Dirac’s Equation in R-Minkowski Spacetime

Abstract

We recently constructed the R-Poincaré algebra from an appropriate deformed Poisson brackets which reproduce the Fock coordinate transformation. We showed then that the spacetime of this transformation is the de Sitter one. In this paper, we derive in the R-Minkowski spacetime the Dirac equation and show that this is none other than the Dirac equation in the de Sitter spacetime given by its conformally flat metric. Furthermore, we propose a new approach for solving Dirac’s equation in the de Sitter spacetime using the Schrödinger picture.

Appendices

Appendix A: Squared Dirac Equation in Curved Spacetime

In a curved spacetime, Dirac’s equation is given by [25]

$$ \left( i \, \gamma^{\mu} \, D_{\mu} - \frac{m c}{\hbar} \right) \psi(x) = 0, $$

(86)

where D _μ is the spinor covariant derivative,

$$ D_{\mu}\psi=(\partial_{\mu}+{\Omega}_{\mu})\psi, $$

(87)

and

$$ {\Omega}_{\mu}(x) \equiv - \frac{i}{4}\,\omega_{ab\mu}(x)\,\sigma^{ab} = \frac{1}{8}\,\omega_{ab\mu}(x) \,[\,\gamma^{a},\,\gamma^{b}\,] $$

(88)

$$ \omega^{a}_{\hspace{.5em}b\mu} = e_{\nu}^{\hspace{.5em}a}\, \left( \partial_{\mu}\,e^{\hspace{.5em}\nu}_{b} + \, e_{b}^{\hspace{.5em}\sigma} \, {\Gamma}^{\nu}_{\sigma\mu} \right), $$

(89)

are respectively the Fock-Ivanenko coefficients [26] and the spin connection, ${\Gamma }^{\nu }_{\sigma \mu }$ being the Christoffel symbols. It follows that the square of Dirac’s equation [16]

$$ \left( i\gamma^{\mu}D_{\mu} + m c/\hbar \right)\,\left( i\gamma^{\nu}D_{\nu} - m c/\hbar \right)\psi =0\, $$

(90)

can be rewritten in the form

$$ \left[g^{\mu\nu}D_{\mu}D_{\nu} -\frac{1}{2} \sigma^{\mu\nu} K_{\mu\nu} + (m c/\hbar)^{2} \right]\psi=0, $$

(91)

where

$$ K_{\mu\nu} \equiv \frac{1}{2}(D_{\nu}\,D_{\mu} -D_{\mu}\,D_{\nu}) =\partial_{\nu}{\Omega}_{\mu} - \partial_{\mu}{\Omega}_{\nu} + [{\Omega}_{\nu},{\Omega}_{\mu}] $$

(92)

is the spin curvature. Equation (91) is the generally covariant extension of the Pauli-Schrödinger equation that describes spin 1/2 particles in a gravitational field [16, 17]. The first term contains, in addition to the Klein-Gordon operator, terms involving the Fock-Ivanenko coefficients Ω _μ

$$\begin{array}{@{}rcl@{}} g^{\mu\nu}D_{\mu}D_{\nu}\psi&=&(g^{\mu\nu}\partial_{\mu}\partial_{\nu} - g^{\mu\nu}{\Gamma}^{\lambda}_{\mu\nu}\partial_{\lambda})\psi +g^{\mu\nu} \left[ (D_{\mu}{\Omega}_{\nu}) +2 {\Omega}_{\nu}\partial_{\mu} \right]\psi \\ &=&\Box_{KG}\psi+g^{\mu\nu} \left[ (D_{\mu}{\Omega}_{\nu}) +2 {\Omega}_{\nu}\partial_{\mu} \right]\psi. \end{array} $$

(93)

In our case, the R-Minkowski space corresponds to the de Sitter spacetime given by the metric

$$ ds^{2}=\displaystyle\frac{1}{(1-x^{0}/R)^{2}} \left[(dx^{0})^{2}-d\vec{x}^{2}\right]=\displaystyle\frac{1}{\xi^{2}}\left[(dx^{0})^{2}-d \vec{x}^{2}\right]. $$

(94)

