topological aspects of abelian gauge theories in algebraic … · 2014. 4. 28. · topological...

Topological aspects of Abelian gauge theoriesin algebraic quantum field theory

Alexander Schenkel

Department of Mathematics, University of Wuppertal

Mathematical Physics Seminar

September 12, 2013, University of York

Based on joint work with Marco Benini and Claudio Dappiaggi:

(i) [arXiv:1303.2515 [math-ph]] to appear in Communications in Mathematical Physics

and with Marco Benini, Claudio Dappiaggi and Thomas-Paul Hack:

(ii) [arXiv:1307.3052 [math-ph]]

A. Schenkel (Wuppertal University) Abelian Quantum Gauge Theories Seminar @ York 13 1 / 22

Outline

1. Locally covariant QFT: Category theory meets QFT

2. Classical Abelian principal connections

3. Quantized Abelian principal connections

4. Topological quantum fields in AQFT

5. Haag-Kastler at last!?!

6. Some last words ...


Locally covariant QFT: Category theory meets QFT


The essentials of locally covariant QFT

The basic idea of locally covariant QFT [a la Brunetti, Fredenhagen, Verch] is toassociate to each spacetime M an algebra of quantum field observablesA(M), such that this association “behaves well” under embeddings ofspacetimes f : M1 →M2.

This can be made precise by using tools from category theory. Let us definethe categories:

Man: Obj(Man) are oriented, time-oriented and globally hyperbolic Lorentzianmanifolds.

Mor(Man) are orientation and time-orientation preserving isometricembeddings, such that the image is causally compatible and open.

C∗Alginj: Obj(C∗Alginj) are unital C∗-algebras over C.

Mor(C∗Alginj) are injective unital C∗-algebra homomorphisms.

Def: A locally covariant QFT is a covariant functor A : Man→ C∗Alginj,satisfying

1. the causality property,

2. the time-slice axiom.

Check out the blackboard for some enlightening pictures!A. Schenkel (Wuppertal University) Abelian Quantum Gauge Theories Seminar @ York 13 3 / 22

Example: The real Klein-Gordon field

The canonical quantization recipe goes as follows:1. Interpret ϕ ∈ C∞0 (M) as smearing functions for abstract Weyl-operators

W (ϕ)“ = ei∫M volϕ Φ ”.

2. Notice that all KGM (h), h ∈ C∞0 (M), lead to trivial observables if Φ ison-shell ⇒ it is better to take E(M) := C∞0 (M)/KGM [C∞0 (M)] for labelingWeyl-operators.

3. For multiplication rule W (ϕ) W (ψ) = e−i τM (ϕ,ψ)/2 W (ϕ+ ψ) (i.e. CCR)follow Peierls and derive from the Lagrangian a symplectic structure τM onE(M), explicitly τM (ϕ,ψ) =

∫M

volϕGM (ψ), where GM = G+M −G

−M .

4. Realize that there exists a unique (up to ∗-isomorphism) C∗-algebra A(M)carrying the CCR stemming from the symp. VS PhSp(M) := (E(M), τM ).

NB1: Canonical quantization is a two step procedure

MPhSp

// PhSp(M)CCR // A(M) = CCR(PhSp(M))

NB2: Since KG and G are natural, we have covariant functors

ManPhSp

// SympVSinj CCR // C∗Alginj

An explicit calculation shows that A = CCR PhSp is locally covariant QFT.


Classical Abelian principal connections


There are different ways to look at electromagnetism!

I. Consider field strength tensor F ∈ Ω2(M) as the quantity of primary interestand demand the equations dF = 0 and δF = 0. [QFT of F is studied by

Dappiaggi, Lang]

Pro: This theory describes electric/magnetic charges.

Contra: No coupling to particles and no Aharonov-Bohm effect.

II. Consider gauge potential A ∈ Ω1(M) as the quantity of primary interest anddemand the equation of motion δdA = 0. [QFT of A is studied by Dimock;

Fewster, Pfenning; Dappiaggi, Siemssen; Sanders, Dappiaggi, Hack]

Pro: This theory describes electric charges and the Aharonov-Bohm effect.

