From time-ordered products in quantum field theory to factorization - - PowerPoint PPT Presentation

(p)AQFT Factorization algebras Comparison

From time-ordered products in quantum field theory to factorization algebras and back

Kasia Rejzner

University of York

Trondheim/York, 24.04.2020

Kasia Rejzner Time-ordered products and factorization algebras 1 / 36

(p)AQFT Factorization algebras Comparison

Outline of the talk

1

(p)AQFT Algebraic quantum field theory and its generalizations pAQFT

2

Factorization algebras

3

Comparison Statement of the main results Time-ordered products

Kasia Rejzner Time-ordered products and factorization algebras 2 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

1

(p)AQFT Algebraic quantum field theory and its generalizations pAQFT

2

Factorization algebras

3

Comparison Statement of the main results Time-ordered products

Kasia Rejzner Time-ordered products and factorization algebras 3 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Algebraic quantum field theory

A convenient framework to investigate conceptual problems in QFT is the Algebraic Quantum Field Theory.

Kasia Rejzner Time-ordered products and factorization algebras 4 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Algebraic quantum field theory

A convenient framework to investigate conceptual problems in QFT is the Algebraic Quantum Field Theory. It started as the axiomatic framework of Haag-Kastler: a model is defined by associating to each region O of Minkowski spacetime the algebra A(O) of observables (a unital C∗-algebra) that can be measured in O. O1 A(O1)

Kasia Rejzner Time-ordered products and factorization algebras 4 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Algebraic quantum field theory

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

Kasia Rejzner Time-ordered products and factorization algebras 4 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Algebraic quantum field theory

A convenient framework to investigate conceptual problems in QFT is the Algebraic Quantum Field Theory. It started as the axiomatic framework of Haag-Kastler: a model is defined by associating to each region O of Minkowski spacetime the algebra A(O) of observables (a unital C∗-algebra) that can be measured in O. The physical notion of subsystems is realized by the condition of isotony, i.e.: O1 ⊂ O2 ⇒ A(O1) ⊂ A(O2). We obtain a net of C∗-algebras. Key idea: algebras of observbles constructed independently of the choice of state (“vacuum”), so allows for degenerate vacuua. This idea can be applied more generally, as we will see later.

Kasia Rejzner Time-ordered products and factorization algebras 4 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Further axioms

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

Kasia Rejzner Time-ordered products and factorization algebras 5 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Further axioms

One can also include two further axioms which are important in QFT: causality and time-slice axiom. Einstein causality: If O1, O2 ⊂ M are spacelike, then [A(O1), A(O2)] = {0}, where [., .] is the commutator in the sense of A(O3), where O3 contains both O1 and O2.

Kasia Rejzner Time-ordered products and factorization algebras 5 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Further axioms

One can also include two further axioms which are important in QFT: causality and time-slice axiom. Einstein causality: If O1, O2 ⊂ M are spacelike, then [A(O1), A(O2)] = {0}, where [., .] is the commutator in the sense of A(O3), where O3 contains both O1 and O2. This encodes the principle of special relativity, that information cannot be exchanged between causally disjoint regions.

Kasia Rejzner Time-ordered products and factorization algebras 5 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Further axioms

One can also include two further axioms which are important in QFT: causality and time-slice axiom. Einstein causality: If O1, O2 ⊂ M are spacelike, then [A(O1), A(O2)] = {0}, where [., .] is the commutator in the sense of A(O3), where O3 contains both O1 and O2. This encodes the principle of special relativity, that information cannot be exchanged between causally disjoint regions. Time-slice axiom: If N is a neighborhood of a Cauchy-surface

• f O, then A(N) is isomorphic to A(O).

Kasia Rejzner Time-ordered products and factorization algebras 5 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Further axioms

One can also include two further axioms which are important in QFT: causality and time-slice axiom. Einstein causality: If O1, O2 ⊂ M are spacelike, then [A(O1), A(O2)] = {0}, where [., .] is the commutator in the sense of A(O3), where O3 contains both O1 and O2. This encodes the principle of special relativity, that information cannot be exchanged between causally disjoint regions. Time-slice axiom: If N is a neighborhood of a Cauchy-surface

• f O, then A(N) is isomorphic to A(O).

This is a QFT version of the initial value problem.

Kasia Rejzner Time-ordered products and factorization algebras 5 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Locally covariant quantum field theory

To include the effects of general relativity

• ne has to be able to describe quantum fields
• n a general class of spacetimes. The

corresponding extension of AQFT is called locally covariant quantum field theory (LCQFT).

Kasia Rejzner Time-ordered products and factorization algebras 6 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Locally covariant quantum field theory

To include the effects of general relativity

• ne has to be able to describe quantum fields
• n a general class of spacetimes. The

corresponding extension of AQFT is called locally covariant quantum field theory (LCQFT). We replace O1 and O2 with smooth Lorentzian manifolds (spacetimes) M, N and require the embedding ψ : M → N to be an isometry, preserving other structure. N M ψ

Kasia Rejzner Time-ordered products and factorization algebras 6 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Locally covariant quantum field theory

To include the effects of general relativity

• ne has to be able to describe quantum fields
• n a general class of spacetimes. The

corresponding extension of AQFT is called locally covariant quantum field theory (LCQFT). We replace O1 and O2 with smooth Lorentzian manifolds (spacetimes) M, N and require the embedding ψ : M → N to be an isometry, preserving other structure. A model in LCQFT is defined by assigning

• bservable algebras A(M) to spacetimes and

algebra morphisms Aψ to embeddings. A(N) A N M ψ A A(M) Aψ

Kasia Rejzner Time-ordered products and factorization algebras 6 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Locally covariant quantum field theory

To include the effects of general relativity

• ne has to be able to describe quantum fields
• n a general class of spacetimes. The

corresponding extension of AQFT is called locally covariant quantum field theory (LCQFT). We replace O1 and O2 with smooth Lorentzian manifolds (spacetimes) M, N and require the embedding ψ : M → N to be an isometry, preserving other structure. A model in LCQFT is defined by assigning

• bservable algebras A(M) to spacetimes and

algebra morphisms Aψ to embeddings. Covariance requirement: A is a functor. A(N) A N M ψ A A(M) Aψ

Kasia Rejzner Time-ordered products and factorization algebras 6 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Categories I

We work over Nuc, the category of nuclear topological vector spaces (alternative: conveninent vector spaces).

Kasia Rejzner Time-ordered products and factorization algebras 7 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Categories I

We work over Nuc, the category of nuclear topological vector spaces (alternative: conveninent vector spaces). Alg(Nuc): unital associative algebras in Nuc.

Kasia Rejzner Time-ordered products and factorization algebras 7 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Categories I

We work over Nuc, the category of nuclear topological vector spaces (alternative: conveninent vector spaces). Alg(Nuc): unital associative algebras in Nuc. CAlg(Nuc): unital commutative algebras in Nuc

Kasia Rejzner Time-ordered products and factorization algebras 7 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Categories I

We work over Nuc, the category of nuclear topological vector spaces (alternative: conveninent vector spaces). Alg(Nuc): unital associative algebras in Nuc. CAlg(Nuc): unital commutative algebras in Nuc PAlg(Nuc): unital Poisson algebras therein.

