Dual photon

For articles about physics of two photons, see Two-photon physics and Two-photon absorption.

In theoretical physics, the dual photon is a hypothetical elementary particle that is a dual of the photon under electric–magnetic duality which is predicted by some theoretical models, including M-theory.

Dual Photon

Composition	Elementary particle
Statistics	Bosonic
Family	Gauge boson
Interactions	Electromagnetic
Status	Hypothetical
Theorized	2000s
Electric charge	0 e
Spin	1

It has been shown that including magnetic monopole in Maxwell's equations introduces a singularity. The only way to avoid the singularity is to include a second four-vector potential, called dual photon, in addition to the usual four-vector potential, photon. Additionally, it is found that the standard Lagrangian of electromagnetism is not dual symmetric (i.e. symmetric under rotation between electric and magnetic charges) which causes problems for the energy–momentum, spin, and orbital angular momentum tensors. To resolve this issue, a dual symmetric Lagrangian of electromagnetism has been proposed, which has a self-consistent separation of the spin and orbital degrees of freedom. The Poincaré symmetries imply that the dual electromagnetism naturally makes self-consistent conservation laws.

Dual electromagnetism

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

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

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

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

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

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

For the electromagnetic field and its dual field, we have

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

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

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

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

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

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

p-form electrodynamics

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

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

which satisfies the equation of motion

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

where $\star$ is the Hodge star operator.

This implies the following action in the spacetime manifold $M$:

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

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

Dark photon

This artist's impression shows the dark photon A′ decays into an electron and a positron.

The dark photon is a spin-1 boson associated with a U(1) gauge field, which could be massless and behaves like electromagnetism. But, it could be unstable and massive, quickly decays into electron–positron pairs, and interact with electrons.

The dark photon was first suggested in 2008 by Lotty Ackerman, Matthew R. Buckley, Sean M. Carroll, and Marc Kamionkowski to explain the 'g–2 anomaly' in experiment E821 at Brookhaven National Laboratory. Nevertheless, it was ruled out in some experiments such as the PHENIX detector at the Relativistic Heavy Ion Collider at Brookhaven.

In 2015, the Hungarian Academy of Sciences's Institute for Nuclear Research in Debrecen, Hungary, suggested the existence of a new, light spin-1 boson, dubbed the X17 particle, 34 times heavier than the electron that decays into a pair of electron and positron with a combined energy of 17 MeV. In 2016, it was proposed that it is an X-boson with a mass of 16.7 MeV that explains the g−2 muon anomaly.

See also

• 't Hooft loop – Magnetic loop operator dual to the Wilson loop
• Covariant Maxwell's equations – Ways of writing certain laws of physics
• Dark radiation – Postulated type of radiation that mediates interactions of dark matter
• Dual graviton – Hypothetical particle found in supergravity
• Electromagnetic mathematics – Formulations of electromagnetism
• Kalb–Ramond field – Two-form field
• p-form electrodynamics – Generalization of electrodynamics
• Photino – Hypothetical superpartner of the photon
• Photon – Elementary particle or quantum of light
• Magnetic monopole – Hypothetical particle with one magnetic pole
• Magnetic photon, a different extension for magnetic monopoles
• Maxwell's equations – Electromagnetism in general relativity