We describe quantum and classical Hamiltonian dynamics in a common Hilbert space framework, that allows the treatment of mixed quantum-classical systems. The analysis of some examples illustrates the possibility of entanglement between classical and quantum systems. We give a summary of the main tools of Berezin-Toeplitz and geometric quantization, that provide a relation between the classical and the quantum models, based essentially on the selection of a subspace of the classical Hilbert space. Coherent states provide a systematic tool for the inverse process, called dequantization, that associates a classical Hamiltonian system to a given quantum dynamics through the choice of a complete set of coherent states.
We describe quantum and classical Hamiltonian dynamics in a common Hilbert space framework, that allows the treatment of mixed quantum-classical systems. The analysis of some examples illustrates the possibility of entanglement between classical and quantum systems. We give a summary of the main tools of Berezin-Toeplitz and geometric quantization, that provide a relation between the classical and the quantum models, based essentially on the selection of a subspace of the classical Hilbert space. Coherent states provide a systematic tool for the inverse process, called dequantization, that associates a classical Hamiltonian system to a given quantum dynamics through the choice of a complete set of coherent states.
The quantum observables used in the case of quantum systems with finite-dimensional Hilbert space are defined either algebraically in terms of an orthonormal basis and discrete Fourier transformation or by using a continuous system of coherent states. We present an alternative approach to these important quantum systems based on the finite frame quantization. Finite systems of coherent states, usually called finite tight frames, can be defined in a natural way in the case of finite quantum systems. Novel examples of such tight frames are presented. The quantum observables used in our approach are obtained by starting from certain classical observables described by functions defined on the discrete phase space corresponding to the system. They are obtained by using a finite frame and a Klauder–Berezin–Toeplitz-type quantization. Semi-classical aspects of tight frames are studied through lower symbols of basic classical observables.
We present a survey of the theory of coherent states (CS) and some of their generalizations, with emphasis on the mathematical structure, rather than on physical applications. Starting from the standard theory of CS over Lie groups, we develop a general formalism, in which CS are associated to group representations which are square integrable over a homogeneous space. A further step allows us to dispense with the group context altogether, and thus obtain the so-called reproducing triples and continuous frames introduced in some earlier work. We discuss in detail a number of concrete examples, namely semisimple Lie groups, the relativity groups and various types of wavelets. Finally we turn to some physical applications, centering on quantum measurement and the quantization/dequantization problem, that is, the transition from the classical to the quantum level and vice versa.
Coherent quantum physics is quantum physics in terms of a coherent space and an appropriate coherent product. The kinematical structure and the meaning of the quantities are given by the symmetries (invertible coherent maps) of the coherent space. If Z is a coherent manifold, a canonical symplectic form is often canonically determined by the coherent product through the coherent action principle discussed in Section 6.1. In a quantum mechanical context, Z is indeed a classical phase space or an extended phase space-typically a symplectic manifold, a Poisson manifold, or a circle or line bundle over such a manifold that incorporates the classical action variable (encoding the Berry phase under quantization). For example, the Aharonov-Bohm effect [3] needs the bundle formulation. A coherent space and its quantum space provide a classical and a quantum view of the same physical system, discussed in Section 6.2. Section 6.3 shows that symmetries of coherent spaces can be quantized, that is, promoted to linear operators on the quantum space. This leads to quantum dynamics, which in special (completely integrable) situations can be solved in closed form in terms of classical motions on the underlying coherent space, if the latter has a compatible manifold structure. Close relations to concepts from geometric quantization and Kähler manifolds are pointed out in Section 6.4. In order that the quantum space of a coherent space Z can describe physics, one needs not only a distinguished class of coherent states but a more general concept of state. In Sections 6.5-6.6, we give a general abstract setting for states in Lie ∗-algebras, emphasize the essential mathematical features and the close analogy between classical and quantum physics. We illustrate these generalities with a number of examples in Section 6.7. The coherent action principle is put to use in Section 6.8 for numerical quantum physics and an application to quantum chaos. Spectral issues can, in favorable cases, be handled in terms of dynamical Lie algebras, discussed in Section 6.9.
