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arXiv:1201.0955 · Journal of Physics A Mathematical and Theoretical

Coherent states and related quantizations for unbounded motions

V G Bagrov, Jean‐Pierre Gazeau, D M Gitman, A D Levin

Published Mar 13, 2012• 9 citations
arXivHTMLPDF

Abstract

We discuss the construction of coherent states (CS) for systems with continuous spectra. First, we propose to adopt the Malkin–Manko approach, developed for systems with discrete spectra, to the case under consideration. Following this approach, we consider two examples, a free particle and a particle in a linear potential. Second, we generalize the approach of action-angle CS to systems with continuous spectra. In the first approach we start with a well-defined quantum formulation (canonical quantization) of a physical system and the construction of CS follows from such a quantization. In the second approach, the quantization procedure is inherent to the CS construction itself.

DOI
10.1088/1751-8113/45/12/125306

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Coherent and semiclassical states of a free particle

Vladislav G. Bagrov, Dmitri M Gitman, A. S. Pereira

Coherent states (CS) were first introduced and studied in detail for bound motion and discrete-spectrum systems like harmonic oscillators and similar systems with a quadratic Hamiltonian.However, the problem of constructing CS has still not been investigated in detail for the simplest and physically important case of a free particle, for which, besides being physically important, the CS problem is of didactic value in teaching quantum mechanics, with the CS regarded as examples of wave packets representing semiclassical motion.In this paper, we essentially follow the Malkin-Dodonov-Man'ko method to construct the CS of a free nonrelativistic particle.We give a detailed discussion of the properties of the CS obtained, in particular, the completeness relations, the minimization of uncertainty relations, and the evolution of the corresponding probability density.We describe the physical conditions under which free-particle CS can be considered semiclassical states.

Uspekhi Fizicheskih Nauk•Sep 1, 2014•7 citations

Coherent and semiclassical states of a free particle

Vladislav G. Bagrov, Dmitri M Gitman, A. S. Pereira

Coherent states (CS) were first introduced and studied in detail for bound motion and discrete-spectrum systems like harmonic oscillators and similar systems with a quadratic Hamiltonian. However, the problem of constructing CS has still not been investigated in detail for the simplest and physically important case of a free particle, for which, besides being physically important, the CS problem is of didactic value in teaching quantum mechanics, with the CS regarded as examples of wave packets representing semiclassical motion. In this paper, we essentially follow the Malkin-Dodonov-Man'ko method to construct the CS of a free nonrelativistic particle. We give a detailed discussion of the properties of the CS obtained, in particular, the completeness relations, the minimization of uncertainty relations, and the evolution of the corresponding probability density. We describe the physical conditions under which free-particle CS can be considered semiclassical states.