Contra: No magnetic charges and unclear structure of gauge transformationsA 7→ A+ dε vs. A 7→ A+ Λ (with dΛ = 0).

III. Consider principal connection ω ∈ Con(P ) ⊂ Ω1(P, g) on a principal

G = U(1)-bundle Pπ→M as the quantity of primary interest (equation of

motion will be described later). [QFT of ω is studied by BDS; BDHS]

Pro: This theory describes electric/magnetic charges and the Aharonov-Bohmeffect. Structure of gauge transformations becomes clear.

“Contra”: Technically quite complicated, however manageable after some study ,A. Schenkel (Wuppertal University) Abelian Quantum Gauge Theories Seminar @ York 13 5 / 22

Three immediate questions!

Q1: Can I expect a QFT of connections to be a functor A : Man→ C∗Alg?

A1: No! Even the association of the configuration space Con(P ) requires a

principal bundle Pπ→M . So what one should look for is a functor

A : G-PrBu→ C∗Alg!

– Obj(G-PrBu) are principal G-bundles P over objects M in Man.

– Mor(G-PrBu) are principal bundle maps f : P1 → P2, such that f : M1 →M2

is a morphism in Man.

Q2: For G = U(1) the field is non-interacting. Can I apply the tools forquantizing linear theories?

A2: No! The configuration space Con(P ) is not a vector space, but affine! So weneed new techniques [BDS: arXiv:1210.3457] for affine QFTs!

Q3: Notice that ω ∈ Con(P )⊂ Ω1(P, g). How does it work with causality and allthat, since P is not equipped with Lorentzian metric?

A3: The affine space Con(P ) has a neat description in terms of an affine bundleover M [Atiyah]! Hence, all fields can be taken to live in bundles overspacetime M , where we have a causal structure and all that!


Atiyah sequence and bundle of connections

Goal: We would like to regard connections as living in a bundle over spacetime.

To any object in G-PrBu we can associate the Atiyah sequence

M × 0 // ad(P )ι // TP/G

π∗++TM

λll

// M × 0

It is not too hard to show that splittings λ : TM → TP/G, i.e.π∗ λ = idTM , are in bijective correspondence with connection forms on P .

These splittings can be described as sections of an affine subbundleC(P ) ⊆ Hom(TM, TP/G), the bundle of connections. The underlying vectorbundle is Hom(TM, ad(P )).

NB: The set of sections Γ∞(C(P )) is an affine space over the vector spaceΓ∞(Hom(TM, ad(P ))) ' Ω1(M, g).

Lem: Con(P ) and Γ∞(C(P )) are naturally isomorphic as affine spaces.

, We have a nice configuration bundle over M for principal connections.


Vector dual bundle and observables

Remember the first point of the canonical quantization recipe:

1. Interpret ϕ ∈ C∞0 (M) as smearing functions for abstract Weyl-operators

W (ϕ)“ = ei∫M volϕ Φ ”.

Heuristically: In gauge theory, Φ gets replaced by a connection quantum field

λ, i.e. a “quantized section of the affine bundle C(P )πC(P )−→ M ”.

→ We can smear λ against a section of the vector dual bundle C(P )†πC(P )†−→ M .

(The fibres C(P )†|x are the affine maps from C(P )|x to R.)

Precisely: To any ϕ ∈ Γ∞0 (C(P )†) we can associate a classical functional (tobe quantized later!)

Wϕ : Γ∞(C(P ))→ C , λ 7→ Wϕ(λ) = ei∫M

vol ϕ(λ)

NB1: Wϕ(λ+ η) =Wϕ(λ) ei∫MϕV ∧∗(η), where ϕV ∈ Ω1

0(M, g∗) linear part of ϕ.

NB2: There exist trivial functionals: Wϕ ≡ 1 iff

ϕ ∈ Triv(P ) :=a1 ∈ Γ∞0 (C(P )†) : a ∈ C∞0 (M) s.t.

∫M

vol a ∈ 2πZ

⇒ A good set of labels is the Abelian group Ekin(P ) := Γ∞0 (C(P )†)/Triv(P ).


