Reference request for a mathematical motivation for the Born rule

Question

I was reading the popular science book The Hidden Reality by Brian Greene. My question is about a part in the notes at the end of the book. It is chapter 8, note 9. Brian Greene describes a mathematical motivation for the probabilistic interpretation (i.e. the Born rule). He does this by defining a frequency operator and then taking the limit by increasing the number of states to infinity.

Greene also states that the majority of the terms in the expansion of $|\phi \rangle ^{\otimes n}$ must be considered as 'non-existent'. However, the challenge is understanding what this 'means', physically speaking (if it already means something at all).

I'm interested in more reading material about these topics. I'm interested in more advanced explanations in the literature (possibly by the authors mentioned by Greene in the note). However, I could not find material for this. Perhaps someone who has read the book or is knowledgeable about this topic could suggest some resources.

Answer 1

Googling on the list of names in the footnote turned up the paper https://arxiv.org/abs/1405.6755, which led to Deutsch and the other references below.

Edward Farhi, Jeffrey Goldstone, and Sam Gutmann, "How Probability Arises in Quantum Mechanics," Annals of Physics, 192 (1989) 368, June 1989. http://www.sciencedirect.com/science/article/pii/0003491689901413, doi:10.1016/0003-4916(89)90141-3

David Deutsch. “Quantum Theory of Probability and Decisions”. 1999. arXiv:quant-ph/9906015

DeWitt, B.S. and Graham, N. 1973 in The Many-Worlds Interpretation of Quantum Mechanics 183-186 (Princeton University Press).

Everett, H. 1957 Rev. Mod. Phys. 29 3 454-462.

Finkelstein, D. 1963 Trans. N.Y. Acad. Sci. 25 621-637.

Hartle, J.B. 1968 Am. Jour. Phys. 36, 704-712.

Ohkuwa, Y. Phys. Rev. 1993 D48 4 1781-1784.

Answer 2

I have found an interesting discussion about this question in the book (in french) "Mécanique quantique, Bases et applications" by Constantin Piron.

He proves a Gleason-like theorem (A.2 Théorème fondamental, p.172) stating something like: if you would like to associate to each state (= vector) and proposition (= closed subspace of the Hilbert space) a real number between $0$ and $1$ behaving like a probability (there are mathematical hypotheses corresponding to this) then the numbers are given by the usual formula.