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By Hideo Kodama

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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∗ )] .

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Cosmological perturbation theory by Hideo Kodama

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