quantization of geometrical structures in locally ...rejzner.com/talks/milano12.pdf · covariant...

IntroductionBV formalism for gravity

Quantization

Quantization of geometrical structures in locallycovariant field theory

Katarzyna Rejzner1

University of Rome “Tor Vergata”INdAM

Milano, 17.12.2012

1based on the joint work with Romeo Brunetti and Klaus FredenhagenKatarzyna Rejzner QG in LCFT 1 / 24


Quantization

Outline of the talk

1 IntroductionEffective quantum gravityLocal covariance

2 BV formalism for gravityKinematical structureDynamics and symmetriesBV complex

3 QuantizationDeformation quantizationApplications


Quantization

Effective quantum gravityLocal covariance

Problems with quantum gravity

Spacetime is dynamical.

"Points" loose their meaning.

It is not clear what should be anobservable.

Need for "backgroundindependance".

What replaces the classicalspacetime structure in Planck scale?

Katarzyna Rejzner QG in LCFT 2 / 24


Quantization


Objectives of our program

Apply the methods of locally covariantquantum field theory to understand some ofthe features of quantum gravity.

Formulate perturbative quantum gravity asan effective theory that is valid in a givenphysical situation.

Answer some interpretational questions.

Find a relation to experiment: QGcorrections to some processes, black holeradiation, cosmology.

Understand the small scale structure ofspacetime: relation to noncommutativegeometry.



Quantization


Intuitive idea

In experiment geometric structure is probed by thelocal observations. We have the following data:

Compact causally convex region O of spacetimewhere the measurement is performed,An observable Φ, which we measure,We don’t measure the scalar curvature at a point,but we have some smearing related to the

experimantal setting: Φ(f ) =

∫f (x)R(x),

supp(f ) ⊂ O.

We can think of the measured observable as aperturbation of the fixed background metric: atentative split into: g̃µν = gµν + hµν .

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

M



Quantization


Intuitive idea


Compact causally convex region O of spacetimewhere the measurement is performed,

An observable Φ, which we measure,We don’t measure the scalar curvature at a point,but we have some smearing related to the


∫f (x)R(x),

supp(f ) ⊂ O.



O

M



Quantization


Intuitive idea


Compact causally convex region O of spacetimewhere the measurement is performed,An observable Φ, which we measure,

We don’t measure the scalar curvature at a point,but we have some smearing related to the


∫f (x)R(x),

supp(f ) ⊂ O.



O

M

Φ



Quantization


Intuitive idea




∫f (x)R(x),

supp(f ) ⊂ O.



M

Φ(f )

f

O



Quantization


Intuitive idea




∫f (x)R(x),

supp(f ) ⊂ O.



(M, g)

Φ(O,g)(f )[h]

f

O



Quantization


Intuitive idea




∫f (x)R(x),

supp(f ) ⊂ O.



O′(M, g)

Φ(O,g)(f )[h]

f

O



Quantization


How to implement it?

To compare Φ(O,g)(f ) and Φ(O′,α∗g)(α∗f ) weneed to know what does it mean to have "thesame observable in a different region".

A good language to formalize it is the categorytheory. We need following categories:

Loc where the objects are all four-dimensional,globally hyperbolic oriented and time-orientedspacetimesM = (M, g). Morphisms: isometricembeddings preserving orientation,time-orientation and the causal structure.Vec with (small) topological vector spaces asobjects and injective continuoushomomorphisms of topological vector spaces asmorphisms.

(M, g)

Φ(O,g)(f ) Φ(O′,α∗g)(α∗f )

f

α∗f

O

O′

α



Quantization



To compare Φ(O,g)(f ) and Φ(O′,α∗g)(α∗f ) weneed to know what does it mean to have "thesame observable in a different region".A good language to formalize it is the categorytheory. We need following categories:


(M, g)

Φ(O,g)(f ) Φ(O′,α∗g)(α∗f )

f

α∗f

O

O′

α



Quantization




Loc where the objects are all four-dimensional,globally hyperbolic oriented and time-orientedspacetimesM = (M, g). Morphisms: isometricembeddings preserving orientation,time-orientation and the causal structure.

Vec with (small) topological vector spaces asobjects and injective continuoushomomorphisms of topological vector spaces asmorphisms.

