The Principles of Newtonian and Quantum Mechanics

, by Gosson, Maurice De; De Gosson, M. A.

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ISBN: 9781860942747 | 1860942741
Cover: Hardcover
Copyright: 10/1/2001

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Deals with the foundations of classical physics from the symplectic point of view, and of quantum mechanics from the metaplectic point of view. A precise form of Feynman's integral is introduced in connection with the extended metaplectic representation.

From Kepler to Schrodinger ... and Beyond

(36)

Classical Mechanics

(4)

Newton's Laws and Mach's Principle

Mass, Force, and Momentum

Symplectic Mechanics

(5)

Hamilton's Equations

Gauge Transformations

Hamiltonian Fields and Flows

The ``Symplectization of Science''

Action and Hamilton-Jacobi's Theory

(2)

Action

Hamilton-Jacobi's Equation

Quantum Mechanics

(6)

Matter Waves

``If There is a Wave, There Must be a Wave Equation!''

Schrodinger's Quantization Rule and Geometric Quantization

The Statistical Interpretation of Ψ

(3)

Heisenberg's Inequalities

Quantum Mechanics in Phase Space

(3)

Schrodinger's ``firefly'' Argument

The Symplectic Camel

Feynman's ''Path Integral``

(2)

The ``Sum Over All Paths''

The Metaplectic Group

Bohmian Mechanics

(4)

Quantum Motion: The Bell-DGZ Theory

Bohm's Theory

Interpretations

(6)

Epistemology or Ontology?

The Copenhagen Interpretation

The Bohmian Interpretation

The Platonic Point of View

Newtonian Mechanics

(40)

Maxwell's Principle and the Lagrange Form

(12)

The Hamilton Vector Field

Force Fields

Statement of Maxwell's Principle

Magnetic Monopoles and the Dirac String

The Lagrange Form

N-Particle Systems

Hamilton's Equations

(9)

The Poincare-Cartan Form and Hamilton's Equations

Hamiltonians for N-Particle Systems

The Transformation Law for Hamilton Vector Fields

The Suspended Hamiltonian Vector Field

Galilean Covariance

(7)

Inertial Frames

The Galilean Group Gal(3)

Galilean Covariance of Hamilton's Equations

Constants of the Motion and Integrable Systems

(5)

The Poisson Bracket

Constants of the Motion and Liouville's Equation

Constants of the Motion in Involution

Liouville's Equation and Statistical Mechanics

(7)

Liouville's Condition

Marginal Probabilities

Distributional Densities: An Example

The Symplectic Group

(50)

Symplectic Matrices and Sp(n)

(3)

Symplectic Invariance of Hamiltonian Flows

(3)

Notations and Terminology

Proof of the Symplectic Invariance of Hamiltonian Flows

Another Proof of the Symplectic Invariance of Flows*

The Properties of Sp(n)

(5)

The Subgroups U(n) and O(n) of Sp(n)

The Lie Algebra sp(n)

Sp(n) as a Lie Group

Quadratic Hamiltonians

(4)

The Linear Symmetric Triatomic Molecule

Electron in a Uniform Magnetic Field

The Inhomogeneous Symplectic Group

(2)

Galilean Transformations and ISp(n)

An Illuminating Analogy

(5)

The Optical Hamiltonian

Paraxial Optics

Gromov's Non-Squeezing Theorem

(9)

Liouville's Theorem Revisited

Gromov's Theorem

The Uncertainty Principle in Classical Mechanics

Symplectic Capacity and Periodic Orbits

108

(5)

The Capacity of an Ellipsoid

Symplectic Area and Volume

Capacity and Periodic Orbits

113

(5)

Periodic Hamiltonian Orbits

Action of Periodic Orbits and Capacity

Cell Quantization of Phase Space

118

(9)

Stationary States of Schrodinger's Equation

Quantum Cells and the Minimum Capacity Principle

Quantization of the N-Dimensional Harmonic Oscillator

Action and Phase

127

(52)

Introduction

127

(1)

The Fundamental Property of the Poincare-Cartan Form

128

(4)

Helmholtz's Theorem: The Case n = 1

Helmholtz's Theorem: The General Case

Free Symplectomorphisms and Generating Functions

132

(5)

Generating Functions

Optical Analogy: The Eikonal

Generating Functions and Action

137

(10)

The Generating Function Determined by H

Action vs. Generating Function

Gauge Transformations and Generating Functions

Solving Hamilton's Equations with W

The Cauchy Problem for Hamilton-Jacobi's Equation

Short-Time Approximations to the Action

147

(9)

