Dual photon

The free electromagnetic field is described by a covariant antisymmetric tensor ${\displaystyle F_{\alpha \beta }}$ of rank 2 by

${\displaystyle F_{\alpha \beta }=\partial _{\alpha }A_{\beta }-\partial _{\beta }A_{\alpha }\,.}$

where ${\displaystyle A_{\alpha }}$ is the electromagnetic potential.

The dual electromagnetic field ${\displaystyle \star F_{\alpha \beta }}$ is defined as

${\displaystyle \star F_{\alpha \beta }={\frac {1}{2}}\epsilon _{\alpha \beta }{}^{\sigma \lambda }F_{\sigma \lambda }.}$

where ${\displaystyle \star }$ is the Hodge star, and ${\displaystyle \epsilon _{\mu \nu \sigma \lambda }}$ is the Levi-Civita tensor

For the electromagnetic field and its dual field, we have

${\displaystyle \partial _{\alpha }F^{\alpha \beta }=0,}$

${\displaystyle \partial _{\alpha }{\star }F^{\alpha \beta }=0.}$

Then, for a given gauge field ${\displaystyle A_{\alpha }}$, the dual configuration ${\displaystyle \star A_{\alpha }}$ is defined as^[2]

${\displaystyle \star F^{\alpha \beta }(A)=F^{\alpha \beta }(\star A),}$

${\displaystyle \star F^{\alpha \beta }(\star A)=-F^{\alpha \beta }(A).}$

where ${\displaystyle \star A_{\alpha }}$ the field potential of the dual photon, and non-locally linked to the original field potential ⁠${\displaystyle A_{\alpha }}$⁠.

A p-form generalization of Maxwell's theory of electromagnetism is described by a gauge-invariant 2-form ${\displaystyle \mathbf {F} }$ defined as

${\displaystyle \mathbf {F} =d\mathbf {A} }$.

which satisfies the equation of motion

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

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

This implies the following action in the spacetime manifold ${\displaystyle M}$:^[7][8]

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

where ${\displaystyle \star \mathbf {F} }$ is the Hodge dual of the gauge-invariant 2-form ${\displaystyle \mathbf {F} }$ for the electromagnetic field.