From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 155

Remark 2.3.42.
(a)A fundamental requirement is that the superselection charges have punctual
spectrum. If insteadR∩R′includes an operatorAwith a continuous part in its
spectrum (Amay also be the strong limit onD(A)of a sequence of elements in
R∩R′), the established proposition does not hold.Hcannot be decomposed into a
direct sum of closed subspaces. In this case, it decomposes into a direct integral and
we find a much more complicated structure whose physical meaning seems dubious.
(b) The representationsRA→A|Hq ∈Rqare not faithful (injective),
because bothIandPs(Q)have the same image under the representation.
(c)The discussed picture is not the most general one though we only deal with
it in these notes. There are quantum physical systems such that theirR′is not
Abelian (think of chromodynamics whereR′includes a faithful representation of
SU(3)) so that the center ofRdoes not contain the full information aboutR′.
In this case, the non-Abelian group of the unitary operators inR′ is called the
gauge groupof the theory. The existence of a gauge group is compatible with
the presence of superselection rules which are completely described by the center
R′∩R. The only difference is that nowRq=B(Hq)cannot be possible for every
coherent subspace otherwise we would haveR′=R∩R′.

2.3.5.5 States in the presence of superselection rules

Let us come to the problem to characterize the states when a superselection struc-
ture is assumed on a complex separable Hilbert spaceHin accordance with(SS1)
and(SS2). In principle we can extend Definition 2.3.21 already given for the
case ofRwith trivial center. As usualLR(H) indicates the lattice of orthogonal
projectors inR, which we know to be bounded by 0 andI, orthocomplemented,σ-
complete, orthomodular and separable, but not atomic and it does not satisfy the
covering property in general. The atoms are one dimensional projectors exactly
as pure sates, so we may expect some difference at that level whenR=B(H).

Definition 2.3.43. LetHbe a complex separable Hilbert space. Aquantum
stateinH, for a quantum sistem with von Neumann algebra of observablesR,is
amapρ:LR(H)→[0,1]such that the following requirement are satisfied.

(1)ρ(I)=1.

(2)If{Qn}n∈N ⊂LR(H),forN at most countable satisfiesQk∧Qh=0

whenh, k∈N,then

ρ(∨k∈NQk)=

∑

k∈N

ρ(Qk). (2.85)

The set of the states will be denoted bySR(H).