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  Casimir operators of de Sitter space

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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,

JAB=i(XAXBXBXA)

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

[JAB,JCD]=i(ηBCJAD+ηADJBCηACJBDηBDJAC)

where ηAB=(1,1,1,1,1). The coordinates on the dS hyperboloid (whose equation is, ηABXAXB=1H2)  are,

X0=12H(Hη1Hη)+x22η

Xi=xiHη

X4=12H(Hη+1Hη)+x22η

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

ds2=dη2+dx2H2η2.

Split the SO(1,4) generators as,

Lij=Jij

D=J04

Pi=J0i+J4i

Ki=J4iJ0i

It can be easily checked that in terms of the η,x coordinates,

Lij=i(xijxji)

D=i(ηη+xii)

Pi=iHi

Ki=i[2Hxi(ηη+xii)+H(η2+x2)i]

These generators follow the standard conformal algebra (check the one given in the big yellow book[1]). The point η,x=0 is left invariant by D,Lij,Ki. Hence, if ϕ(η,x) is a classical field,

Lijϕ(0)=Sijϕ(0)

Dϕ(0)=iΔϕ(0)

Kiϕ(0)=0

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

C1=12JABJAB

C2=WAWA

where WA=18ϵABCDEJBCJDE. In terms of the conformal genertors, 

C1=D212LijLij12{Pi,Ki}

by making C1 act on ϕ(0), the eigenvalues turn out to be,

C1=s(s+1)Δ(Δ3)

In terms of the conformal generators, the components of WA are, 

W0=12ϵijkJijJk4=12Lk(Pk+Kk)

Wi=12ϵijkJjkJ04ϵijkJj4J0k=LiD+14ϵijk{Kj,Pk}

W4=12ϵijkJijJ0k=12Lk(PkKk)

where Lk=12ϵkijLij and it has the following commutation rules,

[Li,Lj]=iϵijkLk

[Li,D]=0

[Li,Pj]=iϵijkPk

[Li,Kj]=iϵijkKk

Now, W2=W20+W2i+W24=(W4+W0)(W4W0)[W0,W4]+W2i, so when it acts on a classical field, the first term does not contribute since it has a Ki on the right which annihilates the field. Also, it is easy to see that Lk commutes with W0 and W4. Therefore,

[W0,W4]=Lk[Jk4,LiJ0i]=iLkWk

In the formula for Wk, commuting the Pi past the Ki gives a factor of iLk so that

Wkϕ(0)=Lk(Di)ϕ(0)

Thus, we get, 

C2ϕ(0)=s(s+1)(Δ+1)(Δ+2)ϕ(0) 

which is incorrect according to this[2] reference (their q is my Δ1). The first Casimir is correct. The correct answer for C2 should be s(s+1)(Δ1)(Δ2). 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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