Born-Jordan Quantization: Theory and Applications

Description

This book presents a comprehensive mathematical study of the operators behind the Born–Jordan quantization scheme. The Schrödinger and Heisenberg pictures of quantum mechanics are equivalent only if the Born–Jordan scheme is used. Thus, Born–Jordan quantization provides the only physically consistent quantization scheme, as opposed to the Weyl quantization commonly used by physicists. In this book we develop Born–Jordan 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. Front Matter....Pages i-xiii Introduction....Pages 1-6 Front Matter....Pages 7-7 On the Quantization Problem....Pages 9-21 Quantization of Monomials....Pages 23-35 Basic Hamiltonian Mechanics....Pages 37-55 Wave Mechanics and the Schrödinger Equation....Pages 57-70 Front Matter....Pages 71-71 The Weyl Correspondence....Pages 73-94 The Cohen Class....Pages 95-111 Born–Jordan Quantization....Pages 113-127 Shubin’s Pseudo-Differential Calculus....Pages 129-145 Born–Jordan Pseudo-Differential Operators....Pages 147-159 Weak Values and the Reconstruction Problem....Pages 161-170 Front Matter....Pages 171-171 Metaplectic Operators....Pages 173-184 Symplectic Covariance Properties....Pages 185-197 Symbol Classes and Function Spaces....Pages 199-215 Back Matter....Pages 217-226