Lokalnie kowariantna kwantowa teoria pola jako podejscie ...

IntroductionMathematical preliminaries

Locally covariant quantum field theoryQuantum gravity

Conclusions

Lokalnie kowariantna kwantowa teoria pola jakopodejscie do kwantowej grawitacji

Katarzyna Rejzner

II. Institute for Theoretical Physics, Hamburg University

Kraków, 08.01.2010

Katarzyna Rejzner Lokalnie kowariantna kwantowa teoria pola

Conclusions

Outline of the talk

1 Introduction

2 Mathematical preliminariesCategory theory

3 Locally covariant quantum field theoryQFT on curved spacetimeLocal covariance

4 Quantum gravityConservative approachBackground independence

Conclusions

Problems with quantum gravity

Spacetime is dynamical

"Points" lost their meaning

It is not clear what should be anobservable

"background independance"

Renormalizability

Conclusions

Renormalizability

Conclusions

Renormalizability

Conclusions

Renormalizability

Conclusions

Renormalizability

Conclusions

Renormalizability

Conclusions

Renormalizability

Conclusions

Renormalizability

Conclusions

Objectives

formulate a consistent theory that is valid in a given physicalsituation

answer some interpretational questions

find a relation to experiment

understand better problems of other approaches

Conclusions

Objectives

Conclusions

Objectives

Conclusions

Objectives

Conclusions

Category theory

Category

A category C consists ofa class Obj(C) of objects,a class hom(C) of morphisms between the objects. Eachmorphism f has a unique source object a and target object bwhere a, b ∈ Obj(C).Notation: if f : a→ b then we write f ∈ hom(a, b)

for a, b, c ∈ Obj(C), a binary operationhom(a, b)× hom(b, c)→ hom(a, c) called composition ofmorphisms. Notation: f ◦ g.

such that the following axioms hold:associativity if f : a→ b, g : b→ c and h : c→ d thenh ◦ (g ◦ f ) = (h ◦ g) ◦ fidentity for every object c, there exists a morphism idc : c→ c,such that for every hom(a, b) 3 f we have: idb ◦ f = f ◦ ida=f.

Conclusions

Category theory

Category

Conclusions

Category theory

Category

Conclusions

Category theory

Category

Conclusions

Category theory

Category

Conclusions

Category theory

Functor

Let C and D be categories. A covariant functor F from C to D is amapping that:

associates to each object c ∈ Obj(C) an object F (c) ∈ Obj(D),

associates to each morphism hom(C) 3 f : a→ b ∈, a morphismhom(D) 3 F (f ) : F (a)→ F (b)

such that the following two conditions hold:

F (idc) = idF (c) for every object c ∈ C.

F (g ◦ f ) = F (g) ◦F (f ) for all morphisms f : a→ b andg : b→ c.

Conclusions

Category theory

Functor

Conclusions

Category theory

Functor

Conclusions

Category theory

Functor

Conclusions

Category theory

Functor

Let C and D be categories. A contravariant functor F from C to D isa mapping that:

associates to each morphism hom(C) 3 f : a→ b, a morphismhom(D) 3 F (f ) : F (a)→ F (b)

F (g ◦ f ) = F (f ) ◦F (g) for all morphisms f : a→ b andg : b→ c.

Conclusions

Category theory

Functor

Conclusions

Category theory

Functor

Conclusions

Category theory

Functor

Conclusions

Category theory

Functor

Covariance:a

f−−−−→ b

F (a)F (f )−−−−→ F (b)

Conclusions

Category theory

Natural transformation

Let F and G be functors between categories C and D, then a naturaltransformation η from F to G associates to every object a ∈ C amorphism hom(D) 3 ηa : F (a)→ G (a), such that for everymorphism C 3 f : a→ b we have:

ηb ◦F (f ) = G (f ) ◦ ηa

This equation can be expressed by the commutative diagram:

F (a)F (f )−−−−→ F (b)

y yηb

G (a)G (f )−−−−→ G (b)

Conclusions

QFT on curved spacetimeLocal covariance

Algebraic formulation

QFT on Minkowski spacetime can be formalized with the use ofHaag-Kastler axioms. Main features of this framework:

Theory formulated in terms of nets ofC∗-algebras (algebras of observables)associated to subsets of Minkowskispacetime: A(O), O ∈ M.