(M, g)

Φ(O,g)(f ) Φ(O′,α∗g)(α∗f )

f

α∗f

O

O′

α



Quantization





(M, g)

Φ(O,g)(f ) Φ(O′,α∗g)(α∗f )

f

α∗f

O

O′

α



Quantization

Kinematical structureDynamics and symmetriesBV complex

Kinematical structure

Having the quantization in mind we formulate already theclassical theory in the perturbative setting.

We work off-shell, so for the effective theory of gravity theconfiguration space is E(M) = Γ((T∗M)2⊗). The space ofcompactly supported configurations is denoted by Ec(M).

We define a contravariant functor E : Loc→ Vec, which assignsto a spacetime the corresponding configuration space and acts onmorphisms χ :M→N as Eχ = χ∗ : E(N )→ E(M).

In a similar way we define a covariant functor Ec : Loc→ Vecby setting Eχ = χ∗, where:

χ∗h.=

{(χ−1)∗h(x) , x ∈ χ(M),0 , else



Quantization


Functionals

We consider the space of smooth functionals on E(M), i.e.C∞(E(M),R).

The support of F ∈ C∞(E(M),R) is defined as:

supp F = {x ∈ M|∀ neighbourhoods U of x ∃h1, h2 ∈ E(M),

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

F is local if it is of the form: F(h) =

∫M

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

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

F(M).= the space of multilocal functionals (products of local).



Quantization


Vector fields

To discuss symmetries we need one more definition.

Vector fields X ∈ Γ(TE(M)) on E(M) (trivial infinitedimensional manifold) can be seen as maps from E(M) toE(M).

We restrict ourselves to smooth maps X with image in Ec(M).They act on F(M) as derivations: ∂XF(h) := 〈F(1)(h),X(h)〉We consider only the multilocal (products of local vector fieldsand local functionals) vector fields with compact spacetimesupport.

V(M).= the space of vector fields with above properties.

V becomes a (covariant) functor after setting:Vχ(X) = Ecχ ◦ X ◦ Eχ .



Quantization


Vector fields



We restrict ourselves to smooth maps X with image in Ec(M).They act on F(M) as derivations: ∂XF(h) := 〈F(1)(h),X(h)〉

We consider only the multilocal (products of local vector fieldsand local functionals) vector fields with compact spacetimesupport.





Quantization


Vector fields



We restrict ourselves to smooth maps X with image in Ec(M).They act on F(M) as derivations: ∂XF(h) := 〈F(1)(h),X(h)〉We consider only the multilocal (products of local vector fieldsand local functionals) vector fields with compact spacetimesupport.





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 ∈ Nat(D,F), where D is the functor oftest function spaces D(M) = C∞c (M) (onecould 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


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 ∈ Nat(D,F), where D is the functor oftest function spaces D(M) = C∞c (M) (onecould 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

ΦM(χ∗f )

ΦO(f )

χ(O)

O

χ∗f

f

χχ−1



Quantization


Dynamics and symmetries

The dynamics is introduced by a generalized Lagrangian Lwhich is a natural transformation between functors D and Floc.The action S(L) is an equivalence class of Lagrangians, whereL1 ∼ L2 if supp(L1,M − L2,M)(f ) ⊂ suppdf ∀f ∈ D(M).

For GR: L(M,g)(f )[h].=

∫R[g̃]f d vol(M,g̃), g̃ = g + h.

The E-L derivative of S(L) is a natural transformationS′ : E→ E′c defined as

⟨S′M(h0), h

⟩=⟨

LM(f )(1)(h0), h⟩

,

where f ≡ 1 on supph. The field equation is: S′M(h0) = 0. Thespace of solutions is denoted by ES(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 ∈ V(M)such that ∀h0 ∈ E(M):0 =

⟨S′M(h0),X(h0)

⟩.



Quantization





∫R[g̃]f d vol(M,g̃), g̃ = g + h.


⟨S′M(h0), h

⟩=⟨

LM(f )(1)(h0), h⟩

,

where f ≡ 1 on supph. The field equation is: S′M(h0) = 0. Thespace of solutions is denoted by ES(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 ∈ V(M)such that ∀h0 ∈ E(M):0 =

⟨S′M(h0),X(h0)

⟩.

