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  Casimir operators of de Sitter space

+ 1 like - 0 dislike

De-Sitter space can be thought of as a 4 dimensional hyperboloid embedded in 5D Minkowski space. Hence, the symmetry group of dS is $SO(1,4)$ whose generators are,

$J_{AB}=i\left(X_A\frac{\partial}{\partial X_B}-X_B\frac{\partial}{\partial X_A}\right)$

where $A,B=0,1,2,3,4$. They follow the standard $SO(1,4)$ commutation relations,


where $\eta_{AB}=(-1,1,1,1,1)$. The coordinates on the dS hyperboloid (whose equation is, $\eta_{AB}X^AX^B=\frac{1}{H^2}$)  are,




Here $i=1,2,3$.  $(\eta,\vec{x})$ are parameters on the hyperboloid. The 5D Minkowski metric restricted to the hyperboloid, in terms of these coordinates, becomes,


Split the $SO(1,4)$ generators as,





It can be easily checked that in terms of the $\eta,\vec{x}$ coordinates,





These generators follow the standard conformal algebra (check the one given in the big yellow book[1]). The point $\eta,\vec{x}=0$ is left invariant by $D,L_{ij},K_i$. Hence, if $\phi(\eta,\vec{x})$ is a classical field,




Now, $SO(1,4)$ has two Casimir operators,



where $W^A=\tfrac18 \epsilon^{ABCDE}J_{BC}J_{DE}$. In terms of the conformal genertors, 


by making $C_1$ act on $\phi(0)$, the eigenvalues turn out to be,


In terms of the conformal generators, the components of $W_A$ are, 




where $L_k=-\tfrac12\epsilon_{kij}L_{ij}$ and it has the following commutation rules,





Now, $W^2=-W_0^2+W_i^2+W_4^2=(W_4+W_0)(W_4-W_0)-[W_0,W_4]+W_i^2$, so when it acts on a classical field, the first term does not contribute since it has a $K_i$ on the right which annihilates the field. Also, it is easy to see that $L_k$ commutes with $W_0$ and $W_4$. Therefore,


In the formula for $W_k$, commuting the $P_i$ past the $K_i$ gives a factor of $iL_k$ so that


Thus, we get, 


which is incorrect according to this[2] reference (their $q$ is my $\Delta-1$). The first Casimir is correct. The correct answer for $C_2$ should be $-s(s+1)(\Delta-1)(\Delta-2)$. Can somebody point out where I made a mistake? 

  [1]: https://www.springer.com/gp/book/9780387947853
  [2]: https://arxiv.org/abs/hep-th/0309104

asked Apr 11, 2020 in Q&A by Sounak Sinha (10 points) [ no revision ]

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