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Born-Jordan Quantization

Part of the Fundamental Theories of Physics Book Series (volume 182)

DOI: 10.1007/978-3-319-27902-2

ISBN: 978-3-319-27900-8

Publisher: Springer International Publishing - Springer Journals

Jan 12, 2016

1 - 10 of 14 Chapters

Available

Introduction

de Gosson, Maurice A.

Pages: 1 –6

[December 14, 1900, is usually regarded as the official date of birth of quantum theory, because on that day Max Planck presented a memoir at a meeting of the Physical Society of Berlin in which he solved the enigma of the blackbody spectrum by introducing a new, fundamental, constant of Nature.]
Available

On the Quantization Problem

de Gosson, Maurice A.

Pages: 9 –21

[In 1925 Max Born and Pascual Jordan set out to give a rigorous mathematical basis to Werner Heisenberg’s newly born “matrix mechanics”. This led them led to state a quantization rule for monomials.]
Available

Quantization of Monomials

de Gosson, Maurice A.

Pages: 23 –35

[In this chapter we begin by collecting some facts about the quantization of monomials and polynomials, with a particular emphasis on the Weyl and Born–Jordan schemes.]
Available

Basic Hamiltonian Mechanics

de Gosson, Maurice A.

Pages: 37 –55

[In order to discuss wave mechanics and the Schrödinger equation in the next chapter we need some basic aspects of classical mechanics in its Hamiltonian formulation.]
Available

Wave Mechanics and the Schrödinger Equation

de Gosson, Maurice A.

Pages: 57 –70

[While we have been concerned in Chap. 3 with the quantization of monomials, we now address the question of how we should quantize an arbitrary dynamical variable (i.e. a Hamiltonian function).]
Available

The Weyl Correspondence

de Gosson, Maurice A.

Pages: 73 –94

[The Weyl correspondence, or Weyl quantization, is well-known both in harmonic analysis and quantum mechanics. It is part of the wider Weyl–Wigner–Moyal theory, where an emphasis on phase space techniques is made.]
Available

The Cohen Class

de Gosson, Maurice A.

Pages: 95 –111

[In this chapter we study the “Cohen class”, which is a family of particular phase space quasi-distributions characterized by two simple properties (continuity, and translation covariance). These quasi-distributions are obtained from the Wigner transform by convolving the latter with a suitable...
Available

Born–Jordan Quantization

de Gosson, Maurice A.

Pages: 113 –127

[This chapter is in a sense the keystone of this book. Using the theory of the Cohen class previously studied we give a first working definition of Born–Jordan quantization by selecting a particular Cohen kernel. We state and prove some important properties of the associated Born–Jordan...
Available

Shubin’s Pseudo-Differential Calculus

de Gosson, Maurice A.

Pages: 129 –145

[In this rather technical chapter we study pseudo-differential calculus from the point of view developed in Shubin [5] (also see Chap. 14 in de Gosson [3] for a review of the essentials of Shubin’s theory).]
Available

Born–Jordan Pseudo-Differential Operators

de Gosson, Maurice A.

Pages: 147 –159

[In Chap. 7 we defined Born–Jordan quantization by using the properties of the Cohen class. In this chapter we use the theory of Shubin operators and define the Born–Jordan operator A^BJ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}...

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