Msupp(f )

supp(h)f ≡ 1



Quantization





∫R[g̃]f d vol(M,g̃), g̃ = g + h.


⟨S′M(h0), h

⟩=⟨

LM(f )(1)(h0), h⟩

,

where f ≡ 1 on supph. The field equation is: S′M(h0) = 0. Thespace of solutions is denoted by ES(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 ∈ V(M)such that ∀h0 ∈ E(M):0 =

⟨S′M(h0),X(h0)

⟩.

Msupp(f )

supp(X)f ≡ 1



Quantization


Diffeomorphism invariance

Since our discussion is local we can concentrate on infinitesimaldiffeomorphisms, i.e. vector fields in X(M)

.= Γ(TM). This

assignment can also be made functorial.

We can now define a Lie algebra X , which provides us with anotion of transforming all the spacetimes in a coherent way.:

X .=

∏M∈Obj(Loc)

X(M)



Quantization



Let ~ξ ∈ X with all the components compactly supported andαM = exp(ξM) a family of diffeomorphisms constructed via theexponential mapping. The action of diffeomorphisms on naturaltransformations is given by:(~αΦ)(M,g)(f )[h] = Φ(M,g)(α

-1M ∗ f )[α∗Mg̃− g].

The derived action reads:(~ξΦ)(M,g)(f )[h] =⟨

(Φ(M,g)(f ))(1)(h),−LξM g̃⟩

+ Φ(M,g)(−LξM f )[h]

The right hand side is well defined also if we drop the compactsupport condition on ~ξ, so we can adapt the above formula as thedefinition of the action of X on Nat(D,F).

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

Example:∫

R[g̃]f d vol(M,g̃) is invariant, but∫

R[g̃]f d vol(M,g)

is not.



Quantization




-1M ∗ f )[α∗Mg̃− g].


(Φ(M,g)(f ))(1)(h),−LξM g̃⟩


The right hand side is well defined also if we drop the compactsupport condition on ~ξ, so we can adapt the above formula as thedefinition of the action of X on Nat(D,F).Diffeomorphism invariance is now the statement that: ~ξΦ = 0.

Example:∫


R[g̃]f d vol(M,g)

is not.



Quantization




-1M ∗ f )[α∗Mg̃− g].


(Φ(M,g)(f ))(1)(h),−LξM g̃⟩


The right hand side is well defined also if we drop the compactsupport condition on ~ξ, so we can adapt the above formula as thedefinition of the action of X on Nat(D,F).Diffeomorphism invariance is now the statement that: ~ξΦ = 0.

Example:∫


R[g̃]f d vol(M,g)

is not.Katarzyna Rejzner QG in LCFT 12 / 24


Quantization


BV complex

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].

Objective: characterize the space of gauge invariant functionalsFinv

S (M) on the space of solutions of EOM’s: ES(M).

Idea: note that ES(M) locally can be seen critical manifold ofthe Lagrangian LM(f ) : E(M)→ R (zero locus of S′M).

We identify ES(M) with its algebra of functions FS(M) andcharacterize it by its Koszul resolution (see [Costello 2011] for afinite dimensional version).

FS(M) = H0

(∧V(M), S′M(.)

).

The underlying algebra of this differential complex is a certaincompletion of the odd cotangent bundle ΠT∗E(M) of E(M).



Quantization


BV complex

To incorporate the gauge invariance we replace the originalconfiguration space E(M) with a graded manifoldE(M)

.= E(M)⊕ X(M) characterized by it’s algebra of

functions F(M)⊗̂∧

X′(M) = C∞ml

(E(M),

∧V(M)

).

The underlying algebra of the Koszu resolution is the oddcotangent bundle ΠT∗E of E and taking into account regularityconditions and topological completion we obtain the BVcomplex:

BV(M) = C∞ml

(E(M),

∧Ec(M) ⊗̂

∧X′(M) ⊗̂ S•Xc(M)

)

Antifields: derivations of F(M), #af = 1, #gh = −1Ghosts: functionals on the symmetry algebra #af = 0, #gh = 1Antifields of ghosts: derivations of ghosts #af = 2, #gh = −2



Quantization


BV complex




X′(M) = C∞ml

(E(M),

∧V(M)

).


