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**Supersymmetry in quantum and classical mechanics.**
*(English)*
Zbl 0970.81021

Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics. 116. Boca Raton, FL: Chapman & Hall/CRC. xiv, 222 p. (2001).

The concept of supersymmetry (boson-fermion symmetry) arose about 30 years ago in the area of quantum field theory. Since then, it has been of importance in various areas of mathematical physics.

The notion of supersymmetric quantum mechanics is due to E. Witten [Nucl. Phys. B 188, 513–554 (1981; Zbl 1258.81046); Constraints on supersymmetry breaking, Nucl. Phys. B 202, 253-316 (1982)]. Nowadays, supersymmetric quantum mechanics has become a separate area of research to which a growing number of journal articles are devoted.

In this monograph, the author summarizes the main developments of the last 15 years. After a short introduction, the basic principles of supersymmetric quantum mechanics are outlined in Chapter 2. The main example of supersymmetry in non-relativistic quantum mechanics is the standard harmonic oscillator (superposition of bosonic and fermionic oscillator). This leads to the notions of superpotential and supersymmetric Hamiltonian. Though the examples are worked out in detail, the author fails to clarify which equations are specific to the example and which are valid in general (and thus should be taken as the defining equations).

In Chapter 3, supersymmetric classical mechanics is considered. This involves the development of classical mechanical systems consisting of ordinary as well as of anti-commuting variables. The treatment is rather short; the notions of Lie algebras or Lie superalgebras are not mentioned. In Chapter 4, the concepts of supersymmetry breaking and Witten index are introduced. Index theory is approached from the physical point of view by counting the zero-energy states. The rest of this chapter deals with \(q\)-deformations (of bosons, fermions, and hence of supersymmetric Hamiltonians), parabosons, and \(q\)-deformations of parabosons. Chapter 5 is devoted to the relation between supersymmetric quantum mechanics and the famous factorization method of Infeld and Hull. Here, the interesting aspects of supersymmetry arise: the possibility of factorizing the SchrĂ¶dinger Hamiltonian and the connection between solvable potentials and “shape invariance”. The remaining chapters deal with more specialized topics : supersymmetric quantum mechanics in three or more dimensions (radial problems and spin-orbit coupling), supersymmetry in nonlinear systems (supersymmetric KdV equation), and parasupersymmetry (models of supersymmetry inspired by parabosons and parafermions).

The book gives an overview of standard and advanced topics where supersymmetric quantum mechanics plays a role. The vocabulary and style of the book is certainly that of a physicist. Mathematical concepts are either not mentioned or incomplete. A reader with a proper physics background gets a good idea of the kind of problems in which supersymmetric quantum mechanics plays a role; moreover, the list of references is fairly complete and up to date. For the reader with a mathematics background the proper definitions and the overall mathematical context are missing.

As a final remark, it is surprising how many typing errors have remained in the text : already in the first two paragraphs, one finds words like “tweory”, “conventicnal” and “tne”. This sort of errors continues throughout the book, and gives the impression that the final text has never been proofread.

The notion of supersymmetric quantum mechanics is due to E. Witten [Nucl. Phys. B 188, 513–554 (1981; Zbl 1258.81046); Constraints on supersymmetry breaking, Nucl. Phys. B 202, 253-316 (1982)]. Nowadays, supersymmetric quantum mechanics has become a separate area of research to which a growing number of journal articles are devoted.

In this monograph, the author summarizes the main developments of the last 15 years. After a short introduction, the basic principles of supersymmetric quantum mechanics are outlined in Chapter 2. The main example of supersymmetry in non-relativistic quantum mechanics is the standard harmonic oscillator (superposition of bosonic and fermionic oscillator). This leads to the notions of superpotential and supersymmetric Hamiltonian. Though the examples are worked out in detail, the author fails to clarify which equations are specific to the example and which are valid in general (and thus should be taken as the defining equations).

In Chapter 3, supersymmetric classical mechanics is considered. This involves the development of classical mechanical systems consisting of ordinary as well as of anti-commuting variables. The treatment is rather short; the notions of Lie algebras or Lie superalgebras are not mentioned. In Chapter 4, the concepts of supersymmetry breaking and Witten index are introduced. Index theory is approached from the physical point of view by counting the zero-energy states. The rest of this chapter deals with \(q\)-deformations (of bosons, fermions, and hence of supersymmetric Hamiltonians), parabosons, and \(q\)-deformations of parabosons. Chapter 5 is devoted to the relation between supersymmetric quantum mechanics and the famous factorization method of Infeld and Hull. Here, the interesting aspects of supersymmetry arise: the possibility of factorizing the SchrĂ¶dinger Hamiltonian and the connection between solvable potentials and “shape invariance”. The remaining chapters deal with more specialized topics : supersymmetric quantum mechanics in three or more dimensions (radial problems and spin-orbit coupling), supersymmetry in nonlinear systems (supersymmetric KdV equation), and parasupersymmetry (models of supersymmetry inspired by parabosons and parafermions).

The book gives an overview of standard and advanced topics where supersymmetric quantum mechanics plays a role. The vocabulary and style of the book is certainly that of a physicist. Mathematical concepts are either not mentioned or incomplete. A reader with a proper physics background gets a good idea of the kind of problems in which supersymmetric quantum mechanics plays a role; moreover, the list of references is fairly complete and up to date. For the reader with a mathematics background the proper definitions and the overall mathematical context are missing.

As a final remark, it is surprising how many typing errors have remained in the text : already in the first two paragraphs, one finds words like “tweory”, “conventicnal” and “tne”. This sort of errors continues throughout the book, and gives the impression that the final text has never been proofread.

Reviewer: Joris Van der Jeugt (Gent)

### MSC:

81Q60 | Supersymmetry and quantum mechanics |

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