Kasia Rejzner Time-ordered products and factorization algebras 7 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Categories I

We work over Nuc, the category of nuclear topological vector spaces (alternative: conveninent vector spaces). Alg(Nuc): unital associative algebras in Nuc. CAlg(Nuc): unital commutative algebras in Nuc PAlg(Nuc): unital Poisson algebras therein. For ∗structures (involution), we use Alg∗(Nuc), CAlg∗(Nuc), and PAlg∗(Nuc), respectively.

Kasia Rejzner Time-ordered products and factorization algebras 7 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Categories I

We work over Nuc, the category of nuclear topological vector spaces (alternative: conveninent vector spaces). Alg(Nuc): unital associative algebras in Nuc. CAlg(Nuc): unital commutative algebras in Nuc PAlg(Nuc): unital Poisson algebras therein. For ∗structures (involution), we use Alg∗(Nuc), CAlg∗(Nuc), and PAlg∗(Nuc), respectively. We use v : PAlg∗(Nuc) → Nuc and v : Alg∗(Nuc) → Nuc to denote forgetful functors to vector spaces and c : PAlg∗(Nuc) → CAlg∗(Nuc) denotes the forgetful functor to commutative algebras.

Kasia Rejzner Time-ordered products and factorization algebras 7 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Categories I

We work over Nuc, the category of nuclear topological vector spaces (alternative: conveninent vector spaces). Alg(Nuc): unital associative algebras in Nuc. CAlg(Nuc): unital commutative algebras in Nuc PAlg(Nuc): unital Poisson algebras therein. For ∗structures (involution), we use Alg∗(Nuc), CAlg∗(Nuc), and PAlg∗(Nuc), respectively. We use v : PAlg∗(Nuc) → Nuc and v : Alg∗(Nuc) → Nuc to denote forgetful functors to vector spaces and c : PAlg∗(Nuc) → CAlg∗(Nuc) denotes the forgetful functor to commutative algebras. If C is an additive category, we write Ch(C) to denote the category of cochain complexes and cochain maps in C.

Kasia Rejzner Time-ordered products and factorization algebras 7 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Categories II

Category of spacetimes Let Locn be the category where

Kasia Rejzner Time-ordered products and factorization algebras 8 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Categories II

Category of spacetimes Let Locn be the category where an object is a connected, (time-)oriented globally hyperbolic spacetime of dimension n

Kasia Rejzner Time-ordered products and factorization algebras 8 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Categories II

Category of spacetimes Let Locn be the category where an object is a connected, (time-)oriented globally hyperbolic spacetime of dimension n and where a morphism χ : M → N is an isometric embedding that preserves orientations and causal structure.

Kasia Rejzner Time-ordered products and factorization algebras 8 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Categories II

Category of spacetimes Let Locn be the category where an object is a connected, (time-)oriented globally hyperbolic spacetime of dimension n and where a morphism χ : M → N is an isometric embedding that preserves orientations and causal structure. The latter means that for any causal curve γ : [a, b] → N, if γ(a), γ(b) ∈ χ(M), then for all t ∈]a, b[, we have γ(t) ∈ χ(M). (χ cannot create new causal links.)

Kasia Rejzner Time-ordered products and factorization algebras 8 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Categories II

Category of spacetimes Let Locn be the category where an object is a connected, (time-)oriented globally hyperbolic spacetime of dimension n and where a morphism χ : M → N is an isometric embedding that preserves orientations and causal structure. The latter means that for any causal curve γ : [a, b] → N, if γ(a), γ(b) ∈ χ(M), then for all t ∈]a, b[, we have γ(t) ∈ χ(M). (χ cannot create new causal links.) With this notation, A is a functor Locn → Alg∗(Nuc).

Kasia Rejzner Time-ordered products and factorization algebras 8 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Categories II

Category of spacetimes Let Locn be the category where an object is a connected, (time-)oriented globally hyperbolic spacetime of dimension n and where a morphism χ : M → N is an isometric embedding that preserves orientations and causal structure. The latter means that for any causal curve γ : [a, b] → N, if γ(a), γ(b) ∈ χ(M), then for all t ∈]a, b[, we have γ(t) ∈ χ(M). (χ cannot create new causal links.) With this notation, A is a functor Locn → Alg∗(Nuc). We consider Caus(M), a subcategory of Locn formed by all causally convex, relatively compact subsets of M. Morphisms are embeddings.

Kasia Rejzner Time-ordered products and factorization algebras 8 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Perturbative AQFT

Building models in AQFT is hard and up to now no 4D interacting model fulfilling the axioms is known. To describe theories like QED or the Standard Model of particle physics we use perturbative methods.

Kasia Rejzner Time-ordered products and factorization algebras 9 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Perturbative AQFT

Building models in AQFT is hard and up to now no 4D interacting model fulfilling the axioms is known. To describe theories like QED or the Standard Model of particle physics we use perturbative methods. pAQFT combines the axiomatic framework

• f Haag-Kastler with formal deformation

quantization and homological algebra.

Kasia Rejzner Time-ordered products and factorization algebras 9 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Perturbative AQFT

Building models in AQFT is hard and up to now no 4D interacting model fulfilling the axioms is known. To describe theories like QED or the Standard Model of particle physics we use perturbative methods. pAQFT combines the axiomatic framework

• f Haag-Kastler with formal deformation

quantization and homological algebra. Contributors: Bahns, Brunetti, Duetsch, Fredenhagen, Hawkins, Hollands, Pinamonti, KR, Wald, . . . .

Kasia Rejzner Time-ordered products and factorization algebras 9 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Perturbative AQFT

Building models in AQFT is hard and up to now no 4D interacting model fulfilling the axioms is known. To describe theories like QED or the Standard Model of particle physics we use perturbative methods. pAQFT combines the axiomatic framework

• f Haag-Kastler with formal deformation

quantization and homological algebra. Contributors: Bahns, Brunetti, Duetsch, Fredenhagen, Hawkins, Hollands, Pinamonti, KR, Wald, . . . . Mathematical foundations of pAQFT have been reviewed in: pAQFT. An Introduction for Mathematicians, KR, Springer 2016.

Kasia Rejzner Time-ordered products and factorization algebras 9 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Physical input

A spacetime M = (M, g) ∈ Obj(Locn).

Kasia Rejzner Time-ordered products and factorization algebras 10 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Physical input

A spacetime M = (M, g) ∈ Obj(Locn). Configuration space E(M) ∈ Obj(Nuc): choice of objects we want to study in our theory (scalars, vectors, tensors,. . . ).

Kasia Rejzner Time-ordered products and factorization algebras 10 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Physical input

A spacetime M = (M, g) ∈ Obj(Locn). Configuration space E(M) ∈ Obj(Nuc): choice of objects we want to study in our theory (scalars, vectors, tensors,. . . ). We denote by E the contravariant functor from Locn to Nuc, which acts on objects as above and the morphisms χ : M → N are mapped to Eχ ≡ χ∗.

