Algebraic approach to QFT Quantum gravity

Locality and beyond: from algebraic quantum field theory to effective quantum gravity

Kasia Rejzner

University of York

Local Quantum Physics and beyond in memoriam Rudolf Haag, 26.09.2016

Kasia Rejzner Locality and beyond 1 / 26 Algebraic approach to QFT Quantum gravity Outline of the talk

1 Algebraic approach to QFT AQFT LCQFT pAQFT

2 Quantum gravity Effective quantum gravity Observables The author of a beautiful book Local Quantum Physics.

One of the fathers of LQP.

We will all miss him. . .

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT The father of Local Quantum Physics

Rudolf Haag (1922 – 2016).

Kasia Rejzner Locality and beyond 2 / 26 One of the fathers of LQP.

We will all miss him. . .

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT The father of Local Quantum Physics

Rudolf Haag (1922 – 2016).

The author of a beautiful book Local Quantum Physics.

Kasia Rejzner Locality and beyond 2 / 26 We will all miss him. . .

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT The father of Local Quantum Physics

Rudolf Haag (1922 – 2016).

The author of a beautiful book Local Quantum Physics.

One of the fathers of LQP.

Kasia Rejzner Locality and beyond 2 / 26 AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT The father of Local Quantum Physics

Rudolf Haag (1922 – 2016).

The author of a beautiful book Local Quantum Physics.

One of the fathers of LQP.

We will all miss him. . .

Kasia Rejzner Locality and beyond 2 / 26 It started as the axiomatic framework of Haag-Kastler[ Haag & Kastler 64]: a model is defined by associating to each region O of Minkowski spacetime the algebra A(O) of observables that can be measured in O. The physical notion of subsystems is realized by the condition of isotony, i.e.: O2 ⊃ O1 ⇒ A(O2) ⊃ A(O1). We obtain a net of algebras.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Local Quantum Physics

Algebraic Quantum Field Theory (Local Quantum Physics) is a mathematically rigorous framework, which allows to investigate conceptual problems in QFT.

Kasia Rejzner Locality and beyond 3 / 26 The physical notion of subsystems is realized by the condition of isotony, i.e.: O2 ⊃ O1 ⇒ A(O2) ⊃ A(O1). We obtain a net of algebras.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Local Quantum Physics

Algebraic Quantum Field Theory (Local Quantum Physics) is a mathematically rigorous framework, which allows to investigate conceptual problems in QFT. It started as the axiomatic framework of Haag-Kastler[ Haag & Kastler 64]: a model is defined by associating to each region O of Minkowski spacetime the algebra A(O) of observables that can be measured in O.

A(O1)

O1

Kasia Rejzner Locality and beyond 3 / 26 AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Local Quantum Physics

Algebraic Quantum Field Theory (Local Quantum Physics) is a mathematically rigorous framework, which allows to investigate conceptual problems in QFT. It started as the axiomatic framework of Haag-Kastler[ Haag & Kastler 64]: a model is defined by associating to each region O of Minkowski spacetime the algebra A(O) of observables that can be measured in O. The physical notion of subsystems A(O2) ⊃ A(O1) is realized by the condition of isotony, i.e.: O2 ⊃ O1 ⇒ A(O2) ⊃ A(O1). We obtain a net of algebras.

O2 ⊃ O1

Kasia Rejzner Locality and beyond 3 / 26 Different aspects of AQFT and relations to physics

Renor- malization QFT, PARTICLE COSMOLOGY PHYSICS Quantum gravity KMS states

Defor- mations PAQFT

MODEL QUANTUM LCQFT STATISTICAL BUILDING QUANTUM MECHANICS CFT in 2D INFORMATION THEORY

QFT ON NONCO- MUTATIVE SPACETIMES AQFT NONCOM- ALGEBRAIC MUTATIVE QIT GEOMETRY CONSTRUC- Foundations Entan- TIVE QFT of quantum glement theory Local algebras of observables A(O) are defined abstractly, the Hilbert space representation comes later (this deals with the non-uniqueness of the vacuum). Algebras A(O) are constructed using only the local data. Local features of the theory (observables) are separated from the global features (states).

Main advantages

The corresponding generalization of AQFT is called locally covariant quantum field theory[ Hollands-Wald 01, Brunetti-Fredenhagen-Verch 01, Fewster-Verch 12,. . . ].

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Algebraic QFT on curved spacetimes

Conceptual problems of QFT on curved spacetimes can be easily solved in the algebraic approach, because of the powerful principle of locality.

Kasia Rejzner Locality and beyond 5 / 26 Local algebras of observables A(O) are defined abstractly, the Hilbert space representation comes later (this deals with the non-uniqueness of the vacuum). Algebras A(O) are constructed using only the local data. Local features of the theory (observables) are separated from the global features (states).

Main advantages

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Algebraic QFT on curved spacetimes

Conceptual problems of QFT on curved spacetimes can be easily solved in the algebraic approach, because of the powerful principle of locality. The corresponding generalization of AQFT is called locally covariant quantum field theory[ Hollands-Wald 01, Brunetti-Fredenhagen-Verch 01, Fewster-Verch 12,. . . ].

Kasia Rejzner Locality and beyond 5 / 26 Local algebras of observables A(O) are defined abstractly, the Hilbert space representation comes later (this deals with the non-uniqueness of the vacuum). Algebras A(O) are constructed using only the local data. Local features of the theory (observables) are separated from the global features (states).

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Algebraic QFT on curved spacetimes

Conceptual problems of QFT on curved spacetimes can be easily solved in the algebraic approach, because of the powerful principle of locality. The corresponding generalization of AQFT is called locally covariant quantum field theory[ Hollands-Wald 01, Brunetti-Fredenhagen-Verch 01, Fewster-Verch 12,. . . ]. Main advantages

Kasia Rejzner Locality and beyond 5 / 26 Algebras A(O) are constructed using only the local data. Local features of the theory (observables) are separated from the global features (states).

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Algebraic QFT on curved spacetimes

Conceptual problems of QFT on curved spacetimes can be easily solved in the algebraic approach, because of the powerful principle of locality. The corresponding generalization of AQFT is called locally covariant quantum field theory[ Hollands-Wald 01, Brunetti-Fredenhagen-Verch 01, Fewster-Verch 12,. . . ]. Main advantages Local algebras of observables A(O) are defined abstractly, the Hilbert space representation comes later (this deals with the non-uniqueness of the vacuum).

Kasia Rejzner Locality and beyond 5 / 26 Local features of the theory (observables) are separated from the global features (states).

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Algebraic QFT on curved spacetimes

Conceptual problems of QFT on curved spacetimes can be easily solved in the algebraic approach, because of the powerful principle of locality. The corresponding generalization of AQFT is called locally covariant quantum field theory[ Hollands-Wald 01, Brunetti-Fredenhagen-Verch 01, Fewster-Verch 12,. . . ]. Main advantages Local algebras of observables A(O) are defined abstractly, the Hilbert space representation comes later (this deals with the non-uniqueness of the vacuum). Algebras A(O) are constructed using only the local data.

Kasia Rejzner Locality and beyond 5 / 26 AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Algebraic QFT on curved spacetimes

Conceptual problems of QFT on curved spacetimes can be easily solved in the algebraic approach, because of the powerful principle of locality. The corresponding generalization of AQFT is called locally covariant quantum field theory[ Hollands-Wald 01, Brunetti-Fredenhagen-Verch 01, Fewster-Verch 12,. . . ]. Main advantages Local algebras of observables A(O) are defined abstractly, the Hilbert space representation comes later (this deals with the non-uniqueness of the vacuum). Algebras A(O) are constructed using only the local data. Local features of the theory (observables) are separated from the global features (states).

Kasia Rejzner Locality and beyond 5 / 26 Replace O1 and O2 with arbitrary spacetimes M = (M, g), N = (N, g0) and require the embedding ψ : M → N to be an isometry. Require that ψ preserves orientations and the causal structure (no new causal links are created by the embedding). Assign to each spacetime M an algebra A(M) and to each admissible embedding ψ a homomorphism of algebras Aψ (notion of subsystems). This has to be done covariantly.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Locally covariant quantum field theory (LCQFT)

In the original AQFT axioms we associate algebras to regions of a fixed spacetimes. Now we go a step further.

