5
votes
2answers
288 views

C*-algebraic representation of observables vs self-adjoint operators one

I am trying to reconcile the "physicist" definition of an observable: self-adjoint operator on a Hilbert space, and the operational one as given by Strocchi in "An introduction to the mathematical ...
7
votes
1answer
361 views

Intuitive meaning of Double Commutant Theorem

Is there any intuitive explanation of the Double Commutant Theorem for Von Neumann Algebras? By intuitive I mean in terms of Quantum Mechanics. For example, duality of states and observables in the ...
5
votes
5answers
1k views

Topology on the space of Schwartz Distributions

If we equip the Schwartz space $\mathcal{S}$ with its usual Fr├ęchet space topology, then the space of continuous linear functionals $\mathcal{S}^\ast$ is known as the space of Schwartz distributions ...
30
votes
3answers
3k views

Quantum mechanics formalism and C*-algebras

Many authors (e.g Landsman, Gleason) have stated that in quantum mechanics, the observables of a system can be taken to be the self-adjoint elements of an appropriate C*-algebra. However, many ...
3
votes
2answers
752 views

Spectral decomposition for an arbitrary linear combination of position and momentum operators

Suppose we have the Hilbert space L2(Rn) and we have n operators Qi and n operators Pi defined in the usual way by: Qi ψ(q1,q2,...,qn) = qi ψ(q1,q2,...,qn) Pi ψ(q1,q2,...,qn) = -i ...
1
vote
0answers
587 views

Tensor products as isomorphic functors in category theory

An earlier question that I posed sought to define a category with a set of quantum channels as arrows and the C$^{*}$-algebra that these channels map from and to as the object. So, for example, my ...
5
votes
4answers
601 views

Quantum channels as categories: question 1.

A quantum channel is a mapping between Hilbert spaces, $\Phi : L(\mathcal{H}_{A}) \to L(\mathcal{H}_{B})$, where $L(\mathcal{H}_{i})$ is the family of operators on $\mathcal{H}_{i}$. In general, we ...
2
votes
3answers
515 views

Are the Gell-Mann matrices extremal when used as Kraus operators for a quantum channel?

Landau and Streater proved that a set of Kraus operators, Ai, is extremal if and only if the set $\{A_{k}^{\dagger}A_{l}\}_{k,l \ldots N}$ are linearly independent. I have seen very convincing ...