Kasia Rejzner Time-ordered products and factorization algebras 10 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Physical input

A spacetime M = (M, g) ∈ Obj(Locn). Configuration space E(M) ∈ Obj(Nuc): choice of objects we want to study in our theory (scalars, vectors, tensors,. . . ). We denote by E the contravariant functor from Locn to Nuc, which acts on objects as above and the morphisms χ : M → N are mapped to Eχ ≡ χ∗. Typically E(M) is a space of smooth sections of some vector bundle E π − → M over M.

Kasia Rejzner Time-ordered products and factorization algebras 10 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Physical input

A spacetime M = (M, g) ∈ Obj(Locn). Configuration space E(M) ∈ Obj(Nuc): choice of objects we want to study in our theory (scalars, vectors, tensors,. . . ). We denote by E the contravariant functor from Locn to Nuc, which acts on objects as above and the morphisms χ : M → N are mapped to Eχ ≡ χ∗. Typically E(M) is a space of smooth sections of some vector bundle E π − → M over M. For the scalar field: E(M) ≡ C∞(M, R).

Kasia Rejzner Time-ordered products and factorization algebras 10 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Physical input

A spacetime M = (M, g) ∈ Obj(Locn). Configuration space E(M) ∈ Obj(Nuc): choice of objects we want to study in our theory (scalars, vectors, tensors,. . . ). We denote by E the contravariant functor from Locn to Nuc, which acts on objects as above and the morphisms χ : M → N are mapped to Eχ ≡ χ∗. Typically E(M) is a space of smooth sections of some vector bundle E π − → M over M. For the scalar field: E(M) ≡ C∞(M, R). We use notation ϕ ∈ E(M), also if it has several components.

Kasia Rejzner Time-ordered products and factorization algebras 10 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Physical input

A spacetime M = (M, g) ∈ Obj(Locn). Configuration space E(M) ∈ Obj(Nuc): choice of objects we want to study in our theory (scalars, vectors, tensors,. . . ). We denote by E the contravariant functor from Locn to Nuc, which acts on objects as above and the morphisms χ : M → N are mapped to Eχ ≡ χ∗. Typically E(M) is a space of smooth sections of some vector bundle E π − → M over M. For the scalar field: E(M) ≡ C∞(M, R). We use notation ϕ ∈ E(M), also if it has several components. Dynamics: we use a modification of the Lagrangian formalism (fully covariant).

Kasia Rejzner Time-ordered products and factorization algebras 10 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Classical observables

Classical observables are modeled as smooth functionals on E(M), i.e. elements of C∞(E(M), C), which is a covariant functor Locn to Nuc.

Kasia Rejzner Time-ordered products and factorization algebras 11 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Classical observables

Classical observables are modeled as smooth functionals on E(M), i.e. elements of C∞(E(M), C), which is a covariant functor Locn to Nuc. For simplicity of notation (and because of functoriality), we drop M, if no confusion arises, i.e. write E, C∞(E, C), etc.

Kasia Rejzner Time-ordered products and factorization algebras 11 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Classical observables

Classical observables are modeled as smooth functionals on E(M), i.e. elements of C∞(E(M), C), which is a covariant functor Locn to Nuc. For simplicity of notation (and because of functoriality), we drop M, if no confusion arises, i.e. write E, C∞(E, C), etc. Localization of functionals governed by their spacetime support: supp F = {x ∈ M|∀ neighbourhoods U of x ∃ϕ, ψ ∈ E, supp ψ ⊂ U such that F(ϕ + ψ) = F(ϕ)} .

Kasia Rejzner Time-ordered products and factorization algebras 11 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Classical observables

Classical observables are modeled as smooth functionals on E(M), i.e. elements of C∞(E(M), C), which is a covariant functor Locn to Nuc. For simplicity of notation (and because of functoriality), we drop M, if no confusion arises, i.e. write E, C∞(E, C), etc. Localization of functionals governed by their spacetime support: supp F = {x ∈ M|∀ neighbourhoods U of x ∃ϕ, ψ ∈ E, supp ψ ⊂ U such that F(ϕ + ψ) = F(ϕ)} . Take home (or rather stay at home) message: pAQFT is a machinery to turn functionals of classical field configurations (classical observables) into quantum observables by means of deformation of algebraic structures. This is done without referring to a Hilbert space representation.

Kasia Rejzner Time-ordered products and factorization algebras 11 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Functionals I

We define Floc, local functionals on E, as functionals that are of the form F(ϕ) =

• f(jk

x(ϕ))dµg ,

for a smooth, compactly supported, function f on the jet bundle.

Kasia Rejzner Time-ordered products and factorization algebras 12 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Functionals I

We define Floc, local functionals on E, as functionals that are of the form F(ϕ) =

• f(jk

x(ϕ))dµg ,

for a smooth, compactly supported, function f on the jet bundle. A functional F is regular, if F(n)(ϕ) is a smooth section (in general it would be distributional). It is called polynomial if there exists N ∈ N such that F(k) ≡ 0 for all k > N

Kasia Rejzner Time-ordered products and factorization algebras 12 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Functionals I

We define Floc, local functionals on E, as functionals that are of the form F(ϕ) =

• f(jk

x(ϕ))dµg ,

for a smooth, compactly supported, function f on the jet bundle. A functional F is regular, if F(n)(ϕ) is a smooth section (in general it would be distributional). It is called polynomial if there exists N ∈ N such that F(k) ≡ 0 for all k > N Let F denote the space of regular, polynomial, compactly supported functionals.

Kasia Rejzner Time-ordered products and factorization algebras 12 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Functionals I

We define Floc, local functionals on E, as functionals that are of the form F(ϕ) =

• f(jk

x(ϕ))dµg ,

for a smooth, compactly supported, function f on the jet bundle. A functional F is regular, if F(n)(ϕ) is a smooth section (in general it would be distributional). It is called polynomial if there exists N ∈ N such that F(k) ≡ 0 for all k > N Let F denote the space of regular, polynomial, compactly supported functionals. Equipped with the pointwise product (F · G)(ϕ) . = F(ϕ)G(ϕ) , F(M) is a commutative algebra and we obtain a covariant functor F : Locn → CAlg(Nuc).

Kasia Rejzner Time-ordered products and factorization algebras 12 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Functionals II

The simplest examples of functionals are smeared fields Φ(f)(ϕ) =

• ϕ(x)f(x)dµ(x) ,

f ∈ D(M) := C∞

c (M, R)

Kasia Rejzner Time-ordered products and factorization algebras 13 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Functionals II

The simplest examples of functionals are smeared fields Φ(f)(ϕ) =

• ϕ(x)f(x)dµ(x) ,

f ∈ D(M) := C∞

c (M, R)

Note that regular, polynomial functionals of degree 2 and higher are not local. Take for example F(ϕ) =

• f(x, y)ϕ(x)ϕ(y)dµ(x)dµ(y) , f ∈ D(M2) := C∞

c (M2, R) .

Kasia Rejzner Time-ordered products and factorization algebras 13 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Functionals II

The simplest examples of functionals are smeared fields Φ(f)(ϕ) =

• ϕ(x)f(x)dµ(x) ,

f ∈ D(M) := C∞

c (M, R)

Note that regular, polynomial functionals of degree 2 and higher are not local. Take for example F(ϕ) =

• f(x, y)ϕ(x)ϕ(y)dµ(x)dµ(y) , f ∈ D(M2) := C∞

c (M2, R) .

