## An Introduction to Hilbert Space and Quantum LogicHistorically, nonclassical physics developed in three stages. First came a collection of "ad hoc" assumptions and then a cookbook of equations known as "quantum mechanics." The equations and their philosophical underpinnings were then collected into a model based on the mathematics of Hilbert space. From the Hilbert space model came the abstaction of "quantum logics." This book explores all three stages, but not in historical order. Instead, in an effort to illustrate how physics and abstract mathematics influence each other we hop back and forth between a purely mathematical development of Hilbert space, and a physically motivated definition of a logic, partially linking the two throughout, and then bringing them together at the deepest level in the last two chapters. This book should be accessible to undergraduate and beginning graduate students in both mathematics and physics. The only strict prerequisites are calculus and linear algebra, but the level of mathematical sophistication assumes at least one or two intermediate courses, for example in mathematical analysis or advanced calculus. No background in physics is assumed. |

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### Contents

Experiments Measure and Integration | 1 |

Hilbert Space Basics | 14 |

The Logic of Nonclassical Physics | 21 |

Copyright | |

6 other sections not shown

### Common terms and phrases

a-algebra associated Borel sets called Cauchy sequence Chapter collector commutes compatible complex numbers complex-valued consider converges in norm countable define denote dimensional Hilbert space dispersion-free eigenvalues electron spin Examples and Projects expected value experiment Figure finite dimensional Hilbert follows from Theorem Gleason's theorem hence Hermitian operator Hilbert space model identity function implies infinite inner product space lattice Lebesgue integral Lebesgue measure Lemma linear manifold linear map linear operator mathematical measurable function nonclassical nonnegative nonzero vector observable ontological uncertainty operational logic operator on H orthonormal set outcome p a q pair pairwise disjoint particle physical system projection logic projection operators properties proposition quantum logic quantum mechanics quantum physics quasimanual real numbers respect Riemann integral satisfying Schroedinger equation space H spectral measure spectral theorem spin manual subspace in H summable unit vector vector space write z-spin zero