Sunday, 12 June 2016

quantum field theory - Dirac Equation in General Relativity



Dirac equation for the massless fermions in curved spase time is γaeμaDμΨ=0, where eμa are the tetrads. I have to show that Dirac spinors obey the following equation: (DμDμ+14R)Ψ=0(1)


where R is the Ricci scalar.


I already know that [Dμ,Dν]Aρ=RμνρσAσ, but a key point is to know what [Dμ,Dν]Ψ is.


(DμΨ=μΨ+AabμΣab is the covariant derivative of the spinor field and Σab the Lorentz generators involving gamma matrices).



Answer



Denoting by γa the Minkowski space gamma matrices with respect to the Lorentz tetrad {ea}, and covariant derivative Da, then the gammas are covariantly constant.


Start with the massless Dirac equation γbDbΨ=0


Act again with the Dirac operator γaDaγbDbΨ=0

So, since D annihilates γ γaγbDaDbΨ=0
so 12{γa,γb}DaDbΨ+12γaγb[Da,Db]Ψ=0  (1)
But {γa,γb}=2ηab
and [Da,Db]Ψ=RabΨ
Where Rab is the spin-curvature (antisymmetric in a and b). Rab satisfies the identity γbRab=Rabγb=12γbRab
where Rab is the Ricci tensor (in the Lorentz tetrad). so (1) becomes [DaDa+14γaγbRab]Ψ=0
i.e. [DaDa14R]Ψ=0


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