Kasia Rejzner Locality and beyond 6 / 26 Require that ψ preserves orientations and the causal structure (no new causal links are created by the embedding). Assign to each spacetime M an algebra A(M) and to each admissible embedding ψ a homomorphism of algebras Aψ (notion of subsystems). This has to be done covariantly.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Locally covariant quantum field theory (LCQFT)

In the original AQFT axioms we associate algebras to regions of a fixed spacetimes. Now we go a step further.

Replace O1 and O2 with arbitrary spacetimes M = (M, g), N = (N, g0) and require the embedding ψ : M → N to be an isometry.

ψ

N M

Kasia Rejzner Locality and beyond 6 / 26 Assign to each spacetime M an algebra A(M) and to each admissible embedding ψ a homomorphism of algebras Aψ (notion of subsystems). This has to be done covariantly.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Locally covariant quantum field theory (LCQFT)

In the original AQFT axioms we associate algebras to regions of a fixed spacetimes. Now we go a step further.

Replace O1 and O2 with arbitrary spacetimes M = (M, g), N = (N, g0) and require the embedding ψ : M → N to be an isometry. Require that ψ preserves orientations and the causal structure (no new causal links are ψ created by the embedding). N M

Kasia Rejzner Locality and beyond 6 / 26 AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Locally covariant quantum field theory (LCQFT)

In the original AQFT axioms we associate algebras to regions of a fixed spacetimes. Now we go a step further.

Replace O1 and O2 with arbitrary spacetimes Aψ M = (M, g), N = (N, g0) and require the embedding ψ : M → N to be an isometry. A(N) A(M) Require that ψ preserves orientations and the causal structure (no new causal links are A ψ A created by the embedding). Assign to each spacetime M an algebra N M A(M) and to each admissible embedding ψ a homomorphism of algebras Aψ (notion of subsystems). This has to be done covariantly.

Kasia Rejzner Locality and beyond 6 / 26 Let D(O) denote the space of test functions supported in O.A locally cov. field is a family of maps ΦM : D(M) → A(M), labeled by spacetimes M such that: Aψ(ΦO(f )) = ΦM(ψ∗f ). This generalizes the notion of Wightman’s operator-valued distributions.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Locally covariant fields

In the framework of LCQFT, locally covariant fields are used to identify (put labels on) observables localized in different M ψ(O) region of spacetime, in the absence of symmetries.

ψ

O

Kasia Rejzner Locality and beyond 7 / 26 This generalizes the notion of Wightman’s operator-valued distributions.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Locally covariant fields

In the framework of LCQFT, locally ΦM(ψ∗f ) covariant fields are used to identify (put ψ∗f labels on) observables localized in different M ψ(O) region of spacetime, in the absence of symmetries. Let D(O) denote the space of test functions −1 supported in O.A locally cov. field is a ψ ψ family of maps Φ : D(M) → A(M), M Φ (f ) labeled by spacetimes M such that: O f Aψ(ΦO(f )) = ΦM(ψ∗f ).

O

Kasia Rejzner Locality and beyond 7 / 26 AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Locally covariant fields

In the framework of LCQFT, locally ΦM(ψ∗f ) covariant fields are used to identify (put ψ∗f labels on) observables localized in different M ψ(O) region of spacetime, in the absence of symmetries. Let D(O) denote the space of test functions −1 supported in O.A locally cov. field is a ψ ψ family of maps Φ : D(M) → A(M), M Φ (f ) labeled by spacetimes M such that: O f Aψ(ΦO(f )) = ΦM(ψ∗f ). This generalizes the notion of Wightman’s operator-valued distributions. O

Kasia Rejzner Locality and beyond 7 / 26 Free theory obtained by the formal deformation quantization of Poisson (Peierls) bracket: ?-product ([Dütsch-Fredenhagen 00, Brunetti-Fredenhagen 00, Brunetti-Dütsch-Fredenhagen 09, . . . ]). Interaction introduced in the causal approach to renormalization due to Epstein and Glaser( [Epstein-Glaser 73]), Generalization to gauge theories using homological algebra ([Hollands 08, Fredenhagen-KR 11]).

It combines Haag’s idea of local quantum physics with methods of perturbation theory. Main contributions:

For a review see the book: Perturbative algebraic quantum field theory. An introduction for mathematicians, KR, Springer 2016.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Perturbative algebraic quantum field theory

Perturbative algebraic quantum field theory (pAQFT) is a mathematically rigorous framework that allows to build interacting LCQFT models.

Kasia Rejzner Locality and beyond 8 / 26 Free theory obtained by the formal deformation quantization of Poisson (Peierls) bracket: ?-product ([Dütsch-Fredenhagen 00, Brunetti-Fredenhagen 00, Brunetti-Dütsch-Fredenhagen 09, . . . ]). Interaction introduced in the causal approach to renormalization due to Epstein and Glaser( [Epstein-Glaser 73]), Generalization to gauge theories using homological algebra ([Hollands 08, Fredenhagen-KR 11]).

Main contributions:

For a review see the book: Perturbative algebraic quantum field theory. An introduction for mathematicians, KR, Springer 2016.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Perturbative algebraic quantum field theory

Perturbative algebraic quantum field theory (pAQFT) is a mathematically rigorous framework that allows to build interacting LCQFT models. It combines Haag’s idea of local quantum physics with methods of perturbation theory.

Kasia Rejzner Locality and beyond 8 / 26 Free theory obtained by the formal deformation quantization of Poisson (Peierls) bracket: ?-product ([Dütsch-Fredenhagen 00, Brunetti-Fredenhagen 00, Brunetti-Dütsch-Fredenhagen 09, . . . ]). Interaction introduced in the causal approach to renormalization due to Epstein and Glaser( [Epstein-Glaser 73]), Generalization to gauge theories using homological algebra ([Hollands 08, Fredenhagen-KR 11]). For a review see the book: Perturbative algebraic quantum field theory. An introduction for mathematicians, KR, Springer 2016.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Perturbative algebraic quantum field theory

Perturbative algebraic quantum field theory (pAQFT) is a mathematically rigorous framework that allows to build interacting LCQFT models. It combines Haag’s idea of local quantum physics with methods of perturbation theory. Main contributions:

Kasia Rejzner Locality and beyond 8 / 26 Interaction introduced in the causal approach to renormalization due to Epstein and Glaser( [Epstein-Glaser 73]), Generalization to gauge theories using homological algebra ([Hollands 08, Fredenhagen-KR 11]). For a review see the book: Perturbative algebraic quantum field theory. An introduction for mathematicians, KR, Springer 2016.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Perturbative algebraic quantum field theory

Perturbative algebraic quantum field theory (pAQFT) is a mathematically rigorous framework that allows to build interacting LCQFT models. It combines Haag’s idea of local quantum physics with methods of perturbation theory. Main contributions: Free theory obtained by the formal deformation quantization of Poisson (Peierls) bracket: ?-product ([Dütsch-Fredenhagen 00, Brunetti-Fredenhagen 00, Brunetti-Dütsch-Fredenhagen 09, . . . ]).

Kasia Rejzner Locality and beyond 8 / 26 Generalization to gauge theories using homological algebra ([Hollands 08, Fredenhagen-KR 11]). For a review see the book: Perturbative algebraic quantum field theory. An introduction for mathematicians, KR, Springer 2016.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Perturbative algebraic quantum field theory

Perturbative algebraic quantum field theory (pAQFT) is a mathematically rigorous framework that allows to build interacting LCQFT models. It combines Haag’s idea of local quantum physics with methods of perturbation theory. Main contributions: Free theory obtained by the formal deformation quantization of Poisson (Peierls) bracket: ?-product ([Dütsch-Fredenhagen 00, Brunetti-Fredenhagen 00, Brunetti-Dütsch-Fredenhagen 09, . . . ]). Interaction introduced in the causal approach to renormalization due to Epstein and Glaser( [Epstein-Glaser 73]),

Kasia Rejzner Locality and beyond 8 / 26 For a review see the book: Perturbative algebraic quantum field theory. An introduction for mathematicians, KR, Springer 2016.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Perturbative algebraic quantum field theory

Perturbative algebraic quantum field theory (pAQFT) is a mathematically rigorous framework that allows to build interacting LCQFT models. It combines Haag’s idea of local quantum physics with methods of perturbation theory. Main contributions: Free theory obtained by the formal deformation quantization of Poisson (Peierls) bracket: ?-product ([Dütsch-Fredenhagen 00, Brunetti-Fredenhagen 00, Brunetti-Dütsch-Fredenhagen 09, . . . ]). Interaction introduced in the causal approach to renormalization due to Epstein and Glaser( [Epstein-Glaser 73]), Generalization to gauge theories using homological algebra ([Hollands 08, Fredenhagen-KR 11]).

