Effective quantum gravity as a locally covariant QFTtheory.fi.infn.it/seminara/Geometry_of_Strings_and_Fieds/Conference... · Effective quantum gravity Local covariance Difﬁculties

IntroductionClassical theory

Quantization

Effective quantum gravity as a locally covariantQFT

Katharina Rejzner1

University of York

Firenze, 10.09.2013

1Based on the joint work with Klaus Fredenhagen and Romeo BrunettiKatharina Rejzner QG from LCQFT 1 / 28


Quantization

Outline of the talk

1 IntroductionEffective quantum gravityLocal covariance

2 Classical theoryKinematical structureDynamics and symmetriesBV complex

3 QuantizationDeformation quantizationQuantum BV formalismBackground independence


QuantizationEffective quantum gravityLocal covariance

Difficulties in quantum gravity

In contrast to QFT on curvedspacetimes, in QG the spacetimestructure is dynamical. Need for"background independance".

"Points" loose their meaning. Thetheory is invariant underdiffeomorphism transformations.

As a QFT, quantum gravity is powercounting non-renormalizable.

Katharina Rejzner QG from LCQFT 2 / 28

















Ways around some of the problems

Non-renormalizability: use Epstein-Glaserrenormalization to obtain finite results for any fixedenergy scale. Think of the theory as an effective theory.Outlook: use the renormalization group flow equations tolook for a UV fixed point (asymptotic safety program).

Dynamical nature of spacetime: make a tentative splitof the metric into background and perturbation, quantizethe perturbation as a quantum field on a curvedbackground, show background independence at the end.

Diffeomorphism invariance: use the BV formalism todo the gauge fixing. Possible difficulties: base manifoldis Lorentzian and non-compact, symmetry group isinfinite dimensional, so is the configuration space.

























The geometry of fields

View the space of field configurations as an infinite dimensionalmanifold.

Symmetries: action of some infinite dimensional Lie group. Goto the Lie algebra action.Look at the derived version of this: infinite dimensional gradedmanifolds (Sachse 2008).To quantize, consider deformations of such structures.





View the space of field configurations as an infinite dimensionalmanifold.Symmetries: action of some infinite dimensional Lie group. Goto the Lie algebra action.

Look at the derived version of this: infinite dimensional gradedmanifolds (Sachse 2008).To quantize, consider deformations of such structures.





View the space of field configurations as an infinite dimensionalmanifold.Symmetries: action of some infinite dimensional Lie group. Goto the Lie algebra action.Look at the derived version of this: infinite dimensional gradedmanifolds (Sachse 2008).

To quantize, consider deformations of such structures.





View the space of field configurations as an infinite dimensionalmanifold.Symmetries: action of some infinite dimensional Lie group. Goto the Lie algebra action.Look at the derived version of this: infinite dimensional gradedmanifolds (Sachse 2008).To quantize, consider deformations of such structures.




Intuitive idea

In experiment, geometric structure is probed bylocal observations. We have the following data:

Some region O of spacetime where themeasurement is performed,An observable Φ, which we measure,We don’t measure the observable curvature at apoint, but we have some smearing related to theexperimantal uncertainty. This is modeled bysmearing with a test function. For example:

Φ(f ) =

∫f (x)R(x).

We can think of the measured observable as afunction of a perturbation of the fixed backgroundmetric: a tentative split into: gµν = gµν + hµν .

Diffeomorphism transformation: move ourexperimental setup to a different region O′.

M




Intuitive idea


Some region O of spacetime where themeasurement is performed,

An observable Φ, which we measure,We don’t measure the observable curvature at apoint, but we have some smearing related to theexperimantal uncertainty. This is modeled bysmearing with a test function. For example:

Φ(f ) =

∫f (x)R(x).



O

M




Intuitive idea


Some region O of spacetime where themeasurement is performed,An observable Φ, which we measure,

We don’t measure the observable curvature at apoint, but we have some smearing related to theexperimantal uncertainty. This is modeled bysmearing with a test function. For example:

Φ(f ) =

∫f (x)R(x).



