### M Theory Lesson 146

Thinking of ordinals $n \in \mathbb{N}$ as finite sets, one notes that the prime numbers don't seem so special any more. What is so special about a set of three oranges, as opposed to a set of four oranges? Still, a composite number of oranges can be arranged into a rectangular shape of dimension equal to the number of prime factors. Primes are then single lines of oranges. At least its nice to see that building blocks for sets are geometrically one dimensional, somehow like space filling curves.

Maybe these sets are equipped with further structure. For instance, they might be the finite fields with $p$ elements. Most fields that physicists play with have the unfortunate property of having no zero divisors, unlike the interesting operator algebras studied by Carl Brannen, where it is quite possible that $\rho_{1} \rho_{2} = 0$. The number $0$ represents an experimental beam stop: the action of allowing no Stern-Gerlach particles through, which is a simple state that one's mathematics really shouldn't ignore.

In the topos Set, the empty set is the object of cardinality zero, but we are not used to breaking the empty set up into pieces. This is a clue that the classical topos set (including set theory) is not the right setting for quantum physics, although we already knew that, because all 1-toposes rely on distributive lattices. Brannen's operator algebras also have the nice feature that the requirement of idempotency (projectors are the natural way to look at quantum lattices) specifies a normalisation for any state, removing the arbitrariness of the usual picture.

Maybe these sets are equipped with further structure. For instance, they might be the finite fields with $p$ elements. Most fields that physicists play with have the unfortunate property of having no zero divisors, unlike the interesting operator algebras studied by Carl Brannen, where it is quite possible that $\rho_{1} \rho_{2} = 0$. The number $0$ represents an experimental beam stop: the action of allowing no Stern-Gerlach particles through, which is a simple state that one's mathematics really shouldn't ignore.

In the topos Set, the empty set is the object of cardinality zero, but we are not used to breaking the empty set up into pieces. This is a clue that the classical topos set (including set theory) is not the right setting for quantum physics, although we already knew that, because all 1-toposes rely on distributive lattices. Brannen's operator algebras also have the nice feature that the requirement of idempotency (projectors are the natural way to look at quantum lattices) specifies a normalisation for any state, removing the arbitrariness of the usual picture.

## 2 Comments:

Marni, I wouldn't call it "Brannen's algebra". He's just another amateur nuthatch. The ideas originated with Julian Schwinger's measurement algebra. the original papers are available on the web and are easy to understand. First paper and second paper.

Thanks for the links, Carl. I am often forgetful of important references in my old age.

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