Abstract
We compute the most general embedding space two-point function in arbitrary Lorentz representations in the context of the recently introduced formalism in [1, 2]. This work provides a first explicit application of this approach and furnishes a number of checks of the formalism. We project the general embedding space two-point function to position space and find a form consistent with conformal covariance. Several concrete examples are worked out in detail. We also derive constraints on the OPE coefficient matrices appearing in the two-point function, which allow us to impose unitarity conditions on the two-point function coefficients for operators in any Lorentz representations.
Highlights
We derive constraints on the operator product expansion (OPE) coefficient matrices appearing in the two-point function, which allow us to impose unitarity conditions on the two-point function coefficients for operators in any Lorentz representations
Expressing all two-point functions in Lorentzian signature is convenient for understanding the unitarity conditions, which can be determined by considering two-point correlators between quasi-primary operators and their conjugates
We have found that the form of the results matches expectations from conformal covariance
Summary
Of interest here is P1N2 which is an embedding space projection operator to the representation N , while (T1N2 Γ) is what we refer to as a half-projector It is evident from (1.2) that the particulars of the projection operators are central in the determination of twopoint correlation functions of quasi-primary operators in general irreducible representations of the Lorentz group. The half-projectors serve to translate the spinor indices carried by each operator to the “dummy” vector and spinor indices that need to be summed over when constructing correlation functions. They earned their name because they square to form projection operators. We discuss some simple algorithms for the construction of hatted projection operators to general irreducible representations of the Lorentz group
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