arXiv:1502.05013•Physics-Uspekhi•Sep 8, 2014•35 citations

Coherent states: a contemporary panorama

S. Twareque Ali, Jean-Pierre Antoine, Fabio Bagarello +1 more

Coherent states (CS) of the harmonic oscillator (also called canonical CS) were introduced in 1926 by Schrödinger in answer to a remark by Lorentz on the classical interpretation of the wave function. They were rediscovered in the early 1960s, first (somewhat implicitly) by Klauder in the context of a novel representation of quantum states, then by Glauber and Sudarshan for the description of coherence in lasers. Since then, CS have grown into an extremely rich domain that pervades almost every corner of physics and have also led to the development of several flourishing topics in mathematics. Along the way, a number of review articles have appeared in the literature, devoted to CS, notably the 1985 reprint volume of Klauder and Skagerstam [1], the 1990 review paper by Zhang et al [2], the 1993 Oak Ridge Conference [3] and the 1995 review paper by Ali et al [4]. Textbooks also have been published, among which one might mention the ground breaking text of Perelomov [5] focusing on the group-theoretical aspects, that of Ali et al [6] 1 analyzing systematically the mathematical structure beyond the group-theoretical approach and also the relation to wavelet analysis, that of Dodonov and Man'ko [7] mostly devoted to quantum optics, that of Gazeau [8] more oriented towards the physical, probabilistic and quantization aspects, and finally the very recent one by Combescure and Robert [9]. In retrospect, one can see that the development of CS has gone through a two-phase transition. First, the (simultaneous) discovery in 1972 by Gilmore and Perelomov that CS were rooted in group theory, then the realization that CS can be defined in a purely algebraic way, as an eigenvalue problem or by a series expansion (Malkin and Man'ko 1969, Barut and Girardello 1971, Gazeau and Klauder 1999; references to the original articles may be found in the textbooks quoted above). Both facts resulted in an explosive expansion of the CS literature. We thought, therefore, that the time was ripe to devote a special issue of Journal of Physics A: Mathematical and Theoretical to CS. However, because of limitations of space and time, it would have been impossible to get a fully representative cross-section of papers, covering all the different facets of the subject. Consequently, we have selected 37 articles, including some by a few of the originators of the field. We thank all the authors for submitting their up-to-date thoughts on this fascinating subject. The contents of this special issue are subdivided into five categories: (1) review papers; (2) physics-oriented CS; (3) physics and quantum information; (4) mathematics, general topics; and (5) mathematics, particular problems. (1) Review papers We start with five review papers. The first paper, by Klauder, surveys the many possible applications of affine variables, both in classical and quantum physics. The second, by Sanders, proposes a grand tour of entangled CS, which are present in many fields, such as quantum optics, quantum information processing, etc. The next paper, by Rowe, surveys the field of vector CS and the attendant group representation problems (including induced representations). Then Oriti et al describe a particular class of CS relevant to (loop) quantum gravity. Finally, Combescure and Robert present a comprehensive review of fermionic CS, including all mathematical details. (2) Physics-oriented CS The six contributions in this section deal with specific physical problems: (i) Dajka–Luczka study Gazeau–Klauder cat states associated with a nonlinear Kerr oscillator, instead of the usual canonical CS leading to Schrödinger cat states; (ii) Angelova et al discuss squeezed CS associated with a 1D Morse potential, used in molecular physics; (iii) Bagrov et al study CS in a magnetic solenoid field and prove their completeness; (iv) Blasone–Jizba treat Nambu–Goldstone dynamics in spontaneously broken symmetries, using CS functional integrals; (v) Calixto et al describe accelerated relativistic particles in the context of spontaneous breakdown of conformal SU(2,2) symmetry, using SU(2,2) CS; and (vi) Mortazavi–Tavassoly study f -deformed charge CS and their physical properties (nonclassical features, sub-Poissonian statistical behavior, etc). (3) Physics and quantum information The second group of physically related CS contains four contributions with a distinct quantum information theoretic flavor. First, Thilagam describes the dynamical behavior of entanglement of a pair of qubits (excitons), using a CS basis. Next, Lavoie–de Guise study SU(3) intelligent states (i.e., minimal uncertainty states), of