C*-Non-Linear Second Quantization

Abstract

We construct an inductive system of C * -algebras each of which is isomorphic to a finite tensor product of copies of the one-mode n-th degree polynomial extension of the usual Weyl algebra constructed in our previous paper (Accardi and Dhahri in Open Syst Inf Dyn 22 (3):1550001, 2015). We prove that the inductive limit C * -algebra is factorizable and has a natural localization given by a family of C * -sub-algebras each of which is localized on a bounded Borel subset of R. Finally, we prove that the corresponding family of Fock states, defined on the inductive family of C * -algebras, is projective if and only if n = 1. This is a weak form of the no-go theorems which emerge in the study of representations of current algebras over Lie algebras.

The generalized Heisenberg group arising from Weyl relations

“…Using density of Q in R and continuity of u, we deduce that the expression (50) holds also for all t ∈ R. Taking z = λu (1) e iλ −1 and using (39), we obtain the expressions of u and w in (29). From (45) and using expressions of u and w in (29), we obtain…”

“…In our case, the group associated to the oscillator algebra is the generalized Heisenberg group. This group will be useful to construct an inductive system of C * -algebras each of which will be isomorphic to a finite tensor product of copies of the one-mode algebra, see [1] for more details. Then the program studies the existence of such factorizable state on this system.…”

$$C^*$$ C ∗ -Quadratic Quantization

“…Even if this example does not require re-normalization, the existence for it of an analogue of the Fock representation for it is still an open problem. As an intermediate step, in the paper [9] an analogue of the Weyl C * -algebra was constructed. In 1-st order case, i.e.…”

“…In the second part of this paper we construct a quadratic Weyl C * -algebra by adapting to the present case the technique, developed in [9]. The main difference between the present paper and [9] is that the construction of the inductive system is realized here at a * -algebra level and the C * -norm is introduced after taking the inductive limit. This simplifies the proof not requiring the proof of the continuity of the inductive embeddings at each step.…”

“…For example since, both in the Fock case [14] and in the equilibrium case [1], quadratic fields can be exponentiated, one can construct quadratic Weyl C * -algebras associated to these representations. In the case of polynomial extensions of the Heisenberg algebra no such representation are known at the moment (even in the quadratic case) and this motivated the inductive construction in [9]. We use a modification of this construction here because the main motivation of the present paper is to understand the structural roots of the main difference between the quadratic Weyl algebra and the algebras based on polynomial extensions of the Heisenberg algebra, namely that the family of the quadratic Fock states, defined on the inductive family of * -algebras, is always projective in the case of quadratic quantization, while for the n-th degree Heisenberg group this is true only for n = 1, i.e.…”