Comment 4 The formulations with � and ˆ � are thus equivalent up to field

Comment 4 the formulations with ? and ˆ ? are thus

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Comment 4 : The formulations with λ and ˆ λ are thus equivalent up to field redefinitions (rescalings). Also the other ambiguities in the Noether method we encountered before can be shown to lead to equivalent formulations up to field redefinitions. We leave this as an exercise. Comment 5 : The local superspace approach will produce a polynomial supergravity action which begins with 1 2 (1 - 2 h )[ . ϕ 2 + . λ ]. We can obtain the complete action and transformation rules by rescaling the results in ( 13.5 ) as follows λ = 1 - 2 h ˆ λ ; ψ = ˆ ψ/ 1 - 2 h ; = ˆ / 1 - 2 h (14.16) 15
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Of course any rescaling will produce an action which is still susy invariant and Einstein invariant, but most rescalings will lead to nonpolynomial actions and/or nonpolynomial transformation laws. However, the rescaling in ( 14.16 ) leads to both a polynomial action and polynomial transformation laws. There is nothing wrong with nonpolynomial actions and/or nonpolynomial transformation rules, but polynomial expresssions are easier to deal with. The result reads L = 1 2 (1 - 2 h )[ . ϕ 2 + i ˆ λ ˆ . λ ] - i ˆ ψ . ϕ ˆ λ δϕ = i ˆ ˆ λ + ξ . ϕ ; δ ˆ λ = - . ϕ ˆ+ ξ ˆ . λ + 1 2 . ξ ˆ λ δ ˆ ψ = (1 - 2 h ) ˆ . + . h ˆ+ ξ ˆ . ψ - 1 2 . ξ ˆ ψ ; δh = - i ˆ ˆ ψ + ξ . h + 1 2 (1 - 2 h ) . ξ (14.17) 15 Rigid Superspace Superspace was introduced by Salam and Strathdee who constructed it as an application of a general group-theoretical approach called the group manifold approach. We shall discuss this approach, but first we want to give a more direct way why we shall use anticommuting as well as commuting coordinates. The superalgebra which followed from our QM model contained the crucial anticommutators { Q, Q } = 2 H . In higher dimensions this generalizes to { Q α , Q β } ∼ γ μ αβ P μ . It seems interesting to treat Q α and P μ (in our case H and Q ) on equal footing. Then, since P μ is a translation generator, also Q should correspond to a translation generator. The coordinates which P μ translates are spacetime coordinates x μ . Then the coordinates which Q α translates must be a new kind of anticommuting coordinates θ α . ( θ α should be anticommuting because Q α are anticommuting.) The natural choice (not the only choice, there are other choices) for anticommuting θ α is as Grassmann variables; if one has two of them, say θ 1 and θ 2 , they satisfy θ 1 θ 2 = - θ 2 θ 1 . In particular θθ = 0. So for our QM model, we are led to consider a two-dimensional space with coordinates t and θ , and fields (called superfields) depending on φ ( t, θ ). An expression in terms of θ has then only two terms φ ( t, θ ) = ϕ 1 ( t ) + θϕ 2 ( t ) (15.1) In the group manifold approach (and, more generally, in the coset manifold approach) one associates with each generator of a (super)algebra a coordinate. The superalgebra of our QM model is { Q, Q } = 2 H , [ Q, H ] = 0, [ H, H ] = 0 (of course). It f t is associated with H, there has to be an anticommuting coordinate θ which is associated with Q. It satisfies θθ = 0. The we can construct group elements as follows g = e itH + θQ ; g = g - 1 (15.2) 16
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The coordinates t and θ parametrize points in superspace. Since H, Q and t
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