Born-Jordan Quantization: Theory and Applications

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Springer, Jan 11, 2016 - Science - 226 pages

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.

 

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Contents

1 Introduction
1
Physical Motivation
7
2 On the Quantization Problem
9
3 Quantization of Monomials
22
4 Basic Hamiltonian Mechanics
37
5 Wave Mechanics and the Schrödinger Equation
56
Part IIMathematical Aspects of BornJordanQuantization
71
6 The Weyl Correspondence
73
9 Shubins PseudoDifferential Calculus
129
10 BornJordan PseudoDifferential Operators
147
11 Weak Values and the Reconstruction Problem
161
Part III Some Advanced Topics
171
12 Metaplectic Operators
172
13 Symplectic Covariance Properties
185
14 Symbol Classes and Function Spaces
199
References
216

7 The Cohen Class
95
8 BornJordan Quantization
112

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About the author (2016)

Maurice de Gosson is a research professor at the Univer-sity of Vienna. He has held various visiting positions at prestigious Universities(Yale, Boulder, Tel Aviv).

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