The Principles of Newtonian and Quantum Mechanics

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

• ISBN13: 9781860942747
• ISBN10: 1860942741
• Format: Hardcover
• Copyright: 2001-10-01
• Publisher: Imperial College Pr

Summary

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.

Table of Contents

From Kepler to Schrodinger ... and Beyond	1	(36)
Classical Mechanics	2	(4)
Newton's Laws and Mach's Principle
Mass, Force, and Momentum
Symplectic Mechanics	6	(5)
Hamilton's Equations
Gauge Transformations
Hamiltonian Fields and Flows
The ``Symplectization of Science''
Action and Hamilton-Jacobi's Theory	11	(2)
Action
Hamilton-Jacobi's Equation
Quantum Mechanics	13	(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	19	(3)
Heisenberg's Inequalities
Quantum Mechanics in Phase Space	22	(3)
Schrodinger's ``firefly'' Argument
The Symplectic Camel
Feynman's ''Path Integral``	25	(2)
The ``Sum Over All Paths''
The Metaplectic Group
Bohmian Mechanics	27	(4)
Quantum Motion: The Bell-DGZ Theory
Bohm's Theory
Interpretations	31	(6)
Epistemology or Ontology?
The Copenhagen Interpretation
The Bohmian Interpretation
The Platonic Point of View
Newtonian Mechanics	37	(40)
Maxwell's Principle and the Lagrange Form	37	(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	49	(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	58	(7)
Inertial Frames
The Galilean Group Gal(3)
Galilean Covariance of Hamilton's Equations
Constants of the Motion and Integrable Systems	65	(5)
The Poisson Bracket
Constants of the Motion and Liouville's Equation
Constants of the Motion in Involution
Liouville's Equation and Statistical Mechanics	70	(7)
Liouville's Condition
Marginal Probabilities
Distributional Densities: An Example
The Symplectic Group	77	(50)
Symplectic Matrices and Sp(n)	77	(3)
Symplectic Invariance of Hamiltonian Flows	80	(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)	83	(5)
The Subgroups U(n) and O(n) of Sp(n)
The Lie Algebra sp(n)
Sp(n) as a Lie Group
Quadratic Hamiltonians	88	(4)
The Linear Symmetric Triatomic Molecule
Electron in a Uniform Magnetic Field
The Inhomogeneous Symplectic Group	92	(2)
Galilean Transformations and ISp(n)
An Illuminating Analogy	94	(5)
The Optical Hamiltonian
Paraxial Optics
Gromov's Non-Squeezing Theorem	99	(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