interest in the quantum information community. Then Muñoz constructs discrete CS for n qubits. Finally, Wagner–Kendon explore the continuous variable Deutsch–Jozsa algorithm known in quantum computing in a discrete formulation. (4) Mathematics, general topics In this subgroup, there are eight papers dealing with general properties of CS, independently of any particular system or application. A whole series discusses the interaction between CS and various mathematical objects: pseudodifferential operators and Weyl calculus (Unterberger); induced representations of the affine group and intertwining operators (Elmabrok–Hutnik); measure-free CS and reproducing kernels (Horzela–Szafraniec); extremal POV measures (Heinosaari–Pellonpää); Hilbert W*-modules (Bhattacharyya–Roy); Toeplitz operators (Hutníková–Hutník); and operator localization and homogeneous structure of nilpotent Lie groups (Kisil). In addition, Balazs et al consider multipliers for continuous frames, including CS or wavelet frames. (5) Mathematics, particular problems The second group of mathematically oriented papers contains 14 contributions, devoted to CS in particular systems. We start with a paper by Gilmore, which explores the (sometimes chaotic) evolution of atomic CS under a time-periodic driving field, using sphere maps S 2 → S 2 . Next, we include a paper on CS on the 2-sphere in a magnetic field (Hall–Mitchell); a paper on CS for a quantum particle on a Möbius strip (Cirilo–Lombardo); a discussion of quantization on the circle (Chadzitaskos et al ); SUSY CS for Pöschl–Teller potentials (Bergeron et al ); generalized Bargmann functions and von Neumann lattices (Vourdas et al ); partial reconstruction for a finite CS system, using the Fock–Bargmann representation (Calixto et al ); phase operators for SU(3) irreps, thus for finite quantum systems (de Guise); semiclassical CS in periodic potentials (Carles–Sparber); complexified CS with non-Hermitian Hamiltonians (Graefe–Schubert); minimal uncertainty states in the context of (semisimple) group representation theory (Oszmaniec); localization operators in the time-frequency domain, i.e., in Gabor analysis (Muzhikyan–Avanesyan); and, finally, two papers about fermionic CS (Daoud–Kibler and Trifonov). This brief description illustrates perfectly the extreme versatility of the CS concept. As already stressed, coherent states constitute nowadays a flourishing research topic, with applications to a wide spectrum of domains. Indeed, CS are everywhere in physics: condensed matter physics, atomic physics, nuclear and particle physics, quantum optics, dynamics—both quantum and classical potentials—quantum gravity, quantization and quantum information theory. On the other hand, CS have grown into a fully-fledged domain in mathematics, incorporating many tools such as group representations, POV measures, frames, holomorphic functions, orthogonal polynomials and so on. Interestingly enough, the majority of contributions to this special issue (22 out of 37) are mathematically minded, demonstrating the widespread interest CS have generated in various areas of mathematics. A third field related to CS (but almost not represented in the present collection) is signal processing. Indeed both Gabor analysis and wavelet analysis derive in the first place from CS theory, namely, CS associated to the Weyl–Heisenberg and the ax + b group, respectively. Here too, a tremendous development has taken place in recent years, another testimony to the richness of the notion of CS. We leave it to the jury of public opinion to judge whether the call for a special issue of the journal, devoted to coherent states, has been justified. References [1] Klauder J R and Skagerstam B S 1985 Coherent States—Applications in Physics and Mathematical Physics (Singapore: World Scientific) [2] Zhang W-M, Feng D H and Gilmore R 1990 Coherent states: theory and some applications Rev. Mod. Phys. 62 867–927 [3] Feng D H, Klauder J R and Strayer M (ed) 1994 Coherent States: Past, Present and Future (Singapore: World Scientific) [4] Ali S T, Antoine J-P, Gazeau J-P and Mueller U A 1995 Coherent states and their generalizations: a mathematical overview Rev. Math. Phys. 7 1013–104 [5] Perelomov A M 1986 Generalized Coherent States and Their Applications (New York: Springer) [6] Ali S T, Antoine J-P and Gazeau J-P 2000 Coherent States, Wavelets and Their Generalizations (New York: Springer) [7] Dodonov V V and Man'ko V I (ed) 2003 Theory of Nonclassical States of Light (London: Taylor & Francis) [8] Gazeau J-P 2009 Coherent States in Quantum Physics (Berlin: Wiley) [9] Combescure M and Robert D 2012 Coherent States and Applications in Mathematical Physics (New York: Springer) 1 A second edition of that volume is in preparation.