O

M

Φ




Intuitive idea



Φ(f ) =

∫f (x)R(x).



M

Φ(f )

f

O




Intuitive idea



Φ(f ) =

∫f (x)R(x).



(M, g)

Φ(O,g)(f )[h]

f

O




Intuitive idea



Φ(f ) =

∫f (x)R(x).



O′(M, g)

Φ(O,g)(f )[h]

f

O




Algebraic quantum field theory (locality)

A convenient framework to investigate conceptual problems inQFT is the Algebraic Quantum Field Theory (recently alsoperturbative AQFT).

A model is defined by associating to each region O ofMinkowski spacetime an algebra A(O) of observables (a unitalinvolutive topological algebra, in the original framework alsoC∗) that can be measured in O.

The physical notion of subsystemsis realized by the condition of isotony,i.e.: O2 ⊃ O1 ⇒ A(O2) ⊃ A(O1).We obtain a net of algebras.








O1

A(O1)








A(O2)

O2 O1

A(O1)⊃

⊃




Locally covariant quantum field theory (LCQFT)

To include effects of general relativity intoQFT, one has to be able to describe quantumfields on a general class of spacetimes. Thecorresponding generalization of AQFT iscalled locally covariant quantum field theoryand it uses the language of category theory.

The category Loc, has globally hyperbolicspacetimes M .

= (M, g) as objects and itsmorphisms are isometric, orientationspreserving, causal embeddings ψ : M→ N.

A model in LCQFT is defined by giving afunctor A from the category of spacetimes tothe category Obs of observables (forexample involutive topological algebras).









N M

ψ









A(N)

A

N M

ψ A

A(M)

Aψ




Building models in LCQFT

One of the methods to build models in LCQFT is the so calledfunctional approach.

The main idea is to model observables as functionals on the thespace E(M) of possible field configurations.

On this space of functionals we introduce first the classicaldynamics by defining a Poisson structure. Next, we use thedeformation quantization to construct the non-commutativequantum algebra.

An important advantage of the deformation quantization is thefact that we work all the time on the same set of functionals, butwe equip it with different algebraic structures (i.e. Poissonbracket, non-commutative ? product).




























Functional approach

There are some mathematical subtleties related with thisapproach. The space of field configurations is infinitedimensional, so the space of all the functionals on it is inprinciple too big.

The first step is to restrict oneself to functionals that are smooth.This requires some tools from calculus on infinite dimensionalvector spaces.

Among all the smooth functionals we can distinguish ones thatare particularly relevant for physics. For example, we canconsider local functionals, i.e. ones that can be written in the

form: F(h) =

∫M

f (jx(h))(x) , where h is a field configuration, f

is a density-valued function on the jet bundle over M and jx(h) isthe jet of h at x.




Functional approach




form: F(h) =

∫M






Functional approach




form: F(h) =

∫M






Spacetime localization of a functional

Another important property of a functional is itsspacetime localization.

For a point x ∈M we want to know if our givenfunctional F is sensitive to fluctuations of fieldconfigurations at this point.

If this is the case, we say that x belongs to thespacetime support of F, i.e. x ∈ supp(F).

More precisely:supp F = {x ∈ M|∀ neighbourhoods U of x ∃h1, h2 configurations,

supp h2 ⊂ U such that F(h1 + h2) 6= F(h1)} .

In the classical theory we will consider functionalsthat are compactly supported and multilocal (i.e.sums of finite products of local functionals).











x

M











xsupp(F)

M











xsupp(F)

M











xsupp(F)

M



Quantization

Kinematical structureDynamics and symmetriesBV complex

Kinematical structure

For the effective theory of gravity the configuration space isE(M) = Γ((T∗M)⊗2). The space of compactly supportedconfigurations is denoted by Ec(M). The assignment of bothE(M) and Ec(M) is functorial.

The space of multilical functionals will be denoted by F(M). Fis a covariant functor from Loc to Vec (the category of locallyconvex topological vector spaces).



Quantization


Kinematical structure

For the effective theory of gravity the configuration space isE(M) = Γ((T∗M)⊗2). The space of compactly supportedconfigurations is denoted by Ec(M). The assignment of bothE(M) and Ec(M) is functorial.