Kasia Rejzner Locality and beyond 8 / 26 AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Perturbative algebraic quantum field theory

Perturbative algebraic quantum field theory (pAQFT) is a mathematically rigorous framework that allows to build interacting LCQFT models. It combines Haag’s idea of local quantum physics with methods of perturbation theory. Main contributions: Free theory obtained by the formal deformation quantization of Poisson (Peierls) bracket: ?-product ([Dütsch-Fredenhagen 00, Brunetti-Fredenhagen 00, Brunetti-Dütsch-Fredenhagen 09, . . . ]). Interaction introduced in the causal approach to renormalization due to Epstein and Glaser( [Epstein-Glaser 73]), Generalization to gauge theories using homological algebra ([Hollands 08, Fredenhagen-KR 11]). For a review see the book: Perturbative algebraic quantum field theory. An introduction for mathematicians, KR, Springer 2016.

Kasia Rejzner Locality and beyond 8 / 26 Configuration space E(M): choice of objects we want to study in our theory (scalars, vectors, tensors,. . . ). Typically this is a space π of smooth sections of some vector bundle E −→ M over M. For ∞ the scalar field: E(M) = C (M, R). Dynamics: we use a modification of the Lagrangian formalism. Since the manifold M is non-compact, we need to introduce a cutoff function into the Lagrangian. For the free scalar field 1 Z L (f )(ϕ) = (∇ ϕ∇µϕ − m2ϕ2)(x)f (x)dµ (x). M 2 µ g Abstractly, a Lagrangian is a locally covariant classical field (another manifestation of the locality principle).

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Physical input

Spacetime M = (M, g): a smooth manifold M with a smooth Lorentzian metric g. Assume M to be oriented, time-oriented and globally hyperbolic (has a Cauchy surface).

Kasia Rejzner Locality and beyond 9 / 26 Dynamics: we use a modification of the Lagrangian formalism. Since the manifold M is non-compact, we need to introduce a cutoff function into the Lagrangian. For the free scalar field 1 Z L (f )(ϕ) = (∇ ϕ∇µϕ − m2ϕ2)(x)f (x)dµ (x). M 2 µ g Abstractly, a Lagrangian is a locally covariant classical field (another manifestation of the locality principle).

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Physical input

Spacetime M = (M, g): a smooth manifold M with a smooth Lorentzian metric g. Assume M to be oriented, time-oriented and globally hyperbolic (has a Cauchy surface). Configuration space E(M): choice of objects we want to study in our theory (scalars, vectors, tensors,. . . ). Typically this is a space π of smooth sections of some vector bundle E −→ M over M. For ∞ the scalar field: E(M) = C (M, R).

Kasia Rejzner Locality and beyond 9 / 26 Abstractly, a Lagrangian is a locally covariant classical field (another manifestation of the locality principle).

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Physical input

Spacetime M = (M, g): a smooth manifold M with a smooth Lorentzian metric g. Assume M to be oriented, time-oriented and globally hyperbolic (has a Cauchy surface). Configuration space E(M): choice of objects we want to study in our theory (scalars, vectors, tensors,. . . ). Typically this is a space π of smooth sections of some vector bundle E −→ M over M. For ∞ the scalar field: E(M) = C (M, R). Dynamics: we use a modification of the Lagrangian formalism. Since the manifold M is non-compact, we need to introduce a cutoff function into the Lagrangian. For the free scalar field 1 Z L (f )(ϕ) = (∇ ϕ∇µϕ − m2ϕ2)(x)f (x)dµ (x). M 2 µ g

Kasia Rejzner Locality and beyond 9 / 26 AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Physical input

Spacetime M = (M, g): a smooth manifold M with a smooth Lorentzian metric g. Assume M to be oriented, time-oriented and globally hyperbolic (has a Cauchy surface). Configuration space E(M): choice of objects we want to study in our theory (scalars, vectors, tensors,. . . ). Typically this is a space π of smooth sections of some vector bundle E −→ M over M. For ∞ the scalar field: E(M) = C (M, R). Dynamics: we use a modification of the Lagrangian formalism. Since the manifold M is non-compact, we need to introduce a cutoff function into the Lagrangian. For the free scalar field 1 Z L (f )(ϕ) = (∇ ϕ∇µϕ − m2ϕ2)(x)f (x)dµ (x). M 2 µ g Abstractly, a Lagrangian is a locally covariant classical field (another manifestation of the locality principle).

Kasia Rejzner Locality and beyond 9 / 26 ∞ The support of F ∈ C (E(M), R) is defined as: supp F = {x ∈ M|∀ neighbourhoods U of x ∃ϕ, ψ ∈ E(M), supp ψ ⊂ U such that F(ϕ + ψ) 6= F(ϕ)} . Z k F is local if it is of the form: F(ϕ) = f (jx(ϕ)) dµ(x) , where f M is a function on the jet bundle over M and k jx(ϕ) = (ϕ(x), ∂ϕ(x), ...) (up to order k) is the k-th jet of ϕ at the point x. Let Floc(M) denote the space of local functionals. Consider functionals that are multilocal, i.e. they are sums of products of local functionals. Denote them by F(M); they play the role of polynomials.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Observables

Classical observables are modeled as smooth functionals on the ∞ configuration space E(M), i.e. elements of C (E(M), R).

Kasia Rejzner Locality and beyond 10 / 26 Z k F is local if it is of the form: F(ϕ) = f (jx(ϕ)) dµ(x) , where f M is a function on the jet bundle over M and k jx(ϕ) = (ϕ(x), ∂ϕ(x), ...) (up to order k) is the k-th jet of ϕ at the point x. Let Floc(M) denote the space of local functionals. Consider functionals that are multilocal, i.e. they are sums of products of local functionals. Denote them by F(M); they play the role of polynomials.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Observables

Classical observables are modeled as smooth functionals on the ∞ configuration space E(M), i.e. elements of C (E(M), R). ∞ The support of F ∈ C (E(M), R) is defined as: supp F = {x ∈ M|∀ neighbourhoods U of x ∃ϕ, ψ ∈ E(M), supp ψ ⊂ U such that F(ϕ + ψ) 6= F(ϕ)} .

Kasia Rejzner Locality and beyond 10 / 26 Consider functionals that are multilocal, i.e. they are sums of products of local functionals. Denote them by F(M); they play the role of polynomials.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Observables

Classical observables are modeled as smooth functionals on the ∞ configuration space E(M), i.e. elements of C (E(M), R). ∞ The support of F ∈ C (E(M), R) is defined as: supp F = {x ∈ M|∀ neighbourhoods U of x ∃ϕ, ψ ∈ E(M), supp ψ ⊂ U such that F(ϕ + ψ) 6= F(ϕ)} . Z k F is local if it is of the form: F(ϕ) = f (jx(ϕ)) dµ(x) , where f M is a function on the jet bundle over M and k jx(ϕ) = (ϕ(x), ∂ϕ(x), ...) (up to order k) is the k-th jet of ϕ at the point x. Let Floc(M) denote the space of local functionals.