In this paper we try to formulate the Berezin quantization on projective Hilbert space P(H) and use its geometric structure. It will be shown that the star product in Berezin quantiza- tion is equivalent to the Poisson bracket on P(H) and the Berezin method to construction a correspondence between a given classical theory and a given quantum theory is used to define a classical limit for geometric quantum mechanics. the covariant symbols. These symbols are expectation values of the observables in terms of coherent states: the holomorphic functions on classical phase space M that is assumed to be aKmanifold. Berezin (2) showed that the covariant symbols form a ∗-algebra which in limit � → 0l eads to the Poisson algebra between the corresponding classical observables: The functions on the phase space M. In this paper we will see that the Berezin ∗-algebra is in fact a Poisson algebra which is induced by the Fubini-Study 2-form on space of coherent states. This space is defined as follows: coherent states span a dense subspace H of Hilbert space H. P( H), which is denoted by M, is aKmanifold with induced symplectic structure from P(H). Therefore the covariant symbols can be considered as functions on M. It is shown (13) that there exists an embedding mapping between the classical phase space M and M, by which to any point z ∈ M is associated ap ointZ ∈M. With this construction to all of the quantum observables are associated their covariant symbols, which form a Poisson algebra on M and since the corresponding classical observables form a Poisson algebra on M , the Berezin quantization is a systematic procedure to relate these two Poisson algebras. Also the relation of Berezin quantization and geometric for- mulation of quantum mechanics will be evident as follows. The geometric quantum mechanics is a formulation of quantum mechanics in projective Hilbert spaces. With our construction one sees that the Berezin quantization is an equivalent formulation and in addition gives a prescription as classical limit for geometric quantum mechanics (9).
A theory of quantization is given, for a model with symmetry, as the inverse of dequantization. Dequanrization is defined as tin� process of extracting the classical mechanics of a model from its quantal counterpart. It is shown that classical dynamics is embedded in quantum dynamics as a constrained subsystem. Thus, classical mechanics is obtained from quantum mechanics without taking limits of large-dimensional representations or letting h �? 0. The embedding shows that, under appropriate conditions, a quantal density can be identified with a classical state. The results are consistent with the conjecture that an algebraic model is quantized by constructing a unitary irrep of its spectrum generating algebra. Quantization of operators not in this algebra is determined by compatibility with dequantization: this method resolves the Dirac problem in a way that avoids the ambiguities of canonical quantization. Geometric quantization is used to express the vector coherent state inducing construction as a theory of quantization. The result provides physically useful quantum models for systems with an algebraic structure and intrinsic degrees of freedom. The results are illustrated by application to the S i (3) model of a rotor. An algebraic model of a rigid rotor is presented, anti the techniques of vector coherent state quantization are applied to give a quantization, including intrinsic degrees of freedom, that is compatible with the geometric structure of the corresponding classical model. An explicit dequantization of the asymmetric top, leading to Euler�s equations, illustrates the consistency of this quantization.
It is well known from the work of Berezin [24] (Commun. Math. Phys. 40:153–174, 1975) in 1975 that the quantization problem of a classical mechanical system is closely related with coherent states. In particular coherent states help to understand the limiting behaviour of a quantum system when the Planck constant ℏ becomes negligible in macroscopic units. This problem is called the semi-classical limit problem. In this chapter we discuss properties of quantum systems when the configuration space is the Euclidean space Rn. So that in the Hamiltonian formalism, the phase space is Rn×Rn with its canonical symplectic form σ. The quantization problem has many solutions, so we choose a convient one, introduced by Weyl [259] (The Classical Groups, 1997) and Wigner [261] (Group Theory and Its Applications to Quantum Mechanics of Atomic Spectra, 1959). We study the symmetries of Weyl quantization, the operationnal calculus and applications to propagation of observables. We show that Wick quantization is a natural bridge between Weyl quantization and coherent states. Applications are given to the semi-classical limit after introducing an efficient modern tool: the semi-classical measures. We illustrate the general results proved in this chapter by explicit computations for the harmonic oscillator. More applications will be given in the following chapters, in particular concerning propagators and trace formulas for a large class of quantum systems.