Abstract
It is known that there are no local scalar Lie fields in more than two dimensions. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. It is demonstrated that these Lie algebras of local observables admit (highly reducible) unitary positive energy representations in a Fock space. The multiplicity of their irreducible components is governed by a compact gauge group. The mutually commuting observable algebra and gauge group form a dual pair in the sense of Howe. In a theory of local scalar fields of conformal dimension two in four space-time dimensions the associated dual pairs are constructed and classified. The talk reviews joint work of B. Bakalov, N. M. Nikolov, K.-H. Rehren, and the author.
Highlights
In his AMS Einstein Lecture [12] Freeman Dyson divides mathematicians into birds and frogs
We note that the oscillator representation of M p(2n) has a minimality property [26, 27] that keeps attracting the attention of both physicists and mathematicians — see, e.g., [18, 33, 34]
Observables are left invariant by gauge transformations. This is, a key property of a gauge symmetry or a superselection rule as defined by Wick, Wightman, and Wigner back in 1952 [54]. It required the nontrivial vision of Rudolf Haag to predict in the 1960s that a local net of observable algebras should determine the compact gauge group that governs the structure of its superselection sectors
Summary
In his (undelivered) AMS Einstein Lecture [12] Freeman Dyson divides mathematicians into birds and frogs. Are for him real, while general theories, “conceptual mathematics” belong to the imaginary axis. If we pretend to play birds, it is legitimate to ask what have to do conformal quantum field theory (QFT) models with real physics? If there are discrete masses in the world (in particular, if we are there), what is conformal invariance good for? Dimensional transmutation, a quantum effect linked to renormalization, which provides a hope of solving the problem with the mass gap in Yang-Mills theory [14]. It is not an accident that some of the most attractive current attempts to understand the generation of mass in the standard model start with a conformally invariant classical Lagrangian [6, 15, 40]
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