Table Of ContentAlgebraic Quantum Field Theory∗
Hans Halvorson†
(with an appendix by Michael Mu¨ger‡)
February 14, 2006
Abstract
Algebraic quantum field theory provides a general, mathematically precise
descriptionofthestructureofquantumfieldtheories,andthendrawsoutconse-
quencesofthisstructurebymeansofvariousmathematicaltools—thetheory
of operator algebras, category theory, etc.. Given the rigor and generality of
AQFT, it is a particularly apt tool for studying the foundations of QFT. This
paper is a survey of AQFT, with an orientation towards foundational topics.
In addition to covering the basics of the theory, we discuss issues related to
nonlocality, the particle concept, the field concept, and inequivalent represen-
tations. We also provide a detailed account of the analysis of superselection
rulesbyDoplicher,Haag,andRoberts(DHR);andwegiveanalternativeproof
of Doplicher and Robert’s reconstruction of fields and gauge group from the
category of physical representations of the observable algebra. The latter is
basedonunpublishedideasduetoJ.E.Robertsandtheabstractdualitytheo-
rem for symmetric tensor ∗-categories, a self-contained proof of which is given
in the appendix.
Contents
1 Algebraic Prolegomena 4
1.1 von Neumann algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 C∗-algebras and their representations. . . . . . . . . . . . . . . . . . . . . . 6
1.3 Type classification of von Neumann algebras . . . . . . . . . . . . . . . . . 8
1.4 Modular theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
∗ForthcominginHandbook of the Philosophy of Physics,editedbyJeremyButterfieldandJohn
Earman. HH wishes to thank: Michael Mu¨ger for teaching him about the Doplicher-Roberts
Theorem;theeditorsfortheirhelpfulfeedbackandpatience;andDavidBaker,TracyLupher,and
David Malament for corrections. MM wishes to thank Julien Bichon for a critical reading of the
appendix and useful comments.
†DepartmentofPhilosophy,PrincetonUniversity,PrincetonNJ,[email protected]
‡Department of Mathematics, Radboud University, Nijmegen, NL. [email protected]
1
2 Structure of the Net of Observable Algebras 13
2.1 Nets of algebras, basic properties . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Existence/uniqueness of vacuum states/representations . . . . . . . . . . . . 14
2.3 The Reeh-Schlieder Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 The funnel property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Type of local algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Nonlocality and Open Systems in AQFT 25
3.1 Independence of C∗ and von Neumann algebras . . . . . . . . . . . . . . . . 26
3.2 Independence of local algebras . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Bell correlation between von Neumann algebras . . . . . . . . . . . . . . . . 29
3.4 Intrinsically entangled states . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4 Prospects for Particles 32
4.1 Particles from Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 Fock space from the algebra of observables. . . . . . . . . . . . . . . . . . . 33
4.3 Nonuniqueness of particle interpretations . . . . . . . . . . . . . . . . . . . 35
4.4 Problems for localized particles . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.5 Particle interpretations generalized: Scattering theory and beyond . . . . . 36
5 The Problem of Value-Definiteness in AQFT 37
5.1 Clifton-Kitajima classification of modal algebras . . . . . . . . . . . . . . . 38
5.2 What is a symmetry in AQFT? . . . . . . . . . . . . . . . . . . . . . . . . . 40
6 Quantum Fields and Spacetime Points 42
6.1 No Go theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.2 Go theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.3 Field interpretations of QFT . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.4 Points of time? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7 The Problem of Inequivalent Representations 54
7.1 Superselection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7.2 Minimal assumptions about the algebra of observables . . . . . . . . . . . . 57
8 The Category ∆ of Localized Transportable Endomorphisms 59
8.1 ∆ is a braided tensor ∗-category . . . . . . . . . . . . . . . . . . . . . . . . 68
8.2 Relation between localized endomorphisms and representations . . . . . . . 74
8.3 Dimension theory in tensor ∗-categories . . . . . . . . . . . . . . . . . . . . 77
8.4 Covariant representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
8.5 Statistics in braided tensor ∗-categories . . . . . . . . . . . . . . . . . . . . 81
9 From Fields to Representations 82
10 From Representations to Fields 90
10.1 Supermathematics and the embedding theorem . . . . . . . . . . . . . . . . 92
10.2 Construction of the field net, algebraic . . . . . . . . . . . . . . . . . . . . . 94
10.3 Completion of the field net . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
10.4 Poincar´e covariance of the field net . . . . . . . . . . . . . . . . . . . . . . . 107
10.5 Uniqueness of the field net . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2
10.6 Further relations between A and F, and a Galois interpretation . . . . . . . 114
