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 $\mathbf {A}$ , a gauge symmetry

{\mathbf {A}}\rightarrow {\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*{\mathbf {J}}=0

where * is the Hodge dual.

Alternatively, we may express $\mathbf {J}$ as a ( $d-1$ )-closed 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*{\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 $\mathbf {B}$ , a gauge symmetry

{\mathbf {B}}\rightarrow {\mathbf {B}}+d{\mathbf {\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*{\mathbf {J}}=0

where * is the Hodge dual.

Alternatively, we may express $\mathbf {J}$ as a (d-p)-closed 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*{\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
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

This article is issued from Wikipedia - version of the 10/2/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.