Now take f ∈ D(M) and consider F(ϕ) =

• fϕ2dµ =
• f(x)δ(x − y)ϕ(x)ϕ(y)dµ(x)dµ(y).

Kasia Rejzner Time-ordered products and factorization algebras 13 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Functionals II

The simplest examples of functionals are smeared fields Φ(f)(ϕ) =

• ϕ(x)f(x)dµ(x) ,

f ∈ D(M) := C∞

c (M, R)

Note that regular, polynomial functionals of degree 2 and higher are not local. Take for example F(ϕ) =

• f(x, y)ϕ(x)ϕ(y)dµ(x)dµ(y) , f ∈ D(M2) := C∞

c (M2, R) .

Now take f ∈ D(M) and consider F(ϕ) =

• fϕ2dµ =
• f(x)δ(x − y)ϕ(x)ϕ(y)dµ(x)dµ(y).

To avoid technical analytic issues, I will formulate the rest of this introduction for F. However, all of this generalizes to Floc.

Kasia Rejzner Time-ordered products and factorization algebras 13 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Dynamics

Dynamics is introduced by a generalized action S, a natural transformation S : D → Floc, where D(M) = C∞

c (M, R) and D

acts on morphisms by pushforward.

Kasia Rejzner Time-ordered products and factorization algebras 14 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Dynamics

Dynamics is introduced by a generalized action S, a natural transformation S : D → Floc, where D(M) = C∞

c (M, R) and D

acts on morphisms by pushforward. For example for the free scalar field: SM(f)[ϕ] =

• M
• 1

2ϕ2 + 1

2∇µϕ∇µϕ

• fdµ .

Kasia Rejzner Time-ordered products and factorization algebras 14 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Dynamics

Dynamics is introduced by a generalized action S, a natural transformation S : D → Floc, where D(M) = C∞

c (M, R) and D

acts on morphisms by pushforward. For example for the free scalar field: SM(f)[ϕ] =

• M
• 1

2ϕ2 + 1

2∇µϕ∇µϕ

• fdµ .

The Euler-Lagrange derivative of S is denoted by dS and defined by dSM(ϕ), ψ =

• SM(f)(1)[ϕ], ψ
• , where f ≡ 1 on suppψ,

ψ ∈ D(M). The field equation is: dSM(ϕ) = 0, so geometrically, the solution space is the zero locus of dSM (seen as a 1-form

• n E(M)).

M

supp(f) supp(ψ) f ≡ 1

Kasia Rejzner Time-ordered products and factorization algebras 14 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Dynamics

Dynamics is introduced by a generalized action S, a natural transformation S : D → Floc, where D(M) = C∞

c (M, R) and D

acts on morphisms by pushforward. For example for the free scalar field: SM(f)[ϕ] =

• M
• 1

2ϕ2 + 1

2∇µϕ∇µϕ

• fdµ .

The Euler-Lagrange derivative of S is denoted by dS and defined by dSM(ϕ), ψ =

• SM(f)(1)[ϕ], ψ
• , where f ≡ 1 on suppψ,

ψ ∈ D(M). The field equation is: dSM(ϕ) = 0, so geometrically, the solution space is the zero locus of dSM (seen as a 1-form

• n E(M)).

Again, we drop M from notation when no confusion arises. M

supp(f) supp(ψ) f ≡ 1

Kasia Rejzner Time-ordered products and factorization algebras 14 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Symmetries

We use the BV framework, where symmetries are identified with vector fields (directions) on E.

E(M)

C ϕ F

Kasia Rejzner Time-ordered products and factorization algebras 15 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Symmetries

We use the BV framework, where symmetries are identified with vector fields (directions) on E. Let V denote regular, polynomial compactly supported vector fields on E. Let PV denote the space of polyvector fields.

E(M)

C ϕ F

Kasia Rejzner Time-ordered products and factorization algebras 15 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Symmetries

We use the BV framework, where symmetries are identified with vector fields (directions) on E. Let V denote regular, polynomial compactly supported vector fields on E. Let PV denote the space of polyvector fields. They act on F as derivations: ∂XF(ϕ) := F(1)(ϕ), X(ϕ)

E(M)

C ϕ F

Kasia Rejzner Time-ordered products and factorization algebras 15 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Symmetries

We use the BV framework, where symmetries are identified with vector fields (directions) on E. Let V denote regular, polynomial compactly supported vector fields on E. Let PV denote the space of polyvector fields. They act on F as derivations: ∂XF(ϕ) := F(1)(ϕ), X(ϕ) A symmetry of S is a direction in E in which the action is constant, i.e. it is a vector field X ∈ V such that ∀ϕ ∈ E: 0 = dS(ϕ), X(ϕ)=: δS(X)(ϕ).

E(M)

C ϕ F

Kasia Rejzner Time-ordered products and factorization algebras 15 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Free scalar field (classical)

E = C∞(M, R) and the equation of motion is dS(ϕ) = Pϕ = 0, where P = −(✷ + m2).

Kasia Rejzner Time-ordered products and factorization algebras 16 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Free scalar field (classical)

E = C∞(M, R) and the equation of motion is dS(ϕ) = Pϕ = 0, where P = −(✷ + m2). Space of solutions: ES ⊂ E. Denote functionals that vanish on ES by F0. They are of the form: δS(X) for some X ∈ V.

Kasia Rejzner Time-ordered products and factorization algebras 16 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Free scalar field (classical)

E = C∞(M, R) and the equation of motion is dS(ϕ) = Pϕ = 0, where P = −(✷ + m2). Space of solutions: ES ⊂ E. Denote functionals that vanish on ES by F0. They are of the form: δS(X) for some X ∈ V. The space of on-shell observables (i.e. functionals on ES) FS is the quotient FS = F/F0.

Kasia Rejzner Time-ordered products and factorization algebras 16 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Free scalar field (classical)

E = C∞(M, R) and the equation of motion is dS(ϕ) = Pϕ = 0, where P = −(✷ + m2). Space of solutions: ES ⊂ E. Denote functionals that vanish on ES by F0. They are of the form: δS(X) for some X ∈ V. The space of on-shell observables (i.e. functionals on ES) FS is the quotient FS = F/F0. δS is called the Koszul differential. Symmetries constitute its kernel.

Kasia Rejzner Time-ordered products and factorization algebras 16 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Free scalar field (classical)

E = C∞(M, R) and the equation of motion is dS(ϕ) = Pϕ = 0, where P = −(✷ + m2). Space of solutions: ES ⊂ E. Denote functionals that vanish on ES by F0. They are of the form: δS(X) for some X ∈ V. The space of on-shell observables (i.e. functionals on ES) FS is the quotient FS = F/F0. δS is called the Koszul differential. Symmetries constitute its kernel. We obtain a sequence: 0 → Sym ֒ → V

δS

− → F → 0.

