Dynamical aspects in the quantizer–dequantizer formalism

Abstract

The use of the quantizer-dequantizer formalism to describe the evolution of a quantum system is reconsidered. We show that it is possible to embed a manifold in the space of quantum states of a given auxiliary system by means of an appropriate quantizerdequantizer system. If this manifold of states is invariant with respect to some unitary evolution, the quantizer-dequantizer system provides a classical-like realization of such dynamics, which in general is non linear. Integrability properties are also discussed. Weyl systems and generalized coherente states are used as a simple illustration of these ideas.

Nonlinear dynamics from linear quantum evolutions

“…While submanifolds deriving from an experimental constraint have to be chosen case-by-case via ad-hoc procedures, there are more general constructions which lead to the introduction of systems of coherent states. One of the first possibilities which has been investigated consists in the definition of suitable displacement operators (see [2] for more details about the use of quantizer-dequantizer formalism for the definition of generalized coherent states). What are nowadays called Weyl systems were introduced by H. Weyl to deal with unbounded operators in the canonical commutation relations.…”

“…Therefore, starting with a normalized vector lying on the unit sphere S(H), equation (1.3) defines an immersion of V into S(H) which is, again, injective. However, it is worth noticing that even though V is a vector space, the immersion does not respect linearity, i.e., ψ v 1 + ψ v 2 = ψ v 1 +v 2 in general, and i(V) is not a linear subspace of H. In summary, generalized coherent states allow us to define injective maps from a given set to the Hilbert space H associated with a quantum system, every point of the set labelling a given vector in H. This is the main property we will exploit in the first part of the paper in order to study how to induce one parameter groups of transformations on subsets of states starting from unitary maps on H. Using coherent states it is possible to interpret these induced maps as classical-like dynamics in the framework of classical-to quantum transition (see [2,3,4]). However, it is worth stressing that the following discussion can be extended to submanifolds which do not possess classical-like properties, representing, therefore, generic constraints.…”

“…From the physical point of view, V has a direct interpretation as a "classical phase-space", while, in general, the manifold M may not have a preferred physical interpretation. However, in all the examples we will propose, we will always choose M to give rise to an immersed submanifold, i(M), of states exhibiting a semi-classical behaviour, for instance the so called squeezed and correlated states which we will address in section (2).…”

Groupoids and Coherent States

“…A. Perelomov [Pe72] showed that families of coherent states can be obtained from irreducible representations of Lie groups, deepening the understanding of the relation between group theory and Quantum Mechanics already started by E. Wigner [Wi39] (see also [Ba47] and references therein) and H. Weyl [We50]. Quite recently Ciaglia et al suggested [Ci17] that the properties of coherent states could be understood in the broader setting of the 'quantizer-dequantizer' formalism put forward by V. Man'ko and G. Marmo [Ma02] showing that, relevant as it is, the group theoretical background is not strictly necessary to provide a background for generalized coherent states.…”

“…Schwinger's observation has deep implications. It shows that the algebraic structure of the quantities describing quantum systems is that of the algebra of a groupoid such groupoid determined by the family of physical transitions between possible outcomes of the system [Ci17].…”