The space of multilical functionals will be denoted by F(M). Fis a covariant functor from Loc to Vec (the category of locallyconvex topological vector spaces).



Quantization


Fields as natural transformations

In the framework of locally covariant fieldtheory [Brunetti-Fredenhagen-Verch 2003] fields arenatural transformation between certainfunctors. For the sake of this talk letΦ ∈ F

.= Nat(D,F), where D is the functor

of test function spaces D(M) = C∞c (M)(one could substitute F with a functor to thecategory of Poisson or C∗ algebras).

The condition for Φ to be a naturaltransformation:

ΦO(f )[χ∗h] = ΦM(χ∗f )[h].

In classical gravity we understand physicalquantities not as pointwise objects but ratheras something defined on all the spacetimesin a coherent way.

M χ(O)

O

χ



Quantization






The condition for Φ to be a naturaltransformation: ΦO(f )[χ∗h] = ΦM(χ∗f )[h].


M

ΦM(χ∗f )

ΦO(f )

χ(O)

O

χ∗f

f

χχ−1



Quantization






The condition for Φ to be a naturaltransformation: ΦO(f )[χ∗h] = ΦM(χ∗f )[h].


M

ΦM(χ∗f )

ΦO(f )

χ(O)

O

χ∗f

f

χχ−1



Quantization


Dynamics and symmetries

To implement dynamics we use a certain generalization of theLagrange formalism of classical mechanics.

For general relativity, we have a Lagrangian of the form:

S(M,g)(f )[h].=

∫R[g]f d vol(M,g), where g = g + h.

We need the cutoff function f because M is not compact.The Euler-Lagrange derivative of S is defined as⟨S′M(h0), h

⟩=⟨

LM(f )(1)(h0), h⟩

, where f ≡ 1 on supph. The

field equation is: S′M(h0) = 0. The space of solutions is denotedby ES(M) and multilocal functionals on this space by FS(M).A symmetry of S is a direction in E(M)in which the action is constant,i.e. it is a vector field X ∈ Γc(TE(M))such that ∀h0 ∈ E(M): 0 =

⟨S′M(h0),X(h0)

⟩.



Quantization



To implement dynamics we use a certain generalization of theLagrange formalism of classical mechanics.For general relativity, we have a Lagrangian of the form:

S(M,g)(f )[h].=



⟩=⟨

LM(f )(1)(h0), h⟩



⟨S′M(h0),X(h0)

⟩.



Quantization




S(M,g)(f )[h].=


We need the cutoff function f because M is not compact.

The Euler-Lagrange derivative of S is defined as⟨S′M(h0), h

⟩=⟨

LM(f )(1)(h0), h⟩



⟨S′M(h0),X(h0)

⟩.



Quantization




S(M,g)(f )[h].=



⟩=⟨

LM(f )(1)(h0), h⟩


field equation is: S′M(h0) = 0. The space of solutions is denotedby ES(M) and multilocal functionals on this space by FS(M).

A symmetry of S is a direction in E(M)in which the action is constant,i.e. it is a vector field X ∈ Γc(TE(M))such that ∀h0 ∈ E(M): 0 =

⟨S′M(h0),X(h0)

⟩.

Msupp(f )

supp(h)f ≡ 1



Quantization




S(M,g)(f )[h].=



⟩=⟨

LM(f )(1)(h0), h⟩



⟨S′M(h0),X(h0)

⟩.

Msupp(f )

supp(X)f ≡ 1



Quantization


Diffeomorphism invariance

Consider Φ ∈ F, which is given by a familyof maps ΦM : D(M)→ F(M) that satisfythe naturality condition.

For each M we can choose somediffeomorphism αM and transform Φ to anew field by relabeling maps ΦM:

(~αΦ)(M,g)[g].= Φ(M,αM∗g)[g] ,

where ~α denotes the family (αM)M∈Obj(Loc).

From the naturality condition follows that(~αΦ)(M,g)(f )[g]=Φ(M,g)(α

−1M ∗ f )[α∗Mg]

always holds (diffeomorphism covariance).