Kasia Rejzner Time-ordered products and factorization algebras 16 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Free scalar field (classical)

E = C∞(M, R) and the equation of motion is dS(ϕ) = Pϕ = 0, where P = −(✷ + m2). Space of solutions: ES ⊂ E. Denote functionals that vanish on ES by F0. They are of the form: δS(X) for some X ∈ V. The space of on-shell observables (i.e. functionals on ES) FS is the quotient FS = F/F0. δS is called the Koszul differential. Symmetries constitute its kernel. We obtain a sequence: 0 → Sym ֒ → V

δS

− → F → 0. In this talk, I discuss only the case where there are no non-trivial (not vanishing on ES) local symmetries,

Kasia Rejzner Time-ordered products and factorization algebras 16 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Free scalar field (classical)

E = C∞(M, R) and the equation of motion is dS(ϕ) = Pϕ = 0, where P = −(✷ + m2). Space of solutions: ES ⊂ E. Denote functionals that vanish on ES by F0. They are of the form: δS(X) for some X ∈ V. The space of on-shell observables (i.e. functionals on ES) FS is the quotient FS = F/F0. δS is called the Koszul differential. Symmetries constitute its kernel. We obtain a sequence: 0 → Sym ֒ → V

δS

− → F → 0. In this talk, I discuss only the case where there are no non-trivial (not vanishing on ES) local symmetries, Introduce the BV complex: BV . = (PV, δS). Then the space of classical on-shell observables is given by FS = H0(BV) and higher cohomology groups vanish.

Kasia Rejzner Time-ordered products and factorization algebras 16 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Antibracket

The space of polyvectors fields is equipped with a graded bracket (called antibracket) which is just the Schouten bracket {., .}.

Kasia Rejzner Time-ordered products and factorization algebras 17 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Antibracket

The space of polyvectors fields is equipped with a graded bracket (called antibracket) which is just the Schouten bracket {., .}. On vector fields it is equal to the commutator {X, Y} = [X, Y], X, Y ∈ V,

Kasia Rejzner Time-ordered products and factorization algebras 17 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Antibracket

The space of polyvectors fields is equipped with a graded bracket (called antibracket) which is just the Schouten bracket {., .}. On vector fields it is equal to the commutator {X, Y} = [X, Y], X, Y ∈ V, For a vector field and a functional we have {X, F} = ∂XF, F ∈ F, X ∈ V,

Kasia Rejzner Time-ordered products and factorization algebras 17 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Antibracket

The space of polyvectors fields is equipped with a graded bracket (called antibracket) which is just the Schouten bracket {., .}. On vector fields it is equal to the commutator {X, Y} = [X, Y], X, Y ∈ V, For a vector field and a functional we have {X, F} = ∂XF, F ∈ F, X ∈ V, It satisfies the graded Leibniz rule: {X, Y ∧ Z} = {X, Y} ∧ Z + (−1)|Y|(|X|+1)Y ∧ {X, Z}.

Kasia Rejzner Time-ordered products and factorization algebras 17 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Antibracket

The space of polyvectors fields is equipped with a graded bracket (called antibracket) which is just the Schouten bracket {., .}. On vector fields it is equal to the commutator {X, Y} = [X, Y], X, Y ∈ V, For a vector field and a functional we have {X, F} = ∂XF, F ∈ F, X ∈ V, It satisfies the graded Leibniz rule: {X, Y ∧ Z} = {X, Y} ∧ Z + (−1)|Y|(|X|+1)Y ∧ {X, Z}. Derivation δS is not inner with respect to {., .}, but locally it can be written as δSX = {X, S(f)} for f ≡ 1 on suppX, X ∈ V.

Kasia Rejzner Time-ordered products and factorization algebras 17 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

dgA(Q)FT: Classical

For convenience, we will from now on restrict our category Locn of spacetimes to its subcategory Caus(M) consisting of causally convex, relatively compact subsets of a fixed spacetime M.

Kasia Rejzner Time-ordered products and factorization algebras 18 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

dgA(Q)FT: Classical

For convenience, we will from now on restrict our category Locn of spacetimes to its subcategory Caus(M) consisting of causally convex, relatively compact subsets of a fixed spacetime M. Definition A dg classical field theory model on a spacetime M is a functor P : Caus(M) → PAlg∗(Ch(Nuc)), so that each P(O) is a locally convex dg Poisson ∗-algebra satisfying Einstein causality: spacelike-separated observables Poisson-commute at the level of cohomology. it satisfies the time-slice axiom if for any N ∈ Caus(M) a neighborhood of a Cauchy surface in the region O ∈ Caus(M), then the map P(N) → P(O) is a quasi-isomorphism.

Kasia Rejzner Time-ordered products and factorization algebras 18 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

dgA(Q)FT: Quantum

Definition A dg QFT model on a spacetime M

Kasia Rejzner Time-ordered products and factorization algebras 19 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

dgA(Q)FT: Quantum

Definition A dg QFT model on a spacetime M is a functor A : Caus(M) → Alg∗(Ch(Nuc)), so that each A(O) is a locally convex unital ∗-dg algebra satisfying Einstein causality: spacelike-separated observables commute at the level

• f cohomology.

Kasia Rejzner Time-ordered products and factorization algebras 19 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

dgA(Q)FT: Quantum

Definition A dg QFT model on a spacetime M is a functor A : Caus(M) → Alg∗(Ch(Nuc)), so that each A(O) is a locally convex unital ∗-dg algebra satisfying Einstein causality: spacelike-separated observables commute at the level

• f cohomology.

it satisfies the time-slice axiom if for any N ∈ Caus(M) a neighborhood of a Cauchy surface in the region O ∈ Caus(M), then the map A(N) → A(O) is a quasi-isomorphism.

Kasia Rejzner Time-ordered products and factorization algebras 19 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Peierls bracket

For M globally hyperbolic, P possesses unique retarded and advanced Green’s functions ∆R, ∆A.

Kasia Rejzner Time-ordered products and factorization algebras 20 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Peierls bracket

For M globally hyperbolic, P possesses unique retarded and advanced Green’s functions ∆R, ∆A. Their difference is the Pauli-Jordan function ∆ . = ∆R − ∆A.

Kasia Rejzner Time-ordered products and factorization algebras 20 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Peierls bracket

For M globally hyperbolic, P possesses unique retarded and advanced Green’s functions ∆R, ∆A. Their difference is the Pauli-Jordan function ∆ . = ∆R − ∆A. The Poisson bracket (Peierls bracket) of the free theory is ⌊F, G⌋ . =

• F(1), ∆G(1)

, for F, G local functions on E(M). supp f supp ∆A(f) supp ∆R(f)

Kasia Rejzner Time-ordered products and factorization algebras 20 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Peierls bracket

For M globally hyperbolic, P possesses unique retarded and advanced Green’s functions ∆R, ∆A. Their difference is the Pauli-Jordan function ∆ . = ∆R − ∆A. The Poisson bracket (Peierls bracket) of the free theory is ⌊F, G⌋ . =

• F(1), ∆G(1)

, for F, G local functions on E(M). supp f supp ∆A(f) supp ∆R(f) This structure extends to BV and for O ∈ Caus(M) we define P(O) := (BV(O), ⌊., .⌋O) as the dg classical filed theory model

• n M. The on-shell classical theory is obtained as

(H0(BV(O)), ⌊., .⌋O, ·), where · is the pointwise product.

