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Born-Jordan quantization. Theory and applications. (English) Zbl 1338.81014

Fundamental Theories of Physics 182. Cham: Springer (ISBN 978-3-319-27900-8/hbk; 978-3-319-27902-2/ebook). xiii, 226 p. (2016).
A quantization is a rule to assign quantum operators to classical quantities. Most of the quantizations agree on the quantization of Cartesian coordinates \(x\) and their associated momenta \(p_x\) as the operators “multiplication by \(x\)” and \(-i\hbar\frac{\partial}{\partial x}\) respectively. For less simple quantities, different rules can be applied. In [Z. Phys. 34, 858–888 (1925; JFM 51.0728.08)], M. Born and P. Jordan proposed a rule for the quantization of monomials \(x^rp_x^s\) fitting with W. Heisenberg’s “matrix mechanics”, generalized later to the multi-dimensional case in a joint work with Heisenberg himself. In the same time, H. Weyl proposed the quantization of classical quantities by introducing certain pseudo-differential operators and this approach is today widely accepted. When restricted to the same monomials, the Weyl’s approach produces different operators than the Born-Jordan’s one. In this book, the Born-Jordan and Weyl quantizations are confronted and discussed into details. In order to do that, the Born-Jordan approach is extended from monomials to any function of coordinates and momenta by defining appropriate pseudo-differential operators in two different ways: via Cohen classes (the classical observable is convoluted with a particular distribution and then quantized by the Weyl formula) and via Shubin pseudo-differential operators (these operators depend on a real parameter \(0\leq \tau \leq 1\) and coincide with Weyl operators for \(\tau=\frac 12\); the Born-Jordan quantization is realized by applying the Shubin operator to the classical observable and then integrating with respect to \(\tau\) between 0 and 1). It is remarkable that the Born-Jordan quantization coincides with the Weyl quantization on the natural Hamiltonians of the \(n\)-dimensional Euclidean space, written in Cartesian coordinates.
The properties of the Born-Jordan quantization are analyzed and compared with those of the Weyl quantization, showing advantages and disadvantages of each one. Among the disadvantages of the Born-Jordan quantization, for example, Moyal identities are no longer verified, some computations become more difficult, the correspondence between classical and quantum quantities is no longer bi-univocal. On the other hand, it is proved that, with an appropriate choice of the short-time action functional, the Born-Jordan quantization is the unique quantization producing the correct quantum Schrödinger Hamiltonian. Indeed, it is claimed that only the Born-Jordan quantization allows the equivalence between Heisenberg’s and Schrödinger’s descriptions of quantum mechanics. It is also shown how the Born-Jordan quantization solves a contradiction in the determination of the energy levels of the orbital angular momentum of the Bohr’s hydrogen atom presented by the Weyl’s approach.
The book is organized into three parts (Chapter 1 being the introduction). The first part (Chapters 2–5) is an introduction to Born-Jordan quantization from a physicist’s point of view; the role played by quantization in the equivalence of Heisenberg’s and Schrödinger’s pictures of quantum mechanics is in particular analyzed. The second part (Chapters 6–11) develops the mathematics of Born-Jordan quantization. Weyl quantization is also reviewed, not only as a comparison, but also as essential ingredient of the definition of Born-Jordan quantization. Cohen classes and Shubin’s pseudo-differential operators are reviewed, providing two different but equivalent ways of defining the Born-Jordan quantization. In the third part (Chapters 12–14), the theory of metaplectic group is introduced in order to study symplectic covariance of Weyl and Born-Jordan quantizations, the boundedness of Born-Jordan operators is also studied (this last topic showing connections with recent researches in functional analysis and time-frequency analysis).
The book is well written and in great part self-contained, it is intended as an introduction to Born-Jordan quantization both for physicists and mathematicians. The subject is analyzed both from the physical and the mathematical point of view.

MSC:

81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
81S10 Geometry and quantization, symplectic methods
35S05 Pseudodifferential operators as generalizations of partial differential operators
53D50 Geometric quantization
53D55 Deformation quantization, star products

Citations:

JFM 51.0728.08
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