## Born-Jordan Quantization: Theory and ApplicationsThis book presents a comprehensive mathematical study of the operators behind the Born–Jordan quantization scheme. The Schrödinger and Heisenberg pictures of quantum mechanics are equivalent only if the Born–Jordan scheme is used. Thus, Born–Jordan quantization provides the only physically consistent quantization scheme, as opposed to the Weyl quantization commonly used by physicists. In this book we develop Born–Jordan quantization from an operator-theoretical point of view, and analyze in depth the conceptual differences between the two schemes. We discuss various physically motivated approaches, in particular the Feynman-integral point of view. One important and intriguing feature of Born-Jordan quantization is that it is not one-to-one: there are infinitely many classical observables whose quantization is zero. |

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### Contents

1 | |

7 | |

9 | |

3 Quantization of Monomials | 22 |

4 Basic Hamiltonian Mechanics | 37 |

5 Wave Mechanics and the Schrödinger Equation | 56 |

Part IIMathematical Aspects of BornJordanQuantization | 71 |

6 The Weyl Correspondence | 73 |

9 Shubins PseudoDifferential Calculus | 129 |

10 BornJordan PseudoDifferential Operators | 147 |

11 Weak Values and the Reconstruction Problem | 161 |

Part III Some Advanced Topics | 171 |

12 Metaplectic Operators | 172 |

13 Symplectic Covariance Properties | 185 |

14 Symbol Classes and Function Spaces | 199 |

References | 216 |