By Mazurov V.D.

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Conversely, the mathematical proof of these equivalences is part of a new argument for these dualities in M theory. G. , 1978; Sternheimer, 1998) is a reformulation of the problem of quantizing a classical mechanical system as follows. One first considers the algebra of observables A of the classical problem; if one starts with a phase space M (perhaps the cotangent space to some configuration space, but of course it can be more general), this will be the algebra of functions C(M). One then finds a deformation of this algebra in the sense of Sec.

A variation on this construction is the twisted group algebra AG, ⑀ , which can be defined if H 2 ͓ G,U(1) ͔ 0. This allows for nontrivial projective representations ␥ characterized by a two-cocycle ⑀, ␥ ͑ g 1 ͒ ␥ ͑ g 2 ͒ ϭ ⑀ ͑ g 1 ,g 2 ͒ ␥ ͑ g 1 g 2 ͒ , (133) and AG, ⑀ is just the group algebra with this multiplication law. Since the phases Eq. (21) are a two-cocycle for Zd , Td itself is an example. A very important source of noncommutative algebras is the crossed product construction. One starts with an algebra A, with a group G acting on it, say on the left: Rev.

38), expands around a pure gauge configu¯ i ϭz i for 1рiрr, and integrates over ration C i ϭz ¯ i and C all variations of this by bounded operators. This type of prescription does make sense in the context of the perturbation theory of Sec. IV with IR and UV cutoffs, and would specify the dimension of space-time there. This dimension can also be inferred from the nature of oneloop divergences; see Connes (1994) and Varilly and Gracia-Bondia (1999) For the reasons we discussed above, this prescription probably does not make sense beyond perturbation theory.