ICCA2020:P000075

A Clifford Bundle Approach to a Dirac-Hestenes Equation in the anti-de Sitter spacetime

*Samuel Wainer (Aeronautics Institute of Technology)

The purpose of this work is to extend the result obtained in [Rodrigues, et al.] in 2016 to the anti-de Sitter spacetime. To this end, we start by considering $M=SO(2,3)/SO(1,3)\simeq S^1\times\mathbb{R}^3$ a submanifold into the structure $(\mathbb{R}^5,\boldsymbol{\mathring{g}})$, where $\boldsymbol{\mathring{g}}$ is a pseudo-Euclidean metric of signature $(2,3)$. Let $\boldsymbol{i}:M\rightarrow\mathbb{R}^{5}$
be the inclusion map and $\boldsymbol{g}=\boldsymbol{i}^{\ast
}\boldsymbol{\mathring{g}}$ as the pullback metric on $M$ with signature $(1,3)$. Let $\boldsymbol{D}$ be the Levi-Civita connection of $\boldsymbol{g}%
$. The structure $(M,\boldsymbol{g})$ is called the anti-de Sitter manifold of signature $(2,3)$ and
$M^{adSL}=(M,\boldsymbol{g},\boldsymbol{D},\tau
_{\boldsymbol{g}},\uparrow)$ is called as the anti-de Sitter Lorentzian space of signature $(2,3)$, which is oriented by $\tau_{\boldsymbol{g}}\in\sec{\textstyle\bigwedge\nolimits^{4}}T^{\ast}M$ and time-oriented by $\uparrow$.  In this work, we show that if the movement of a particle restricted to $M^{adSL}$ without the action of any force detectable by observers in $M$ happens under constant angular momentum seen in $\mathbb{R}^{2,3},$ then this movement in $ M^{adSL} $ happens in a geodesic. Also any movement in a geodesic in the structure $ M^{adSL} $ implies that this movement of the particle happens under constant angular momentum in $ \mathbb {R} ^ {2,3} $. Using the Clifford Bundle approach, we derive a wave equation, called Dirac-Hestenes equation in the anti-de Sitter spacetime 2, of a free spin $1/2$ fermion in the de anti-de Sitter spacetime $M^{adSL},$ which is analogous to the Dirac equation in Minkowski spacetime. Furthermore, considering the quantum angular momentum operator $\boldsymbol{L}$ and factorizing the two Casimir invariants $C_{1}:=-\frac{1}{\ell}\boldsymbol{L}\cdot\boldsymbol{L} $ and $C_{2}:=-\frac{1}{64\ell^2}(\boldsymbol{L}\wedge \boldsymbol{L})(\boldsymbol{L}\wedge \boldsymbol{L})$ of the Lie algebra of the de Sitter group, we obtain an equation that we
called Dirac-Hestenes equation in de anti-de Sitter spacetime 1. Finally, in our conclusion we compare these equations. 


Reference: Rodrigues, W.A., Wainer, S.A., Rivera-Tapia, M., Notte-Cuello, E.A., and Kondrashuk, I., A Clifford Bundle Approach to the Wave Equation of a Spin 1/2 Fermion in the de Sitter Manifold, Adv. Appl. Clifford Algebras, vol. $\boldsymbol{26}$, 253-277 (2016).

The purpose of this work is to extend the result obtained in [Rodrigues, et al.] in 2016 to the anti-de Sitter spacetime. To this end, we start by considering $M=SO(2,3)/SO(1,3)\simeq S^1\times\mathbb{R}^3$ a submanifold into the structure $(\mathbb{R}^5,\boldsymbol{\mathring{g}})$, where $\boldsymbol{\mathring{g}}$ is a pseudo-Euclidean metric of signature $(2,3)$. Let $\boldsymbol{i}:M\rightarrow\mathbb{R}^{5}$ be the inclusion map and $\boldsymbol{g}=\boldsymbol{i}^{\ast }\boldsymbol{\mathring{g}}$ as the pullback metric on $M$ with signature $(1,3)$. Let $\boldsymbol{D}$ be the Levi-Civita connection of $\boldsymbol{g}% $. The structure $(M,\boldsymbol{g})$ is called the anti-de Sitter manifold of signature $(2,3)$ and $M^{adSL}=(M,\boldsymbol{g},\boldsymbol{D},\tau _{\boldsymbol{g}},\uparrow)$ is called as the anti-de Sitter Lorentzian space of signature $(2,3)$, which is oriented by $\tau_{\boldsymbol{g}}\in\sec{\textstyle\bigwedge\nolimits^{4}}T^{\ast}M$ and time-oriented by $\uparrow$. In this work, we show that if the movement of a particle restricted to $M^{adSL}$ without the action of any force detectable by observers in $M$ happens under constant angular momentum seen in $\mathbb{R}^{2,3},$ then this movement in $ M^{adSL} $ happens in a geodesic. Also any movement in a geodesic in the structure $ M^{adSL} $ implies that this movement of the particle happens under constant angular momentum in $ \mathbb {R} ^ {2,3} $. Using the Clifford Bundle approach, we derive a wave equation, called Dirac-Hestenes equation in the anti-de Sitter spacetime 2, of a free spin $1/2$ fermion in the de anti-de Sitter spacetime $M^{adSL},$ which is analogous to the Dirac equation in Minkowski spacetime. Furthermore, considering the quantum angular momentum operator $\boldsymbol{L}$ and factorizing the two Casimir invariants $C_{1}:=-\frac{1}{\ell}\boldsymbol{L}\cdot\boldsymbol{L} $ and $C_{2}:=-\frac{1}{64\ell^2}(\boldsymbol{L}\wedge \boldsymbol{L})(\boldsymbol{L}\wedge \boldsymbol{L})$ of the Lie algebra of the de Sitter group, we obtain an equation that we called Dirac-Hestenes equation in de anti-de Sitter spacetime 1. Finally, in our conclusion we compare these equations. Reference: Rodrigues, W.A., Wainer, S.A., Rivera-Tapia, M., Notte-Cuello, E.A., and Kondrashuk, I., A Clifford Bundle Approach to the Wave Equation of a Spin 1/2 Fermion in the de Sitter Manifold, Adv. Appl. Clifford Algebras, vol. $\boldsymbol{26}$, 253-277 (2016).

The 12th International Conference on Clifford Algebras and Their Applications in Mathematical Physics

A Clifford Bundle Approach to a Dirac-Hestenes Equation in the anti-de Sitter spacetime