
By Hideo Kodama
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Extra resources for Cosmological perturbation theory
Example text
21) We will return to this more general case in Chapter 4, and show that interesting new terms may appear. The supersymmetry algebra has several important consequences. First from (II) one infers [Qr , P µ Pµ ] = 0. 22) Let us then consider two states |b and |f of respective mass mb and mf which are obtained from one another by supersymmetry: Qr |b = |f . Then since P µ Pµ |b = m2b |b and P µ Pµ |f P µ Pµ Qr |b = P µ Pµ |f = m2f |f , = m2f |f = Qr P µ Pµ |b = m2b Qr |b = m2b |f , hence mb = mf .
And thus Qr conservation follows. Although the Coleman–Mandula no-go theorem can be evaded, it imposes severe constraints on the possible ways out. For example, the anticommutator {Qr , Qs } is an operator which is a symmetric combination of two spin 1/2 objects. It is thus of bosonic type and obeys commutation relations. Moreover, being a symmetric combination of two Lorentz spinors, it is a vector under Lorentz transformations ([(1/2) ⊗ (1/2)]s = (1)). Coleman–Mandula restrictions then apply to it and impose it to be proportional to the generator of translations Pµ .
27) (ε = εc ) 2 √ ∂W ∂ W ∂W 1 δS Fi i − Ψc iR ΨjL = ∂µ −i 2 ε¯γ µ ΨiL . 31) i j ∂φ 2 ij ∂φ ∂φ ∂φi i i This transformation law is actually a consequence of the fact that this combination may be considered as the auxiliary field of a composite chiral supermultiplet5 . We thus note, for any analytic function W of the fields φi , [W (φ)]F ≡ Fi i ∂W 1 − ∂φi 2 ij ∂2W Ψc iR ΨjL . 32) We end this section by discussing some subtleties associated with the derivation of the supersymmetric current. 28) which we write, using the notation just introduced, ¯ iγ µ ∂µ Ψ + F ∗ F + [W (φ)] + [W ∗ (φ∗ )] .
Cosmological perturbation theory by Hideo Kodama
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