By E.S. Fradkin, Mark Ya. Palchik

Our top hindrance during this publication is to debate a few best prosppcts that experience happened lately in conformally invariant quantum box conception in a D-diuwnsional area. the most promising traits is developing an pxact answer for a cprtain type of types. This activity seems fairly possible within the mild of contemporary resllits. the location here's to some degree just like what used to be happening some time past ypars with the two-dimensional quantum box thought. Our research of conformal Ward identities in a D-dimensional area, performed as a ways hack because the overdue H. J7Gs, confirmed that during the D-dimensional quantum box idea, regardless of the kind of interartion, there exists a different set of states of the sector with the next estate: if we rpqllire that this kind of states should still vanish, this determines a precise answer of three. definite box version. those states are analogous to null-vectors which verify the minimum types within the two-dimensional box concept. nevertheless, the hot resparches provided us with a couple of 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 versions are box theoretical realizations of the representations of those new symmetries for D 2: ;3.

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Global field transformations, local commutativity and spectrality condition for the states 'PIO),'P'PIO) are also the subject of the discussion in Chapter II. ')0, 76]. Global conformal transformations of fields have the dynamical nature in this case [47, 50], a definite specification of global special conformal transformation laws for the fields is to a some degree equivalent to the setting of the model. These results are tightly connected with the problem of the global causal order in conformal theory, which is solved [96,97] (see also [51]) by the introduction of the universal (infinite-sheeted) covering of Minkowski space.

S2 ::; . . ::; 8k, a S1 ,... ,Sk P;1, ... 54) Sl"",Sk 81 + 82 + ... + Sk ::; 8 - 1. The coefficients a S1 .... 52) for each of the fields No more principal differences with the prpvious model exist. These results are obtained in sections 4,5 of Chapter VIII. 55) A possible conjecture (discussl~d in Chapter VIII)is that the three-diI]lensional Ising model corresponds to one of the solutions of a model 18This expression is formal, see footnote on the page 32. 5 Conformal Group and its Representations .

Their choice defines different types of models (see below). It will be shown that these two statements, supplied with the requirement of conformal symmetry, would result in a quite specific structure of a Hilbert space resembling that of two-dimensional conformal theories. There exists a special subspace in a total Hilbert space, which is begot by the energy-momentum tensor and the current. We call this subspace the dynamical sector. The next step consists in a formulation of the dynamical principle that fixes a model.