Kasia Rejzner Locality and beyond 10 / 26 AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Observables

Classical observables are modeled as smooth functionals on the ∞ configuration space E(M), i.e. elements of C (E(M), R). ∞ The support of F ∈ C (E(M), R) is defined as: supp F = {x ∈ M|∀ neighbourhoods U of x ∃ϕ, ψ ∈ E(M), supp ψ ⊂ U such that F(ϕ + ψ) 6= F(ϕ)} . Z k F is local if it is of the form: F(ϕ) = f (jx(ϕ)) dµ(x) , where f M is a function on the jet bundle over M and k jx(ϕ) = (ϕ(x), ∂ϕ(x), ...) (up to order k) is the k-th jet of ϕ at the point x. Let Floc(M) denote the space of local functionals. Consider functionals that are multilocal, i.e. they are sums of products of local functionals. Denote them by F(M); they play the role of polynomials.

Kasia Rejzner Locality and beyond 10 / 26 The Euler-Lagrange derivative of an ation SM is a map 0 0 SM : E(M) → D (M) defined as (thanks to locality): 0 D (1) E SM(ϕ), h = LM(f ) (ϕ), h , where f ≡ 1 on supph.

supp(f ) M supp(h) f ≡ 1

0 The equation of motion (EOM) is the equation SM(ϕ) ≡ 0 for an unknown function ϕ ∈ E(M) and it determines a subspace of E(M) denoted by ES(M) (on-shell configurations).

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Equations of motion I

Actions S are equivalence classes of Lagrangians L1 ∼ L2 if supp(L1 − L2)(f ) ⊂ supp df .

Kasia Rejzner Locality and beyond 11 / 26 0 The equation of motion (EOM) is the equation SM(ϕ) ≡ 0 for an unknown function ϕ ∈ E(M) and it determines a subspace of E(M) denoted by ES(M) (on-shell configurations).

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Equations of motion I

Actions S are equivalence classes of Lagrangians L1 ∼ L2 if supp(L1 − L2)(f ) ⊂ supp df .

The Euler-Lagrange derivative of an ation SM is a map 0 0 SM : E(M) → D (M) defined as (thanks to locality): 0 D (1) E SM(ϕ), h = LM(f ) (ϕ), h , where f ≡ 1 on supph.

supp(f ) M supp(h) f ≡ 1

Kasia Rejzner Locality and beyond 11 / 26 AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Equations of motion I

Actions S are equivalence classes of Lagrangians L1 ∼ L2 if supp(L1 − L2)(f ) ⊂ supp df .

The Euler-Lagrange derivative of an ation SM is a map 0 0 SM : E(M) → D (M) defined as (thanks to locality): 0 D (1) E SM(ϕ), h = LM(f ) (ϕ), h , where f ≡ 1 on supph. F E(M) ϕ R

ES(M)

0 The equation of motion (EOM) is the equation SM(ϕ) ≡ 0 for an unknown function ϕ ∈ E(M) and it determines a subspace of E(M) denoted by ES(M) (on-shell configurations).

Kasia Rejzner Locality and beyond 11 / 26 Let V(M) denote the space of multilocal vector fields on E(M) . and let δX = ∂XSM. The kernel of δ consists of vector fields X ∈ V(M), which 0 satisfy SM(ϕ), X(ϕ) = 0 for all ϕ ∈ E(M), i.e. ∂XSM ≡ 0. These characterize directions in the configuration space E(M) in which the action S is constant, we call them local symmetries.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Equations of motion II

In the algebraic spirit, characterize ES(M) by its space of functions FS(M), given by the quotient FS(M) = F(M)/F0(M), where F0(M) is generated by the elements of the form: 0 ϕ 7→ SM(ϕ), X(ϕ) = ∂XSM(ϕ), with X a vector field on E(M), i.e. X ∈ Γ(TE(M)).

Kasia Rejzner Locality and beyond 12 / 26 The kernel of δ consists of vector fields X ∈ V(M), which 0 satisfy SM(ϕ), X(ϕ) = 0 for all ϕ ∈ E(M), i.e. ∂XSM ≡ 0. These characterize directions in the configuration space E(M) in which the action S is constant, we call them local symmetries.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Equations of motion II

In the algebraic spirit, characterize ES(M) by its space of functions FS(M), given by the quotient FS(M) = F(M)/F0(M), where F0(M) is generated by the elements of the form: 0 ϕ 7→ SM(ϕ), X(ϕ) = ∂XSM(ϕ), with X a vector field on E(M), i.e. X ∈ Γ(TE(M)). Let V(M) denote the space of multilocal vector fields on E(M) . and let δX = ∂XSM.

Kasia Rejzner Locality and beyond 12 / 26 These characterize directions in the configuration space E(M) in which the action S is constant, we call them local symmetries.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Equations of motion II

In the algebraic spirit, characterize ES(M) by its space of functions FS(M), given by the quotient FS(M) = F(M)/F0(M), where F0(M) is generated by the elements of the form: 0 ϕ 7→ SM(ϕ), X(ϕ) = ∂XSM(ϕ), with X a vector field on E(M), i.e. X ∈ Γ(TE(M)). Let V(M) denote the space of multilocal vector fields on E(M) . and let δX = ∂XSM. The kernel of δ consists of vector fields X ∈ V(M), which 0 satisfy SM(ϕ), X(ϕ) = 0 for all ϕ ∈ E(M), i.e. ∂XSM ≡ 0.

Kasia Rejzner Locality and beyond 12 / 26 AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Equations of motion II

In the algebraic spirit, characterize ES(M) by its space of functions FS(M), given by the quotient FS(M) = F(M)/F0(M), where F0(M) is generated by the elements of the form: 0 ϕ 7→ SM(ϕ), X(ϕ) = ∂XSM(ϕ), with X a vector field on E(M), i.e. X ∈ Γ(TE(M)). Let V(M) denote the space of multilocal vector fields on E(M) . and let δX = ∂XSM. The kernel of δ consists of vector fields X ∈ V(M), which 0 satisfy SM(ϕ), X(ϕ) = 0 for all ϕ ∈ E(M), i.e. ∂XSM ≡ 0. These characterize directions in the configuration space E(M) in which the action S is constant, we call them local symmetries.

Kasia Rejzner Locality and beyond 12 / 26 Note that FS(M) = H0(ΛV(M), δ) and H1 characterizes non-trivial local symmetries.

Working with ΛV(M) instead of FS(M) allows us to quantize the theory off-shell. In the presence of non-trivial symmetries, one has to further extend the configuration space and replace δ with s, the classical BV operator.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Symmetries and the kernel of δ

The space of symmetries includes elements of the form δ(Λ2V(M)), where Λ2V(M) is the second exterior power of V(M). Such symmetries are called trivial, because they vanish on ES(M). Consider a complex δ δ ... → Λ2V(M) −→ V(M) −→ F(M) → 0,

Kasia Rejzner Locality and beyond 13 / 26 Working with ΛV(M) instead of FS(M) allows us to quantize the theory off-shell. In the presence of non-trivial symmetries, one has to further extend the configuration space and replace δ with s, the classical BV operator.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Symmetries and the kernel of δ

The space of symmetries includes elements of the form δ(Λ2V(M)), where Λ2V(M) is the second exterior power of V(M). Such symmetries are called trivial, because they vanish on ES(M). Consider a complex δ δ ... → Λ2V(M) −→ V(M) −→ F(M) → 0,

Note that FS(M) = H0(ΛV(M), δ) and H1 characterizes non-trivial local symmetries.

Kasia Rejzner Locality and beyond 13 / 26 In the presence of non-trivial symmetries, one has to further extend the configuration space and replace δ with s, the classical BV operator.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Symmetries and the kernel of δ

The space of symmetries includes elements of the form δ(Λ2V(M)), where Λ2V(M) is the second exterior power of V(M). Such symmetries are called trivial, because they vanish on ES(M). Consider a complex δ δ ... → Λ2V(M) −→ V(M) −→ F(M) → 0,

Note that FS(M) = H0(ΛV(M), δ) and H1 characterizes non-trivial local symmetries.

Working with ΛV(M) instead of FS(M) allows us to quantize the theory off-shell.

Kasia Rejzner Locality and beyond 13 / 26 AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Symmetries and the kernel of δ

The space of symmetries includes elements of the form δ(Λ2V(M)), where Λ2V(M) is the second exterior power of V(M). Such symmetries are called trivial, because they vanish on ES(M). Consider a complex δ δ ... → Λ2V(M) −→ V(M) −→ F(M) → 0,

Note that FS(M) = H0(ΛV(M), δ) and H1 characterizes non-trivial local symmetries.

