# Was Von Neumann and Birkhoff's original formulation of Quantum Logic related with projective geometry?

I was looking at how did von Neumann and Birkhoff formulate their Quantum Logic formalism back in 1936. To solve some questions, I contacted via email a philosopher who studied this topic.

I thought that Von Neumann and Birkhoff's original formalism was strongly related to projective geometry (and that basically, Quantum Logic was fundamentally based on projective geometry, as they said in their original paper:

Hence, we conclude that the propositional calculus of quantum mechanics has the same structure as an abstract projective geometry

I also thought that Von Neumann disliked the Hilbert space formulation of quantum mechanics before publishing his Quantum Logic with Birkhoff, since in a letter to Birkhoff in 1935, Von Neumann said:

I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space any more.

— John von Neumann, letter to Garrett Birkhoff, 1935

Finally, in 1954, almost two decades after his original paper with Birkhoff, he still studied and considered the non-boolean/non-commutative/non-distributive characteristics of quantum logic, since according to Jeffrey Bub:

They (probabilities) are “uniquely given from the start” as a feature of the non-Boolean structure, to quote von Neumann,[10] related to the angles in Hilbert space, not measures over states as they are in a classical or Boolean theory.

[10] John von Neumann, “Unsolved problems in mathematics,” an address to the International Mathematical Congress, Amsterdam, September 2, 1954.

The thing is that the philosopher I contacted with contradicted all of this. He said:

In the strict sense, quantum logic is a non-distributive consequence relation, not a geometry or an algebra. In this sense of quantum logic, von Neumann abandoned his interest in it (and in the Hilbert state formulation of quantum mechanics more generally), after 1936, and shifted it to type II factor algebras (which we now call von Neumann algebras).

I have a few questions about this:

1. Is this philosopher right? Or on the contrary, Von Neumann and Birkhoff's quantum logic is fundamentally based in an abstract projective geometry?

2. Didn't Von Neumann already dislike or abandon the Hilbert space formalism before 1936 (according to the letter that von Neumann sent to Birkhoff in 1935)?

• The article Continuous Geometry came out in January 1936, before The Logic of Quantum Mechanics, and this suggests that von Neumann was motivated by projective geometry considerations (specifically the projective geometry he had discovered that is described by the lattice of projections in a von Neumann algebra). – Robert Furber Mar 18 at 3:40
• Von Neumann also produced Continuous Geometries with a Transition Probability in 1937, though it wasn't completed and published until the 80s, with the aid of Israel Halperin. In it, he has two consequence relations, $\leq$ and the transition probability. I've heard that he got discouraged when he realized that it wouldn't produce anything new that didn't occur in von Neumann algebras, but have no source for this. – Robert Furber Mar 18 at 3:43

## 1 Answer

In the strict sense, quantum logic is a non-distributive consequence relation, not a geometry or an algebra. In this sense of quantum logic, von Neumann abandoned his interest in it (and in the Hilbert state formulation of quantum mechanics more generally), after 1936, and shifted it to type II factor algebras (which we now call von Neumann algebras).

I'd say that your philosopher hasn't really understood what von Neumann & Birkhoff were aiming at; their notion of a Quantum Logic was merely a preliminary attempt to ask questions on the foundations of classical physics; the physics of which - before quantum mechanics - were understood to hold without exception.

Its undoubtedly this that gave a strong motivation and impetus to investigate non-classical logics and geometry; that this has been a fruitful line of inquiry one only needs to look at how toposes, which can be considered as a non-classical set theory with an underlying intuitionistic logic and are moreover geometric have shown their usefulness in physics, and moreover in quantum physics; for example, take the reformulation of the Kochen-Specker statement in the foundations of quantum mechanics: the spectral presheaf of the algebra of bounded operators has no global elements when the dimension of the underlying Hilbert space is greater than 2.

"I also thought that Von Neumann disliked the Hilbert space formulation of quantum mechanics before publishing his Quantum Logic with Birkhoff, since in a letter to Birkhoff in 1935, Von Neumann said:

"I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space any more."

First, a couple of points; one should realise that von Neumann, was primarily a mathematician rather than a physicist, thus his confession above; secondly, it was von Neumann himself that had promoted the notion of a Hilbert space in his book-length treatment in axiomatising Quantum Mechanics. Why Von Neumann thought to be 'immoral', I don't know; most likely by then, he understood that his formalism of the Hilbert space was not sufficient for the purposes of QM, never mind QFT? But then it would have been 'immoral' to keep quiet; to make a confession would then be moral, especially if one was the main protagonist in pushing forward this concept.

Paul Dirac of course made no such confession; to him, mathematics was a tool used in order to discover more physics; this is why in QM physicists use his bra-ket notation - which superficially looks like the inner product of Hilbert spaces but is not - because it works - we can't always wait for mathematicians to play catch up (in fact, they did catch up - but much later, after the discovery of distributions).

.. von Neumann abandoned his interest in it (and in the Hilbert state formulation of quantum mechanics more generally), after 1936, and shifted it to type II factor algebras (which we now call von Neumann algebras).and shifted it to type II factor algebras (which we now call von Neumann algebras).

These are merely particular types of C*-algebras - those that have a predual (and in fact are also called W*-algebras); and since the interest in C*-algebras primarily arose from their use in Quantum Physics, it seems merely that von Neumann was still investigating the foundations of QM, but not as a physicist, but as a mathematician would by looking at their 'prime' factors.