BV(M) = C∞ml

(E(M),

∧Ec(M) ⊗̂

∧X′(M) ⊗̂ S•Xc(M)

)Antifields: derivations of F(M), #af = 1, #gh = −1

Ghosts: functionals on the symmetry algebra #af = 0, #gh = 1Antifields of ghosts: derivations of ghosts #af = 2, #gh = −2



Quantization


BV complex




X′(M) = C∞ml

(E(M),

∧V(M)

).


BV(M) = C∞ml

(E(M),

∧Ec(M) ⊗̂

∧X′(M) ⊗̂ S•Xc(M)

)Antifields: derivations of F(M), #af = 1, #gh = −1Ghosts: functionals on the symmetry algebra #af = 0, #gh = 1

Antifields of ghosts: derivations of ghosts #af = 2, #gh = −2



Quantization


BV complex




X′(M) = C∞ml

(E(M),

∧V(M)

).


BV(M) = C∞ml

(E(M),

∧Ec(M) ⊗̂

∧X′(M) ⊗̂ S•Xc(M)

)Antifields: derivations of F(M), #af = 1, #gh = −1Ghosts: functionals on the symmetry algebra #af = 0, #gh = 1Antifields of ghosts: derivations of ghosts #af = 2, #gh = −2



Quantization


BV complex extended to natural transformations

The assignment of the BV complex to the manifold is a functorfrom Loc to the category of graded algebras.

Extending the algebra of functionals to the BV complex impliesthat the generalized fields should be also extended:

Fld =∞⊕

k=0

Nat(Ekc,BV) ,

where Ekc be a functor from the category Loc to the product

category Veck, that assigns to a spacetimeM a k-fold product ofthe test section spaces Ec(M)× . . .× Ec(M).The set Fld becomes a graded algebra if we set:

(ΦΨ)M(f1, ..., fp+q) =

=1

p!q!

∑π∈Pp+q

ΦM(fπ(1), ..., fπ(p))ΨM(fπ(p+1), ..., fπ(p+q)) .



Quantization



The assignment of the BV complex to the manifold is a functorfrom Loc to the category of graded algebras.Extending the algebra of functionals to the BV complex impliesthat the generalized fields should be also extended:

Fld =

∞⊕k=0

Nat(Ekc,BV) ,


category Veck, that assigns to a spacetimeM a k-fold product ofthe test section spaces Ec(M)× . . .× Ec(M).

The set Fld becomes a graded algebra if we set:

(ΦΨ)M(f1, ..., fp+q) =

=1

p!q!

∑π∈Pp+q




Quantization



The assignment of the BV complex to the manifold is a functorfrom Loc to the category of graded algebras.Extending the algebra of functionals to the BV complex impliesthat the generalized fields should be also extended:

Fld =

∞⊕k=0

Nat(Ekc,BV) ,


category Veck, that assigns to a spacetimeM a k-fold product ofthe test section spaces Ec(M)× . . .× Ec(M).The set Fld becomes a graded algebra if we set:

(ΦΨ)M(f1, ..., fp+q) =

=1

p!q!

∑π∈Pp+q




Quantization



Since BV(M) is the algebra of vector fields on the gradedmanifold E(M), we can equip it with the Schouten bracket:

{X,F} = ∂XF for X a vector field and F function,{X,Y} = [X,Y] for X,Y a vector fields,graded Leibniz rule.

The BV-differential on Fld is given by:(sΦ)M(f ) = {ΦM(f ), S + γ}+ ΦM(LC f ),where C ∈ X(M) is the ghost and γ is the Chevalley-Eilenbergdifferential, which acts on Fld via infinitesimal diffeomorphismtransformations along the ghost fields C. For Φ ∈ Nat(Ec,F):

(γΦ)M(f )(h) :=⟨

(ΦM(f ))(1)(h),LCg̃⟩

,The gauge invariant observables are given by:

Fldinv := H0(Fld, s)

For example: Φ(M,g)(f )(h) =

∫M

Rµναβ [g̃]Rµναβ [g̃]fdvol(M,g̃).



Quantization




{X,F} = ∂XF for X a vector field and F function,

{X,Y} = [X,Y] for X,Y a vector fields,graded Leibniz rule.


