Observables and states in the Hilbert space formalism of quantum mechanics

Eduard Prugovečki

Another axiom has to be introduced to make the formalism given in Part I physically equivalent to the conventional Hilbert-space formalism. Then it is shown that, given a certain fundamental set of observables, a B*-algebra A can be built into which the set 𝒪b of all bounded observables can be mapped injectively. The closure of the algebra generated by the image of 𝒪b into A is A itself. A Hilbert space ℋ exists into which the set 𝒪0 of all pure physical states can be mapped injectively. The closure of the subset of ℋ which is the image of this mapping is ℋ itself. The algebra A can be mapped in such a fashion into the C*-algebra B(ℋ) of all bounded observables that these mappings provide, essentially, just a translation of the original formalism in a Hilbert-space formalism.

Jaromir Tosiek, P. Brzykcy

Holger Neumann

Christoph Schulte, Reidun Twarock, Arno Böhm

Quantum mechanics in the Rigged Hilbert Space formulation describes quasistationary phenomena mathematically rigorously in terms of Gamow vectors. We show that these vectors exhibit microphysical irreversibility, related to an intrinsic quantum mechanical arrow of time, which states that preparation of a state has to precede the registration of an observable in this state. Moreover, the Rigged Hilbert Space formalism allows the derivation of an exact golden rule describing the transition of a pure Gamow state into a mixture of interaction-free decay products.

Peter Gibbins

Quantum mechanics comes in a variety of different forms and formalisms. There is wave mechanics, matrix mechanics, Dirac's version of quantum mechanics, von Neumann's version of quantum mechanics. The philosopher of physics has to ask not only what quantum mechanics means, but also what it is. Orthodoxy in the philosophy of physics treats quantum mechanics as the general theory of microphysics which takes Hilbert space as its state space and which associates observables with certain special operators on that space. This amounts to a decision as to what quantum mechanics is. However it is a far from arbitrary decision. For the Hilbert space, or von Neumann, formalism summarizes and captures everything that is in the other versions of quantum mechanics, without any of the dubious mathematics of, for example, Dirac's version. Contemporary philosophers of physics take a very formal view of what quantum mechanics is, a view which has many advantages and some disadvantages. Whatever the disadvantages of focussing on the formalism of quantum mechanics - and it may lead to losing sight of the ‘physics' and possibly to a bias against Bohrian Copenhagenism - it is certainly true that the literate philosopher of physics must have some acquaintance with Hilbert space. There are in fact several topics we have to deal with, though we deal with them only very cursorily. (This chapter is no substitute for Jauch's classic text Foundations of Quantum Mechanics .)

Peter Mittelstaedt