10.7 Spontaneous symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . 116
11 Foundational Implications of the Reconstruction Theorem 117
11.1 Algebraic Imperialism and Hilbert Space Conservatism . . . . . . . . . . . . 118
11.2 Explanatory relations between representations. . . . . . . . . . . . . . . . . 120
11.3 Fields as theoretical entities, as surplus structure . . . . . . . . . . . . . . . 122
11.4 Statistics, permutation symmetry, and identical particles . . . . . . . . . . . 125
Bibliography 133
Appendix (by Michael Mu¨ger) 144
A Categorical Preliminaries 144
A.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
A.2 Tensor categories and braidings . . . . . . . . . . . . . . . . . . . . . . . . . 145
A.3 Graphical notation for tensor categories . . . . . . . . . . . . . . . . . . . . 149
A.4 Additive, C-linear and ∗-categories . . . . . . . . . . . . . . . . . . . . . . . 149
A.5 Abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
A.6 Commutative algebra in abelian symmetric tensor categories . . . . . . . . 157
A.7 Inductive limits and the Ind-category . . . . . . . . . . . . . . . . . . . . . . 161
B Abstract Duality Theory for Symmetric Tensor ∗-Categories 162
B.1 Fiber functors and the concrete Tannaka theorem. Part I . . . . . . . . . . 163
B.2 Compact supergroups and the abstract Tannaka theorem . . . . . . . . . . 165
B.3 Certain algebras arising from fiber functors . . . . . . . . . . . . . . . . . . 168
B.4 Uniqueness of fiber functors . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
B.5 The concrete Tannaka theorem. Part II . . . . . . . . . . . . . . . . . . . . 175
B.6 Making a symmetric fiber functor ∗-preserving . . . . . . . . . . . . . . . . 177
B.7 Reduction to finitely generated categories . . . . . . . . . . . . . . . . . . . 181
B.8 Fiber functors from monoids . . . . . . . . . . . . . . . . . . . . . . . . . . 183
B.9 Symmetric group action, determinants and integrality of dimensions . . . . 186
B.10 The symmetric algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
B.11 Construction of an absorbing commutative monoid . . . . . . . . . . . . . . 194
B.12 Addendum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
Introduction
From the title of this Chapter, one might suspect that the subject is some idiosyn-
cratic approach to quantum field theory (QFT). The approach is indeed idiosyn-
cratic in the sense of demographics: only a small proportion of those who work on
QFT work on algebraic QFT (AQFT). However, there are particular reasons why
philosophers, and others interested in foundational issues, will want to study the
“algebraic” approach.
In philosophy of science in the analytic tradition, studying the foundations of a
theory T has been thought to presuppose some minimal level of clarity about the
3
referent of T. (Moreover, to distinguish philosophy from sociology and history, T is
not taken to refer to the activities of some group of people.) In the early twentieth
century,itwasthoughtthatthereferentofT mustbeasetofaxiomsofsomeformal,
preferably first-order, language. It was quickly realized that not many interesting
physical theories can be formalized in this way. But in any case, we are no longer
in the grip of axiomania, as Feyerabend called it. So, the standards were loosened
somewhat—butonlytotheextentthatthestandardsweresimultaneouslyloosened
within the community of professional mathematicians. There remains an implicit
working assumption among many philosophers that studying the foundations of a
theory requires that the theory has a mathematical description. (The philosopher’s
working assumption is certainly satisfied in the case of statistical mechanics, special
andgeneralrelativity,andnonrelativisticquantummechanics.) Inanycase,whether
or not having a mathematical description is mandatory, having such a description
greatly facilitates our ability to draw inferences securely and efficiently.
So, philosophersofphysicshavetakentheirobjectofstudytobetheories, where
theories correspond to mathematical objects (perhaps sets of models). But it is not
so clear where “quantum field theory” can be located in the mathematical universe.
In the absence of some sort of mathematically intelligible description of QFT, the
philosopher of physics has two options: either find a new way to understand the
task of interpretation, or remain silent about the interpretation of quantum field
theory.1
It is for this reason that AQFT is of particular interest for the foundations of
quantum field theory. In short, AQFT is our best story about where QFT lives
in the mathematical universe, and so is a natural starting point for foundational
inquiries.
1 Algebraic Prolegomena
This first section provides a minimal overview of the mathematical prerequisites of
the remainder of the Chapter.
1.1 von Neumann algebras
The standard definition of a von Neumann algebra involves reference to a topology,
and it is then shown (by von Neumann’s double commutant theorem) that this
topological condition coincides with an algebraic condition (condition 2 in the Def-
inition 1.2). But for present purposes, it will suffice to take the algebraic condition
as basic.
