##
**Generalized coherent states and their applications.**
*(English)*
Zbl 0605.22013

Texts and Monographs in Physics. Berlin etc.: Springer-Verlag. XI, 320 p. DM 123.00 (1986).

The concept of coherent states can be traced back to the early times of quantum mechanics. In 1926, Erwin Schrödinger introduced a system of non-orthogonal wave functions in order to describe nonspreading wave packets for quantum oscillators. In a contemporary context, however, the appellation ”coherent states” was first used by Roy Glauber (1964) in quantum optics to deal with coherent laser light beams by the states invented earlier by Schrödinger. The standard system of coherent states is intimately related to the real Heisenberg nilpotent Lie group. Subsequently, the concept of coherent state was applied to cases well outside its original meaning and to families of quantum-mechanical states quite different from the original one. Therefore, some authors have attempted to preserve a distinction between the original definition and the expanded versions, calling the coherent states related to linear representations of ’arbitrary’ Lie groups ”generalized coherent states”.

The monograph under review treats the theory of generalized coherent states and their applications to various physical problems. Part I contains a study of properties of the coherent state system for the Heisenberg group. An explanation of the powerful Kirillov coadjoint orbit picture is postponed to Part II. Moreover, generalized coherent state systems are considered for the case of the three-dimensional rotation group and the Lorentz group. Part II considers generalized coherent states for nilpotent Lie groups, compact semisimple Lie groups and the automorphism groups of homogeneous symmetric domains. Finally, Part III is concerned with applications to physics (quantum oscillators, particles in external electromagnetic fields, relaxation to thermodynamic equilibrium, Landau diamagnetism, synchrotron radiation,...). Unfortunately, applications to electrical engineering like signal processing [cf. the reviewer: Harmonic analysis on the Heisenberg nilpotent Lie group, with application to signal theory. Pitman Res. Notes Math. 147, Longman 1986] and laser opto-electronics are missing.

A great deal of the material included in the book is due to the author. For the reader who is not a specialist in the field of noncommutative harmonic analysis, parts of the monograph, however, are not elaborated enough. In particular, the rôle played in the context of coherent states theory by the oscillator representation of the metaplectic group and the closely related group theoretic version of the Maslov index becomes not clear. Therefore it is doubtful whether it will help to fill the regrettable communications gap between mathematics and theoretical physics. In any case, the reader should consult the recent collection of articles dealing with coherent states [J. R. Klauder and B.-S. Skagerstam ”Coherent States”. World Scientific, Singapore 1985] for further information on the subject.

The monograph under review treats the theory of generalized coherent states and their applications to various physical problems. Part I contains a study of properties of the coherent state system for the Heisenberg group. An explanation of the powerful Kirillov coadjoint orbit picture is postponed to Part II. Moreover, generalized coherent state systems are considered for the case of the three-dimensional rotation group and the Lorentz group. Part II considers generalized coherent states for nilpotent Lie groups, compact semisimple Lie groups and the automorphism groups of homogeneous symmetric domains. Finally, Part III is concerned with applications to physics (quantum oscillators, particles in external electromagnetic fields, relaxation to thermodynamic equilibrium, Landau diamagnetism, synchrotron radiation,...). Unfortunately, applications to electrical engineering like signal processing [cf. the reviewer: Harmonic analysis on the Heisenberg nilpotent Lie group, with application to signal theory. Pitman Res. Notes Math. 147, Longman 1986] and laser opto-electronics are missing.

A great deal of the material included in the book is due to the author. For the reader who is not a specialist in the field of noncommutative harmonic analysis, parts of the monograph, however, are not elaborated enough. In particular, the rôle played in the context of coherent states theory by the oscillator representation of the metaplectic group and the closely related group theoretic version of the Maslov index becomes not clear. Therefore it is doubtful whether it will help to fill the regrettable communications gap between mathematics and theoretical physics. In any case, the reader should consult the recent collection of articles dealing with coherent states [J. R. Klauder and B.-S. Skagerstam ”Coherent States”. World Scientific, Singapore 1985] for further information on the subject.

Reviewer: W.Schempp

### MSC:

22E70 | Applications of Lie groups to the sciences; explicit representations |

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

81R30 | Coherent states |

22E43 | Structure and representation of the Lorentz group |