Gauge invariant observables

Goal: Identify those observables Wϕ, ϕ ∈ Ekin(P ), which are gauge invariant!

Rem: The gauge group Gau(P ) is isomorphic to C∞(M,G), and f ∈ C∞(M,G)

acts on λ ∈ Γ∞(C(P )) by λ+ f∗(µG).

Def: (i) dC∞(M, g) ⊆ B(P ) := f∗(µG) : f ∈ C∞(M,G) ⊆ Ω1d(M, g).

(ii) ϕ ∈ Ekin(P ) is gauge invariant iff Wϕ(λ+B(P )) =Wϕ(λ).

Notation: E inv(P ) ⊂ Ekin(P ).

Prop: a) B(P ) = η ∈ Ω1d(M, g) : [η] ∈ H1(M,Z).

b) E inv(P ) =ϕ ∈ Ekin(P ) : δϕV = 0 and [ϕV ] ∈ 2πH1(M,Z)∗

⇒ Gauge invariant observables are labeled by the Abelian group E inv(P ).

Thm: The set Wϕ : ϕ ∈ E inv(P ) of gauge invariant observables is separating ongauge equivalence classes of connections. This means that, for any twoλ, λ′ ∈ Γ∞(C(P )) which are not gauge equivalent, there exists ϕ ∈ E inv(P ),such that Wϕ(λ′) 6=Wϕ(λ).


Aharonov-Bohm effect

Remember: E inv(P ) =ϕ ∈ Ekin(P ) : δϕV = 0 and [ϕV ] ∈ 2πH1(M,Z)∗

is an Abelian group, not a vector space!

⇒ Can’t construct for all Wϕ, ϕ ∈ E inv(P ), a “generator” Oϕ := ddsWsϕ

∣∣s=0

.

⇒ There is a huge difference in studying gauge invariant polynomial algebras(generated by the Oϕ) or Weyl algebras (generated by the Wϕ).

Indeed, the polynomial algebras are i.g. not separating! [see BDS]

Q: Is this physically reasonable?

A: Indeed! Even though the configuration space Γ∞(C(P )) is affine, the spaceof gauge equivalence classes Γ∞(C(P ))/B(P ) is rather complicated.

For example, gauge equivalence classes of flat connections on R3 × S1 arelabeled by eiΦ ∈ S1 (the Aharonov-Bohm phase).

Our observables with discrete ϕ ∈ E inv(P ) measure exactly this phase, whichis in agreement with the experimental results.

Heuristically: Compare this to the discrete momenta k ∈ Z on the circle S1!


Maxwell’s equations

The second part in the canonical quantization recipe is to implement thedynamics by a quotient of the labeling space E inv(P ).

Here we are looking for Maxwell’s equations.

Reminder: For a connection form ω ∈ Con(P ) ⊂ Ω1(P, g) we can define thecurvature FP (ω) := dPω + 1

2 [ω, ω]g = dPω ∈ Ω2hor(P, g)G-eqv.

We have natural isomorphism Ω2hor(P, g)G-eqv ' Ω2(M, g), hence we can

regard FP (ω) ∈ Ω2(M, g).

Notice that dMFP (ω) = 0, for all connections ω.

Rem: Maxwell’s equations MWP (λ) := δMFP (ωλ) = 0 can be regarded as kernelof Maxwell’s affine differential operator MWP : Γ∞(C(P ))→ Ω1(M, g).

Prop: Given affine differential operator MWP : Γ∞(C(P ))→ Ω1(M, g) there existsa unique formal adjoint MW∗P : Ω1

0(M, g∗)→ Ekin(P ). (determined by duality)

Furthermore, MW∗P[Ω1

0(M, g∗)]⊆ E inv(P ).

Def: On-shell labeling by the Abelian group E(P ) := E inv(P )/MW∗P[Ω1

0(M, g∗)].


The multiplication rule (aka presymplectic structure)

The third step in the canonical quantization recipe is to equip E(P ) with a(pre)symplectic structure τP in order to define later the multiplication rule for

Weyl operators W (ϕ) W (ψ) = e−iτP (ϕ,ψ)/2 W (ϕ+ ψ).

