The principles of Newtonian and quantum mechanics : the need for Planck's constant, h / M A de Gosson.

Includes bibliographical references (pages 343-351) and index.

Print version record.

1. From Kepler to Schrödinger ... and beyond. 1.1. Classical mechanics -- 1.2. Symplectic mechanics -- 1.3. Action and Hamilton-Jacobi's theory -- 1.4. Quantum mechanics -- 1.5. The statistical interpretation of [symbol] -- 1.6. Quantum mechanics in phase space -- 1.7. Feynman's "Path integral" -- 1.8. Bohmian mechanics -- 1.9. Interpretations -- 2. Newtonian mechanics. 2.1. Maxwell's principle and the lagrange form -- 2.2. Hamilton's equations -- 2.3. Galilean covariance -- 2.4. Constants of the motion and integrable systems -- 2.5. Liouville's equation and statistical mechanics -- 3. THE symplectic group. 3.1. Symplectic matrices and Sp(n) -- 3.2. Symplectic invariance of Hamiitonian flows -- 3.3. The properties of Sp(n) -- 3.4. Quadratic Hamiltonians -- 3.5. The inhomogeneous symplectic group -- 3.6. An illuminating analogy -- 3.7. Gromov's non-squeezing theorem -- 3.8. Symplectic capacity and periodic orbits -- 3.9. Capacity and periodic orbits -- 3.10. Cell quantization of phase space -- 4. Action and phase. 4.1. Introduction -- 4.2. The fundamental property of the Poincaré-Cartan form -- 4.3. Free symplectomorphisms and generating functions -- 4.4. Generating functions and action -- 4.5. Short-time approximations to the action -- 4.6. Lagrangian manifolds -- 4.7. The phase of a Lagrangian manifold -- 4.8. Keller-Maslov quantization -- 5. Semi-classical mechanics. 5.1. Bohmian motion and half-densities -- 5.2. The Leray index and the signature function -- 5.3. De Rham forms -- 5.4. Wave-forms on a Lagrangian manifold -- 6. The metaplectic group and the Maslov index. 6.1. Introduction -- 6.2. Free symplectic matrices and their generating functions -- 6.3. The metaplectic group Mp(n) -- 6.4. The projections II and II[symbol] -- 6.5. The Maslov index on Mp(n) -- 6.6. The cohomological meaning of the Maslov index -- 6.7. The inhomogeneous metaplectic group -- 6.8. The metaplectic group and wave optics -- 6.9. The groups Symp(n) and Ham(n) -- 7. Schrödinger 's equation and the metatron. 7.1. Schrödinger 's equation for the free particle -- 7.2. Van Vleck's determinant -- 7.3. The continuity equation for Van Vleck's density -- 7.4. The short-time propagator -- 7.5. The case of quadratic Hamiltonians -- 7.6. Solving Schrödinger 's equation: general case -- 7.7. Metatrons and the implicate order -- 7.8. Phase space and Schrödinger 's equation.

This book 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.

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