1.1 Definition. Let H be a Hilbert space. Let B(H) be the set of bounded linear
operators on H in the sense that for each A ∈ B(H) there is a smallest nonnegative
1For the first option, see [Wallace, forthcoming].
4
number kAk such that hAx,Axi1/2 ≤ kAk for all unit vectors x ∈ H. [Subsequently
we use k · k ambiguously for the norm on H and the norm on B(H).] We use
juxtaposition AB to denote the composition of two elements A,B of B(H). For
each A ∈ B(H) we let A∗ denote the unique element of B(H) such that hA∗x,yi =
hx,Ayi, for all x,y ∈ R.
1.2 Definition. Let R be a ∗-subalgebra of B(H), the bounded operators on the
Hilbert space H. Then R is a von Neumann algebra if
1. I ∈ R,
2. (R0)0 = R,
where R0 = {B ∈ B(H) : [B,A] = 0,∀A ∈ R}.
1.3 Definition. We willneedfourstandardtopologies ontheset B(H)ofbounded
linear operators on H. Each of these topologies is defined in terms of a family of
seminorms — see [Kadison and Ringrose, 1997, Chaps. 1,5] for more details.
• The uniform topology on B(H) is defined in terms of a single norm:
kAk = sup{kAvk : v ∈ H,kvk ≤ 1},
wherethenormontherightisthegivenvectornormonH. Hence,anoperator
A is a limit point of the sequence (Ai)i∈N iff (kAi−Ak)i∈N converges to 0.
• The weak topology on B(H) is defined in terms of the family {p : u,v ∈ H}
u,v
of seminorms where
p (A) = hu,Avi.
u,v
The resulting topology is not generally first countable, and so the closure of a
subsetS ofB(H)isgenerallylargerthanthesetofalllimitpointsofsequences
inS. Rather,theclosureofS isthesetoflimitpointsofgeneralizedsequences
(nets) in S — see [Kadison and Ringrose, 1997, Chap. 1] for more details. A
net(A ) inB(H)convergesweaklytoAjustincase(p (A )) converges
i i∈I u,v i i∈I
to p (A) for all u,v ∈ H.
u,v
• The strong topology on B(H) is defined in terms of the family {p : v ∈ H}
v
of seminorms where
p (A) = kAvk.
v
Thus, anet(A ) converges stronglyto A iff(p (A )) converges to p (A),
i i∈I v i i∈I v
for all v ∈ H.
• The ultraweak topology on B(H) is defined in terms of the family {p : ρ ∈
ρ
T(H)} where T(H) is the set of positive, trace 1 operators (“density opera-
tors”) on H and
p (A) = Tr(ρA).
ρ
5
Thus a net (A ) converges ultraweakly to A just in case (Tr(ρA )) con-
i i∈I i i∈I
verges to Tr(ρA), for all ρ ∈ T(H).
1.4 Fact. The topologies are ordered as follows:
norm
⊆
⊇
ultraweak strong
⊆
⊇
weak
Sinceclosedsetsarejustthecomplementsofopensets,thismeansthataweakly
closed set is ultraweakly closed, and an ultraweakly closed subset is norm closed.
Furthermore, the four topologies on B(H) coincide iff H is finite dimensional.
1.5 Fact. If S is a bounded, convex subset of B(H), then the weak, ultraweak, and
norm closures of S are the same.
1.6 Fact. For a ∗-algebra R on H that contains I, the following are equivalent:
(i) R is weakly closed; (ii) R00 = R. This is von Neumann’s double commutant
theorem.
1.7 Definition. Let R be a subset of B(H). A vector x ∈ H is said to be cyclic for
R just in case [Rx] = H, where Rx = {Ax : A ∈ R}, and [Rx] is the closed linear
span of Rx. A vector x ∈ H is said to be separating for R just in case Ax = 0 and
A ∈ R entails A = 0.
1.8 Fact. Let R be a von Neumann algebra on H, and let x ∈ H. Then x is cyclic
for R iff x is separating for R0.
1.9 Definition. A normal state of a von Neumann algebra R is an ultraweakly
continuous state. We let R denote the normal state space of R.
∗
1.2 C∗-algebras and their representations
1.10 Definition. A C∗-algebra is a pair consisting of a ∗-algebra A and a norm
k·k : A → C such that
kABk ≤ kAk·kBk, kA∗Ak = kAk2,
for all A,B ∈ A. We usually use A to denote the algebra and its norm.
