Born-Jordan Quantization Theory and Applications [Hardcover]

This book presents a comprehensive mathematical study of the operators behind the BornJordan quantization scheme. The Schr?dinger and Heisenberg pictures of quantum mechanics are equivalent only if the BornJordan scheme is used. Thus, BornJordan quantization provides the only physically consistent quantization scheme, as opposed to the Weyl quantization commonly used by physicists. In this book we develop BornJordan quantization from an operator-theoretical point of view, and analyze in depth the conceptual differences between the two schemes. We discuss various physically motivated approaches, in particular the Feynman-integral point of view. One important and intriguing feature of Born-Jordan quantization is that it is not one-to-one: there are infinitely many classical observables whose quantization is zero.

Born-Jordan Quantization: Physical Motivation: On the Quantization Problem.- Quantization of Monomials.- Basic Hamiltonian Mechanics.- Wave Mechanics and the Schr?dinger Equation.- Mathematical Aspects of Born-Jordan Quantization: The Weyl Correspondence.- The Cohen Class.- Born-Jordan Quantization.- Shubin's Pseudo-Differential Calculus.- Born-Jordan Pseudo-Differential Operators.- Weak Values and the Reconstruction Problem.- Some Advanced Topics: Metaplectic Operators.- Symplectic Covariance Properties.- Symbol Classes and Function Spaces.

In this book, the Born-Jordan and Weyl quantizations are confronted and discussed into details. & 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. (Giovanni Rastelli, zbMATH 1338.81014, 2016)