The diffeomorphism invariance is acondition: (~αΦ)(M,g)(f )[g] = Φ(M,g)(f )[g].



Quantization





(~αΦ)(M,g)[g].= Φ(M,αM∗g)[g] ,



−1M ∗ f )[α∗Mg]



(M, g)

α−1M ∗ f

f

O

O′

αM



Quantization





(~αΦ)(M,g)[g].= Φ(M,αM∗g)[g] ,



−1M ∗ f )[α∗Mg]



(M, g)

Φ(M,g)(α−1M ∗ f )[α−1

M ∗g]

Φ(M,αM∗g)(f )[g]

α−1M ∗ f

f

O

O′

αM



Quantization





(~αΦ)(M,g)[g].= Φ(M,αM∗g)[g] ,



−1M ∗ f )[α∗Mg]



(M, g)

Φ(M,g)(α−1M ∗ f )[α−1

M ∗g]

Φ(M,αM∗g)(f )[g]

α−1M ∗ f

f

O

O′

αM



Quantization



Let us now look at the infinitesimal version, i.e. considerαM = exp(ξM), ξM ∈ X(M)

.= Γ(TM). The family ~ξ of “gauge”

parameters acts on a field Φ by

(~ξΦ)(M,g)(f )[g] =⟨(Φ(M,g)(f ))(1)[g],−LξM g

⟩+ Φ(M,g)(−LξM f )[g]

Diffeomorphism invariance is the statement that: ~ξΦ = 0.

Example:∫

R[g]f d vol(M,g) is diffeomorphism invariant, but∫R[g]f d vol(M,g) is not.



Quantization







⟩+ Φ(M,g)(−LξM f )[g]


Example:∫




Quantization







⟩+ Φ(M,g)(−LξM f )[g]


Example:∫




Quantization


BV complex I

A general method to quantize theories with local symmetries isthe so called Batalin-Vilkovisky (BV) formalism. Here wepresent its version proposed by [K. Fredenhagen, K.R., CMP 2011].

Differences to the approaches presented up to now: we work inLorentzian QFT, the base manifold is non-compact, we takeseriously the infinite dimensional character of the configurationspace.

Objective: characterize FS, the space of gauge invariant fields onthe space of solutions of EOM’s. More precisely, we considerelements of F := Nat(D,F), which are invariant underdiffeomorphisms (~ξΦ = 0) and take quotient by the idealconsisting of fields satisfying ΦM(f )(h) = 0, for all M,f ∈ D(M), h ∈ ES(M).



Quantization


BV complex I






Quantization


BV complex I






Quantization


BV complex II

Idea: note that ES(M) locally can be seen as the critical manifoldof the Lagrangian SM(f ) : E(M)→ R (zero locus of S′M).

We can apply standard methods and characterize the space ofon-shell fields by its Koszul-Tate resolution. However, one has tobe a little bit careful about the topologies and completions, sincewe work with infinite dimensional spaces!

The space of invariants under a lie algebra action is given by the0th cohomology of the Chevalley-Eilenberg complex, so we canuse this structure to characterize fields that are gauge invariant.

Combining the Koszul-Tate and the Chevalley-Eilenbergcomplex we obtain the BV complex. Its 0th cohomologycharacterizes then the space of gauge invariant on-shell fields.



Quantization


BV complex II







Quantization


BV complex II







Quantization


BV complex II







Quantization


BV complex III

We will denote the underlying algebra of the BV complex byBV. Recall that we are working with fields, so elements of BV

are in particular functions from D(M) to a graded algebraBV(M), constructed in a “standard” way.

BV(M) can be seen as the algebra of functions on T∗E(M),where E(M) is a certain graded manifold (in the simplest case:E(M) = E(M)⊕ X(M)[1]).We call E(M) the extended configuration space.On BV(M) we have a natural structure of a Schouten bracket{., .} (the antibracket), which extends to BV.The classical BV differential can be written as

sΦM(f ) = {ΦM(f ), SM}+ ΦM(LCf ) ,

where S ∈ BV is the so called extended action and C is a ghost.



