An Introduction to Hilbert Space and Quantum Logic
Published on 1. May 1989
Book
Hardback
XII, 149 pages
978-0-387-96870-4 (ISBN)
Description
Historically, 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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David W. Cohen
An Introduction to Hilbert Space and Quantum Logic
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10/2011
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Content
1. Experiments, Measure, and Integration.- A. Measures.- Experiments and weight functions, expected value of a weight function, measures, Lebesgue measure, signed measures, complex measures, measurable functions, almost everywhere equality..- B. Integration.- Simple functions, simple integrals, general integrals, Lebesgue integrals, properties of integrals, expected values as integrals, complex integrals..- 2. Hilbert Space Basics.- Inner product space, norm, orthogonality, Pythagorean theorem, Bessel and Cauchy-Schwarz and triangle inequalities, Cauchy sequences, convergence in norm, completeness, Hilbert space, summability, bases, dimension..- 3. The Logic of Nonclassical Physics.- A. Manuals of Experiments and Weights.- Manuals, outcomes, events, orthogonality, refinements, compatibility, weights on manuals, electron spin, dispersion-free weights, uncertainty..- B. Logics and State Functions.- Implication in manuals, logical equivalence, operational logic, implication and orthocomplementation in the logic, lattices, general logics (quantum logics), propositions, compatibility, states on logics, pure states, epistemic and ontological uncertainty..- 4. Subspaces in Hilbert Space.- Linear manifolds, closure, subspaces, spans, orthogonal complements, the subspace logic, finite projection theorem, compatibility of subspaces..- 5. Maps on Hilbert Spaces.- A. Linear Functional and Function Spaces.- Linear maps, continuity, boundedness, linear functional, Riesz representation theorem, dual spaces, adjoints, Hermitian operators..- B. Projection Operators and the Projection Logic.- Projection operators, summability of operators, the projection logic, compatibility and commutativity..- 6. State Space and Gleason's Theorem.- A. The Geometry of State Space.- State space, convexity, faces, extreme points, properties, detectability, pure states, observables, spectrum, expected values, exposed faces..- B. Gleason's Theorem.- Vector state, mixture, resolution of an operator into projection operators, expected values of operators, Gleason's theorem..- 7. Spectrality.- A. Finite Dimensional Spaces, the Spectral Resolution Theorem.- Eigenvalues, point spectrum, eigenspaces, diagonalization, the spectral resolution theorem..- B. Infinite Dimensional Spaces, the Spectral Theorem.- Spectral values, spectral measures, the spectral theorem, functions of operators, commutativity and functional relationships between operators, commutativity and compatibility of operators..- 8. The Hilbert Space Model for Quantum Mechanics and the EPR Dilemma.- A. A Brief History of Quantum Mechanics.- B. A Hilbert Space Model for Quantum Mechanics.- Schroedinger's equation, probability measures, stationary states, the harmonic oscillator, the assumptions of quantum mechanics, position and momentum operators, compatibility..- C. The EPR Experiment and the Challenge of the Realists.- Electron spin, spin states, singlet state, EPR apparatus, the EPR dilemma..- Index of Definitions.