From the Lagrangian L(λ) = − 12F (λ) ∧ ∗

(F (λ)

)we obtain by generalizing

Peierls construction (be careful: gauge invariance!!!)

τP (ϕ,ψ) =∫MϕV ∧ ∗

(GM (ψV )

), where GM corresponds to M .

Prop: PhSp(P ) := (E(P ), τP ) is a presymplectic Abelian group.

Def: Define the null Abelian groups:

(i) ψ ∈ N (P ) ⊆ E(P ) iff τP (E(P ), ψ) ⊆ 2πZ.

(ii) ψ ∈ N 0(P ) ⊆ E(P ) iff τP (E(P ), ψ) = 0.

Prop: a) N (P ) =ψ ∈ E inv(P ) : ψV ∈ δΩ2

0,d(M, g∗)

and [GP (ψV )] ∈ 2πH1(M,Z)/MW∗P [Ω1

0(M, g∗)].

b) N 0(P ) =ψ ∈ E inv(P ) : ψV ∈ δ

(Ω2

0(M, g∗) ∩ dΩ1tc(M, g∗)

)/MW∗P [Ω1

0(M, g∗)].

NB: The null Abelian groups N 0(P ) ⊆ N (P ) are never empty and they dependrather non-trivially on the topology of M via H2

0 (M, g∗).


Functoriality of the classical theory

We have seen that we can associate to any object P in G-PrBu apresymplectic Abelian group PhSp(P ) = (E(P ), τP ).

Q: Do we have a covariant functor PhSp : G-PrBu→ PreSympAG?

Thm: The association

– P 7→ PhSp(P )

–(f : P1 → P2

)7→(PhSp(f) : PhSp(P1)→ PhSp(P2)

)given by the

push-forward PhSp(f)(ϕ) = f∗(ϕ), for all ϕ ∈ E(P1),

is a covariant functor PhSp : G-PrBu→ PreSympAG.

Sketch: All quotients and restrictions entering the construction of E(P ) from Γ∞0 (C(P )†),

as well as the presymplectic structure τP , are natural.

Prop: The covariant functor PhSp : G-PrBu→ PreSympAG does not satisfy thelocality property, i.e. it is not a functor to PreSympAGinj.

Sketch: Take P2 such that M2 = Rm, m ≥ 4. Take M1 := M2 \ JM2(0) and induce P1

by pull-back. The morphism f : P1 → P2 pushes forward the elements

F ∗P1(ζ) ∈ N 0(P1), ζ ∈ Ω2

0,d(M1, g∗), [ζ] 6= 0, to the class 0.


Properties of the functor PhSp

We have already seen that one axiom of locally covariant classical field theory isviolated, that is locality, i.e. PhSp is not a functor to PreSympAGinj.

An interpretation follows later.

How about the classical causality property and the classical time-slice axiom?

Thm: The covariant functor PhSp : G-PrBu→ PreSympAG satisfies the classicalcausality property and the classical time-slice axiom.

Sketch: The causality property follows easily, since τP is constructed from the causal

propagator GM of M .

The time-slice axiom requires some technical tricks which we have developed in

[BDS: arXiv:1210.3457] in order to treat the constant parts a1 of ϕ.

⇒ We get almost a locally covariant classical field theory, however theimportant locality property is missing.

Q: Can we modify the functor PhSp in some way to get the locality property?


Are there modifications of PhSp?

Looking again at the example showing that PhSp(f) is not injective, we seethat the problematic elements in PhSp(P ) are of the form F ∗P (ζ) ∈ N 0(P ),with ζ ∈ Ω2

0,d(M, g∗), [ζ] 6= 0.

The corresponding classical observables yield, for all solutions λ,

WF∗P

(ζ)(λ) = ei∫M vol (F∗P (ζ))(λ) = ei

∫M ζ∧∗(FP (λ)) = ei 〈[ζ],[∗(FP (λ))]〉

Physical argument: If I live in a world of pure electromagnetism, all chargemeasurements yield zero. Hence, I should identify the electric chargeobservables with zero.