Quantization


BV complex III


are in particular functions from D(M) to a graded algebraBV(M), constructed in a “standard” way.BV(M) can be seen as the algebra of functions on T∗E(M),where E(M) is a certain graded manifold (in the simplest case:E(M) = E(M)⊕ X(M)[1]).

We call E(M) the extended configuration space.On BV(M) we have a natural structure of a Schouten bracket{., .} (the antibracket), which extends to BV.The classical BV differential can be written as





Quantization


BV complex III


are in particular functions from D(M) to a graded algebraBV(M), constructed in a “standard” way.BV(M) can be seen as the algebra of functions on T∗E(M),where E(M) is a certain graded manifold (in the simplest case:E(M) = E(M)⊕ X(M)[1]).We call E(M) the extended configuration space.

On BV(M) we have a natural structure of a Schouten bracket{., .} (the antibracket), which extends to BV.The classical BV differential can be written as





Quantization


BV complex III


are in particular functions from D(M) to a graded algebraBV(M), constructed in a “standard” way.BV(M) can be seen as the algebra of functions on T∗E(M),where E(M) is a certain graded manifold (in the simplest case:E(M) = E(M)⊕ X(M)[1]).We call E(M) the extended configuration space.On BV(M) we have a natural structure of a Schouten bracket{., .} (the antibracket), which extends to BV.

The classical BV differential can be written as





Quantization


BV complex III


are in particular functions from D(M) to a graded algebraBV(M), constructed in a “standard” way.BV(M) can be seen as the algebra of functions on T∗E(M),where E(M) is a certain graded manifold (in the simplest case:E(M) = E(M)⊕ X(M)[1]).We call E(M) the extended configuration space.On BV(M) we have a natural structure of a Schouten bracket{., .} (the antibracket), which extends to BV.The classical BV differential can be written as





Quantization


Equations of motion and Poisson bracket

As an output of the classical theory we have the extendedconfiguration space E and the extended action S. Now we applyto this data the deformation quantization.

We can Taylor expand the action around an arbitrary backgroundmetric g and obtain SM = S0g + Vg, where S0g is an at mostquadratic function on E(M).For each globally hyperbolic background g, we have the retardedand advanced Green’s functions ∆

R/Ag for the EOM’s derived

from S0g.Using this input, we define the free Poisson bracket on BV(M):

{F,G}g0.=⟨

F(1),∆gG(1)⟩

∆g = ∆Rg −∆A

g ,

This Poisson structure can be naturally extended to a Poissonbracket {., .}0 on BV.



Quantization



As an output of the classical theory we have the extendedconfiguration space E and the extended action S. Now we applyto this data the deformation quantization.We can Taylor expand the action around an arbitrary backgroundmetric g and obtain SM = S0g + Vg, where S0g is an at mostquadratic function on E(M).

For each globally hyperbolic background g, we have the retardedand advanced Green’s functions ∆



{F,G}g0.=⟨

F(1),∆gG(1)⟩

∆g = ∆Rg −∆A

g ,




Quantization



As an output of the classical theory we have the extendedconfiguration space E and the extended action S. Now we applyto this data the deformation quantization.We can Taylor expand the action around an arbitrary backgroundmetric g and obtain SM = S0g + Vg, where S0g is an at mostquadratic function on E(M).For each globally hyperbolic background g, we have the retardedand advanced Green’s functions ∆


from S0g.

Using this input, we define the free Poisson bracket on BV(M):

{F,G}g0.=⟨

F(1),∆gG(1)⟩

∆g = ∆Rg −∆A

g ,




Quantization






{F,G}g0.=⟨

F(1),∆gG(1)⟩

∆g = ∆Rg −∆A

g ,




Quantization






{F,G}g0.=⟨

F(1),∆gG(1)⟩

∆g = ∆Rg −∆A

g ,




Quantization

Deformation quantizationQuantum BV formalismBackground independence

Deformation quantization

We start with the deformation quantization of (BV, {., .}0),which is done in the standard way and provides a ?-product withthe following properties:

F ? G ~→0−−−→ F · G,[F,G]? := F ? G− F ? G ~→0−−−→ {F,G}0.

