The Principles of Newtonian and Quantum Mechanics

The following sections are included:

• FOREWORD BY BASIL HILEY

• PREFACE

• CONTENTS

The mathematical structure underlying Newtonian mechanics is symplectic geometry, which contains a classical form of Heisenberg's uncertainty principle. Quantum mechanics is based on de Broglie's theory of matter waves, whose evolution is governed by Schrödinger's equation. The latter emerges from classical mechanics using the metaplectic representation of the symplectic group.

A basic physical postulate, the Maxwell principle, implies that Newton's Second Law can be expressed in terms of the Poincaré-Cartan differential form. This leads to a Galilean covariant Hamiltonian mechanics.

Symplectic matrices form a connected Lie group. Hamiltonian flows consist of symplectomorphisms. Gromov's theorem shows that Hamiltonian flows preserve symplectic capacities. This leads to a topological version of Heisenberg's uncertainty principle in classical mechanics. It also leads to a topological quantization of phase space in cells, and to the Maslov quantization of Lagrangian manifolds.

The gain in action of a system is the integral of its Poincaré-Cartan form along the phase-space trajectory. For completely integrable systems, that gain is related to the phase of a Lagrangian manifold. This yields a geometric interpretation of Hamilton-Jacobi's equation. The amount of action needed to go from one point in physical space to another is the value of the generating function.

The time-evolution of the wave functions of quantum mechanics is essentially the Bohmian motion of half-densities. Semi-classical mechanics is the study of wave forms on Lagrangian manifolds; their definition requires the Leray index. The shadows on configuration space of wave forms are the usual semi-classical wave functions.

The symplectic group has a double covering which can be realized as a group of unitary operators on L²(ℝⁿ), the metaplectic group Mp(n). Each element of Mp(n) is the product of two "quadratic Fourier transforms". This property allows the definition of the Maslov index on Mp(n), which can be expressed in terms of the Leray index.

The metaplectic representation yields an algorithm allowing to calculate the solutions of Schrödinger's equation from the classical trajectories. Conversely, the classical trajectories can be recovered from the knowledge of the wave function. Both classical and quantum motion are thus deduced from the same mathematical object, the Hamiltonian flow.

The following sections are included:

• Appendix A SYMPLECTIC LINEAR ALGEBRA

• Appendix B THE LIE-TROTTER FORMULA FOR FLOWS

• Appendix C THE HEISENBERG GROUPS

• Appendix D THE BUNDLE OF S-DENSITIES

• Appendix E THE LAGRANGIAN GRASSMANNIAN

• BIBLIOGRAPHY

• INDEX