By E.S. Fradkin, Mark Ya. Palchik
Our major challenge during this publication is to debate a few finest prosppcts that experience happened lately in conformally invariant quantum box thought in a D-diuwnsional area. probably the most promising tendencies is developing an pxact resolution for a cprtain classification of versions. This activity looks rather possible within the mild of modern resllits. the placement this is to a point just like what was once happening long ago ypars with the two-dimensional quantum box thought. Our research of conformal Ward identities in a D-dimensional house, conducted as a ways hack because the past due H. J7Gs, confirmed that during the D-dimensional quantum box concept, without reference to the kind of interartion, there exists a different set of states of the sphere with the subsequent estate: if we rpqllire that the sort of states should still vanish, this determines an actual resolution of three. convinced box version. those states are analogous to null-vectors which ascertain the minimum types within the two-dimensional box concept. nonetheless, 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 kind 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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Extra info for Conformal Quantum Field Theory in D-dimensions
This result explains the fact that the invariant Wightman functions for two or three fields are completely determined by conformal symmetry. Another consequence of this fact is the vacuum operator expansions The latter arise as a result of expansion lO of the states 'P'PIO) into irreducible representations of conformal group'. The basis vectors of any irreducible representation may be chosen to be the statps m(x)IO) of a definite conformal field m(x). This leads to the idea of operator expansions for the products of conformal fields m which, as already mentioned, are the direct consequences of the Schwinger-Dyson system equations.
Of mathematical statements. They are discussed briefly and are commented only up to a degree sufficient to provide a "physical" understanding of the structure of conformal theory. We rely on the mathematical results [109-112]. Besides this, tIlt-' results of unpublished works [33,34,46,77] by one of the authors (in co-authorship with Konopelchenko) are used, the latter dealing with representations of the universal covering group of the conformal group for D = 2,4 (a brief sketch of these works is given in Appendix I, see also [42,43,47]).
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 . 37 The constraints on the states of dynamical sector, viewed as a means to define a model, were first studied by the authors in 1978. It was shown [49,50] that the solutions of trivial models (the Thirring model and gradient model for D = 4 [67-69]) are defined by these conditions (Chapter VIII reproduces these results).
Conformal Quantum Field Theory in D-dimensions by E.S. Fradkin, Mark Ya. Palchik