##
**Supersymmetric methods in quantum and statistical physics.**
*(English)*
Zbl 0867.00011

Texts and Monographs in Physics. Berlin: Springer-Verlag. xiv, 172 p. (1996).

We witnessed great enthusiasm in recent years for the idea that elementary particle interactions should possess global supersymmetry (SUSY). It involves the introduction of spinor generators, the so-called supercharges, to supplement the Lie algebra \({\mathfrak g}\) of the Poincaré group and to form what is now called a super Lie algebra. The author embarks on a study of supersymmetry within the context of ordinary quantum mechanics where the Lie algebra \({\mathfrak g}\) reduces to the multiples of the Hamiltonian.

He starts with the definition of \(N\)-extended supersymmetric theory. While often \(N=2\) in concrete cases, the general concept is due to E. Witten. The simplest example however – an \(N=1\) model – is provided by the Pauli Hamiltonian for an electron in a magnetic field. After two introductory chapters the author studies Witten’s \(N=2\) model of 1981 and the breaking mechanism of SUSY in some detail. The expository work is supplemented by another chapter where he studies the classical version of that theory leading to a class of so-called pseudoclassical models describing spin degrees on a classical level by Grassmann variables. This then prepares the ground for a stationary-phase approximation of the path integral for Witten’s model and a quasi-classical quantization procedure giving the exact bound-state energies for those models that are known to be exactly solvable: the relation to the Duistermaat-Heckmann localization formula in mathematics is still open and remains to be investigated. Next, the author focusses on classical stochastic differential equations originating from a supersymmetric Schrödinger equation with imaginary time, and the last chapter is devoted to studying the Pauli paramagnetism of a non-interacting electron gas. An interesting consequence of broken SUSY may be the existence of counterexamples to the paramagnetic conjecture of Hogreve, Schrader, and Seiler.

In summary, the open-end text, though giving a preliminary account of important contemporary research activities, is very inspiring to students and researchers alike.

He starts with the definition of \(N\)-extended supersymmetric theory. While often \(N=2\) in concrete cases, the general concept is due to E. Witten. The simplest example however – an \(N=1\) model – is provided by the Pauli Hamiltonian for an electron in a magnetic field. After two introductory chapters the author studies Witten’s \(N=2\) model of 1981 and the breaking mechanism of SUSY in some detail. The expository work is supplemented by another chapter where he studies the classical version of that theory leading to a class of so-called pseudoclassical models describing spin degrees on a classical level by Grassmann variables. This then prepares the ground for a stationary-phase approximation of the path integral for Witten’s model and a quasi-classical quantization procedure giving the exact bound-state energies for those models that are known to be exactly solvable: the relation to the Duistermaat-Heckmann localization formula in mathematics is still open and remains to be investigated. Next, the author focusses on classical stochastic differential equations originating from a supersymmetric Schrödinger equation with imaginary time, and the last chapter is devoted to studying the Pauli paramagnetism of a non-interacting electron gas. An interesting consequence of broken SUSY may be the existence of counterexamples to the paramagnetic conjecture of Hogreve, Schrader, and Seiler.

In summary, the open-end text, though giving a preliminary account of important contemporary research activities, is very inspiring to students and researchers alike.

Reviewer: G.Roepstorff (Aachen)

### MSC:

00A79 | Physics |

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

81Q60 | Supersymmetry and quantum mechanics |