Emergence of the Quantum from the Classical

Mathematical Aspects of Quantum Processes

Maurice de Gosson

• Publisher: WSPC (EUROPE)
• Year: 2017
• ISBN:
  9781786344168
• Topic: Physical Sciences
• Subtopic: Mechanics

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The emergence of quantum mechanics from classical world mechanics is now a well-established theme in mathematical physics. This book demonstrates that quantum mechanics can indeed be viewed as a refinement of Hamiltonian mechanics, and builds on the work of George Mackey in relation to their mathematical foundations. Additionally when looking at the differences with classical mechanics, quantum mechanics crucially depends on the value of Planck's constant h. Recent cosmological observations tend to indicate that not only the fine structure constant α but also h might have varied in both time and space since the Big Bang. We explore the mathematical and physical consequences of a variation of h; surprisingly we see that a decrease of h leads to transitions from the quantum to the classical.

Emergence of the Quantum from the Classical provides help to undergraduate and graduate students of mathematics, physics and quantum theory looking to advance into research in the field.

--> Contents:

• Hamiltonian Mechanics
• Hamilton–Jacobi Theory
• Matter Waves, Schrödinger's Equation, and Bohm's Theory
• The Metatron
• Uncertainties and Quantum Blobs
• Quantum States and the Density Matrix
• Varying Planck's Constant
• Appendices:
  • The Symplectic Group
  • The Metaplectic Representation
  • Born–Jordan Quantization
  • Twisted Product and Convolution

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--> Readership: Undergraduate and graduate students of mathematics and physics, interested in analysis and differential equations, probability and statistics, geometry and topology and mathematical physics. -->
Keywords:Classical Mechanics;Hamiltonian Mechanics;Quantum Mechanics;Planck's Constant;Mathematical Physics;Algebraic GeometryReview:0

Chapter 1

Hamiltonian Mechanics

Hamiltonian mechanics is perhaps the most powerful classical theory ever; it allows the prediction of the motion of celestial bodies, of aeroplanes, and of particles in fluids. While Hamilton’s equations of motion are easily derived, in the simplest cases, from Newton’s second law, Hamiltonian mechanics is more than just a fancy way of doing Newtonian mechanics. Hamiltonian mechanics could already be found in disguise in the work of Lagrange in celestial mechanics. Namely, Lagrange discovered that the equations expressing the perturbation of elliptical planetary motion due to interactions could be written down as a simple system of partial differential equations (known today as Hamilton’s equations, but Hamilton was only six years old when Lagrange made his discovery!). It is however undoubtedly Hamilton who realized, some twenty four years later, the theoretical importance of Lagrange’s discovery and exploited it fully. We mention in passing that the notation H for a Hamiltonian function was proposed by Lagrange to honor Christiaan Huygens, and not Hamilton!

Like the movement of a symphony, a Hamiltonian flow involves a total ordering which implies the whole movement: Past, present, and future are actively present in any one movement. When we are listening to music we are actually directly perceiving an implicate order. This order is active because it is continuously flowing in emotional responses which are inseparable from the flow itself. Similarly, the solutions of Hamilton’s equations are uniquely determined for all bounded times and all locations close to the original one, exactly as in the symphony metaphor: if we observe during a tiny time interval the motion of a particle moving under the influence of a Hamiltonian flow, we see an unfoldment of the totality of the flow, which is uniquely determined by the past — and the future!

There are nowadays many texts, at various levels, presenting Hamiltonian mechanics from the symplectic point of view; for instance the books by Arnold [5], Abraham et al. [2], Guillemin and Sternberg [139, 140] are classical references; for good introductions to symplectic geometry from the modern point of view see [23].

1.1. Hamilton’s Equations

1.1.1. The origins; examples

Most physical systems can be studied by using two specific theories originating from Newtonian mechanics, and having overlapping — but not identical — domains of validity. The first of these theories is “Lagrangian mechanics”, which essentially uses variational principles (e.g., the “least action principle”); it will not be discussed at all in this book; we refer to Souriau [268] (especially p. 140) for an analysis of some of the drawbacks of the Lagrangian approach. The second theory, “Hamiltonian mechanics”, is based on Hamilton’s equations of motion

where the Hamiltonian f...

Table of contents

• Cover page
• Title page
• Copyright
• Dedication
• Contents
• Preface
• Introduction
• Chapter 1. Hamiltonian Mechanics
• Chapter 2. Hamilton–Jacobi Theory
• Chapter 3. Matter Waves, Schrödinger’s Equation, and Bohm’s Theory
• Chapter 4. The Metatron
• Chapter 5. Uncertainties and Quantum Blobs
• Chapter 6. Quantum States and the Density Matrix
• Chapter 7. Varying Planck’s Constant
• Appendix A. The Symplectic Group
• Appendix B. The Metaplectic Representation
• Appendix C. Born–Jordan Quantization
• Appendix D. Twisted Product and Convolution
• Bibliography
• Index