Mathematical Foundations of Quantum Mechanics 1472.3.5 von Neumann algebra of observables, superselection rules
The aim of this section is to focus on the class of observables of a quantum system,
described in the complex Hilbert spaceH, exploiting some elementary results
of the theory of von Neuman algebras. Up to now, we have tacitly supposed
thatallselfadjoint operators inHrepresent observables,allorthogonal projectors
represent elementary observables, allnormalized vectors represent pure states.
This is not the case in physics due to the presence of the so-calledsuperselection
rules. Within the Hilbert space approach the modern tool to deal with this notion
is the mathematical structure of avon Neumann algebra. For this reason we spend
the initial part of this section to introduce this mathematical tool.
2.3.5.1 von Neumann algebras
Before we introduce it, let us first define thecommutantof an operator algebra and
state an important preliminary theorem. IfM⊂B(H) is a subset in the algebra
of bounded operators on the complex Hilbert spaceB(H), thecommutantof
Mis:
M′:={T∈B(H)|TA−AT=0 foranyA∈M}. (2.81)IfMis closed under the adjoint operation (i.e.A†∈MifA∈M) the commutant
M′is certaintly a∗-algebra with unit. In general: M′ 1 ⊂M′ 2 ifM 2 ⊂M 1 and
M⊂(M′)′, which implyM′=((M′)′)′. Hence we cannot reach beyond the second
commutant by iteration.
The continuity of the product of operators in the uniform topology says that
the commutantM′is closed in the uniform topology, so ifMis closed under the
adjoint operation, its commutantM′is aC∗-algebra (C∗-subalgebra) inB(H).
M′has other pivotal topological properties in this general setup. It is easy to
prove thatM′is both strongly and weakly closed. This holds, despite the product
of operators is not continuous with respect to the strong operator topology, because
separate continuity in each variable is sufficient.
In the sequel, we shall adopt the standard convention used for von Neumann
algebras and writeM′′in place of (M′)′etc. The next crucial result is due to von
Neumann (see e.g.[5; 6]).
Theorem 2.3.30(von Neumann’s double commutant theorem).IfHis a complex
Hilbert space andAan unital∗-subalgebra inB(H), the following statements are
equivalent.
(a)A=A′′;
(b)Ais weakly closed;
(c)Ais strongly closed.