Kasia Rejzner Time-ordered products and factorization algebras 20 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Free scalar field quantization

We define a ⋆-product (deformation quantization of the classical Poisson algebra): (F ⋆ G)(ϕ) . =

∞

• n=0

n n!

• F(n)(ϕ), (∆+)⊗nG(n)(ϕ)
• ,

where ∆+ = i 2∆ + H is of positive type and H is symmetric. Different choices of H correspond to different normal ordering.

Kasia Rejzner Time-ordered products and factorization algebras 21 / 36

(p)AQFT Factorization algebras Comparison Algebraic quantum field theory and its generalizations pAQFT

Free scalar field quantization

We define a ⋆-product (deformation quantization of the classical Poisson algebra): (F ⋆ G)(ϕ) . =

∞

• n=0

n n!

• F(n)(ϕ), (∆+)⊗nG(n)(ϕ)
• ,

where ∆+ = i 2∆ + H is of positive type and H is symmetric. Different choices of H correspond to different normal ordering. The free dg QFT model on M is defined by assigning to O ∈ Caus(M) the algebra A(O) := (v ◦ BV(O)[[]], ⋆, ∗), where ∗ is the complex conjugation.

Kasia Rejzner Time-ordered products and factorization algebras 21 / 36

(p)AQFT Factorization algebras Comparison

1

(p)AQFT Algebraic quantum field theory and its generalizations pAQFT

2

Factorization algebras

3

Comparison Statement of the main results Time-ordered products

Kasia Rejzner Time-ordered products and factorization algebras 22 / 36

(p)AQFT Factorization algebras Comparison

Prefactorization algebras I

A prefactorization algebra A on M with values in a symmetric monoidal category C⊗ consists of the following data: for each open U ⊂ M, an object A(U) ∈ C,

Kasia Rejzner Time-ordered products and factorization algebras 23 / 36

(p)AQFT Factorization algebras Comparison

Prefactorization algebras I

A prefactorization algebra A on M with values in a symmetric monoidal category C⊗ consists of the following data: for each open U ⊂ M, an object A(U) ∈ C, for each finite collection of pairwise disjoint opens U1, . . . , Un, with n > 0, and an open V containing every Ui, a morphism A({Ui}; V) : A(U1) ⊗ · · · ⊗ A(Un) → A(V),

Kasia Rejzner Time-ordered products and factorization algebras 23 / 36

(p)AQFT Factorization algebras Comparison

Prefactorization algebras II

. . . and satisfying the following conditions: composition is associative, so that the triangle

• i
• j

A(Tij)

• i

A(Ui) A(V) commutes for any collection {Ui}, as above, contained in V and for any collections {Tij}j where for each i, the opens {Tij}j are pairwise disjoint and each contained in Ui,

Kasia Rejzner Time-ordered products and factorization algebras 24 / 36

(p)AQFT Factorization algebras Comparison

Prefactorization algebras II

. . . and satisfying the following conditions: composition is associative, so that the triangle

• i
• j

A(Tij)

• i

A(Ui) A(V) commutes for any collection {Ui}, as above, contained in V and for any collections {Tij}j where for each i, the opens {Tij}j are pairwise disjoint and each contained in Ui, the morphisms A({Ui}; V) are equivariant under permutation of labels.

Kasia Rejzner Time-ordered products and factorization algebras 24 / 36

(p)AQFT Factorization algebras Comparison

Factorization algebras

A factorization algebra is a prefactorization algebra for which the value on bigger opens is determined by the values on smaller

• pens (local-to-global property)

Kasia Rejzner Time-ordered products and factorization algebras 25 / 36

(p)AQFT Factorization algebras Comparison

Factorization algebras

A factorization algebra is a prefactorization algebra for which the value on bigger opens is determined by the values on smaller

• pens (local-to-global property)

A key point is that we need to be able to reconstruct the “multiplication maps” from the local data, and so we an appropriate notion of cover that encodes the notion of being

• multilocal. These are the Weiss covers.

Kasia Rejzner Time-ordered products and factorization algebras 25 / 36

(p)AQFT Factorization algebras Comparison

Factorization algebras

A factorization algebra is a prefactorization algebra for which the value on bigger opens is determined by the values on smaller

• pens (local-to-global property)

A key point is that we need to be able to reconstruct the “multiplication maps” from the local data, and so we an appropriate notion of cover that encodes the notion of being

• multilocal. These are the Weiss covers.

The local to global property defining factorization algebras is essentially the property of being a sheaf with respect to Weiss covers.

Kasia Rejzner Time-ordered products and factorization algebras 25 / 36

(p)AQFT Factorization algebras Comparison

Models

A classical field theory model is a 1-shifted Poisson (aka P0) algebra P in factorization algebras FA(M, Ch(Nuc)). That is, to each open U ⊂ M, the cochain complex P(U) is equipped with a commutative product · and a degree 1 Poisson bracket {−, −}; moreover, each structure map is a map of shifted Poisson algebras.

Kasia Rejzner Time-ordered products and factorization algebras 26 / 36

(p)AQFT Factorization algebras Comparison

Models

A classical field theory model is a 1-shifted Poisson (aka P0) algebra P in factorization algebras FA(M, Ch(Nuc)). That is, to each open U ⊂ M, the cochain complex P(U) is equipped with a commutative product · and a degree 1 Poisson bracket {−, −}; moreover, each structure map is a map of shifted Poisson algebras. A quantum field theory model is a BD algebra A in factorization algebras FA(M, Ch(Nuc)). That is, to each open U ⊂ M, the cochain complex A(U) is flat over C[[]] and equipped with

Kasia Rejzner Time-ordered products and factorization algebras 26 / 36

(p)AQFT Factorization algebras Comparison

Models

A classical field theory model is a 1-shifted Poisson (aka P0) algebra P in factorization algebras FA(M, Ch(Nuc)). That is, to each open U ⊂ M, the cochain complex P(U) is equipped with a commutative product · and a degree 1 Poisson bracket {−, −}; moreover, each structure map is a map of shifted Poisson algebras. A quantum field theory model is a BD algebra A in factorization algebras FA(M, Ch(Nuc)). That is, to each open U ⊂ M, the cochain complex A(U) is flat over C[[]] and equipped with an -linear commutative product ·, an -linear, degree 1 Poisson bracket {−, −}, and a differential such that

Kasia Rejzner Time-ordered products and factorization algebras 26 / 36

(p)AQFT Factorization algebras Comparison

Models

A classical field theory model is a 1-shifted Poisson (aka P0) algebra P in factorization algebras FA(M, Ch(Nuc)). That is, to each open U ⊂ M, the cochain complex P(U) is equipped with a commutative product · and a degree 1 Poisson bracket {−, −}; moreover, each structure map is a map of shifted Poisson algebras. A quantum field theory model is a BD algebra A in factorization algebras FA(M, Ch(Nuc)). That is, to each open U ⊂ M, the cochain complex A(U) is flat over C[[]] and equipped with an -linear commutative product ·, an -linear, degree 1 Poisson bracket {−, −}, and a differential such that d(a · b) = d(a) · b + (−1)aa · d(b) + {a, b}

Kasia Rejzner Time-ordered products and factorization algebras 26 / 36

(p)AQFT Factorization algebras Comparison

Models

A classical field theory model is a 1-shifted Poisson (aka P0) algebra P in factorization algebras FA(M, Ch(Nuc)). That is, to each open U ⊂ M, the cochain complex P(U) is equipped with a commutative product · and a degree 1 Poisson bracket {−, −}; moreover, each structure map is a map of shifted Poisson algebras. A quantum field theory model is a BD algebra A in factorization algebras FA(M, Ch(Nuc)). That is, to each open U ⊂ M, the cochain complex A(U) is flat over C[[]] and equipped with an -linear commutative product ·, an -linear, degree 1 Poisson bracket {−, −}, and a differential such that d(a · b) = d(a) · b + (−1)aa · d(b) + {a, b} Moreover, each structure map is a map of BD algebras.

