Groupoids and Coherent States

Abstract

Schwinger's algebra of selective measurements has a natural interpretation in terms of groupoids. This approach is pushed forward in this paper to show that the theory of coherent states has a natural setting in the framework of groupoids. Thus given a quantum mechanical system with associated Hilbert space determined by a representation of a groupoid, it is shown that any invariant subset of the group of invertible elements in the groupoid algebra determines a family of generalized coherent states provided that a completeness condition is satisfied. The standard coherent states for the harmonic oscillator as well as generalized coherent states for f -oscillators are exemplified in this picture.

Schwinger’s picture of quantum mechanics II: Algebras and observables

“…It is interesting to observe that, by means of the fundamental representation, the groupoid functions q, p become the standard position and momentum operators q = π(q), p = π(p), which are affiliated to the C * -algebra A ∞ . It is also noticeable that other significant aspects of the harmonic oscillator, like the construction of coherent states, can also be nicely described in this setting (see for instance [23]).…”

“…. , in the expansion (23) as defining an infinite matrix F whose entries F nm are the numbers f (n, m). In doing so, the convolution product on the algebra F alg (A ∞ ) becomes the matrix product of the matrices F and G corresponding to f and g respectively (notice that the product is well defined as there are only finitely many non zero entries on both matrices).…”