## An Introduction to the Mathematical Structure of Quantum Mechanics: A Short Course for MathematiciansThis book arises out of the need for Quantum Mechanics (QM) to be part of the common education of mathematics students. The mathematical structure of QM is formulated in terms of the C*-algebra of observables, which is argued on the basis of the operational definition of measurements and the duality between states and observables, for a general physical system. The Dirac???von Neumann axioms are then derived. The description of states and observables as Hilbert space vectors and operators follows from the GNS and Gelfand???Naimark Theorems. The experimental existence of complementary observables for atomic systems is shown to imply the noncommutativity of the observable algebra, the distinctive feature of QM; for finite degrees of freedom, the Weyl algebra codifies the experimental complementarity of position and momentum (Heisenberg commutation relations) and Schr??dinger QM follows from the von Neumann uniqueness theorem. The existence problem of the dynamics is related to the self-adjointness of the Hamiltonian and solved by the Kato???Rellich conditions on the potential, which also guarantee quantum stability for classically unbounded-below Hamiltonians. Examples are discussed which include the explanation of the discreteness of the atomic spectra. Because of the increasing interest in the relation between QM and stochastic processes, a final chapter is devoted to the functional integral approach (Feynman???Kac formula), to the formulation in terms of ground state correlations (the quantum mechanical analog of the Wightman functions) and their analytic continuation to imaginary time (Euclidean QM). The quantum particle on a circle is discussed in detail, as an example of theinterplay between topology and functional integral, leading to the emergence of superselection rules and ??? sectors. |

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### Contents

Introduction | 1 |

Mathematical description of a physical system | 7 |

Mathematical description of a quantum system | 39 |

The quantum particle | 59 |

Quantum dynamics The Schrodinger equation | 87 |

Examples | 105 |

Quantum mechanics and stochastic processes | 119 |

169 | |

177 | |

### Common terms and phrases

abelian C∗-algebra Academic Press algebra of observables Appendix atomic axioms Banach Borel bounded brieﬂy Chap characterized classical mechanics commutation relations compact continuous functions convergence corresponding deﬁned denoted dense domain described diﬀerence diﬀerential Dirac Dirac-von Neumann discussion eﬀect eigenvalue elements energy equation equivalent Euclidean experimental fact Feynman path integral ﬁnite ﬁrst formula functional integral Gaussian Gelfand given GNS representation Hamiltonian Hilbert space implies inﬁnite irreducible representations isomorphic kernel lattice limit Math mathematical description mathematical structure non-commutative norm perturbative Phys physical system polynomial positive linear functional potential probability distribution problem Proof Proposition quantization quantum mechanics quantum particle Quantum Physics quantum systems random variables Schwinger functions Sect self-adjoint extensions self-adjoint operator Simon solution spectral spectrum Springer stochastic processes Strocchi symmetric theorem topology trajectories uncertainty relations unitary vector wave function Weyl algebra Wightman Wightman functions