## 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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action allows arbitrary argument associated assume begin calculated called capacity Chapter classical condition connected Consider constant construct coordinates covering deﬁned deﬁnition denote density depends determined differential discussion energy equality equation equivalent Example exists expression fact ﬁrst ﬂow follows force formula function geometric given Hamilton’s equations Hamiltonian hence immediately initial instance integral Lagrangian manifold Lagrangian plane leads Lemma Leray index loop mapping Maslov index mass mathematical matrix Maxwell metaplectic metatron momentum motion Newton’s observation obtained operator optical orbits oriented particle path periodic phase space physical position potential principle projection Proof Proposition prove quadratic quantization quantum mechanics representation result satisﬁes simply solution Sp(n Subsection Suppose symplectic symplectic matrices symplectomorphism theorem theory trajectory variables vector write