Working with ΛV(M) instead of FS(M) allows us to quantize the theory off-shell. In the presence of non-trivial symmetries, one has to further extend the configuration space and replace δ with s, the classical BV operator.

Kasia Rejzner Locality and beyond 13 / 26 If M is globally hyperbolic, then P posses the retarded and advanced Green’s functions ∆R, ∆A. They satisfy:

R R/A supp ∆ (f ) P ◦ ∆ = idD(M), R/A ∆ ◦ (P D(M)) = idD(M), supp f

supp ∆A(f ) . Their difference is the causal propagator ∆ = ∆R − ∆A.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Free scalar field

For the free scalar field the equation of motion is of the form 0 2 SM(ϕ) = Pϕ = 0, where P = −(2 + m ) is the Klein-Gordon operator.

Kasia Rejzner Locality and beyond 14 / 26 . Their difference is the causal propagator ∆ = ∆R − ∆A.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Free scalar field

For the free scalar field the equation of motion is of the form 0 2 SM(ϕ) = Pϕ = 0, where P = −(2 + m ) is the Klein-Gordon operator. If M is globally hyperbolic, then P posses the retarded and advanced Green’s functions ∆R, ∆A. They satisfy:

R R/A supp ∆ (f ) P ◦ ∆ = idD(M), R/A ∆ ◦ (P D(M)) = idD(M), supp f

supp ∆A(f )

Kasia Rejzner Locality and beyond 14 / 26 AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Free scalar field

For the free scalar field the equation of motion is of the form 0 2 SM(ϕ) = Pϕ = 0, where P = −(2 + m ) is the Klein-Gordon operator. If M is globally hyperbolic, then P posses the retarded and advanced Green’s functions ∆R, ∆A. They satisfy:

R R/A supp ∆ (f ) P ◦ ∆ = idD(M), R/A ∆ ◦ (P D(M)) = idD(M), supp f

supp ∆A(f ) . Their difference is the causal propagator ∆ = ∆R − ∆A.

Kasia Rejzner Locality and beyond 14 / 26 Next: decompose ∆ into negative nad positive energy parts (in curved spacetimes use the notion of wavefront sets), i.e. i 2 ∆ = ∆+ − H. Define the ?-product (deformation of the pointwise product) by:

∞ . X n D E (F ? G)(ϕ) = ~ F(n)(ϕ), (∆ )⊗nG(n)(ϕ) , n! + n=0

where F, G belong to Fµc(M), a larger class of functionals, which contains the multilocal ones.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Classical theory and deformation

The classical Poisson bracket is given by . D E bF, Gc = F(1), ∆G(1) .

Kasia Rejzner Locality and beyond 15 / 26 Define the ?-product (deformation of the pointwise product) by:

∞ . X n D E (F ? G)(ϕ) = ~ F(n)(ϕ), (∆ )⊗nG(n)(ϕ) , n! + n=0

where F, G belong to Fµc(M), a larger class of functionals, which contains the multilocal ones.

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Classical theory and deformation

The classical Poisson bracket is given by . D E bF, Gc = F(1), ∆G(1) . Next: decompose ∆ into negative nad positive energy parts (in curved spacetimes use the notion of wavefront sets), i.e. i 2 ∆ = ∆+ − H.

Kasia Rejzner Locality and beyond 15 / 26 AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Classical theory and deformation

The classical Poisson bracket is given by . D E bF, Gc = F(1), ∆G(1) . Next: decompose ∆ into negative nad positive energy parts (in curved spacetimes use the notion of wavefront sets), i.e. i 2 ∆ = ∆+ − H. Define the ?-product (deformation of the pointwise product) by:

∞ . X n D E (F ? G)(ϕ) = ~ F(n)(ϕ), (∆ )⊗nG(n)(ϕ) , n! + n=0

where F, G belong to Fµc(M), a larger class of functionals, which contains the multilocal ones.

Kasia Rejzner Locality and beyond 15 / 26 The time-ordering operator T is defined as: ∞ . X n D E TF(ϕ) = ~ F(2n)(ϕ), (∆F)⊗n , n! n=0 F i R A where ∆ = 2 (∆ + ∆ ) + H. Formally it would correspond to the operator of convolution with F the oscillating Gaussian measure “with covariance i~∆ ”, formal Z TF(ϕ) = F(ϕ − φ) dµi~∆F (φ) .

We define the time-ordered product ·T on Freg(M)[[~]] by: . −1 −1 F ·T G = T(T F · T G)

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Time-ordered product

Let Freg(M) be the space of functionals whose derivatives are (n) ∞ n test functions, i.e. F (ϕ) ∈ Cc (M , R),

Kasia Rejzner Locality and beyond 16 / 26 Formally it would correspond to the operator of convolution with F the oscillating Gaussian measure “with covariance i~∆ ”, formal Z TF(ϕ) = F(ϕ − φ) dµi~∆F (φ) .

We define the time-ordered product ·T on Freg(M)[[~]] by: . −1 −1 F ·T G = T(T F · T G)

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Time-ordered product

Let Freg(M) be the space of functionals whose derivatives are (n) ∞ n test functions, i.e. F (ϕ) ∈ Cc (M , R), The time-ordering operator T is defined as: ∞ . X n D E TF(ϕ) = ~ F(2n)(ϕ), (∆F)⊗n , n! n=0 F i R A where ∆ = 2 (∆ + ∆ ) + H.

Kasia Rejzner Locality and beyond 16 / 26 We define the time-ordered product ·T on Freg(M)[[~]] by: . −1 −1 F ·T G = T(T F · T G)

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Time-ordered product

Let Freg(M) be the space of functionals whose derivatives are (n) ∞ n test functions, i.e. F (ϕ) ∈ Cc (M , R), The time-ordering operator T is defined as: ∞ . X n D E TF(ϕ) = ~ F(2n)(ϕ), (∆F)⊗n , n! n=0 F i R A where ∆ = 2 (∆ + ∆ ) + H. Formally it would correspond to the operator of convolution with F the oscillating Gaussian measure “with covariance i~∆ ”, formal Z TF(ϕ) = F(ϕ − φ) dµi~∆F (φ) .

Kasia Rejzner Locality and beyond 16 / 26 AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Time-ordered product

Let Freg(M) be the space of functionals whose derivatives are (n) ∞ n test functions, i.e. F (ϕ) ∈ Cc (M , R), The time-ordering operator T is defined as: ∞ . X n D E TF(ϕ) = ~ F(2n)(ϕ), (∆F)⊗n , n! n=0 F i R A where ∆ = 2 (∆ + ∆ ) + H. Formally it would correspond to the operator of convolution with F the oscillating Gaussian measure “with covariance i~∆ ”, formal Z TF(ϕ) = F(ϕ − φ) dµi~∆F (φ) .

We define the time-ordered product ·T on Freg(M)[[~]] by: . −1 −1 F ·T G = T(T F · T G)

Kasia Rejzner Locality and beyond 16 / 26 Interaction is a functional V (for a moment we assume that it belongs to Freg(M)). Using the commutative product ·T we define the formal S-matrix:  −1  . iV/~ T (iV/~) S(V) = eT = T e . Interacting fields are defined by the formula of Bogoliubov:  ?−1   . iV/~ iV~ RV (F) = eT ? eT ·T F . Renormalization problem: extend these structures to local non-linear functionals (these are not regular. . . ).

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Interaction

We now have two products on Freg(M)[[~]]: non-commutative ? and commutative ·T . They are related by a relation:

F ·T G = F ? G , if the support of F is later than the support of G.

Kasia Rejzner Locality and beyond 17 / 26 Interacting fields are defined by the formula of Bogoliubov:  ?−1   . iV/~ iV~ RV (F) = eT ? eT ·T F . Renormalization problem: extend these structures to local non-linear functionals (these are not regular. . . ).