In the chart with x ⁰∈(−∞,0],the tetrad field is given by $e_{\ a}^{\mu } =\xi \,\delta _{\ a}^{\mu }$ and the spin connection by $\omega _{\mu ab}=\displaystyle {{(R\xi )^{-1}} }\left [ \eta _{\mu b}{\delta _{a}^{0}}-\eta _{\mu a}{\delta _{b}^{0}}\right ]$. Then, the covariant derivative reads

$$ D_{\mu} =\partial_{\mu} +\displaystyle{\frac{1}{4R\xi} }\eta_{\mu a}\left[ \gamma^{0},\gamma^{a}\right], $$

(95)

and a simple calculus gives for the sum of the terms involving the Fock-Ivanenko coefficients Ω

$$\begin{array}{@{}rcl@{}} g^{\mu\nu} \left[ (D_{\mu}{\Omega}_{\nu}) + 2{\Omega}_{\nu}\partial_{\mu} \right] &=& g^{\mu\nu} \left[ (\partial_{\mu}{\Omega}_{\nu}) - {\Gamma}^{\sigma}_{\nu\mu}{\Omega}_{\sigma}+{\Omega}_{\mu}{\Omega}_{\nu} +2{\Omega}_{\nu}\partial_{\mu} \right] \\ &=&-\frac{1}{R}\xi\gamma^{0}\gamma^{i}\partial_{i}- \frac{3}{4R^{2}}. \end{array} $$

(96)

On the other hand, it is known that

$$ - \frac{1}{2} \sigma^{\mu\nu} K_{\mu\nu}= \frac{\mathcal R}{4}, $$

(97)

where $\mathcal {R}$ is the Ricci scalar and that in the de Sitter space, we have $\mathcal {R}=12/R^{2}$. Taking into account (93), (96) and (97), relation (91) can be rewritten as

$$ \left[\Box_{KG} -\frac{1}{R}\xi\gamma^{0}\gamma^{i}\partial_{i} + \frac{9}{4R^{2}} + (m c/\hbar)^{2} \right]\psi=0. $$

(98)

Comparing to (29), it is clear that expressions of terms which make the square of Dirac’s operator different from the one of Klein-Gordon are established.

Appendix B: Remark on the Integration Measure

In the natural representation (NP), the normalization of the wave function, expressed through the canonical variable X ^μ = x ^μ/ξ, is given by the usual condition in special relativity

$$\begin{array}{@{}rcl@{}} <{\Psi}_{NP},{\Psi}_{NP}>&=&{\int}_{\Sigma} d^{3}X \bar{\Psi}_{NP}(X)\gamma^{0} {\Psi}_{NP}(X) \\ &=&{\int}_{\Sigma} d^{3}X {\Psi}^{\dagger}_{NP}(X){\Psi}_{NP}(X) = 1, \end{array} $$

(99)

where the integration must be done over the spacelike hypersurface Σ, determined by X ⁰ = c s t meaning that ξ = c s t. So, the spacelike hypersurface at a constant time in the R-Minkowski spacetime is given by

$$ d^{3}X|_{T=cst}=dX\wedge dY\wedge dZ\,|_{t=cst}=d^{3}x \xi^{-3}. $$

(100)

Thus, the normalization condition in the R-Minkowski spacetime is given by

$$ <{\Psi}_{NP},{\Psi}_{NP}>={\int}_{\Sigma} d^{3}x \xi^{-3}{\Psi}^{\dagger}_{NP}(x){\Psi}_{NP}(x)=1, $$

(101)

where d ³ x = r ² d r dΩ. Taking into account relation (33), one can check that in the R-Minkowski spacetime the wave function, Ψ≡Ψ _{S P} , in the Schrödinger representation, is normalized with respect to the usual condition of special relativity

$$\begin{array}{@{}rcl@{}} <{\Psi}_{SP},{\Psi}_{SP}>& = &<{\Psi}_{NP},{\Psi}_{NP}> \\ & =& {\int}_{\Sigma} d^{3}x \xi^{-3}{\Psi}^{\dagger}_{NP}(x){\Psi}_{NP}(x) \\ & =& {\int}_{\Sigma} d^{3}x {\Psi}^{\dagger}_{SP}(x){\Psi}_{SP}(x)=1. \end{array} $$

(102)