(γΦ)M(f )(h) :=⟨

(ΦM(f ))(1)(h),LCg̃⟩




∫M




Quantization




{X,F} = ∂XF for X a vector field and F function,{X,Y} = [X,Y] for X,Y a vector fields,

graded Leibniz rule.The BV-differential on Fld is given by:(sΦ)M(f ) = {ΦM(f ), S + γ}+ ΦM(LC f ),where C ∈ X(M) is the ghost and γ is the Chevalley-Eilenbergdifferential, which acts on Fld via infinitesimal diffeomorphismtransformations along the ghost fields C. For Φ ∈ Nat(Ec,F):

(γΦ)M(f )(h) :=⟨

(ΦM(f ))(1)(h),LCg̃⟩




∫M




Quantization






(γΦ)M(f )(h) :=⟨

(ΦM(f ))(1)(h),LCg̃⟩




∫M




Quantization






(γΦ)M(f )(h) :=⟨

(ΦM(f ))(1)(h),LCg̃⟩

,

The gauge invariant observables are given by:Fldinv := H0(Fld, s)


∫M




Quantization






(γΦ)M(f )(h) :=⟨

(ΦM(f ))(1)(h),LCg̃⟩




∫M




Quantization


Gauge fixing

Gauge fixing is implemented by means of the so called gaugefixing fermion Ψ ∈ Fld with ghost number #gh = 1.

We define an automorphism of BV(M) by

αΨ(X) :=∞∑

n=0

1n!{ΨM(f ), . . . , {ΨM(f )︸︷︷︸

n

,X} . . . } ,

where f ≡ 1 on the support of X. This automorphism in a simpleway extends to Fld.

We obtain a new extended action S̃ .= αΨ(S + γ) and

gauge-fixed BV differential sΨ = αΨ ◦ s ◦ α−1Ψ

It holds: H0(sΨ, αΨ(Fld)) = H0(s,Fld) = Fldinv.



Quantization


Gauge fixing

Gauge fixing is implemented by means of the so called gaugefixing fermion Ψ ∈ Fld with ghost number #gh = 1.

We define an automorphism of BV(M) by

αΨ(X) :=

∞∑n=0

1n!{ΨM(f ), . . . , {ΨM(f )︸︷︷︸

n

,X} . . . } ,

where f ≡ 1 on the support of X. This automorphism in a simpleway extends to Fld.

We obtain a new extended action S̃ .= αΨ(S + γ) and

gauge-fixed BV differential sΨ = αΨ ◦ s ◦ α−1Ψ

It holds: H0(sΨ, αΨ(Fld)) = H0(s,Fld) = Fldinv.



Quantization


Equations of motion and Poisson bracket

As an output of classical field theory we have a graded manifoldE and an extended action S̃. Now we apply to this data thedeformation quantization.

We can Taylor expand the gauge fixed action around an arbitrarybackground metric g and obtain S̃ = S0

g + Vg, where S0 isquadratic in fields and has #af = 0.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 Fld.



Quantization



As an output of classical field theory we have a graded manifoldE and an extended action S̃. Now we apply to this data thedeformation quantization.We can Taylor expand the gauge fixed action around an arbitrarybackground metric g and obtain S̃ = S0

g + Vg, where S0 isquadratic in fields and has #af = 0.

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






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

Deformation quantizationApplications

Deformation quantization

We start with the deformation quantization of (Fld, {., .}0).

We need to include into the space of functionals on E(M) somemore singular objects. The right notion of regularity is related toa certain wavefront set property of Hadamard 2-point functions(microlocal spectrum condition). The resulting space will bedenoted by BVµc(M).The deformation quantization of (BVµc(M), {., .}g

0) can beperformed in the standard way, by introducing a ?-product:

(F ?H G).= m ◦ exp(i~ΓωH )(F ⊗ G) ,

where ΓωH

.=

12

∫dx dyωH(x, y)

δ

δϕ(x)⊗ δ

δϕ(y)and

ωH =i2

∆g + H is the Hadamard 2-point function (satisfies thelinearized EOM’s in both arguments and the µSC).



Quantization



We start with the deformation quantization of (Fld, {., .}0).We need to include into the space of functionals on E(M) somemore singular objects. The right notion of regularity is related toa certain wavefront set property of Hadamard 2-point functions(microlocal spectrum condition). The resulting space will bedenoted by BVµc(M).

