The Principles of Newtonian and Quantum Mechanics : The Need for Planck's Constant, H

The Principles of Newtonian and Quantum Mechanics : The Need for Planck's Constant, H

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  • 製本 Hardcover:ハードカバー版/ページ数 320 p.
  • 言語 ENG,ENG
  • 商品コード 9781860942747
  • DDC分類 530.15564

Full Description


This work deals with the foundations of classical physics from the "symplectic" point of view, and of quantum mechanics from the "metaplectic" point of view. The Bohmian interpretation of quantum mechanics is discussed. Phase space quantization is achieved using the "principle of the symplectic camel", which is a recently discovered deep topological property of Hamiltonian flows. The mathematical tools developed in this book are the theory of the metaplectic group, the Maslov index in a precise form, and the Leray index of a pair of Lagrangian planes. The concept of the "metatron" is introduced, in connection with the Bohmian theory of motion. 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 65 (5)
Systems
The Poisson Bracket
Constants of the Motion and Liouville's
Equation
Constants of the Motion in Involution
Liouville's Equation and Statistical 70 (7)
Mechanics
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 128(4)
Poincare-Cartan Form
Helmholtz's Theorem: The Case n = 1
Helmholtz's Theorem: The General Case
Free Symplectomorphisms and Generating 132(5)
Functions
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 186(15)
Function*
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 225(6)
Generating Functions
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 247(6)
Index*
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 280(4)
Density
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