Mathematical realization: Look for a natural association of Abeliansubgroups Q(P ) contained in N 0(P ). Then the association PhSp(P )/Q(P )is a covariant functor PhSp/Q : G-PrBu→ PreSympAG.

Thm: Neither N nor N 0 is a covariant functor G-PrBu→ PreSympAG.

Sketch: Take P2, such that M2 = R× S1 × Sm−2, m ≥ 4. M1 is Cauchy development of

I × Sm−2 and P1 is obtained by pull-back. The morphism f : P1 → P2 pushes

forward the elements F ∗P1(ζ) ∈ N 0(P1), ζ ∈ Ω2

0,d(M1, g∗), [ζ] 6= 0, to elements in

E(P2) that are not in N (P2).

Cor: Neither PhSp/N nor PhSp/N 0 is a covariant functor.


A no-go theorem

TheoremThere exists no covariant functor Q : G-PrBu→ PreSympAG together with aninjective natural transformation ι : Q⇒ PhSp satisfying ιP (Q(P )) ⊆ N 0(P ) forany object P in G-PrBu, such that PhSp/Q : G-PrBu→ PreSympAGinj.

Rem: This means that there is no natural way of modifying the presymplecticAbelian groups PhSp(P ) by quotienting out electric charges, such that thelocality property holds!

Sketch: This follows essentially from the sketches we had before. In pictures and takingeverywhere the trivial principal G-bundle over the M ’s:

M1 = R× S1 × S2

g1 = −dt⊗ dt+ dφ⊗ dφ+ gS2

M2 = R4

g2 = −dt⊗ dt+ +α(r2) dxi ⊗ dxi+ β(r2) dr ⊗ dr

M3 = DM1

(0 × I × S2

)g3 = g1|M3

f1is incompatible with quotient by N0(P3)

OO

f2

injectivity requires quotient by N0(P3)

44


Quantized Abelian principal connections


The CCR-functor for PreSympAG

Everybody knows the covariant functor CCR : SympVS(inj) → C∗Alg(inj), butdoes it extend to presymplectic Abelian groups?

Based on the results of [Manuceau et al.] this is indeed the case by thefollowing construction:

– to any object (B, τ) in PreSympAG we associate the unital ∗-algebra ∆(B, τ),that is the vector space spanned by the basis W (b) : b ∈ B with productW (b)W (c) = e−iτ(b,c)/2 W (b+ c) and involution W (b)∗ = W (−b).

to any morphism φ : (B1, τ1)→ (B2, τ2) in PreSympAG we associate themorphism ∆(φ) : ∆(B1, τ1)→ ∆(B2, τ2) via ∆(φ)(W1(b)) = W2(φ(b)).

– on each ∆(B, τ) there is a ∗-norm ‖∑ni=1 αiW (bi)‖Ban :=

∑ni=1 |αi|. The

completion ∆Ban(B, τ) is unital ∗-Banach algebra, the association ∆Ban

functorial (all ∆(φ) are bounded, hence extend to completion).

– on ∆Ban(B, τ) there exists C∗-norm ‖a‖ := supω∈F√ω(a∗ a). The

completion CCR(B, τ) is unital C∗-algebra. All morphisms ∆Ban(φ) arebounded, hence extend to C∗-completion.

Thm: The above construction gives covariant functor CCR : PreSympAG→ C∗Alg.

It restricts to a covariant functor CCR : PreSympAGinj → C∗Alginj.


Main result on algebras for Abelian gauge theories

TheoremThere exists a covariant functor A := CCR PhSp : G-PrBu→ C∗Alg describingthe C∗-algebras of observables for quantized principal U(1)-connections.

The covariant functor A satisfies the quantum causality property and the quantumtime-slice axiom.

Furthermore, for each object P in G-PrBu the C∗-algebra A(P ) is a quantizationof an algebra of functionals on Γ∞(C(P )) that separates gauge equivalenceclasses of connections.

However, A does not satisfy the locality property, i.e. A is not a covariant functorto the subcategory C∗Alginj. There exists no further quotient of PhSp, such thatthe locality property holds.


