and we denote with A0 the quasilocal algebra of this net, i.e., the inductive limit A0:=[
O
A(O)
k·k
,
and assume that it acts irreducibly onH0, i.e., A00 =C1H0.
(For any set B of bounded operators on some Hilbert space, B0 denotes its commutant, that is the set of all bounded linear operators which commute with each element ofB.)
The last structural assumption to be listed here is calledweak additivity and encodes the idea that there is some field theory which underlies the given netA: for each open set O, one has
A000= [
x∈R1+s
A(O+x)00 .
By the arguments collected nicely in [11] it follows from these assumptions that A0 is a simple C*-algebra, i.e., that is has no nontrivial *-ideals; in particular, every representation of A0 is faithful.
In contrast to more specific additional assumption made only in certain parts, the above assumptions shall be valid throughout all of the following work.
For later use, let us introduce here some more terminology and notations: The identical representation π0 :A0 −→ B(H0) is also called the vacuum representation since it is the GNS representation of the vacuum state ω0 :A0 −→ C : a7−→ ω0(a) = hΩ0, aΩ0i. The spectral family associated with the translationsU0 is denoted byE0.
A very important criterion which selects other representations of interest for particle physics at zero temperature is what is called positivity of the energy: A representation π :A0−→ B(H) of the quasilocal algebra on some Hilbert spaceHis said to havepositive energy if there exists a strongly continuous unitary representation U :R1+s −→ B(H) of the translation group satisfying
π◦α0x = AdU(x)◦π and spU ⊂V+.
In this situation, it is known [12] that the representation U can be chosen such that U(x) ∈ π(A0)00, and moreover [13] that there is a unique such choice for which, on any subspace H0 ⊂ H invariant under π(A0), the spectrum spU|H0 has a Lorentz-invariant lower boundary. This choice is called the minimal representation of the translations for the representation (H, π), and we will reserve for it the notationUπ (and, correspondingly, for its spectral family the notationEπ). It has been shown in [14] that spU|H0 is actually a Lorentz-invariant subset ofV+. The monograph [15] is a standard reference for a thorough discussion of these and related topics.
1.3 Structure of This Thesis
The structure of this thesis is as follows. InChapter 2a notion of infravacuum represen-tations is presented and some properties of such represenrepresen-tations are described. Chapter 3 contains an account of the DHR theory of superselection sectors for charges with general
localisation properties. We pay particular attention to the categorical aspects of the the-ory and to the description of covariant objects by means of charge transporting cocycles.
In Chapter 4 we investigate the properties of energy-momentum spectra associated to translation covariant charges in front of an infravacuum background. We make a step towards showing that the properties of the spectra known in front of the vacuum continue to hold in the infravacuum case as well. In Chapter 5 we discuss in the example of the free massless scalar field a class of charges which become much better localised in front of a suitable infravacuum background than in front of the vacuum. Chapter 6, finally, is devoted to conclusions and perspectives.
The appendices discuss several mathematical topics. Appendix A represents the mathematical background of Chapter 3; we collect numerous definitions and notions in the context of monoidal C*-categories. It is intended to be a reasonably self-contained and pedagogical overview. In Appendix B we prove a technical lemma in connection with cocycles used at several places in Section 3.7. In Appendix C we give the proof in categorical terms of a formula crucial to Section 4.4. Appendix D too is related to Chapter 4, since it contains the rigorous derivation of an argument necessary there for connecting position and momentum space properties of distributions.
The content of Chapter 2 and parts of Chapter 4 represent the content of [16], whereas the example treated in Chapter 5 has been published in [17].
Chapter 2
Infravacua
In this chapter we will introduce the notion of infravacuum representations of the algebra A0. The idea behind this notion is that a vector state in such a representation should describe a physical situation where some infrared cloud but no total charge is present.
Originally, infrared clouds appear in quantum electrodynamics and consist (for any energy threshold > 0) of a finite number N of “hard” photons and an infinite number of
“soft” photons with total energy less than . They can thus be regarded as a certain finite energy limits (outside the vacuum sector) of states which do belong to the vacuum sector. The definition of infravacuum representations presented below tries to translate this characteristic feature into the model-independent framework. It is based upon the notion of energy components of positive energy representations.
2.1 Energy Components
Energy components are certain sets of states associated to positive energy representations of A0. They have been introduced by Borchers and Buchholz [13] and discussed further by Wanzenberg [18]. This last reference being difficultly accessible, we repeat some of its results here.