Physical interpretation through states(functionals on observables’ algebras).

There is a special state, that respectssymmetries of M: vacuum state.

Conclusions

QFT on curved spacetime

Important insights:

Dimock: Haag-Kastler axioms for globally hyperbolicspacetimes, covariance for isometric diffeomorphisms

Kay: Hadamard condition as a local characterization ofadmissible states

Radzikowski (followed by Köhler): Hadamard conditionformulated in terms of wave-front sets.

Methods of microlocal analysis applied to QFT on curvedspacetime (Fredenhagen, Brunetti, . . . )

Conclusions

Important insights:

Conclusions

Important insights:

Conclusions

Important insights:

Conclusions

Local covariance

Ideas developed recently by: Brunetti-Fredenhagen-Verch,Hollands-Wald.

Application of category theory provides a formulation whichdoesn’t fix the background

One constructs the theory simultaneously on all spacetimes (of agiven class) in a coherent way

The theory is fixed by a covariant functor between certaincategories

Fields are natural transformations

Conclusions

Local covariance

Conclusions

Local covariance

Conclusions

Local covariance

Conclusions

Local covariance

Conclusions

Categories

Man Obj(Man): all four-dimensional, globally hyperbolic orientedand time-oriented spacetimes (M, g).Morphisms: Isometric embeddings that fulfill:• Given (M1, g1), (M2, g2) ∈ Obj(Man), for any causal curveγ : [a, b]→ M2, if γ(a), γ(b) ∈ ψ(M1) then for all t ∈]a, b[ wehave: γ(t) ∈ ψ(M1).

• Preserving orientation and time-orientation of the embeddedspacetime.

Alg Obj(Alg): C∗ unital algebrasMorphisms: faithful, unit-preserving ∗-homomorphisms.

Conclusions

Categories

Conclusions

Categories

Conclusions

Categories

Conclusions

Categories

Conclusions

Categories

Conclusions

Categories

Conclusions

Categories

Conclusions

Locally covariant quantum field theory

A locally covariant quantum field theory is defined as a covariantfunctor A between Man and Alg. This means that the followingdiagram commutes for all morphismsψ ∈ homMan((M1, g1), (M2, g2)) and all objects of Man:

(M1, g)ψ−−−−→ (M2, g′)

A (M1, g)A (ψ)−−−−→ A (M2, g′)

Denoting αψ ≡ A (ψ), the covariance property reads:

αψ′ ◦ αψ = αψ′◦ψ , αidM = idA (M,g) ,

for all morphisms ψ′ from homMan((M2, g2), (M3, g3)),ψ ∈ homMan((M1, g1), (M2, g2)) and all objects of Man.

Conclusions

Further axioms

One can also include two further axioms which are important in QFT:causality and time-slice axiom.

Causality: If there exist morphismsψj ∈ homMan((Mj, gj), (M, g)), j = 1, 2, such that the setsψ1(M1) and ψ2(M2) are causally separated in (M, g), then:

[αψ1(A (M1, g1)), αψ2(A (M2, g2))] = {0},

where [., .] is the commutator of given C∗ algebras.

Time-slice axiom: If the morphismψ ∈ homMan((M, g), (M′, g′)) is such that ψ(M) contains aCauchy-surface in (M′, g′), then αψ is an isomorphism.

Conclusions

Further axioms

One can also include two further axioms which are important in QFT:causality and time-slice axiom.

Causality: If there exist morphismsψj ∈ homMan((Mj, gj), (M, g)), j = 1, 2, such that the setsψ1(M1) and ψ2(M2) are causally separated in (M, g), then:

[αψ1(A (M1, g1)), αψ2(A (M2, g2))] = {0},

where [., .] is the commutator of given C∗ algebras.

