The Principles of Newtonian and Quantum Mechanics

The Need for Planck's Constant, h

The second edition of this book deals, as the first, with the foundations of classical physics from the "symplectic" point of view, and of quantum mechanics from the "metaplectic" point of view. We have revised and augmented the topics studied in the first edition in the light of new results, and added several new sections. The Bohmian interpretation of quantum mechanics is discussed in detail. Phase space quantization is achieved using the "principle of the symplectic camel", which is a deep topological property of Hamiltonian flows. We introduce the notion of "quantum blob", which can be viewed as the fundamental phase space unit. The mathematical tools developed in this book are the theory of the symplectic and metaplectic group, the Maslov index in a rigorous 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. The short-time behavior of the propagator is studied and applied to the quantum Zeno effect.

Contents:

• From Kepler to Schrödinger ... and Beyond

• Newtonian Mechanics

• The Symplectic Group

• Action and Phase

• Semi-Classical Mechanics

• Metaplectic Group, Maslov Index and Quantization

• Schrödinger's Equation and the Metatron

• Appendices:

  • Symplectic Linear Algebra
  • The Lie-Trotter Formula for Flows
  • The Heisenberg Groups
  • The Bundle of s-Densities
  • The Lagrangian Grassmannian

Readership: Reserchers and graduate students in mathematical physics and physics.