In this Chapter, we will only use C∗-algebras that contain a multiplicative iden-
tity I.
1.11 Definition. A state ω on A is a linear functional such that ω(A∗A) ≥ 0 for
all A ∈ A, and ω(I) = 1.
6
1.12 Definition. Astateω ofAissaidtobemixed ifω = 1(ω +ω )withω 6= ω .
2 1 2 1 2
Otherwise ω is said to be pure.
1.13 Definition. Let A be a C∗-algebra. A representation of A is a pair (H,π),
where H is a Hilbert space and π is a ∗-homomorphism of A into B(H). A rep-
resentation (H,π) is said to be irreducible if π(A) is weakly dense in B(H). A
representation (H,π) is said to be faithful if π is an isomorphism.
1.14 Definition. Let(H,π)and(K,φ)berepresentationsofaC∗-algebraA. Then
(H,π) and (K,φ) are said to be:
1. unitarily equivalent ifthereisaunitaryU : H → K suchthatUπ(A) = φ(A)U
for all A ∈ A.
2. quasiequivalent ifthevonNeumannalgebrasπ(A)00andφ(A)00are∗-isomorphic.
3. disjoint if they are not quasiequivalent.
1.15 Definition. Arepresentation(K,φ)issaidtobeasubrepresentation of(H,π)
justincasethereisanisometryV : K → Hsuchthatπ(A)V = Vφ(A)forallA ∈ A.
1.16 Fact. Two representations are quasiequivalent iff they have unitarily equiva-
lent subrepresentations.
ThefamousGelfand-Naimark-Segal(GNS)theoremshowsthateveryC∗-algebraic
state can be represented by a vector in a Hilbert space.
1.17 Theorem (GNS). Let ω be a state of A. Then there is a representation
(H,π) of A, and a unit vector Ω ∈ H such that:
1. ω(A) = hΩ,π(A)Ωi, for all A ∈ A;
2. π(A)Ω is dense in H.
Furthermore, the representation (H,π) is the unique one (up to unitarily equiva-
lence) satisfying the two conditions.
Since we will later need to invoke the details of the GNS construction, we sketch
the outlines of its proof here.
Sketch of proof. We construct the Hilbert space H from equivalence classes of ele-
ments in A, and the representation π is given by the action of left multiplication.
In particular, define a bounded sesquilinear form on A by setting
hA,Bi = ω(A∗B), A,B ∈ A.
ω
7
1/2
Let H be the quotient of A induced by the norm kAk = hA,Ai . Let H be the
0 ω ω
unique completion of the pre-Hilbert space H . Thus there is an inclusion mapping
0
j : A → H with j(A) dense in H. Define the operator π(A) on H by setting
π(A)j(B) = j(AB), B ∈ A.
Onemustverifythatπ(A)iswell-defined, andextendsuniquelytoaboundedlinear
operator on H. One must also then verify that π is a ∗-homomorphism. Finally, if
we let Ω = j(I), then Ω is obviously cyclic for π(A).
1.18 Proposition. Let ω be a state of A. The GNS representation (H,π) of A
induced by ω is irreducible iff ω is pure.
Notes: StandardreferencesonC∗-algebrasinclude[KadisonandRingrose,1997]and[Take-
saki, 2002].
1.3 Type classification of von Neumann algebras
1.19 Definition. Two projections E,F in a von Neumann algebra R are said to
be equivalent, written E ∼ F just in case there is a V ∈ R such that V∗V = E and
VV∗ = F.
1.20 Remark. If we were being really careful, we would replace “equivalent” in
thepreviousdefinitionwith“equivalencemoduloR”, andsimilarly“∼”with“∼ .”
R
But we will not run into trouble by omitting the reference to R. The operator V
in the previous definition is called a partial isometry with initial projection E and
final projection F.
1.21 Definition. ForvonNeumannalgebrasR andR , weletR ∧R = R ∩R .
1 2 1 2 1 2
We let R ∨R denote the von Neumann algebra generated by R and R , i.e. the
1 2 1 2
intersection of all von Neumann algebras containing R and R .
1 2
1.22Definition. Z(R) = R∧R0 iscalledthecenter ofthevonNeumannalgebraR.
A von Neumann algebra R is called a factor just in case Z(R) = CI, equivalently,
R∨R0 = B(H). A projection E ∈ Z(R) is called a central projection in R.
1.23 Definition. Let E ∈ R be a projection, and let ERE = {EAE : A ∈ R}.
Then clearly, ERE is a linear subspace of R. Furthermore, since for A,B ∈ R,
AEB ∈ R and A∗ ∈ R, it follows that ERE is closed under products, as well as
under ∗. It is also not difficult to see that ERE is weakly closed, and hence is a
von Neumann algebra on EH.
