Gleason's Theorem

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Recorded: 2022/01/18 Released: 2022/03/20

Jim discusses Gleason's Theorem with Blake C. Stacey of the University of Massachuesetts - Boston. Gleason's Theorem is a theorem in the foundations of quantum mechanics that, for a system meets some simple requirements, you can find a set of valid staes and a rule for calculating probailities, a la the Born Rule. This is the first part of the interview, the next will be on Blake's discussion of how people are trying to reformulate the Born Rule.
------------------------------------------- Notes:

1. Papers we both read for this program:

• Gleason, A.M., "Measures on the Closed Subspaces of a Hilbert Space." Indiana Univ. Math. J. 6, 855 (1957). [arXiv]
• Stacey, B., "On Two Recent Approaches to the Born Rule." (2021) [arXiv]
• Busch, P., "Quantum States and Generalized Observables: A Simple Proof of Gleason’s Theorem." Phys. Rev. Lett. 91, 120403 (2003) [arXiv]
• Hossenfelder, S., "Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements" Found. Phys. 34 193 (2004). [arXiv]

2. Other papers one or the other of us read:

• Cabello, A. "Interpretations of Quantum Theory: A Map of Madness." What is Quantum Information?, 38 (2017). [arXiv]

3. Books mentioned in the discussion:

• David W. Cohen, An Introduction to Hilbert Space and Quantum Mechanics[Amazon] The book I found in a college book store in the 1990's. Short, written for mathemtics undergraduate students (at the upper division, high-end SLAC level), but also accessible to philosophy and physics students who can read math. Video review here.
• John von Neumann, Mathematical Foundations of Quantum Mechanics[Amazon] A classic text that started a lot of this quantum philosophy nonsense in the first place. Very difficult read.
• Michaal Neilson and Isaac Chuang, Quantum Computation and Quantum Information.[Amazon] An early, well known book on quantum information. This is one of three technical books that I've had to replace because my original copy wandered off with a graduate student to postdocs unknown.
• John Bell, Speakable and Unspeakable in Quantum Mechanics. [Amazon]
This is a collection of essays by John Bell, some of which are very important, and others of which you might not think are important until you start reading the philosophy of physics literature. Blake mentioned two papers that I think are in this volume. One is "On the Einstein-Poldosky-Rosen Paradox," which was originally published in the small jounal below.

4. Physics-Physique-Fisika, the journal Blake Stacey mentioned in our discussion.

5. Related Episodes of Physics Frontiers:

• Physics Frontiers 46: Wigner's Friend
• Physics Frontiers 44: Spooky Action at a Distance

6. Please visit and comment on our subreddit, YouTube Channel and if you can help us keep this going by contributing to our Patreon, we'd be grateful.

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