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Interaction

We now have two products on Freg(M)[[~]]: non-commutative ? and commutative ·T . They are related by a relation:

F ·T G = F ? G , if the support of F is later than the support of G. Interaction is a functional V (for a moment we assume that it belongs to Freg(M)). Using the commutative product ·T we define the formal S-matrix:  −1  . iV/~ T (iV/~) S(V) = eT = T e .

Kasia Rejzner Locality and beyond 17 / 26 Renormalization problem: extend these structures to local non-linear functionals (these are not regular. . . ).

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Interaction

We now have two products on Freg(M)[[~]]: non-commutative ? and commutative ·T . They are related by a relation:

F ·T G = F ? G , if the support of F is later than the support of G. Interaction is a functional V (for a moment we assume that it belongs to Freg(M)). Using the commutative product ·T we define the formal S-matrix:  −1  . iV/~ T (iV/~) S(V) = eT = T e . Interacting fields are defined by the formula of Bogoliubov:  ?−1   . iV/~ iV~ RV (F) = eT ? eT ·T F .

Kasia Rejzner Locality and beyond 17 / 26 AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Interaction

We now have two products on Freg(M)[[~]]: non-commutative ? and commutative ·T . They are related by a relation:

F ·T G = F ? G , if the support of F is later than the support of G. Interaction is a functional V (for a moment we assume that it belongs to Freg(M)). Using the commutative product ·T we define the formal S-matrix:  −1  . iV/~ T (iV/~) S(V) = eT = T e . Interacting fields are defined by the formula of Bogoliubov:  ?−1   . iV/~ iV~ RV (F) = eT ? eT ·T F . Renormalization problem: extend these structures to local non-linear functionals (these are not regular. . . ).

Kasia Rejzner Locality and beyond 17 / 26 Investigation of topological aspects of LCQFT ([Becker-Schenkel-Szabo 14]) is a first step in generalizing the framework towards “LCQFT up to homotopy”. Recent results on the Sine Gordon model ([Bahns-KR 16, to appear soon...]) go beyond perturbation theory in showing convergence of the formal S-matrix in the finite regime of the theory. Now to quantum gravity. . .

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Perspectives in pAQFT

Recent results and perspectives: Construction of thermal states and introduction of the mollified Hamiltonian formalism ([Fredenhagen-Lindner 13]) opens up a perspective for applications to problems like Bose-Einstein condensation, Lamb shift, etc.

Kasia Rejzner Locality and beyond 18 / 26 Recent results on the Sine Gordon model ([Bahns-KR 16, to appear soon...]) go beyond perturbation theory in showing convergence of the formal S-matrix in the finite regime of the theory. Now to quantum gravity. . .

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Perspectives in pAQFT

Recent results and perspectives: Construction of thermal states and introduction of the mollified Hamiltonian formalism ([Fredenhagen-Lindner 13]) opens up a perspective for applications to problems like Bose-Einstein condensation, Lamb shift, etc. Investigation of topological aspects of LCQFT ([Becker-Schenkel-Szabo 14]) is a first step in generalizing the framework towards “LCQFT up to homotopy”.

Kasia Rejzner Locality and beyond 18 / 26 Now to quantum gravity. . .

AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Perspectives in pAQFT

Recent results and perspectives: Construction of thermal states and introduction of the mollified Hamiltonian formalism ([Fredenhagen-Lindner 13]) opens up a perspective for applications to problems like Bose-Einstein condensation, Lamb shift, etc. Investigation of topological aspects of LCQFT ([Becker-Schenkel-Szabo 14]) is a first step in generalizing the framework towards “LCQFT up to homotopy”. Recent results on the Sine Gordon model ([Bahns-KR 16, to appear soon...]) go beyond perturbation theory in showing convergence of the formal S-matrix in the finite regime of the theory.

Kasia Rejzner Locality and beyond 18 / 26 AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Perspectives in pAQFT

Recent results and perspectives: Construction of thermal states and introduction of the mollified Hamiltonian formalism ([Fredenhagen-Lindner 13]) opens up a perspective for applications to problems like Bose-Einstein condensation, Lamb shift, etc. Investigation of topological aspects of LCQFT ([Becker-Schenkel-Szabo 14]) is a first step in generalizing the framework towards “LCQFT up to homotopy”. Recent results on the Sine Gordon model ([Bahns-KR 16, to appear soon...]) go beyond perturbation theory in showing convergence of the formal S-matrix in the finite regime of the theory. Now to quantum gravity. . .

Kasia Rejzner Locality and beyond 18 / 26 Non-renormalizability: use Epstein-Glaser renormalization to obtain finite results for any fixed energy scale. Think of the theory as an effective theory. Dynamical nature of spacetime: make a split of the metric into background and perturbation, quantize the perturbation as a quantum field on a curved background, show background independence at the end. Diffeomorphism invariance: use the BV formalism to perform the quantization.

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Ways around some problems in QG

Based on: R. Brunetti, K. Fredenhagen, KR, Quantum gravity from the point of view of locally covariant quantum field theory, [arXiv:1306.1058], CMP 2016.

Kasia Rejzner Locality and beyond 19 / 26 Dynamical nature of spacetime: make a split of the metric into background and perturbation, quantize the perturbation as a quantum field on a curved background, show background independence at the end. Diffeomorphism invariance: use the BV formalism to perform the quantization.

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Ways around some problems in QG

Based on: R. Brunetti, K. Fredenhagen, KR, Quantum gravity from the point of view of locally covariant quantum field theory, [arXiv:1306.1058], CMP 2016. Non-renormalizability: use Epstein-Glaser renormalization to obtain finite results for any fixed energy scale. Think of the theory as an effective theory.

Kasia Rejzner Locality and beyond 19 / 26 Diffeomorphism invariance: use the BV formalism to perform the quantization.

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Ways around some problems in QG

Based on: R. Brunetti, K. Fredenhagen, KR, Quantum gravity from the point of view of locally covariant quantum field theory, [arXiv:1306.1058], CMP 2016. Non-renormalizability: use Epstein-Glaser renormalization to obtain finite results for any fixed energy scale. Think of the theory as an effective theory. Dynamical nature of spacetime: make a split of the metric into background and perturbation, quantize the perturbation as a quantum field on a curved background, show background independence at the end.

Kasia Rejzner Locality and beyond 19 / 26 Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Ways around some problems in QG

Based on: R. Brunetti, K. Fredenhagen, KR, Quantum gravity from the point of view of locally covariant quantum field theory, [arXiv:1306.1058], CMP 2016. Non-renormalizability: use Epstein-Glaser renormalization to obtain finite results for any fixed energy scale. Think of the theory as an effective theory. Dynamical nature of spacetime: make a split of the metric into background and perturbation, quantize the perturbation as a quantum field on a curved background, show background independence at the end. Diffeomorphism invariance: use the BV formalism to perform the quantization.

Kasia Rejzner Locality and beyond 19 / 26 M Some region O of spacetime where the measurement is performed, An observable Φ, which we measure, We don’t measure the observable (e.g. curvature) at a point. This is modeled by smearing with a test function f . For example: Z Φ(f ) = f (x)R(x)dµ(x). Think of the measured observable as a function of a perturbation of the fixed background metric: ∗ ⊗s2 g = g0 + h. Hence E(M) = Γ((T M) ). Diffeomorphism transformation: move our experimental setup to a different region O0.

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Intuitive idea

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

Kasia Rejzner Locality and beyond 20 / 26 An observable Φ, which we measure, We don’t measure the observable (e.g. curvature) at a point. This is modeled by smearing with a test function f . For example: Z Φ(f ) = f (x)R(x)dµ(x). Think of the measured observable as a function of a perturbation of the fixed background metric: ∗ ⊗s2 g = g0 + h. Hence E(M) = Γ((T M) ). Diffeomorphism transformation: move our experimental setup to a different region O0.

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Intuitive idea

In experiment, geometric structure is probed by local observations. We have the following data: M Some region O of spacetime where the measurement is performed, O

Kasia Rejzner Locality and beyond 20 / 26 We don’t measure the observable (e.g. curvature) at a point. This is modeled by smearing with a test function f . For example: Z Φ(f ) = f (x)R(x)dµ(x). Think of the measured observable as a function of a perturbation of the fixed background metric: ∗ ⊗s2 g = g0 + h. Hence E(M) = Γ((T M) ). Diffeomorphism transformation: move our experimental setup to a different region O0.