The deformation quantization of (BVµc(M), {., .}g0) can be

performed in the standard way, by introducing a ?-product:

(F ?H G).= m ◦ exp(i~ΓωH )(F ⊗ G) ,

where ΓωH

.=

12

∫dx dyωH(x, y)

δ

δϕ(x)⊗ δ

δϕ(y)and

ωH =i2




Quantization



We start with the deformation quantization of (Fld, {., .}0).We need to include into the space of functionals on E(M) somemore singular objects. The right notion of regularity is related toa certain wavefront set property of Hadamard 2-point functions(microlocal spectrum condition). The resulting space will bedenoted by BVµc(M).The deformation quantization of (BVµc(M), {., .}g

0) can beperformed in the standard way, by introducing a ?-product:

(F ?H G).= m ◦ exp(i~ΓωH )(F ⊗ G) ,

where ΓωH

.=

12

∫dx dyωH(x, y)

δ

δϕ(x)⊗ δ

δϕ(y)and

ωH =i2




Quantization


Interaction

With some technical considerations the deformation quantizationon eachM leads to a deformation quantization on the space offields and we obtain {Fldµc[[~, λ]], ?}.

In the next step we have to introduce the interaction, i.e. considerthe algebras {BVµc(M)[[~, λ]], ?H} and define on them therenormalized time-ordered products ·T by the Epstein-Glasermethod.Time ordered products on differentM can be defined in acovariant way, which allows to extend it to a product onFldµc[[~, λ]].One can define the formal S-matrix as: S(Vg)

.= eVg

T .Interacting fields are obtained from free ones in {Fldµc[[~, λ]], ?}by the Bogoliubov formula:

(RV(Φ))M(f ).=

ddλ

∣∣∣λ=0S(Vg)?−1 ? S(Vg + λΦM(f )) .



Quantization


Interaction

With some technical considerations the deformation quantizationon eachM leads to a deformation quantization on the space offields and we obtain {Fldµc[[~, λ]], ?}.In the next step we have to introduce the interaction, i.e. considerthe algebras {BVµc(M)[[~, λ]], ?H} and define on them therenormalized time-ordered products ·T by the Epstein-Glasermethod.

Time ordered products on differentM can be defined in acovariant way, which allows to extend it to a product onFldµc[[~, λ]].One can define the formal S-matrix as: S(Vg)

.= eVg


(RV(Φ))M(f ).=

ddλ

∣∣∣λ=0S(Vg)?−1 ? S(Vg + λΦM(f )) .



Quantization


Interaction

With some technical considerations the deformation quantizationon eachM leads to a deformation quantization on the space offields and we obtain {Fldµc[[~, λ]], ?}.In the next step we have to introduce the interaction, i.e. considerthe algebras {BVµc(M)[[~, λ]], ?H} and define on them therenormalized time-ordered products ·T by the Epstein-Glasermethod.Time ordered products on differentM can be defined in acovariant way, which allows to extend it to a product onFldµc[[~, λ]].

One can define the formal S-matrix as: S(Vg).= eVg


(RV(Φ))M(f ).=

ddλ

∣∣∣λ=0S(Vg)?−1 ? S(Vg + λΦM(f )) .



Quantization


Interaction

With some technical considerations the deformation quantizationon eachM leads to a deformation quantization on the space offields and we obtain {Fldµc[[~, λ]], ?}.In the next step we have to introduce the interaction, i.e. considerthe algebras {BVµc(M)[[~, λ]], ?H} and define on them therenormalized time-ordered products ·T by the Epstein-Glasermethod.Time ordered products on differentM can be defined in acovariant way, which allows to extend it to a product onFldµc[[~, λ]].One can define the formal S-matrix as: S(Vg)

.= eVg

T .

Interacting fields are obtained from free ones in {Fldµc[[~, λ]], ?}by the Bogoliubov formula:

(RV(Φ))M(f ).=

ddλ

∣∣∣λ=0S(Vg)?−1 ? S(Vg + λΦM(f )) .



