D. 83 (12): 125015. arXiv:1103.3621. Bibcode:2011PhRvD..83l5015B. doi:10.1103/PhysRevD.83.125015. S2CID 119268081.
• Navarro; Sancho (2012), "Energy and electromagnetism of a differential k-form ", J. Math. Phys. 53, 102501 (2012) doi:10.1063/1.4754817