To introduce the interaction one has to define the so called timeordered products. Formally, they are the coefficients in expansionof the S-matrix in powers of the interaction term Vg, i.e.:

S(Vg) =∞∑

n=0

1n!Tn(V⊗n

g ).

Because of the singularity structure of the Feynman propagator,time ordered products of local non-linear functionals are welldefined only for arguments with pairwise disjoint supports. Inparticular the above formula would not make sense for local Vg.



Quantization




F ? G ~→0−−−→ F · G,[F,G]? := F ? G− F ? G ~→0−−−→ {F,G}0.


S(Vg) =∞∑

n=0

1n!Tn(V⊗n

g ).




Quantization




F ? G ~→0−−−→ F · G,[F,G]? := F ? G− F ? G ~→0−−−→ {F,G}0.


S(Vg) =∞∑

n=0

1n!Tn(V⊗n

g ).




Quantization


Interaction

Since most of interaction terms relevant in physics are local, weneed to extend maps Tn to local arguments with arbitrarysupports. To this end we use the so called Epstein-Glaserrenormalization. Mathematically it reduces to extension ofcertain distributions.

As a result, we obtain a family of maps Tn from BV⊗nloc to a

certain completion of BV. We have shown that these maps canbe seen as arising from a binary product ·T defined on a certaindomain containing Floc and S(Vg) = eVg

T is a time-orderedexponential with respect to this product.This allows to define the interacting fields by means of theBogoliubov formula:

(RV(Φ))(M,g)(f ).=(eVgT

)?−1?(eVgT ·T Φ(M,g)(f )

).



Quantization


Interaction

Since most of interaction terms relevant in physics are local, weneed to extend maps Tn to local arguments with arbitrarysupports. To this end we use the so called Epstein-Glaserrenormalization. Mathematically it reduces to extension ofcertain distributions.As a result, we obtain a family of maps Tn from BV⊗n

loc to acertain completion of BV. We have shown that these maps canbe seen as arising from a binary product ·T defined on a certaindomain containing Floc and S(Vg) = eVg

T is a time-orderedexponential with respect to this product.

This allows to define the interacting fields by means of theBogoliubov formula:


)?−1?(eVgT ·T Φ(M,g)(f )

).



Quantization


Interaction

Since most of interaction terms relevant in physics are local, weneed to extend maps Tn to local arguments with arbitrarysupports. To this end we use the so called Epstein-Glaserrenormalization. Mathematically it reduces to extension ofcertain distributions.As a result, we obtain a family of maps Tn from BV⊗n

loc to acertain completion of BV. We have shown that these maps canbe seen as arising from a binary product ·T defined on a certaindomain containing Floc and S(Vg) = eVg

T is a time-orderedexponential with respect to this product.This allows to define the interacting fields by means of theBogoliubov formula:


)?−1?(eVgT ·T Φ(M,g)(f )

).



Quantization


Quantum master equation

In the framework of [K. Fredenhagen, K.R., CMP 2013], the gaugeinvariance of the S-matrix is guaranteed by the so called quantummaster equation (QME):

{eVgT , S0g} = 0 ,

where {., .} is the Schouten bracket.

With the use of Master Ward Identity [F.Brennecke, M.Duetsch, RMP

2008], this condition can be rewritten as

12{S0g + Vg, S0g + Vg} = i~4Vg ,

where4Vg is a certain local linear operator, which we identifywith the renormalized BV Laplacian.



Quantization


Quantum master equation

In the framework of [K. Fredenhagen, K.R., CMP 2013], the gaugeinvariance of the S-matrix is guaranteed by the so called quantummaster equation (QME):

{eVgT , S0g} = 0 ,

where {., .} is the Schouten bracket.

With the use of Master Ward Identity [F.Brennecke, M.Duetsch, RMP

2008], this condition can be rewritten as

12{S0g + Vg, S0g + Vg} = i~4Vg ,

where4Vg is a certain local linear operator, which we identifywith the renormalized BV Laplacian.