Quantization


Interaction

With some technical considerations the deformation quantizationon eachM leads to a deformation quantization on the space offields and we obtain {Fldµc[[~, λ]], ?}.In the next step we have to introduce the interaction, i.e. considerthe algebras {BVµc(M)[[~, λ]], ?H} and define on them therenormalized time-ordered products ·T by the Epstein-Glasermethod.Time ordered products on differentM can be defined in acovariant way, which allows to extend it to a product onFldµc[[~, λ]].One can define the formal S-matrix as: S(Vg)

.= eVg


(RV(Φ))M(f ).=

ddλ

∣∣∣λ=0S(Vg)?−1 ? S(Vg + λΦM(f )) .



Quantization


Quantized geometrical structures

In general relativity the basic physical objects are fields (naturaltransformations), since they are defined not on a fixedbackground but rather on a class of spacetimes in a coherent way.

The BV construction can be applied the algebra of fields Fld andgives a homological interpretation to the notion of gaugeinvariant physical quantities in general relativity.

The algebra Fldµc can be equipped with the noncommutative?-product, which provides the deformation quantization of thefree theory. The interaction is next introduced in the perturbativeway and we obtain a notion of interacting quantum fields RV(Φ),where Φ is a classical field constructed covariantly from the

metric. For example:∫

R[g̃]f d vol(M,g̃).



Quantization


Physical interpretation

Although we parametrize Fldµc with spacetimes, theconstruction is fully covariant and physical quantities areinvariant under reparametrization.

The background independence would mean that the algebraicstructure on Fldµc doesn’t depend on the split into the free andinteracting part. This can be obtained as a certainrenormalization condition called perturbative agreement([Hollands, Wald, 2004] for the scalar field). Work in progress.

The physical interpretation of the theory is provided byconstructing states on {Fldµc[[~, λ]], ?}. This problem is notentirely solved, since one needs to prove the existence of “gaugeinvariant” states on arbitraryM for the linearized theory. Up tonow states can be explicitly given only on some special classesof spaetimes (for example ultrastatic).



Quantization


Spacetime in Planck scale

Even without solving the question of states in full generalitythere are some effects that can be calculated in our framework.For example one can investigate the localization of events.

The principle of gravitational stability against localization ofevents was proposed in [Doplicher, Fredenhagen, Roberts, CMP 95]. Itstates that a physical reason for Planck scale noncommutativityof spacetime is the fact that we cannot measure all the spacetimecoordinates with arbitrary precision, because this would result informing a trapped surface. This principle was used in [DFR 95] toderive the STUR in a simplified model.In a recent work of [Doplicher, Morsella, Pinamonti 2012] this problemis studied for a model of scalar field coupled to gravity in thesemiclassical approximation. We hope that using our notions ofquantized metric and curvature one can move this one stepfurther and include quantum gravity corrections.



Quantization



Even without solving the question of states in full generalitythere are some effects that can be calculated in our framework.For example one can investigate the localization of events.The principle of gravitational stability against localization ofevents was proposed in [Doplicher, Fredenhagen, Roberts, CMP 95]. Itstates that a physical reason for Planck scale noncommutativityof spacetime is the fact that we cannot measure all the spacetimecoordinates with arbitrary precision, because this would result informing a trapped surface. This principle was used in [DFR 95] toderive the STUR in a simplified model.

In a recent work of [Doplicher, Morsella, Pinamonti 2012] this problemis studied for a model of scalar field coupled to gravity in thesemiclassical approximation. We hope that using our notions ofquantized metric and curvature one can move this one stepfurther and include quantum gravity corrections.



Quantization



Even without solving the question of states in full generalitythere are some effects that can be calculated in our framework.For example one can investigate the localization of events.The principle of gravitational stability against localization ofevents was proposed in [Doplicher, Fredenhagen, Roberts, CMP 95]. Itstates that a physical reason for Planck scale noncommutativityof spacetime is the fact that we cannot measure all the spacetimecoordinates with arbitrary precision, because this would result informing a trapped surface. This principle was used in [DFR 95] toderive the STUR in a simplified model.In a recent work of [Doplicher, Morsella, Pinamonti 2012] this problemis studied for a model of scalar field coupled to gravity in thesemiclassical approximation. We hope that using our notions ofquantized metric and curvature one can move this one stepfurther and include quantum gravity corrections.


Appendix

Thank you for your attention


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