Dalton, Bryan J.; Jeffers, John; Barnett, Stephen M. Phase space methods for degenerate quantum gases. (English) Zbl 1311.81003 International Series of Monographs on Physics 163. Oxford: Oxford University Press (ISBN 978-0-19-956274-9/hbk). xiv, 417 p. (2015). The book is basically a technical text, with the aim to introduce the reader to phase-space computational methods in quantum optics (specifically with multi-mode systems like e.g. boson or fermion condensates in mind). The major novelty in addition to more or less standard coherent state methods is a heavy exploitation of exterior algebras. Here the Grassmann algebra is viewed as the fermion analogue of c-numbers (commuting, employed in boson calculations). The Grassmann numbers are discussed at length, although it is rather hard to give them a direct physical meaning. Like in the S-matrix and path integral approaches to quantum field theory of fermions their role is basically reduced to a book-keeping device (encoding of the Pauli principle on the classical-looking level). Major arguments are inherited from quantum field theory monographs (I would recommend [J. Rzewuski, “Field Theory”, London: Iliffe Books (1969)]). The introduction of transport equations (Fokker-Planck analogues) and their Langevin-type bacground is basically formal (a good reading in this context would be [A. Rogers, Commun. Math. Phys. 113, 353–368 (1987; Zbl 0642.60035)]). Since the authors attempt to develop and generalize in the field of atom quantum optics methods introduced in the past by K. E. Cahill and R. J. Glauber [“Density operators for fermions”, Phys . Rev. A 59, No. 2, 1538–1555, (1999; doi:10.1103/PhysRevA.59.1538)], the major advantege of the text is that a broad technical material is gathered and explained in a single comprehensive source-book. Possible quantum optics applications are mentioned. A number of methodological inputs (including the usage of Grassmann numbers) is justified by verifying that model computations are in agreement with those obtained by traditional (and possibly more cumbersome) techniques. Reviewer: Piotr Garbaczewski (Opole) Cited in 2 Documents MSC: 81-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory 81V80 Quantum optics 81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices 81R30 Coherent states 81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics 81S40 Path integrals in quantum mechanics 82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics 82C10 Quantum dynamics and nonequilibrium statistical mechanics (general) 15A75 Exterior algebra, Grassmann algebras 35Q84 Fokker-Planck equations Keywords:phase space methods; bosons; fermions; Grassmann calculus; fermion and boson coherent states; phase space distributions; Fokker-Planck equations; Langevin equations; few-mode systems; functional calculus for c-number and Grassmann fields; distribution functionals in quantum atom physics; functional Fokker-Planck equations; Langevin field equations; multi-mode systems; condensates Citations:Zbl 0642.60035 PDFBibTeX XMLCite \textit{B. J. Dalton} et al., Phase space methods for degenerate quantum gases. Oxford: Oxford University Press (2015; Zbl 1311.81003) Full Text: DOI