Kasia Rejzner Time-ordered products and factorization algebras 26 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

1

(p)AQFT Algebraic quantum field theory and its generalizations pAQFT

2

Factorization algebras

3

Comparison Statement of the main results Time-ordered products

Kasia Rejzner Time-ordered products and factorization algebras 27 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Comparison I

Comparing nets and factorization algebras of observables: the free scalar field, O. Gwilliam, KR, CMP 2020.

Kasia Rejzner Time-ordered products and factorization algebras 28 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Comparison I

Comparing nets and factorization algebras of observables: the free scalar field, O. Gwilliam, KR, CMP 2020. There is a natural transformation ιcl : c ◦ P|Caus(M) ⇒ c ◦ P

• f functors to commutative dg algebras CAlg(Ch(Nuc)).

Kasia Rejzner Time-ordered products and factorization algebras 28 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Comparison I

Comparing nets and factorization algebras of observables: the free scalar field, O. Gwilliam, KR, CMP 2020. There is a natural transformation ιcl : c ◦ P|Caus(M) ⇒ c ◦ P

• f functors to commutative dg algebras CAlg(Ch(Nuc)).

This natural transformation is a quasi-isomorphism.

Kasia Rejzner Time-ordered products and factorization algebras 28 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Comparison I

Comparing nets and factorization algebras of observables: the free scalar field, O. Gwilliam, KR, CMP 2020. There is a natural transformation ιcl : c ◦ P|Caus(M) ⇒ c ◦ P

• f functors to commutative dg algebras CAlg(Ch(Nuc)).

This natural transformation is a quasi-isomorphism. There is a relation between the shifted Poisson bracket {., .} (identified with the antibracket) and the Peierls bracket ⌊., .⌋: ⌊Φ(f), Φ(g)⌋ = {σ(Φ(f)), Φ(g)} , where σ : PV0 → PV1 is given by: σ(Φ(f)) = ιdΦ(f)∆ =

• ∆(y, x)f(x)

δ δϕ(y) .

Kasia Rejzner Time-ordered products and factorization algebras 28 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Comparison II

There is a natural transformation ιq : v ◦ A|Caus(M) ⇒ v ◦ A

• f functors to Ch(Nuc), and this natural transformation is a

quasi-isomorphism.

Kasia Rejzner Time-ordered products and factorization algebras 29 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Comparison II

There is a natural transformation ιq : v ◦ A|Caus(M) ⇒ v ◦ A

• f functors to Ch(Nuc), and this natural transformation is a

quasi-isomorphism. In fact, on each O ∈ Caus(M), the map ιq is given in terms of an isomorphism of cochain complexes T : A(O)

∼ =

− → v ◦ P(O)[[]] .

Kasia Rejzner Time-ordered products and factorization algebras 29 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Comparison II

There is a natural transformation ιq : v ◦ A|Caus(M) ⇒ v ◦ A

• f functors to Ch(Nuc), and this natural transformation is a

quasi-isomorphism. In fact, on each O ∈ Caus(M), the map ιq is given in terms of an isomorphism of cochain complexes T : A(O)

∼ =

− → v ◦ P(O)[[]] . T is the time-ordering operator, related to time-ordered products. Reconstruction of the star product from the factorisation product (provided A is locally constant in time direction) proceeds by the way of time-ordered products.

Kasia Rejzner Time-ordered products and factorization algebras 29 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Time-ordering

In pAQFT, the time-ordering operator T is defined by: T F(ϕ) . =

∞

• n=0
• F(2n)(ϕ), (∆F)⊗n

, where ∆F = i 2(∆A + ∆R) + H.

Kasia Rejzner Time-ordered products and factorization algebras 30 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Time-ordering

In pAQFT, the time-ordering operator T is defined by: T F(ϕ) . =

∞

• n=0
• F(2n)(ϕ), (∆F)⊗n

, where ∆F = i 2(∆A + ∆R) + H. Formally T corresponds to the operator of convolution with the

• scillating Gaussian measure “with covariance i∆F”,

T F(ϕ) formal =

• F(ϕ − φ) dµi∆F(φ) .

Kasia Rejzner Time-ordered products and factorization algebras 30 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Time-ordering

In pAQFT, the time-ordering operator T is defined by: T F(ϕ) . =

∞

• n=0
• F(2n)(ϕ), (∆F)⊗n

, where ∆F = i 2(∆A + ∆R) + H. Formally T corresponds to the operator of convolution with the

• scillating Gaussian measure “with covariance i∆F”,

T F(ϕ) formal =

• F(ϕ − φ) dµi∆F(φ) .

Define the time-ordered product ·T on PV[[]] by: F ·T G . = T (T −1F · T −1G)

Kasia Rejzner Time-ordered products and factorization algebras 30 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Interacting regular observables

Interaction is a functional V ∈ PV. Using the commutative product ·T we define the S-matrix: S(V) . = eiV/

T

= T (eT −1iV/) .

Kasia Rejzner Time-ordered products and factorization algebras 31 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Interacting regular observables

Interaction is a functional V ∈ PV. Using the commutative product ·T we define the S-matrix: S(V) . = eiV/

T

= T (eT −1iV/) . Interacting fields are defined by the formula of Bogoliubov: RV(F) . = (eiV/

T

)⋆−1 ⋆ (eiV/

T

·T F) .

Kasia Rejzner Time-ordered products and factorization algebras 31 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Interacting regular observables

Interaction is a functional V ∈ PV. Using the commutative product ·T we define the S-matrix: S(V) . = eiV/

T

= T (eT −1iV/) . Interacting fields are defined by the formula of Bogoliubov: RV(F) . = (eiV/

T

)⋆−1 ⋆ (eiV/

T

·T F) . We define the interacting star product as: F ⋆int G . = R−1

V (RV(F) ⋆ RV(G)) ,

Kasia Rejzner Time-ordered products and factorization algebras 31 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Details of the comparison result I

Classical theory is straightforward, since in this case the underlying algebra of both P|Caus(M) and P is PV and the differentials coincide, so ιcl is a trivial identification.

Kasia Rejzner Time-ordered products and factorization algebras 32 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Details of the comparison result I

Classical theory is straightforward, since in this case the underlying algebra of both P|Caus(M) and P is PV and the differentials coincide, so ιcl is a trivial identification. In the quantum case, for the free theory we have A = (v ◦ PV[[]], ⋆, δS), together with ·T .