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Intuitive idea

In experiment, geometric structure is probed by local observations. We have the following data: M Some region O of spacetime where the measurement is performed, O An observable Φ, which we measure,

Φ

Kasia Rejzner Locality and beyond 20 / 26 Think of the measured observable as a function of a perturbation of the fixed background metric: ∗ ⊗s2 g = g0 + h. Hence E(M) = Γ((T M) ). Diffeomorphism transformation: move our experimental setup to a different region O0.

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Intuitive idea

In experiment, geometric structure is probed by local observations. We have the following data: M Some region O of spacetime where the f measurement is performed, An observable Φ, which we measure, O We don’t measure the observable (e.g. curvature) at a point. This is modeled by smearing with a test function f . For example: Φ(f ) Z Φ(f ) = f (x)R(x)dµ(x).

Kasia Rejzner Locality and beyond 20 / 26 Diffeomorphism transformation: move our experimental setup to a different region O0.

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Intuitive idea

In experiment, geometric structure is probed by local observations. We have the following data: (M, g) Some region O of spacetime where the f measurement is performed, An observable Φ, which we measure, O We don’t measure the observable (e.g. curvature) at a point. This is modeled by smearing with a test function f . For example: Φ O g (f )[h] Z ( , ) Φ(f ) = f (x)R(x)dµ(x). Think of the measured observable as a function of a perturbation of the fixed background metric: ∗ ⊗s2 g = g0 + h. Hence E(M) = Γ((T M) ).

Kasia Rejzner Locality and beyond 20 / 26 Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Intuitive idea

In experiment, geometric structure is probed by local observations. We have the following data: (M, g) O0 Some region O of spacetime where the f measurement is performed, An observable Φ, which we measure, O We don’t measure the observable (e.g. curvature) at a point. This is modeled by smearing with a test function f . For example: Φ O g (f )[h] Z ( , ) Φ(f ) = f (x)R(x)dµ(x). Think of the measured observable as a function of a perturbation of the fixed background metric: ∗ ⊗s2 g = g0 + h. Hence E(M) = Γ((T M) ). Diffeomorphism transformation: move our experimental setup to a different region O0.

Kasia Rejzner Locality and beyond 20 / 26 A locally covariant field A is called diffeomorphism equivariant Z

if it is realized as A(M,g0)(f )[h] = Ag(x)f (x), where Ag is a M scalar depending locally and covariantly on the full metric, so Aχ∗g = Ag ◦ χ for all formal diffeomorphisms χ. One can characterize the space of classical gauge invariant on-shell observables as the 0th cohomology of the classical BV operator s acting on functionals that satisfy a weaker notion of locality.

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Local symmetries in gravity

In gravity symmetries arise from vector fields on M i.e. we have ρ : X(M) → Γ(TE(M)), defined by

. D (1) E ∂ρ(ξ)F(h) = F (h), Lξ(g0 + h) ,

where ξ(h) ∈ X(M).

Kasia Rejzner Locality and beyond 21 / 26 One can characterize the space of classical gauge invariant on-shell observables as the 0th cohomology of the classical BV operator s acting on functionals that satisfy a weaker notion of locality.

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Local symmetries in gravity

In gravity symmetries arise from vector fields on M i.e. we have ρ : X(M) → Γ(TE(M)), defined by

. D (1) E ∂ρ(ξ)F(h) = F (h), Lξ(g0 + h) ,

where ξ(h) ∈ X(M). A locally covariant field A is called diffeomorphism equivariant Z

if it is realized as A(M,g0)(f )[h] = Ag(x)f (x), where Ag is a M scalar depending locally and covariantly on the full metric, so Aχ∗g = Ag ◦ χ for all formal diffeomorphisms χ.

Kasia Rejzner Locality and beyond 21 / 26 Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Local symmetries in gravity

In gravity symmetries arise from vector fields on M i.e. we have ρ : X(M) → Γ(TE(M)), defined by

. D (1) E ∂ρ(ξ)F(h) = F (h), Lξ(g0 + h) ,

where ξ(h) ∈ X(M). A locally covariant field A is called diffeomorphism equivariant Z

if it is realized as A(M,g0)(f )[h] = Ag(x)f (x), where Ag is a M scalar depending locally and covariantly on the full metric, so Aχ∗g = Ag ◦ χ for all formal diffeomorphisms χ. One can characterize the space of classical gauge invariant on-shell observables as the 0th cohomology of the classical BV operator s acting on functionals that satisfy a weaker notion of locality.

Kasia Rejzner Locality and beyond 21 / 26 Idea: Take an equivariant field Ag and express it in terms of a coordinate system that depends on other fields. We realize the choice of a coordinate system by constructing µ four equivariant fields Xg , µ = 0,..., 3 which will parametrize points of spacetime.

We choose a background g0 such that the map

X : x 7→ (X0 ,..., X3 ) g0 g0 g0 is injective.

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Construction of gauge-invariant observables I

We fix M = (M, g0). We want to construct functionals that describe relations between classical fields (relational observables).

Kasia Rejzner Locality and beyond 22 / 26 We realize the choice of a coordinate system by constructing µ four equivariant fields Xg , µ = 0,..., 3 which will parametrize points of spacetime.

We choose a background g0 such that the map

X : x 7→ (X0 ,..., X3 ) g0 g0 g0 is injective.

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Construction of gauge-invariant observables I

We fix M = (M, g0). We want to construct functionals that describe relations between classical fields (relational observables).

Idea: Take an equivariant field Ag and express it in terms of a coordinate system that depends on other fields.

Kasia Rejzner Locality and beyond 22 / 26 We choose a background g0 such that the map

X : x 7→ (X0 ,..., X3 ) g0 g0 g0 is injective.

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Construction of gauge-invariant observables I

We fix M = (M, g0). We want to construct functionals that describe relations between classical fields (relational observables).

Idea: Take an equivariant field Ag and express it in terms of a coordinate system that depends on other fields. We realize the choice of a coordinate system by constructing µ four equivariant fields Xg , µ = 0,..., 3 which will parametrize points of spacetime.

Kasia Rejzner Locality and beyond 22 / 26 Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Construction of gauge-invariant observables I

We fix M = (M, g0). We want to construct functionals that describe relations between classical fields (relational observables).

Idea: Take an equivariant field Ag and express it in terms of a coordinate system that depends on other fields. We realize the choice of a coordinate system by constructing µ four equivariant fields Xg , µ = 0,..., 3 which will parametrize points of spacetime.

We choose a background g0 such that the map

X : x 7→ (X0 ,..., X3 ) g0 g0 g0 is injective.

Kasia Rejzner Locality and beyond 22 / 26 −1 αg transforms as αχ∗g = χ ◦ αg. Let Ag be an equivariant field. Then −1 Ag := Ag ◦ αg = Ag ◦ Xg ◦ Xg0 . is invariant under formal diffeo.’s. −1 [Ag ◦ Xg ](Y) corresponds to the value of the quantity Ag provided that the quantity Xg has the value Xg = Y. Thus it is a partial or relational observable (cf. Rovelli, Dittrich, Thiemann). −1 By considering Ag = Ag ◦ Xg ◦ Xg0 and choosing a test density f , we identify this observable with a field on spacetime: Z F = Ag(x)f (x) .

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Construction of gauge-invariant observables II

Take g = g0 + h sufficiently near to g0 and set −1 αg = Xg ◦ Xg0 .

Kasia Rejzner Locality and beyond 23 / 26 Let Ag be an equivariant field. Then −1 Ag := Ag ◦ αg = Ag ◦ Xg ◦ Xg0 . is invariant under formal diffeo.’s. −1 [Ag ◦ Xg ](Y) corresponds to the value of the quantity Ag provided that the quantity Xg has the value Xg = Y. Thus it is a partial or relational observable (cf. Rovelli, Dittrich, Thiemann). −1 By considering Ag = Ag ◦ Xg ◦ Xg0 and choosing a test density f , we identify this observable with a field on spacetime: Z F = Ag(x)f (x) .