Journal of Physics A Mathematical and Theoretical•May 30, 2012•9 citations

COHERENT STATES AND THEIR GENERALIZATIONS: A MATHEMATICAL OVERVIEW

S. Twareque Ali, Jean-Pierre Antoine, Jean‐Pierre Gazeau +1 more

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.

Reviews in Mathematical Physics•Oct 1, 1995•149 citations

6. Coherent quantum physics

Arnold Neumaier

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.

De Gruyter eBooks•Oct 24, 2019•0 citations

ON SOME ASPECTS OF COHERENT STATES QUANTIZATION WITH RELATED EXAMPLES

Isiaka Aremua, Komi Sodoga, P.K. Tchakpélé

This work addresses the general procedure of quantization also known as the Berezin-Klauder-Toeplitz quantization, or as coherent state (CS) (anti-Wick) quantization. The method is first illustrated by the motion of a particle on the circle. Then, we take as second example, a set of generalized photon-added coherent states related to associated hypergeometric functions. The nonclassical behaviour of this set of coherent states is also investigated.

Advances in Mathematics Scientific Journal•Nov 21, 2021•1 citation
We discuss the construction of coherent states (CS) for systems with continuous spectra. First, we propose to adopt the Malkin–Manko approach, developed for systems with discrete spectra, to the case under consideration. Following this approach, we consider two examples, a free particle and a particle in a linear potential. Second, we generalize the approach of action-angle CS to systems with continuous spectra. In the first approach we start with a well-defined quantum formulation (canonical quantization) of a physical system and the construction of CS follows from such a quantization. In the second approach, the quantization procedure is inherent to the CS construction itself.
Coherent states (CS) were first introduced and studied in detail for bound motion and discrete-spectrum systems like harmonic oscillators and similar systems with a quadratic Hamiltonian.However, the problem of constructing CS has still not been investigated in detail for the simplest and physically important case of a free particle, for which, besides being physically important, the CS problem is of didactic value in teaching quantum mechanics, with the CS regarded as examples of wave packets representing semiclassical motion.In this paper, we essentially follow the Malkin-Dodonov-Man'ko method to construct the CS of a free nonrelativistic particle.We give a detailed discussion of the properties of the CS obtained, in particular, the completeness relations, the minimization of uncertainty relations, and the evolution of the corresponding probability density.We describe the physical conditions under which free-particle CS can be considered semiclassical states.
Coherent states (CS) were first introduced and studied in detail for bound motion and discrete-spectrum systems like harmonic oscillators and similar systems with a quadratic Hamiltonian. However, the problem of constructing CS has still not been investigated in detail for the simplest and physically important case of a free particle, for which, besides being physically important, the CS problem is of didactic value in teaching quantum mechanics, with the CS regarded as examples of wave packets representing semiclassical motion. In this paper, we essentially follow the Malkin-Dodonov-Man'ko method to construct the CS of a free nonrelativistic particle. We give a detailed discussion of the properties of the CS obtained, in particular, the completeness relations, the minimization of uncertainty relations, and the evolution of the corresponding probability density. We describe the physical conditions under which free-particle CS can be considered semiclassical states.
Coherent states (CS) of the harmonic oscillator (also called canonical CS) were introduced in 1926 by Schrödinger in answer to a remark by Lorentz on the classical interpretation of the wave function. They were rediscovered in the early 1960s, first (somewhat implicitly) by Klauder in the context of a novel representation of quantum states, then by Glauber and Sudarshan for the description of coherence in lasers. Since then, CS have grown into an extremely rich domain that pervades almost every corner of physics and have also led to the development of several flourishing topics in mathematics. Along the way, a number of review articles have appeared in the literature, devoted to CS, notably the 1985 reprint volume of Klauder and Skagerstam [1], the 1990 review paper by Zhang et al [2], the 1993 Oak Ridge Conference [3] and the 1995 review paper by Ali et al [4]. Textbooks also have been published, among which one might mention the ground breaking text of Perelomov [5] focusing on the group-theoretical aspects, that of Ali et al [6] 1 analyzing systematically the mathematical structure beyond the group-theoretical approach and also the relation to wavelet analysis, that of Dodonov and Man'ko [7] mostly devoted to quantum optics, that of Gazeau [8] more oriented towards the physical, probabilistic and quantization aspects, and finally the very recent one by Combescure and Robert [9]. In retrospect, one can see that the development of CS has gone through a two-phase transition. First, the (simultaneous) discovery in 1972 by Gilmore and Perelomov that CS were rooted in group theory, then the realization that CS can be defined in a purely algebraic way, as an eigenvalue problem or by a series expansion (Malkin and Man'ko 1969, Barut and Girardello 1971, Gazeau and Klauder 1999; references to the original articles may be found in the textbooks quoted above). Both facts resulted in an explosive expansion of the CS literature. We thought, therefore, that the time was ripe to devote a special issue of Journal of Physics A: Mathematical and Theoretical to CS. However, because of limitations of space and time, it would have been impossible to get a fully representative cross-section of papers, covering all the different facets of the subject. Consequently, we have selected 37 articles, including some by a few of the originators of the field. We thank all the authors for submitting their up-to-date thoughts on this fascinating subject. The contents of this special issue are subdivided into five categories: (1) review papers; (2) physics-oriented CS; (3) physics and quantum information; (4) mathematics, general topics; and (5) mathematics, particular problems. (1) Review papers We start with five review papers. The first paper, by Klauder, surveys the many possible applications of affine variables, both in