An Introduction to Hilbert Space and Quantum Logic

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.

1. Experiments, Measure, and Integration.- A. Measures.- B. Integration.- 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.- B. Logics and State Functions.- 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.- B. Projection Operators and the Projection Logic.- 6. State Space and Gleason’s Theorem.- A. The Geometry of State Space.- B. Gleason’s Theorem.- 7. Spectrality.- A. Finite Dimensional Spaces, the Spectral Resolution Theorem.- B. Infinite Dimensional Spaces, the Spectral Theorem.- 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.- C. The EPR Experiment and the Challenge of the Realists.- Index of Definitions.