Auxiliary Conformally Invariant Higher-Spin Field in de Sitter Space

Abstract

We employ de Sitter isometry to study a mixed symmetric rank-3 tensor field in de Sitter space by finding the field equation, solution and two-point function which are conformally invariant. It is proved that such a tensor field plays a key role in conformal theory of linear gravity (Binegar et al., Phys. Rev. D 27, 2249, 1983) . In de Sitter space from the group theoretical point of view this kind of tensor could associate with two unitary irreducible representations (UIR) of the de Sitter group (Takook et al., J.Math. Phys. 51, 032503, 2010), which one representation has a flat limit, namely, in zero curvature coincides to the UIR of Poincaré group, however, the second one which is named as auxiliary field, becomes significant in the study of conformal gravity in de Sitter background. We show that the rank-3 tensor solution can be written in terms of a massless minimally coupled scalar field and also the related two-point function is a function of a massless minimally coupled scalar two-point function.

Appendix A: Some useful relations

In this appendix, some useful relations are given. The action of Casimir operators of de Sitter group Q ₁ (vector Casimir operator), Q ₂ and Q ₃) (Casimir operator for the rank-2 and rank-3 tensor field respectively) can be written in the more explicit form as [27]

$$ Q_{1}K_{\alpha}=(Q_{0}-2)K_{\alpha}+2x_{\alpha}\partial\cdot{K}-2\partial_{\alpha} x\cdot{K}, $$

(A.1)

$$ Q_{2}\mathcal{ K}_{\alpha\beta}=(Q_{0}-6)\mathcal{ K}_{\alpha\beta}+2\mathcal{ S}x_{\alpha}\partial\cdot{\mathcal{ K}_{\beta}}-2\mathcal{ S}\partial_{\alpha} x\cdot{\mathcal{ K}_{\beta}}+2\eta_{\alpha\beta}\mathcal{ K}^{\prime}, $$

(A.2)

where $\mathcal { K}^{\prime }=\mathcal { K}_{\alpha }^{\alpha }$.

$$\begin{array}{@{}rcl@{}} Q_{3}F_{\alpha\beta\gamma}&&=(Q_{0}-6) F_{\alpha\beta\gamma}+2 \left( \eta_{\alpha\beta} F_{..\gamma}+ \eta_{\alpha\gamma}F_{.\beta.}+ \eta_{\beta\gamma} F_{\alpha..}\right)\\ &&+2\left( x_{\alpha}\Bar \partial\cdot {F_{.\beta\gamma}}+ x_{\beta}\Bar\partial\cdot{F_{\alpha.\gamma}}+x_{\gamma}\Bar \partial\cdot {F_{\alpha\beta.}}\right)\\ &&-2\left( \partial_{\alpha}x\cdot{F_{.\beta\gamma}}+\partial_{\beta}x\cdot{F_{\alpha.\gamma}}+\partial_{\gamma}x\cdot{F_{\alpha\beta.}}\right)-2\left( F_{\alpha\gamma\beta}+ F_{\beta\alpha\gamma}+F_{\gamma\beta\alpha} \right),\\ \end{array} $$

(A.3)

where F _..γ = F _{α α γ} . After imposing mixed symmetry condition on F _{α β γ} , above relation becomes

$$\begin{array}{@{}rcl@{}} Q_{3}F_{\alpha\beta\gamma}&&=(Q_{0}-4) F_{\alpha\beta\gamma}-2F_{\beta\gamma\alpha}-2F_{\gamma\alpha\beta}+2 \left( \eta_{\alpha\beta} F_{..\gamma}+ \eta_{\alpha\gamma}F_{.\beta.}+ \eta_{\beta\gamma} F_{\alpha..}\right)\\ &&+2\left( x_{\alpha}\Bar \partial\cdot {F_{.\beta\gamma}}+ x_{\beta}\Bar\partial\cdot{F_{\alpha.\gamma}}+x_{\gamma}\Bar \partial\cdot {F_{\alpha\beta.}}\right). \end{array} $$

(A.4)

To obtain the two-point function, the following identities are used

$$ \Bar\partial_{\alpha} f(\mathcal{ Z})=-(x^{\prime}\cdot{\theta_{\alpha}})\frac{df(\mathcal{ Z})}{d(\mathcal{ Z})}, $$

(A.5)

$$ \theta^{\alpha\beta}\theta^{\prime}_{\alpha\beta}=\theta\cdot\cdot{\theta^{\prime}}=3+\mathcal{ Z}^{2}, $$

(A.6)

$$ (x\cdot\theta^{\prime}_{\alpha^{\prime}})(x\cdot\theta^{\prime\alpha^{\prime}})=\mathcal{ Z}^{2}-1, $$

(A.7)

$$ (x\cdot{\theta^{\prime}_{\alpha}})(x^{\prime}\cdot{\theta^{\alpha}})=\mathcal{ Z}(1-\mathcal{ Z}^{2}), $$

(A.8)

$$ \Bar\partial_{\alpha}(x\cdot\theta^{\prime\beta^{\prime}})=\theta_{\alpha}\cdot\theta^{\prime\beta^{\prime}}, $$

(A.9)

$$ \Bar\partial_{\alpha}(x^{\prime}\cdot{\theta_{\beta}})=x_{\beta}(x^{\prime}\cdot{\theta_{\alpha}})-\mathcal{ Z}\theta_{\alpha\beta}, $$

(A.10)

$$ \Bar\partial_{\alpha}(\theta_{\beta}\cdot{\theta^{\prime}_{\beta^{\prime}}})=x_{\beta}(\theta_{\alpha}\cdot{\theta^{\prime}_{\beta^{\prime}}})+ \theta_{\alpha\beta}(x\cdot{\theta^{\prime}_{\beta^{\prime}}}), $$

(A.11)

$$ \theta^{\prime\beta}_{\alpha^{\prime}}(x^{\prime}\cdot{\theta_{\beta}})=-\mathcal{ Z}(x\cdot\theta^{\prime\alpha^{\prime}}), $$

(A.12)

$$ \theta^{\prime\gamma}_{\alpha^{\prime}}(\theta_{\gamma}\cdot\theta^{\prime}_{\beta^{\prime}})=\theta^{\prime}_{\alpha^{\prime}\beta^{\prime}}+(x\cdot\theta^{\prime\alpha^{\prime}})(x\cdot\theta^{\prime\beta^{\prime}}), $$

(A.13)

$$ Q_{0}f(\mathcal{ Z})=(1-\mathcal{ Z}^{2})\frac{d^{2}f(\mathcal{ Z})}{d\mathcal{ Z}^{2}}-4\mathcal{ Z}\frac{df(\mathcal{ Z})}{d(\mathcal{ Z})}. $$

(A.14)