2014/01/21 by Luigi Accardi, Accardi, Luigi, Ameur Dhahri +1
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #math-ph #math.MP #math.OA
paper · pdf · doi:10.48550/arxiv.1401.5500
29 pages
arxiv created 2014/09/12 · arxiv updated 2014/09/15
Recently, we have constructed a nonlinear (polynomial) extension of the 1-mode Heisenberg group and the corresponding Fock and Weyl representations. The transition from the 1-mode case to the current algebra level, in which the operators are indexed by elements of an appropriate test function space (second quantization), can be done at Lie algebra level. A way to bypass the difficulties of constructing a (non trivial) Hilbert space representation is to try and construct directly a C^*-algebra rep- resentation and then to look for its Hilbert space representations. In usual (linear) quantization, this corresponds to the construction of the Weyl C^*-algebra. In this paper, we produce such a construction for the above mentioned polynomial extension of the Weyl C^*-algebra. The result of this construction is a factorizable system of local alge- bras localized on bounded Borel subsets of ℝ and obtained as induc- tive limit of tensor products of finite sets of copies of the one mode C^*-algebra. The C^*-embeddings of the inductive system require some nontrivial rescaling of the generators of the algebras involved. These rescalings are responsible of a C^*-analogue of the "no-go" theorems, first met at the level of Fock second quantization, namely the proof that the family of Fock states defined on the inductive family of C^*-algebras is projective only in the linear case (i.e. the case of the usual Weyl algebra). Thus the solution of the representa- tion problem at C^*-level does not automatically imply its solution at Hilbert space level.