Topological quantum fields in AQFT


Topological quantum fields

According to [Brunetti, Fredenhagen, Verch] a locally covariant quantum field is anatural transformation Ψ = ΨM : D⇒ A from a covariant functorD : Man→ Vec describing test sections to A : Man→ C∗Alg.

Def: (i) A generally covariant topological quantum field is a natural transformationΨ : T⇒ A from a covariant functor T : G-PrBu→ Monoid describing“topological information” to the QFT functor A : G-PrBu→ Monoid.

(ii) Denote by H2 : G-PrBu→ Monoid the singular homology functor withH2(P ) = H2(M, g∗).

(iii) Let KP : Hp(M, g∗)→ Hp0 dR(M, g∗) be the natural isomorphism induced by

Poincare duality and the de Rham isomorphism.

Thm: The collection Ψmag := ΨmagP of monoid morphisms

ΨmagP : H2(P )→ A(P ) , σ 7→

[W(F ∗P(KP (σ)

))]is a generally covariant topological quantum field.

NB: Ψmag maps to the center of the C∗-algebras! Looking at the correspondingclassical observables Wϕ we see that Ψmag measures the Chern class of P !


Haag-Kastler at last!?!


Haag-Kastler works... and this is why!

Goal: Understand in detail where the violation of locality comes from!

The problem comes from central observables on a small spacetime M1, whichcan be mapped on the one hand to trivial observables on a larger spacetimeM2 and on the other hand to non-central observables on another largerspacetime M3.

These observables measure cohomology class [∗(F(λ))] ∈ Hm−2dR (M1, g)

(electric charge in the language of [Sanders,Dappiaggi,Hack]) of a solution λ.

Important: The solution λ on M1 can be extended to M3, but not to M2!

⇒ It depends on if we live in M3 or M2 in order to decide if the observable inM1 is physical or not.

⇒ We have to know the whole universe in order to construct physical localalgebras of pure electromagnetism!

! Open problem: Show that electromagnetism coupled to some dynamicalmodel for charge densities does not have this issue!


Haag-Kastler works... and this is how!

Restricting the functor PhSp : G-PrBu→ PreSympAG to subcategory with

terminal object P we have for any morphism a commuting diagram

PhSp(P )

PhSp(P1)

88

// PhSp(P2)

ff

Lem: Defining Ker(P ) as the kernel of the canonical morphism

PhSp(P )→ PhSp(P ) we obtain (with the induced morphisms) a covariant

functor Ker : G-PrBuP → PreSympAG. Together with the canonical

injections ιP it is a quotient of PhSp : G-PrBuP → PreSympAG.

Thm: PhSp0 := PhSp/Ker : G-PrBuP → PreSympAGinj satisfies the causalityproperty, the time-slice axiom and the locality property.

Composing with the CCR-functor for presymplectic Abelian groups we get a

QFT functor A0 := CCR PhSp0 : G-PrBuP → C∗Alginj.

NB: Repeating [Brunetti,Fredenhagen,Verch] A0 can be shown to be aHaag-Kastler-type QFT. It is not locally covariant due to arguments before!


Some last words ...


Some last words ...

The Abelian gauge field is strange!

We have to know the global spacetime (terminal object) in order to give aconstruction of the local algebras of pure electromagnetism!

Reason: We can not disentangle locally the effects induced by physicalcharges (e.g. electron outside my region) from a non-trivial topology (bothlead to non-zero de Rham classes [∗(FP (λ))] ∈ Hdim(M)−2(M, g)).

For the functor PhSp we even have a no-go theorem!

So what should we do about it?

b Work on generalizations of the locally covariant QFT axioms that include thespecial non-local features present in gauge theories.

b Work out interacting Dirac-Maxwell system and check if the conjecture thatin the interacting theory we have locality is true.

Hard task! Strong people required!

I think that it is worth it, since electromagnetism is very important and it wouldbe shameful if we, the AQFT community, could not understand it!


topological aspects of abelian gauge theories in algebraic … · 2014. 4. 28. · topological...

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