Let (H, π) be a positive energy representation of A0 andEπ the spectral family of the associated minimal representationUπ of the translations (cf. Section 1.2). The naturality of Uπ implies that there is a natural notion of energy contents of π-normal states: If D⊂V+ is a compact subset of the forward light cone,1 denote by
Sπ(D) :=
n
trρπ(·)
ρ∈ I1(H), ρ≥0, trρ= 1, ρEπ(D) =ρ o
the set of allπ-normal states which have energy-momentum inD. ThenSπ =S
DSπ(D)k·k is the folium of π. A larger set ˜Sπ of states, called theenergy component of π, is defined by
S˜π:=[
D
S˜π(D)
k·k
,
where ˜Sπ(D) stands for the set of all locallyπ-normal states in the weak closure ofSπ(D).
It can be shown that, like Sπ, also ˜Sπ is a folium, that is, a closed convex subset of the
1In this chapter, D will always stand for a compact subset ofV+. Moreover we will use the notation Dq:=V+∩(q+V−) for the double cone in momentum space with apices 0 andq∈V+.
11
set of all states over A0 stable under the operationsω 7−→ ωA:=ω(A∗·A), A ∈A0 such that ω(A∗A) = 1. Physically, ˜Sπ is interpreted as the set of states which can be reached from states in Sπ by operations requiring only a finite amount of energy. Some important properties of the elements of ˜Sπ are collected in the following lemma, Parts ii and iii of which are due to Wanzenberg [18].
Lemma 2.1 Let ω ∈S˜π and denote by (Hω, πω,Ωω) its GNS triple. Then:
i. (Hω, πω) is a locallyπ-normal positive energy representation of A0.
ii. Moreover, if ω∈S˜π(Dq) for some q∈V+, then one has Ωω ∈Eπω(Dq)Hω. iii. In the situation of ii, one has S˜πω(D)⊂S˜π(D+Dq−Dq) for anyD.
Proof: i. Using the positivity of the energy in the representation π, Part i follows from the fact that ω is locally π-normal by arguments similar to those in [19]. (In [19], these arguments are only applied to states ω ∈ S˜π(Dq), but they carry over to norm limits of such states as well.)
ii. For simplicity, Part ii will only be proved in the case where πω is factorial. For the general case, see [18]. Let Dq be given. Then the idea is to show that Ωω does not have momentum outside Dq. To this end, fix some p ∈V+\Dq and choose a neighbourhood Np⊂V+\Dq ofpand an open setN ⊂spUπω such that (Dq+N − Np)∩V+=∅. (Such a choice is always possible because spUπω is Lorentz invariant.) Now choose a test function f satisfying supp ˜f ⊂ N −Npand take an arbitraryA∈A0. ThenA(f) := R
dx f(x)αx(A) is an element ofA0 and satisfiesπ(A(f))Eπ(Dq)Hπ ={0}. Sinceω∈S˜π(Dq),this implies ω(A(f)∗A(f)) = 0,hence πω(A(f))Ωω= 0. This means that Ωω is orthogonal to
D:= spann
πω(A(f))∗Ψ | Ψ∈ Hω, A∈A0,supp ˜f ⊂ N − Np
o .
Since πω is factorial andN − Np is open, it follows by an argument explained in [9] (see, in particular, the proof of Prop. 2.2 therein) that the closure of D equals Eπω(spUπω + Np− N)Hω. Thus, Ωω∈ D⊥ yields
{Ωω}⊥⊃ D⊥⊥=D=Eπω(spUπω +Np− N)Hω⊃Eπω(Np)Hω,
where the last inclusion holds because spUπω − N 30. From this, we getEπω(Np)Ωω = 0 or, as p∈V+\Dq was arbitrary, Ωω ∈Eπω(Dq)Hω.
iii. To prove the last part, let Ψ ∈ Eπω(D)Hω. Part ii and the cyclicity of Ωω imply that there exists in A0 a sequence (An)n∈N of operators with energy-momentum support in D−Dq and normalised to ω(A∗nAn) = 1 such that Ψ = limn→∞πω(An)Ωω. From ω ∈S˜π(Dq) it follows ω(A∗n·An)∈S˜π(Dq+ (D−Dq)) = ˜Sπ(D+Dq−Dq), i.e.,
hΨ, πω(·)Ψi= lim
n→∞ω(A∗n·An)∈S˜π(D+Dq−Dq).
Thus any vector state from Sπω(D) lies in ˜Sπ(D+Dq−Dq). One now gets the assertion by taking convex combinations, norm limits and locally normal weak limits.
Lemma 2.1 has shown that positivity of the energy is a property which “survives”
the process of going from the representation π to the GNS representation of a state in S˜π. Other properties survive as well, as for instance the compactness condition C] of Fredenhagen and Hertel [20, 21] which can be formulated as follows: