Duality as a method to derive a gauge invariant massive electrodynamics and new interactions

Abstract

Taking into account the recent developments associated with duality in physics, this article is focused on investigating the properties of a tensor generalization of the electrodynamics dual to the standard vector model even considering the full radiative corrections. A discussion on duality is extended for nonlinearly interacting models. The alternative tensor structure allows a new local self-interaction and also a partial gauge invariant mass term. In this way, the renormalization conditions, a new interaction, Ward–Takahashi identities for all the sets of gauge symmetries, and the Schwinger–Dyson quantum equations are carefully provided to consistently characterize the propagating modes of the quantum system. The possibility of this gauge invariant massive extension is considered from the perspective of a generalized Stueckelberg procedure for higher derivative systems.

Appendices

Appendix 1

The spin 1 projectors, from which the rank 2 ones are constructed, are listed below

$$\begin{aligned} \theta _{\mu \nu }(k)=\eta _{\mu \nu }-\frac{k_\mu k_\nu }{k^2}\quad , \quad \omega _{\mu \nu }= \frac{k_\mu k_\nu }{k^2} \end{aligned}$$

(91)

With the properties

$$\begin{aligned} \theta _{\mu \nu }(k)\theta ^{\mu \alpha }(k)=\theta ^{\alpha }_\nu (k)\,\ \omega _{\mu \nu }(k)\omega ^{\mu \alpha }(k)=\omega ^{\alpha }_\nu (k)\ ,\ \theta _{\mu \nu }(k)\omega ^{\mu \alpha }(k)=0\ ,\ \theta _{\mu \nu }(k)+\omega _{\mu \nu }(k)=\eta _{\mu \nu } \end{aligned}$$

(92)

Then, the spin 2 projectors are of the following

$$\begin{aligned}{} & {} {P^{(2)}_{ss} }_{\ \ \rho \sigma }^{\mu \nu }=\frac{1}{2}\left( \theta ^{\mu }_{\ \rho }\theta ^{\nu }_{\ \sigma }+\theta ^{\mu }_{\ \sigma }\theta ^{\nu }_{\ \rho } \right) -\frac{\theta ^{\mu \nu }\theta _{\rho \sigma }}{(D-1)}\quad ,\quad {P^{(1)}_{ss} }_{\ \ \rho \sigma }^{\mu \nu }=\frac{1}{2}\left( \theta ^{\mu }_{\ \rho }\omega ^{\nu }_{\ \sigma }+\theta ^{\mu }_{\ \sigma }\omega ^{\nu }_{\ \rho }+\theta ^{\nu }_{\ \rho }\omega ^{\mu }_{\ \sigma }+\theta ^{\nu }_{\ \sigma }\omega ^{\mu }_{\ \rho } \right) \end{aligned}$$

(93)

$$\begin{aligned}{} & {} {P^{(0)}_{s\omega } }_{\ \ \rho \sigma }^{\mu \nu }=\frac{\theta ^{\mu \nu }\omega _{\rho \sigma }}{\sqrt{(D-1)}} \quad ,\quad {P^{(0)}_{ss} }_{\ \ \rho \sigma }^{\mu \nu }=\frac{\theta ^{\mu \nu }\theta _{\rho \sigma }}{(D-1)} \end{aligned}$$

(94)

$$\begin{aligned}{} & {} {P^{(0)}_{\omega \omega } }_{\ \ \rho \sigma }^{\mu \nu }=\omega ^{\mu \nu }\omega _{\rho \sigma } \quad ,\quad {P^{(0)}_{\omega s} }_{\ \ \rho \sigma }^{\mu \nu }=\frac{\omega ^{\mu \nu }\theta _{\rho \sigma }}{\sqrt{(D-1)}} \end{aligned}$$

(95)

$$\begin{aligned}{} & {} {P^{(0)}_{s\omega } }_{\ \ \rho \sigma }^{\mu \nu }=\frac{\theta ^{\mu \nu }\omega _{\rho \sigma }}{\sqrt{(D-1)}} \end{aligned}$$

(96)

The completeness relations are the following

$$\begin{aligned} {P^{(2)}_{ss} }_{\ \ \rho \sigma }^{\mu \nu }+{P^{(1)}_{ss} }_{\ \ \rho \sigma }^{\mu \nu }+{P^{(0)}_{ss} }_{\ \ \rho \sigma }^{\mu \nu }+{P^{(0)}_{\omega \omega } }_{\ \ \rho \sigma }^{\mu \nu }=\frac{1}{2}\left( \delta ^\mu _\rho \delta ^\nu _\sigma +\delta ^\nu _\rho \delta ^\mu _\sigma \right) \end{aligned}$$

(97)

The projector algebra reads $P^{(s)}_{IL}P^{(\tilde{s}}_{JK}=\delta ^{s\tilde{s}}P^{(s)}_{IK}$

Appendix 2

The gamma matrices obey the relations

$$\begin{aligned} \left\{ \gamma ^\mu ,\gamma ^\nu \right\} =2I_{4\times 4}\eta ^{\mu \nu } \end{aligned}$$

(98)

For the traces, we have

$$\begin{aligned} tr(\gamma ^\mu )&\,=\,0\,=\,tr(\gamma ^{\mu \ 1 }...\gamma ^{\mu n+1 }) \ , \ tr(\gamma ^\mu \gamma ^\nu )=4\eta ^{\mu \nu }\nonumber \\ tr(\gamma ^\mu \gamma ^\nu \gamma ^\rho \gamma ^\sigma )&\,=\,4\left( \eta ^{\mu \nu }\eta ^{\rho \sigma }-\eta ^{\mu \rho }\eta ^{\nu \sigma }+\eta ^{\mu \sigma }\eta ^{\nu \rho } \right) \end{aligned}$$

(99)