The Case of a Scalar Potential

One Particle in a Gauge (A, U)

Many-Particle Systems in a Gauge (A, U)

Lagrangian Manifolds

156

(5)

Definitions and Basic Properties

Lagrangian Manifolds in Mechanics

The Phase of a Lagrangian Manifold

161

(7)

The Phase of an Exact Lagrangian Manifold

The Universal Covering of a Manifold*

The Phase: General Case

Phase and Hamiltonian Motion

Keller-Maslov Quantization

168

(11)

The Maslov Index for Loops

Quantization of Lagrangian Manifolds

Illustration: The Plane Rotator

Semi-Classical Mechanics

179

(42)

Bohmian Motion and Half-Densities

179

(7)

Wave-Forms on Exact Lagrangian Manifolds

Semi-Classical Mechanics

Wave-Forms: Introductory Example

The Leray Index and the Signature Function*

186

(15)

Cohomological Notations

The Leray Index: n = 1

The Leray Index: General Case

Properties of the Leray Index

More on the Signature Function

The Reduced Leray Index

De Rham Forms

201

(11)

Volumes and their Absolute Values

Construction of De Rham Forms on Manifolds

De Rham Forms on Lagrangian Manifolds

Wave-Forms on a Lagrangian Manifold

212

(9)

Definition of Wave Forms

The Classical Motion of Wave-Forms

The Shadow of a Wave-Form

The Metaplectic Group and the Maslov Index

221

(46)

Introduction

221

(4)

Could Schrodinger have Done it Rigorously?

Schrodinger's Idea

Sp(n)'s ''Big Brother`` Mp(n)

Free Symplectic Matrices and their Generating Functions

225

(6)

Free Symplectic Matrices

The Case of Affine Symplectomorphisms

The Generators of Sp(n)

The Metaplectic Group Mp(n)

231

(6)

Quadratic Fourier Transforms

The Operators ML, m and VP

The Projections Π and Π

237

(5)

Construction of the Projection Π

The Covering Groups Mp(n)

The Maslov Index on Mp(n)

242

(5)

Maslov Index: A ``Simple'' Example

Definition of the Maslov Index on Mp(n)

The Cohomological Meaning of the Maslov Index*

247

(6)

Group Cocycles on Sp(n)

The Fundamental Property of m(.)

The Inhomogeneous Metaplectic Group

253

(5)

The Heisenberg Group

The Group IMp(n)

The Metaplectic Group and Wave Optics

258

(2)

The Passage from Geometric to Wave Optics

The Groups Symp(n) and Ham(n)*

260

(7)

A Topological Property of Symp(n)

The Group Ham(n) of Hamiltonian Symplectomorphisms

The Groenewold-Van Hove Theorem

Schrodinger's Equation and the Metatron

267

(56)

Schrodinger's Equation for the Free Particle

267

(10)

The Free Particle's Phase

The Free Particle Propagator

An Explicit Expression for G

The Metaplectic Representation of the Free Flow

More Quadratic Hamiltonians

Van Vleck's Determinant

277

(3)

Trajectory Densities

The Continuity Equation for Van Vleck's Density

280

(4)

A Property of Differential Systems

The Continuity Equation for Van Vleck's Density

The Short-Time Propagator

284

(4)

Properties of the Short-Time Propagator

The Case of Quadratic Hamiltonians

288

(2)

Exact Green Function

Exact Solutions of Schrodinger's Equation

Solving Schrodinger's Equation: General Case

290

(10)

The Short-Time Propagator and Causality

Statement of the Main Theorem

The Formula of Stationary Phase

Two Lemmas - and the Proof

Metatrons and the Implicate Order

300

(13)

Unfolding and Implicate Order

Prediction and Retrodiction

The Lie-Trotter Formula for Flows

The ``Unfolded'' Metatron

The Generalized Metaplectic Representation

Phase Space and Schrodinger's Equation

313

(10)

Phase Space and Quantum Mechanics

Mixed Representations in Quantum Mechanics

Complementarity and the Implicate Order

A Symplectic Linear Algebra

323

(4)

B The Lie-Trotter Formula for Flows

327

(4)

C The Heisenberg Groups

331

(4)

D The Bundle of s-Densities

335

(4)

E The Lagrangian Grassmannian

339

(4)

Bibliography

343

(10)

Index

353

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The Principles of Newtonian and Quantum Mechanics

Summary

Table of Contents

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Summary

Table of Contents

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