Quantization


Quantum observables

If the QME holds, then gauge invariant quantum observables arerecovered as the 0th cohomology of the quantum BV operator s,which acts on quantum fields by

(sΦ)M(f ) = e−VgT ·T {eVg

T ·T ΦM(f ), S0g}+ ΦM(LCf ) ,

where C is the ghost field.

Again, using the MWI, this can be rewritten as

sΦM(f ) = {ΦM(f ), S0g + Vg}+ ΦM(LCf )− i~4Vg (ΦM(f )) .



Quantization


Quantum observables

If the QME holds, then gauge invariant quantum observables arerecovered as the 0th cohomology of the quantum BV operator s,which acts on quantum fields by

(sΦ)M(f ) = e−VgT ·T {eVg

T ·T ΦM(f ), S0g}+ ΦM(LCf ) ,

where C is the ghost field.

Again, using the MWI, this can be rewritten as

sΦM(f ) = {ΦM(f ), S0g + Vg}+ ΦM(LCf )− i~4Vg (ΦM(f )) .



Quantization


Relative Cauchy evolution

Let N+ and N− be two spacetimesthat embed into two otherspacetimes M1 and M2 aroundCauchy surfaces, via causalembeddings given by χk,±, k = 1, 2.

Denote αχi±.= Aχk±, k = 1, 2.

From the time-slice axiom followsthat β = αχ1+α

−1χ2+

αχ2−α−1χ1− is an

automorphism of A(M1).

It depends only on the spacetimebetween the two Cauchy surfaces

M1 M2

N+

N−

χ1+ χ2+

χ1− χ2−



Quantization






−1χ2+




M1 M2

N+

N−

χ1+ χ2+

χ1− χ2−



Quantization






−1χ2+




M1 M2

N+

N−

χ1+ χ2+

χ1− χ2−



Quantization






−1χ2+




M1 M2

N+

N−

χ1+ χ2+

χ1− χ2−



Quantization


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−) = ∅,

Θµν(x).=

δβh

δhµν(x)

∣∣∣h=0

is a derivation

valued distribution which is covariantlyconserved.

The infinitesimal version of thebackground independence is acondition: Θµν = 0.

(M, g1) (M, g2)

N+

N−

supp(h)

χ1+ χ2+

χ1− χ2−



Quantization




Θµν(x).=

δβh

δhµν(x)

∣∣∣h=0

is a derivation



(M, g1) (M, g2)

N+

N−

supp(h)

χ1+ χ2+

χ1− χ2−



Quantization




Θµν(x).=

δβh

δhµν(x)

∣∣∣h=0

is a derivation



(M, g1) (M, g2)

N+

N−

supp(h)

χ1+ χ2+

χ1− χ2−



Quantization



Theorem [Brunetti, Fredenhagen, K.R. 2013]The functional derivative Θµν of the relative Cauchy evolution can beexpressed as

Θµν(ΦM1(f ))o.s.= [RV1(ΦM1(f )),RV1(Tµν)]? ,

where Tµν is the stress-energy tensor of the extended action and onecan define the time-ordered products in such a way that Tµν = 0holds, so the interacting theory is background independent.


Appendix

Conclusions

We have constructed a consistent model of perturbative quantumgravity within the framework of locally covariant quantum fieldstheory.

In our framework, physical diffeomorphism invariant quantitiesare constructed as natural transformations between certainfunctors. We have proposed a quantization prescription for suchobjects, which makes use of the BV formalism.

To quantize the theory, we make a tentative split into a free andinteracting theory. We quantize the free theory first and then usethe Epstein-Glaser renormalization to introduce the interaction.

We have shown, using the relative Cauchy evolution, that ourtheory is background independent, i.e. independent of the splitinto free and interacting part.


Appendix

Conclusions






Appendix

Conclusions






Appendix

Conclusions






Appendix

Thank you for your attention!


Documents

Effective quantum gravity as a locally covariant QFTtheory.fi.infn.it/seminara/Geometry_of_Strings_and_Fieds/Conference... · Effective quantum gravity Local covariance Difﬁculties