By E.S. Fradkin, Mark Ya. Palchik

Our top hindrance during this publication is to debate a few finest prosppcts that experience happened lately in conformally invariant quantum box idea in a D-diuwnsional house. some of the most promising tendencies is developing an pxact answer for a cprtain type of types. This activity seems fairly possible within the mild of modern resllits. the placement this is to a point just like what was once occurring long ago ypars with the two-dimensional quantum box conception. Our research of conformal Ward identities in a D-dimensional area, performed as some distance hack because the past due H. J7Gs, confirmed that during the D-dimensional quantum box thought, without reference to the kind of interartion, there exists a different set of states of the sphere with the next estate: if we rpqllire that this sort of states should still vanish, this determines a precise answer of three. convinced box version. those states are analogous to null-vectors which confirm the minimum versions within the two-dimensional box idea. nevertheless, the new resparches provided us with a few symptoms at the existencp of an intinite-parampter algebra analogous to the Virasoro algebra in areas of upper dimensions D 2: :~. It has additionally been proven that this algebra admits an operator rentral growth. it sort of feels to us that the above-mentioned types are box theoretical realizations of the representations of those new symmetries for D 2: ;3.

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Under that, a conformal partial wave expansion is introduced for each of the Wightman functions, which are treated as invariant scalar products of these states. In particular, the analog of Kallen-Lehmann representation 9 is derived for the Wightman function while the states *(x) 10), gIn this case the one expands in dimensions and spins of intermediate states. 4 A Review of Branches and Results in Conformal Theory 25 where (x) is any conformal field, are proved to transform by an irreducible representation of the conformal group. *

N - 1; The oY derivatives act on the argument of a field PLm (y) only. The if operators are the sums of terms of the type (tJx ) ",-r+l o(x - y) (oYr-m+r, 7· = 0,1, ... lculated for any class of models. 5' = 0,1, ... 5-l. ( X2) ... + (X2n)X( x2n+d ... ( xd ... + (X2n )X( X2n+l) ... X( X2n+rn)), aX) -_ pAT (. L] ... 'i + 1 found All the above remains true for the fields Pl. 20) ':'X2n+~) ... 5. 21 ) are treated by analogy. 20). 12) belong. The dynamical principle which governs the effective interaction, is formulated in this subspace, called below the dynamical sector of a Hilbert space.

The contents of these Chapters is commented helow in a greater detail. Chapters IX,X deal with the conformally invariant version of gauge theories in 1-dimensional space [98-107]. It is shown that the non-linear (with respect to Ai') realization of conformal transformations takes place for non-Abelian models. In the Abelian case (conformal QED) the two variants of realization of the symmetry are discussed: the first having linear conformal transformations, and the second, where the transformations of spinor fields havp a non-linear form.