classical and quantum physics. The second, by Sanders, proposes a grand tour of entangled CS, which are present in many fields, such as quantum optics, quantum information processing, etc. The next paper, by Rowe, surveys the field of vector CS and the attendant group representation problems (including induced representations). Then Oriti et al describe a particular class of CS relevant to (loop) quantum gravity. Finally, Combescure and Robert present a comprehensive review of fermionic CS, including all mathematical details. (2) Physics-oriented CS The six contributions in this section deal with specific physical problems: (i) Dajka–Luczka study Gazeau–Klauder cat states associated with a nonlinear Kerr oscillator, instead of the usual canonical CS leading to Schrödinger cat states; (ii) Angelova et al discuss squeezed CS associated with a 1D Morse potential, used in molecular physics; (iii) Bagrov et al study CS in a magnetic solenoid field and prove their completeness; (iv) Blasone–Jizba treat Nambu–Goldstone dynamics in spontaneously broken symmetries, using CS functional integrals; (v) Calixto et al describe accelerated relativistic particles in the context of spontaneous breakdown of conformal SU(2,2) symmetry, using SU(2,2) CS; and (vi) Mortazavi–Tavassoly study f -deformed charge CS and their physical properties (nonclassical features, sub-Poissonian statistical behavior, etc). (3) Physics and quantum information The second group of physically related CS contains four contributions with a distinct quantum information theoretic flavor. First, Thilagam describes the dynamical behavior of entanglement of a pair of qubits (excitons), using a CS basis. Next, Lavoie–de Guise study SU(3) intelligent states (i.e., minimal uncertainty states), of interest in the quantum information community. Then Muñoz constructs discrete CS for n qubits. Finally, Wagner–Kendon explore the continuous variable Deutsch–Jozsa algorithm known in quantum computing in a discrete formulation. (4) Mathematics, general topics In this subgroup, there are eight papers dealing with general properties of CS, independently of any particular system or application. A whole series discusses the interaction between CS and various mathematical objects: pseudodifferential operators and Weyl calculus (Unterberger); induced representations of the affine group and intertwining operators (Elmabrok–Hutnik); measure-free CS and reproducing kernels (Horzela–Szafraniec); extremal POV measures (Heinosaari–Pellonpää); Hilbert W*-modules (Bhattacharyya–Roy); Toeplitz operators (Hutníková–Hutník); and operator localization and homogeneous structure of nilpotent Lie groups (Kisil). In addition, Balazs et al consider multipliers for continuous frames, including CS or wavelet frames. (5) Mathematics, particular problems The second group of mathematically oriented papers contains 14 contributions, devoted to CS in particular systems. We start with a paper by Gilmore, which explores the (sometimes chaotic) evolution of atomic CS under a time-periodic driving field, using sphere maps S 2 → S 2 . Next, we include a paper on CS on the 2-sphere in a magnetic field (Hall–Mitchell); a paper on CS for a quantum particle on a Möbius strip (Cirilo–Lombardo); a discussion of quantization on the circle (Chadzitaskos et al ); SUSY CS for Pöschl–Teller potentials (Bergeron et al ); generalized Bargmann functions and von Neumann lattices (Vourdas et al ); partial reconstruction for a finite CS system, using the Fock–Bargmann representation (Calixto et al ); phase operators for SU(3) irreps, thus for finite quantum systems (de Guise); semiclassical CS in periodic potentials (Carles–Sparber); complexified CS with non-Hermitian Hamiltonians (Graefe–Schubert); minimal uncertainty states in the context of (semisimple) group representation theory (Oszmaniec); localization operators in the time-frequency domain, i.e., in Gabor analysis (Muzhikyan–Avanesyan); and, finally, two papers about fermionic CS (Daoud–Kibler and Trifonov). This brief description illustrates perfectly the extreme versatility of the CS concept. As already stressed, coherent states constitute nowadays a flourishing research topic, with applications to a wide spectrum of domains. Indeed, CS are everywhere in physics: condensed matter physics, atomic physics, nuclear and particle physics, quantum optics, dynamics—both quantum and classical potentials—quantum gravity, quantization and quantum information theory. On the other hand, CS have grown into a fully-fledged domain in mathematics, incorporating many tools such as group representations, POV measures, frames, holomorphic functions, orthogonal polynomials and so on. Interestingly enough, the majority of contributions to this special issue (22 out of 37) are mathematically minded, demonstrating the widespread interest CS have generated in various areas of mathematics. A third field related to CS (but almost not represented in the present collection) is signal processing. Indeed both Gabor analysis and wavelet analysis derive in the first place from CS theory, namely, CS associated to the Weyl–Heisenberg and the ax + b group, respectively. Here too, a tremendous development has taken place in recent years, another testimony to the richness of the notion of CS. We leave it to the jury of public opinion to judge whether the call for a special issue of the journal, devoted to coherent states, has been justified. References [1] Klauder J R and Skagerstam B S 1985 Coherent States—Applications in Physics and Mathematical Physics (Singapore: World Scientific) [2] Zhang W-M, Feng D H and Gilmore R 1990 Coherent states: theory and some applications Rev. Mod. Phys. 62 867–927 [3] Feng D H, Klauder J R and Strayer M (ed) 1994 Coherent States: Past, Present and Future (Singapore: World Scientific) [4] Ali S T, Antoine J-P, Gazeau J-P and Mueller U A 1995 Coherent states and their generalizations: a mathematical overview Rev. Math. Phys. 7 1013–104 [5] Perelomov A M 1986 Generalized Coherent States and Their Applications (New York: Springer) [6] Ali S T, Antoine J-P and Gazeau J-P 2000 Coherent States, Wavelets and Their Generalizations (New York: Springer) [7] Dodonov V V and Man'ko V I (ed) 2003 Theory of Nonclassical States of Light (London: Taylor & Francis) [8] Gazeau J-P 2009 Coherent States in Quantum Physics (Berlin: Wiley) [9] Combescure M and Robert D 2012 Coherent States and Applications in Mathematical Physics (New York: Springer) 1 A second edition of that volume is in preparation.
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.