An expansion of the type (@(&ij &(&n~&() = (fr(' (& pl&(&2l&0(@(&p~' @(&n~&p (x, ) Xi is derived, where yi -fl, ci] are labels for infinite-dimensional symmetric tensor representations of the Euclidean conformal group 0 (2@+1, 1), X, =[ l, , -ci], the constants C (y i) are real, and Q)( and u)x have the properties of vacuum expectation values of field products. The starting point is an infinite set of coupled nonlinear integral equations for Euclidean Green's functions in 2h space-time dimensions of the type written some 15 years ago by Fradkin and Symanzik. The Green's functions of the corresponding Gell-Mann-Low limit theory are expanded in conformal partial waves. The dynamical equations imply the existence of poles and factorization of residues in the partial waves as functions of the representation parameters. In proving the validity of the expansion we use some differential relations between partially equivalent exceptional representations of 0& (2 h + 1, 1), established in an earlier paper. This work completes the group-theoretical derivation of the vacuum operator-product expansion undertaken by Mack in 1973.
We give function space realizations of all representations of the conformal superalgebra su(2,2/N) and of the supergroup s U ( 2 , 2 / N ) induced from irreducible finite-dimensional Lorentz and X U ( N ) representations realized without spin and isospin indices. We use the lowest weight module structure of our su(2,2/N) representations to present a general procedure (adapted from the semisimple Lie algebra case) for the canonical construction of invariant differential operators closely related to the reducible (indecomposable) representations. All conformal supercovariant derivatives are obtained in this way. Exa.mples of higher order invariant differential operators are given. l ) A. v. Humboldt Foundation Pellow. 2) On leave of absence from Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria. 538 V. K. DOBREV, V. B. PETKOVA, Group Theoretical Approach We construct and study all representations of su(2,2/N) and of the supergroup XU(2,Si N ) induced from irreducible finite-dimensional Lorentz and XU( N/O) representations.These are called (as in the semisimple case) elementary representations (ER). We give a more detailed derivation of the reducibility conditions of the E R for arbitrary N announced in [a] (for N = 1 see [9, 101). As for SU (2,2) [7] these are obtained purely algebraically since the E R contain the structure of lowest weight modules (LWM) over the complexification gc = s1(4/N; C) of g. Whenever such a reducibility condition for the LWM or E R x is satisfied there arises an invariant map from the representation space C, to the space C,) where x' is obtained by the action of a definite Weyl reflection on the lowest weight corresponding to x. The existence of such a map is equivalent to the partial equivalence of the LWM (or ER) x and 2'. These results were used in the purely algebraic setting in [4] for the classification of the physically important multiplets3) of LWM.The algebraic approach exploited in [a] is more economical and rather powerful (see . K. DOBREV and V. B. PETKOVA, in preparation.
1974, in Russian).
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