Quantum mechanics without quantum logic Authors: D.A. Slavnov Comments: 24 pages, no figures, Latex
We describe a scheme of quantum mechanics in which the Hilbert space and linear operators are only secondary structures of the theory. As primary structures we consider observables, elements of noncommutative algebra, and the physical states, the nonlinear functionals on this algebra, which associate with results of single measurement. We show that in such scheme the mathematical apparatus of the standard quantum mechanics does not contradict a hypothesis on existence of an objective local reality, a principle of a causality and Kolmogorovian probability theory.
To talk frankly it seems to me that Slavnov is a little bit unfair about Quantum Logic (QL), confusing it with some Hilbert Space idolatry. It looks still more unfair when you remember that, in the process of writing the QL founding 1936 paper (ref in my thesis), von Neumann wrote to Birkhoff and said:
"I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space any more." (quoted at length in the formidable book by Miklos Redei : Quantum Logic in Algebraic Approach , Kluwer, 1998).
And so we can say that QL has been literally born from a first skeptical move with respect to the Hilbert space worship. And as far as I understand Slavnov his move seems similar to von Neumann's one. Which I think is not a bad move at all. The reason why von Neumann has abandonned the "obvious" orthomodular lattice of the closed linear subspaces of an (infinite dimensional) Hilbert space was that he wanted to keep *modularity* which is closer to the distributivity (of the 'and' and the 'or') axioms of a Boolean Algebra, ... so close that it makes it possible to define the unique probabilities from the "probability one" logic, that is from Quantum Logic (there would be some universal density operator).
I do believe this has no bearing at all with any magical trick capable of making vanishing the other relative worlds, histories, minds, maximal consistent extensions, possibilities ... That seems to me the most preposterous part of Slavnov paper. In 1939 von Neumann still invokes a "magical" role of consciousness in his singling out a "collapsed" reality.
That Quantum logic *can* be a formidable tools is exemplified in my thesis where I show that if we are turing-emulable then physics (as a science of correct prediction) is necessarily redefined as a measure on all the "computational histories" going through our relatively actual states. The "all" is managed by explicit appeal to Church thesis. And then, translating this in the language of a sound universal (lobian) machine I extract the logic of the probability one (from and on all the maximal consistent extensions) and got an (arithmetical) quantum logic (AQL*) Is it modular, orthomodular? Open problems!
Of course modularity would help for the sequel (the derivation of physics from arithmetics/machine 'psychology'). You can look at the last pages of the following document for the precise definition of the arithmetical quantum logic which I call AQL* now but is named QuelQL* in the following document: http://iridia.ulb.ac.be/~marchal/bxlthesis/Volume1CC/4Recapitulation.pdf