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Construction of gauge-invariant observables II

Take g = g0 + h sufficiently near to g0 and set −1 αg = Xg ◦ Xg0 . −1 αg transforms as αχ∗g = χ ◦ αg.

Kasia Rejzner Locality and beyond 23 / 26 −1 [Ag ◦ Xg ](Y) corresponds to the value of the quantity Ag provided that the quantity Xg has the value Xg = Y. Thus it is a partial or relational observable (cf. Rovelli, Dittrich, Thiemann). −1 By considering Ag = Ag ◦ Xg ◦ Xg0 and choosing a test density f , we identify this observable with a field on spacetime: Z F = Ag(x)f (x) .

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Construction of gauge-invariant observables II

Take g = g0 + h sufficiently near to g0 and set −1 αg = Xg ◦ Xg0 . −1 αg transforms as αχ∗g = χ ◦ αg. Let Ag be an equivariant field. Then −1 Ag := Ag ◦ αg = Ag ◦ Xg ◦ Xg0 . is invariant under formal diffeo.’s.

Kasia Rejzner Locality and beyond 23 / 26 −1 By considering Ag = Ag ◦ Xg ◦ Xg0 and choosing a test density f , we identify this observable with a field on spacetime: Z F = Ag(x)f (x) .

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Construction of gauge-invariant observables II

Take g = g0 + h sufficiently near to g0 and set −1 αg = Xg ◦ Xg0 . −1 αg transforms as αχ∗g = χ ◦ αg. Let Ag be an equivariant field. Then −1 Ag := Ag ◦ αg = Ag ◦ Xg ◦ Xg0 . is invariant under formal diffeo.’s. −1 [Ag ◦ Xg ](Y) corresponds to the value of the quantity Ag provided that the quantity Xg has the value Xg = Y. Thus it is a partial or relational observable (cf. Rovelli, Dittrich, Thiemann).

Kasia Rejzner Locality and beyond 23 / 26 Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Construction of gauge-invariant observables II

Take g = g0 + h sufficiently near to g0 and set −1 αg = Xg ◦ Xg0 . −1 αg transforms as αχ∗g = χ ◦ αg. Let Ag be an equivariant field. Then −1 Ag := Ag ◦ αg = Ag ◦ Xg ◦ Xg0 . is invariant under formal diffeo.’s. −1 [Ag ◦ Xg ](Y) corresponds to the value of the quantity Ag provided that the quantity Xg has the value Xg = Y. Thus it is a partial or relational observable (cf. Rovelli, Dittrich, Thiemann). −1 By considering Ag = Ag ◦ Xg ◦ Xg0 and choosing a test density f , we identify this observable with a field on spacetime: Z F = Ag(x)f (x) .

Kasia Rejzner Locality and beyond 23 / 26 More examples: [Bergmann 61, Bergmann-Komar 60]. When matter fields are present in the considered model, also these can serve as coordinates, e.g. the dust fields in the Brown-Kucharˇ model [Brown-Kucharˇ 95]; the scalar field in the Einstein-Klein-Gordon system. For an explicit construction on a cosmological background see the recent work by R. Brunetti, K. Fredenhagen, T.-P. Hack, N. Pinnamonti and myself: Cosmological perturbation theory and quantum gravity [arXiv:gr-qc/1605.02573], JHEP 2016.

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Examples:

On generic backgrounds g0 one can use traces of the powers of the Ricci operator:

a a Xg := Tr(R ), a ∈ {1, 2, 3, 4}

Kasia Rejzner Locality and beyond 24 / 26 When matter fields are present in the considered model, also these can serve as coordinates, e.g. the dust fields in the Brown-Kucharˇ model [Brown-Kucharˇ 95]; the scalar field in the Einstein-Klein-Gordon system. For an explicit construction on a cosmological background see the recent work by R. Brunetti, K. Fredenhagen, T.-P. Hack, N. Pinnamonti and myself: Cosmological perturbation theory and quantum gravity [arXiv:gr-qc/1605.02573], JHEP 2016.

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Examples:

On generic backgrounds g0 one can use traces of the powers of the Ricci operator:

a a Xg := Tr(R ), a ∈ {1, 2, 3, 4}

More examples: [Bergmann 61, Bergmann-Komar 60].

Kasia Rejzner Locality and beyond 24 / 26 For an explicit construction on a cosmological background see the recent work by R. Brunetti, K. Fredenhagen, T.-P. Hack, N. Pinnamonti and myself: Cosmological perturbation theory and quantum gravity [arXiv:gr-qc/1605.02573], JHEP 2016.

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Examples:

On generic backgrounds g0 one can use traces of the powers of the Ricci operator:

a a Xg := Tr(R ), a ∈ {1, 2, 3, 4}

More examples: [Bergmann 61, Bergmann-Komar 60]. When matter fields are present in the considered model, also these can serve as coordinates, e.g. the dust fields in the Brown-Kucharˇ model [Brown-Kucharˇ 95]; the scalar field in the Einstein-Klein-Gordon system.

Kasia Rejzner Locality and beyond 24 / 26 Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Examples:

On generic backgrounds g0 one can use traces of the powers of the Ricci operator:

a a Xg := Tr(R ), a ∈ {1, 2, 3, 4}

More examples: [Bergmann 61, Bergmann-Komar 60]. When matter fields are present in the considered model, also these can serve as coordinates, e.g. the dust fields in the Brown-Kucharˇ model [Brown-Kucharˇ 95]; the scalar field in the Einstein-Klein-Gordon system. For an explicit construction on a cosmological background see the recent work by R. Brunetti, K. Fredenhagen, T.-P. Hack, N. Pinnamonti and myself: Cosmological perturbation theory and quantum gravity [arXiv:gr-qc/1605.02573], JHEP 2016.

Kasia Rejzner Locality and beyond 24 / 26 It turned out to be very successful in QFT on curved spacetime and it pointed a direction towards effective theory of quantum gravity. Combined with perturbative methods it gave rise to pAQFT. It is also interesting and challenging to look at possible generalizations: non-local quantities in QG, LCQFT up to homotopy. More to come!

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Conclusions

More that 60 years after the paper of Haag and Kastler the locality principle is a powerful paradigm in QFT.

Kasia Rejzner Locality and beyond 25 / 26 Combined with perturbative methods it gave rise to pAQFT. It is also interesting and challenging to look at possible generalizations: non-local quantities in QG, LCQFT up to homotopy. More to come!

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Conclusions

More that 60 years after the paper of Haag and Kastler the locality principle is a powerful paradigm in QFT. It turned out to be very successful in QFT on curved spacetime and it pointed a direction towards effective theory of quantum gravity.

Kasia Rejzner Locality and beyond 25 / 26 It is also interesting and challenging to look at possible generalizations: non-local quantities in QG, LCQFT up to homotopy. More to come!

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Conclusions

More that 60 years after the paper of Haag and Kastler the locality principle is a powerful paradigm in QFT. It turned out to be very successful in QFT on curved spacetime and it pointed a direction towards effective theory of quantum gravity. Combined with perturbative methods it gave rise to pAQFT.

Kasia Rejzner Locality and beyond 25 / 26 More to come!

Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Conclusions

More that 60 years after the paper of Haag and Kastler the locality principle is a powerful paradigm in QFT. It turned out to be very successful in QFT on curved spacetime and it pointed a direction towards effective theory of quantum gravity. Combined with perturbative methods it gave rise to pAQFT. It is also interesting and challenging to look at possible generalizations: non-local quantities in QG, LCQFT up to homotopy.

Kasia Rejzner Locality and beyond 25 / 26 Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables Conclusions

More that 60 years after the paper of Haag and Kastler the locality principle is a powerful paradigm in QFT. It turned out to be very successful in QFT on curved spacetime and it pointed a direction towards effective theory of quantum gravity. Combined with perturbative methods it gave rise to pAQFT. It is also interesting and challenging to look at possible generalizations: non-local quantities in QG, LCQFT up to homotopy. More to come!

Kasia Rejzner Locality and beyond 25 / 26 Algebraic approach to QFT Effective quantum gravity Quantum gravity Observables

Thank you for your attention!

Kasia Rejzner Locality and beyond 26 / 26