1.24 Definition. Let R be a von Neumann algebra. A projection E ∈ R is said
to be:
1. minimal just in case R contains no proper subprojection of E.
8
2. abelian just in case the algebra ERE is abelian.
3. infinite just in case there is a projection E ∈ R such that E < E and
0 0
E ∼ E .
0
4. finite just in case it is not infinite.
5. properly infinite just in case E is infinite and for each central projection P of
R, either PE = 0 or PE is infinite.
1.25 Fact. We have the following relations for projections:
minimal =⇒ abelian =⇒ finite
properly infinite =⇒ infinite ⇐⇒ ¬finite
For factors, the first arrows on both lines can be reversed.
We now give the Murray-von Neumann type classification of factors (for more
on this, see [Kadison and Ringrose, 1997, Chap. 7] or [Sunder, 1987, Chap. 1]).
1.26 Definition. A von Neumann factor R is said to be:
1. type I if it contains an abelian projection;
2. type II if it contains a finite projection, but no abelian projection;
3. type III if it is neither type I nor type II.
The type I factors were already completely classified by Murray and von Neu-
mann: for each cardinal number κ there is a unique (up to isomorphism) type I
κ
factor, namely B(H) where H is a Hilbert space of dimension κ. The type II factors
can be further subdivided according to whether or not the identity projection I is
finite (type II ) or infinite (type II ). The type III factors can be subdivided into
1 ∞
types III with λ ∈ [0,1], although the basis for this subclassification depends on
λ
Tomita-Takesaki modular theory (see Section 1.4).
For general von Neumann algebras, the type classification must be just a bit
more sophisticated: a type I algebra is defined as an algebra that has an abelian
projection E such that no nontrivial projection in Z(R) majorizes E. Similarly, a
type II algebra is defined as an algebra having a finite projection E such that no
nontrivial projection in Z(R) majorizes E. Thus we have:
1.27 Proposition. Let R be a von Neumann algebra. Then R = R ⊕R ⊕R ,
I II III
where R is type X for X=I,II,III.
X
Proof. See [Kadison and Ringrose, 1997, Thm. 6.5.2].
9
We will soon see that the local algebras in QFT are “typically” type III, and
this has many interesting implications. The fact that type III algebras do not have
abelianprojectionsisconnectedtoquestionsoflocalityinSection3.3. Thefactthat
the state space of type III factors is homogeneous is also connected to questions of
1
localityinSection3.3. ThefactthattypeIIIalgebrasdonotcontainrepresentatives
of their states (i.e. density operators) is connected to the modal interpretation of
QFT in Section 5.
The following classification of von Neumann algebras is also natural, but it cuts
across the Murray-von Neumann classification.
1.28 Definition. A von Neumann algebra R is said to be:
• of infinite type if I is infinite in R;
• properly infinite if I is properly infinite in R.
• semi-finite if the central projection E in R (defined in Prop. 1.27) is zero.
III
The finite factors include the type I and type II factors. The infinite factors
n 1
include the type I factors as well as the type II and III factors. The distinction
∞ ∞
between finite and infinite factors coincides with the existence of a tracial state.
1.29 Definition. A faithful normalized trace on a von Neumann algebra R is a
state ρ on R such that:
1. ρ is tracial; i.e. ρ(AB) = ρ(BA), for all A,B ∈ R,
2. ρ is faithful; i.e. ρ(A∗A) = 0 only if A = 0.
1.30 Fact. A von Neumann factor R is finite iff there is a faithful normal tracial
state ρ on R. A von Neumann factor R is semifinite iff there is a “faithful normal
semifinite trace” on R; but we do not pause here to define this notion.
1.4 Modular theory
WestateherewithoutproofsomeofthebasicfactsaboutTomita-Takesakimodular
theory. ThesefactsarenecessaryinordertounderstandtheclassificationoftypeIII
vonNeumannalgebras,whichinturnisessentialtounderstandingthemathematical
structure of AQFT.
1.31 Definition. Let R be a von Neumann algebra acting on a Hilbert space H,
and suppose that Ω ∈ H is cyclic and separating for R. In such a case, we say that
(R,Ω) is in standard form. Define an operator S on H by setting
0
S AΩ = A∗Ω, A ∈ R.
0
Then S extends to a closed anti-linear operator S on H. Let S = J∆1/2 be the
0
polar decomposition of S, so that ∆ is positive (but generally unbounded), and J is
10
