- 662

- 2

yeah, I think they were worried about the "completeness" part of Hlibert Space not being a fundamental requirement (to describe reality), they'll probably be proved right soon.Algebraic Quantum Mechanics, Algebraic Spinors and Hilbert Space.

http://www.bbk.ac.uk/tpru/BasilHiley/Algebraic Quantum Mechanic 5.pdf

.....These results have only intensified my curiosity as to why most if not all of the results

can be obtained without seemingly the need to resort to Hilbert space. This goes against

the prevailing orthodoxy that appears to insist that quantum mechanics cannot be done

except in the context of a Hilbert space. Yet there have been other voices raised against

the necessity of Hilbert space. Von Neumann himself wrote to Birkoff (1966) writing "I

would like to make a confession which may seem immoral: I do not believe absolutely in

Hilbert space any more." (A detailed discussion of why von Neumann made this

comment can be found in Rédei 1996).

But there are more important reasons why an algebraic approach has advantages. As

Dirac (1965) has stressed, when algebraic methods are used for systems with an infinite

number of degrees of freedom.....

....We have shown how an approach to quantum mechanics can be built from the algebraic

structure of the Clifford algebra and the discrete Weyl algebra (or the generalised

Clifford algebra). These algebras can be treated by the same techniques that do not

require Hilbert space yet enable us to calculating mean values required in quantum

mechanics.......

------------

Algebra approach to Quantum Mechanics A: The Schroedinger and Pauli Particles.

http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.4031v1.pdf

.