Time-slice axiom: If the morphismψ ∈ homMan((M, g), (M′, g′)) is such that ψ(M) contains aCauchy-surface in (M′, g′), then αψ is an isomorphism.

Conclusions

Fields

Let D be a functor that associates to each spacetimeM a space of testfunctions D(M) ∈ Obj(Test). A field Φ is defined as a naturaltransformation between D and A . To any object (M, g) ∈Man itassociates a morphism Φ(M,g) : D(M, g)→ A (M, g) in such a way,that for each given isometric embedding χ : (M1, g1) −→ (M2, g2)following diagram commutes

D(M1, g1)Φ(M1,g1)−−−−−→ A (M1, g1)

y yαχD(M2, g2)

Φ(M2,g2)−−−−−→ A (M2, g2)

where χ∗ is the push forward under D . This means:

αχ ◦ Φ(M1,g1) = Φ(M2,g2) ◦ χ∗ .

Conclusions

Conservative approachBackground independence

Perturbative quantum gravity

splitting a metric gµν into backgroundmetric ηµν and fluctuation hµν :

gµν = ηµν + hµν

The fluctuation metric is to be quantized

The renormalization scheme:Epstein-Glaser renormalization (valid infinite order).

Conclusions

Relative Cauchy evolution

Let N+ and N− be two spacetimes thatembed into two other spacetimes M1 andM2 around Cauchy surfaces, via causalembeddings given by χk,±, k = 1, 2.

Then β = αχ1+α−1χ2+

αχ2−α−1χ1− is an

automorphism of A (M1).

Conclusions

Background independence

Let M1 = (M, g1) and M2 = (M, g2),where (g1)µν and (g2)µν differ by a(compactly supported) symmetric tensorhµν withsupp(h) ∩ J+(N+) ∩ J−(N−) = ∅,the automorphism depends only on thespacetime between the two Cauchysurfaces

Θµν(x).=

δhµν(x)|h=0 is a derivation

valued distribution which is covariantlyconserved

background independance condition:Θµν = 0

Conclusions

Θµν(x).=

Conclusions

Θµν(x).=

Conclusions

Θµν(x).=

Conclusions

Θµν(x).=

Conclusions

Θµν(x).=

Conclusions

Θµν(x).=

Conclusions

Problems

Nonrenormalizability: In every order new counter terms. Onehas to show that theory can be applied when QG effects aresmall.

Constraints have to be imposed. Best developped withinperturbation theory: BRST

BRST cohomology has to be formulated for global quantities:Fields (natural transformations)

Conclusions

Problems

Conclusions

Problems

Conclusions

Locally covariant quantum field theory turned out to besuccessful in describing QFT on CS.

New mathematical tools of LCQFT can be applied toperturbative quantum gravity.

Main mathematical difficulties: BRST cohomology,renormalization procedure.

Relations to other approaches: TQFT, perturbative QG, loops,strings . . .

Conclusions

Appendix References

References I

K. Fredenhagen, R. Brunetti, Towards a Background IndependentFormulation of Perturbative Quantum Gravity,arXiv:gr-qc/0603079v3

R. Brunetti, K. Fredenhagen, R. Verch, The generally covariantlocality principle - A new paradigm for local quantum fieldtheory, Commun. Math. Phys. 237 (2003) 31-68

Haag, R., Local Quantum Physics, 2nd ed. Springer-Verlag,Berlin, Heidelberg, New York, 1996

Radzikowski, M.J., Micro-local approach to the Hadamardcondition in quantum field theory in curved spacetime, Commun.Math. Phys. 179, 529 (1996)

Appendix References

References II

R.Brunetti, K.Fredenhagen, Microlocal analysis and interactingquantum field theories: Renormalization on physicalbackgrounds, Commun. Math. Phys. 208, 623 (2000)

Segal G., Two-dimensional conformal field theory and modularfunctors,Proc. IXth Intern. Congr. Math. Phys. (Bristol,Philadelphia). Eds. B.Simon, A.Truman and I.M.Davies, IOPPubl. Ltd, 1989, 22-37

Appendix References

Thank you for your attention

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