Kasia Rejzner Time-ordered products and factorization algebras 32 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Details of the comparison result I

Classical theory is straightforward, since in this case the underlying algebra of both P|Caus(M) and P is PV and the differentials coincide, so ιcl is a trivial identification. In the quantum case, for the free theory we have A = (v ◦ PV[[]], ⋆, δS), together with ·T . We already know that T −1 intertwines the products, so we upgrade it to a map of chain complexes: (PV[[]], ·T , δS) T −1 − − → (PV[[]], ·,ˆ s0) , where ˆ s0 . = T −1 ◦ δS ◦ T is the quantum BV operator.

Kasia Rejzner Time-ordered products and factorization algebras 32 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Details of the comparison result I

Classical theory is straightforward, since in this case the underlying algebra of both P|Caus(M) and P is PV and the differentials coincide, so ιcl is a trivial identification. In the quantum case, for the free theory we have A = (v ◦ PV[[]], ⋆, δS), together with ·T . We already know that T −1 intertwines the products, so we upgrade it to a map of chain complexes: (PV[[]], ·T , δS) T −1 − − → (PV[[]], ·,ˆ s0) , where ˆ s0 . = T −1 ◦ δS ◦ T is the quantum BV operator. It can be written as ˆ s0 = {., S} − i△ , where △ is the BV Laplacian (divergence on V, extended to PV with appropriate signs) and {., .} is the antibracket.

Kasia Rejzner Time-ordered products and factorization algebras 32 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Details of the comparison result II

We have ˆ s0(X ∧ Y) = (−1)deg Yˆ s0X ∧ Y + X ∧ ˆ s0Y − i{X, Y}

Kasia Rejzner Time-ordered products and factorization algebras 33 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Details of the comparison result II

We have ˆ s0(X ∧ Y) = (−1)deg Yˆ s0X ∧ Y + X ∧ ˆ s0Y − i{X, Y} (PV[[]], ·,ˆ s0, {., .}) is the BD (Beilinson-Drinfeld) algebra A considered by Costello and Gwilliam as the space of quantum

• bservables for the free scalar field.

Kasia Rejzner Time-ordered products and factorization algebras 33 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Details of the comparison result II

We have ˆ s0(X ∧ Y) = (−1)deg Yˆ s0X ∧ Y + X ∧ ˆ s0Y − i{X, Y} (PV[[]], ·,ˆ s0, {., .}) is the BD (Beilinson-Drinfeld) algebra A considered by Costello and Gwilliam as the space of quantum

• bservables for the free scalar field.

We have thus demonstrated that T −1 : v ◦ P[[]]

∼ =

− → v ◦ A .

Kasia Rejzner Time-ordered products and factorization algebras 33 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Details of the comparison result II

We have ˆ s0(X ∧ Y) = (−1)deg Yˆ s0X ∧ Y + X ∧ ˆ s0Y − i{X, Y} (PV[[]], ·,ˆ s0, {., .}) is the BD (Beilinson-Drinfeld) algebra A considered by Costello and Gwilliam as the space of quantum

• bservables for the free scalar field.

We have thus demonstrated that T −1 : v ◦ P[[]]

∼ =

− → v ◦ A . Since the underlying algebras of P|Caus(M) and P coincide and v ◦ P[[]] coincides with v ◦ A, we obtain: v ◦ A|Caus(M)

T

− → v ◦ A .

Kasia Rejzner Time-ordered products and factorization algebras 33 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Details of the comparison result II

We have ˆ s0(X ∧ Y) = (−1)deg Yˆ s0X ∧ Y + X ∧ ˆ s0Y − i{X, Y} (PV[[]], ·,ˆ s0, {., .}) is the BD (Beilinson-Drinfeld) algebra A considered by Costello and Gwilliam as the space of quantum

• bservables for the free scalar field.

We have thus demonstrated that T −1 : v ◦ P[[]]

∼ =

− → v ◦ A . Since the underlying algebras of P|Caus(M) and P coincide and v ◦ P[[]] coincides with v ◦ A, we obtain: v ◦ A|Caus(M)

T

− → v ◦ A . Hence we have explicitly constructed ιq ≡ T .

Kasia Rejzner Time-ordered products and factorization algebras 33 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Details of the comparison result III

In the interacting theory, with interaction V, we have (PV[[]], ⋆int) as a further deformation of (PV[[]], ⋆) by means

• f RV.

Kasia Rejzner Time-ordered products and factorization algebras 34 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Details of the comparison result III

In the interacting theory, with interaction V, we have (PV[[]], ⋆int) as a further deformation of (PV[[]], ⋆) by means

• f RV.

Defining the interacting BV differential by ˆ sV . = R−1

V ◦ δS ◦ RV,

we obtain: (PV[[]], ⋆, δS)

R−1

V

− − → (PV[[]], ⋆int,ˆ sV)

Kasia Rejzner Time-ordered products and factorization algebras 34 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Details of the comparison result III

In the interacting theory, with interaction V, we have (PV[[]], ⋆int) as a further deformation of (PV[[]], ⋆) by means

• f RV.

Defining the interacting BV differential by ˆ sV . = R−1

V ◦ δS ◦ RV,

we obtain: (PV[[]], ⋆, δS)

R−1

V

− − → (PV[[]], ⋆int,ˆ sV) Assume the following: δS(S(V)) = 0. This can be also written as 1 2{S + V, S + V} − i △ (S + V) = 0 and is called quantum master equation (QME).

Kasia Rejzner Time-ordered products and factorization algebras 34 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Details of the comparison result III

In the interacting theory, with interaction V, we have (PV[[]], ⋆int) as a further deformation of (PV[[]], ⋆) by means

• f RV.

Defining the interacting BV differential by ˆ sV . = R−1

V ◦ δS ◦ RV,

we obtain: (PV[[]], ⋆, δS)

R−1

V

− − → (PV[[]], ⋆int,ˆ sV) Assume the following: δS(S(V)) = 0. This can be also written as 1 2{S + V, S + V} − i △ (S + V) = 0 and is called quantum master equation (QME). Assuming QME, we have ˆ sV = {., S + V} − i△, so (PV[[]], ⋆int,ˆ sV, △) is the BD algebra that Costello and Gwilliam assign to the interacting quantum theory.

Kasia Rejzner Time-ordered products and factorization algebras 34 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Relation between the approaches (summary)

Bottom line: In pAQFT we deform the product, while in CG approach one deforms the differential. Both viewpoints are shown to be equivalent, using the maps:

Kasia Rejzner Time-ordered products and factorization algebras 35 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Relation between the approaches (summary)

Bottom line: In pAQFT we deform the product, while in CG approach one deforms the differential. Both viewpoints are shown to be equivalent, using the maps: T in the free case.

Kasia Rejzner Time-ordered products and factorization algebras 35 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Relation between the approaches (summary)

Bottom line: In pAQFT we deform the product, while in CG approach one deforms the differential. Both viewpoints are shown to be equivalent, using the maps: T in the free case. RV in the interacting case.

Kasia Rejzner Time-ordered products and factorization algebras 35 / 36

(p)AQFT Factorization algebras Comparison Statement of the main results Time-ordered products

Thank you very much for your attention! (and see you in person one day...)

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