Projective fourier duality and Weyl quantization

Abstract

The Weyl-Wigner correspondence prescription, which makes great use of Fourier duality, is reexamined from the point of view of Kac algebras, the most general background for noncommutative Fourier analysis allowing for that property. It is shown how the standard Kac structure has to be extended in order to accommodate the physical requirements. Both an Abelian and a symmetric projective Kac algebra are shown to provide, in close parallel to the standard case, a new dual framework and a well-defined notion of projective Fourier duality for the group of translations on the plane. The Weyi formula arises naturally as an irreducible component of the duality mapping between these projective algebras.

Weyl quantization and a symbol calculus for abelian groups

“…The significance of this factor 2 was already mentioned in Howe [6,7]. For quite another approach using Kac algebras, see Aldrovandi and Saeger [1].…”

“…Then 5c consists of all complex numbers of the form y\ • • • Yi w ' t n Yi a n " i t n r o o t °f unity. This is clearly a subgroup of S1 .D Any finite subgroup of S 1 is cyclic. Let |5c| = m, so that S G consists of all mth roots of unity.…”

Fourier duality as a quantization principle

“…By the Stone-von Neumann theorem [25], the former is equal to (Z − {0}) ∪ R 2 , while the latter is just Z − {0}. We have shown in a separate paper [2] that the Weyl-Wigner formalism can be described in terms of duality of projective Kac algebras. In such a projective duality framework for R 2 , Weyl's formula comes from an expression analogous to (69) for the decomposition of the respective Fourier representation, namely…”

“…By the Stone-yon Neumann theorem (Taylor, 1986), the former is equal to (Z -{0}) U R 2, while the latter is just Z -{0}. We have shown (Aldrovandi and Saeger, 1996) where q,/~ are the usual position and momentum operators. Comparing (81) with Weyl's formula, we get immediately v = h -l. Actually, the only formal difference between (68) and the original Weyl formula,or (81), is that the latter is written in terms of unitary irreducible projective operators instead of the linear ones which appear in formula (68).…”

“…We must abstract from groups to Kac algebras in order to have a Fourier duality. In this sense the usual Weyl correspondence is part of a highly nontrivial projective duality for the Abelian group R 2 , where an algebra generated no more by linear, but by projective operators comes into play [2].…”