Quantization Of Massless Conformally Vector Field In de Sitter Space-Time

Transcription

1 6th Interntionl Worshop on Pseudo Hermitin Hmiltonin In Quntum Physics London 6th -8th July 7 Quntition Of Mssless Conformlly Vector Field In de Sitter Spce-Time Mohmd Re Tnhyi Islmic Ad University-Centrl Tehrn Brnch M_tnhyi@iuctb.c.ir Sr Ftemi Islmic Ad University-Centrl Tehrn Brnch

2 An open problem of theoreticl physics : Unifying theory tht include ll fundmentl forces. The mjor difficulty : Unifiying guge interctions nd mssless fields. A solution tht mybe wor : Indefinite metric.

3 Why de Sitter spce time? Astrophysicl dt indicte tht the universe is in de Sitter phse. Einstein s eqution: R T µν µν g Λ H µν Λg > µν KT µν

4 How cn we quntie field? Streter nd wightmn s method iomtic field theory: w Ω Ω w the two point function Ω the vcuum stte the mssless vector field. Ref.[] R.F.Streter nd A.S.WightmnBenjminNewYor 964 PCTSpin And Sttistics

5 Wht eqution does stisfy? So4 de Sitter group leves this form invrint: X X X X 3 X 4 constnt 3 So4 conforml group leves the following form invrint: u u u u3 u4 + u5 constnt 4

6 de Sitter group conforml group represnttion rep. positive energy rep. negtive energy [] A.O. Brut nd A. Bohm J.Mth.Phys E. Angelopoulos nd M. Loues Rev.Mth.Phys. 7998

7 Conformlly invrint field eqution Dirc s method finds conformlly invrint eqution. Dirc' s cone physicl spce conformlly projection conformlly invrint equtions equtions invrint [3] P.A.M. Dirc Ann.Mth

8 Dirc s cone: A 5- dimensionl supersurfce in 6 R η b u η b u u dig b 5

9 Conformlly invrint system on the cone: N 5 p Ψ p N Ψ 5 u Ψ 6 [4] S.Behrooi S.Rouhni M.V.Too M.R.Tnhyi Phys.Rev.D

10 The projection on de Sitter spce: [4] S.Behrooi S.Rouhni M.V.Too M.R.Tnhyi Phys.Rev.D Ψ Ψ Ψ N u u u

11 Mssless conformlly invrint vector field in de Sitter spce-time: Ref.[4] 9 + Q D D Q. 3. prmeter constn rbitrry + σ ξ ξ σ ζ ε ξ σ ζ ε σ

12 Spce of solutions nd the indefinite metric ε ζ σ. ξ σ de Sitter invrint inner product on the spce of solutions: i [ ] Ω ρ c ρ d H ρ [5] J.P.Geu M.Hns R.Mureni Clss.Qun.Grv

13 : < > indefinite V in c ] [ : > Ω definite semi d c H i inv ρ ρ ρ. : V in g definite positive is V V physiclspce g

14 It hs been proven tht the use of n indefinite metric is unvoidble if one insists on the preservtion of cuslity nd covrince in guge quntum field theories. [6] F.Strochi Phys.Rev.D

15 The two-point function 3 Ω Ω w 4 ζ µ ζ ζ σ ξ ε σ ξ ε σ σ λ λ λ d Z Z c w s

16 w bv w 5 [7] J.Bros J.P.Geu nd U. Moschell Phy.Rev.Lett ibid. Rev.Mth.Phys

17 The cuslity condition s in Ref. [] 6 w w 7 4 ] [ Z D H ε < > H Z ε

18 Conclusion It ws pointed out tht Einstein s theory of grvittion cn be interpreted s theory of metric field. In the bcground field method B. G gµν gµν + hµν it cn consider s mssless symmetric tensor field on fied bcground. We hve shown to obtin conformlly invrint wve eqution for grviton mied-symmetry rn-3 tensor is needed. [8] S.Rouhni M.V.Too M.R.Tnhyi Conformlly invrint wve eqution for mssless spin- field in de Sitter spce ppers in Phy.Rev.D

19 So it seems tht the use of n indefinite metric is unvoidble for quntition of grvition.

20 We would lie to thn Dr. Too for his very useful discussions.