## The Principles of Newtonian and Quantum Mechanics: The Need for Planck's Constant, HThis book deals with the foundations of classical physics from the OC symplecticOCO point of view, and of quantum mechanics from the OC metaplecticOCO point of view. The Bohmian interpretation of quantum mechanics is discussed. Phase space quantization is achieved using the OC principle of the symplectic camelOCO, 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 OC metatronOCO 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. Contents: From Kepler to SchrAdinger OC and Beyond; Newtonian Mechanics; The Symplectic Group; Action and Phase; Semi-Classical Mechanics; The Metaplectic Group and the Maslov Index; SchrAdinger''s Equation and the Metatron. Readership: Researchers and graduate students in mathematical physics." |

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### Contents

1 FROM KEPLER TO SCHRODINGER AND BEYOND | 1 |

2 NEWTONIAN MECHANICS | 37 |

3 THE SYMPLECTIC GROUP | 77 |

4 ACTION AND PHASE | 127 |

5 SEMICLASSICAL MECHANICS | 179 |

6 THE METAPLECTIC GROUP AND THE MASLOV INDEX | 221 |

7 SCHRODINGERS EQUATION AND THE METATRON | 267 |

A Symplectic Linear Algebra | 323 |

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### Common terms and phrases

action arbitrary associated Bohmian calculated Chapter classical mechanics condition conﬁguration space constant coordinates deﬁned deﬁnition denote density diffeomorphism differential form energy shell equivalent fact ﬁnal ﬁnd ﬁrst ﬁxed follows formula Fourier transform free particle free symplectic matrices gauge geometric geometric quantization Gosson Gromov’s Hamilton’s equations Hamiltonian function Hamiltonian mechanics harmonic oscillator Heisenberg hence homotopy identiﬁed integral Jacobian Lag(n Lagrangian manifold Lagrangian plane Lemma Leray index Liouville’s loop mapping Maslov index mathematical matrix Maxwell Hamiltonian metaplectic group metaplectic representation momentum motion Newton’s second law notation operator optical oriented periodic orbits phase space physical Planck’s constant potential projection Proof Proposition quadratic Fourier transforms quantization quantum mechanics Rham form satisﬁes Schr6dinger’s equation semi-classical solution subgroup Subsection symmetric symplectic capacity symplectic geometry symplectic group symplectic matrices symplectomorphism theorem theory trajectory unitary universal covering variables vector ﬁeld velocity wave function