(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Categories II Category of spacetimes Let Loc n be the category where Kasia Rejzner Time-ordered products and factorization algebras 8 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Categories II Category of spacetimes Let Loc n 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Categories II Category of spacetimes Let Loc n 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Categories II Category of spacetimes Let Loc n 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Categories II Category of spacetimes Let Loc n 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 Loc n → Alg ∗ ( Nuc ) . Kasia Rejzner Time-ordered products and factorization algebras 8 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Categories II Category of spacetimes Let Loc n 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 Loc n → Alg ∗ ( Nuc ) . We consider Caus ( M ) , a subcategory of Loc n 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 of Haag-Kastler with formal deformation quantization and homological algebra. Kasia Rejzner Time-ordered products and factorization algebras 9 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 of 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 of 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Physical input A spacetime M = ( M , g ) ∈ Obj ( Loc n ) . Kasia Rejzner Time-ordered products and factorization algebras 10 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Physical input A spacetime M = ( M , g ) ∈ Obj ( Loc n ) . 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Physical input A spacetime M = ( M , g ) ∈ Obj ( Loc n ) . 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 Loc n to Nuc , which acts on objects as above and the morphisms χ : M → N are mapped to E χ ≡ χ ∗ . Kasia Rejzner Time-ordered products and factorization algebras 10 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Physical input A spacetime M = ( M , g ) ∈ Obj ( Loc n ) . 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 Loc n to Nuc , which acts on objects as above and the morphisms χ : M → N are mapped to E χ ≡ 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Physical input A spacetime M = ( M , g ) ∈ Obj ( Loc n ) . 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 Loc n to Nuc , which acts on objects as above and the morphisms χ : M → N are mapped to E χ ≡ 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Physical input A spacetime M = ( M , g ) ∈ Obj ( Loc n ) . 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 Loc n to Nuc , which acts on objects as above and the morphisms χ : M → N are mapped to E χ ≡ 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Physical input A spacetime M = ( M , g ) ∈ Obj ( Loc n ) . 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 Loc n to Nuc , which acts on objects as above and the morphisms χ : M → N are mapped to E χ ≡ 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 Loc n to Nuc . Kasia Rejzner Time-ordered products and factorization algebras 11 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 Loc n 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 Loc n 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 Loc n 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Functionals I We define F loc , local functionals on E , as functionals that are of the form � f ( j k F ( ϕ ) = x ( ϕ )) d µ g , for a smooth, compactly supported, function f on the jet bundle. Kasia Rejzner Time-ordered products and factorization algebras 12 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Functionals I We define F loc , local functionals on E , as functionals that are of the form � f ( j k F ( ϕ ) = x ( ϕ )) d µ g , 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Functionals I We define F loc , local functionals on E , as functionals that are of the form � f ( j k F ( ϕ ) = x ( ϕ )) d µ g , 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Functionals I We define F loc , local functionals on E , as functionals that are of the form � f ( j k F ( ϕ ) = x ( ϕ )) d µ g , 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 : Loc n → CAlg ( Nuc ) . Kasia Rejzner Time-ordered products and factorization algebras 12 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Functionals II The simplest examples of functionals are smeared fields � f ∈ D ( M ) := C ∞ Φ( f )( ϕ ) = ϕ ( x ) f ( x ) d µ ( x ) , c ( M , R ) Kasia Rejzner Time-ordered products and factorization algebras 13 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Functionals II The simplest examples of functionals are smeared fields � f ∈ D ( M ) := C ∞ Φ( f )( ϕ ) = ϕ ( x ) f ( x ) d µ ( x ) , c ( M , R ) Note that regular, polynomial functionals of degree 2 and higher are not local. Take for example � f ( x , y ) ϕ ( x ) ϕ ( y ) d µ ( x ) d µ ( y ) , f ∈ D ( M 2 ) := C ∞ c ( M 2 , R ) . F ( ϕ ) = Kasia Rejzner Time-ordered products and factorization algebras 13 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Functionals II The simplest examples of functionals are smeared fields � f ∈ D ( M ) := C ∞ Φ( f )( ϕ ) = ϕ ( x ) f ( x ) d µ ( x ) , c ( M , R ) Note that regular, polynomial functionals of degree 2 and higher are not local. Take for example � f ( x , y ) ϕ ( x ) ϕ ( y ) d µ ( x ) d µ ( y ) , f ∈ D ( M 2 ) := C ∞ c ( M 2 , R ) . F ( ϕ ) = Now take f ∈ D ( M ) and consider � � f ϕ 2 d µ = f ( x ) δ ( x − y ) ϕ ( x ) ϕ ( y ) d µ ( x ) d µ ( y ) . F ( ϕ ) = Kasia Rejzner Time-ordered products and factorization algebras 13 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Functionals II The simplest examples of functionals are smeared fields � f ∈ D ( M ) := C ∞ Φ( f )( ϕ ) = ϕ ( x ) f ( x ) d µ ( x ) , c ( M , R ) Note that regular, polynomial functionals of degree 2 and higher are not local. Take for example � f ( x , y ) ϕ ( x ) ϕ ( y ) d µ ( x ) d µ ( y ) , f ∈ D ( M 2 ) := C ∞ c ( M 2 , R ) . F ( ϕ ) = Now take f ∈ D ( M ) and consider � � f ϕ 2 d µ = f ( x ) δ ( x − y ) ϕ ( x ) ϕ ( y ) d µ ( x ) d µ ( y ) . F ( ϕ ) = To avoid technical analytic issues, I will formulate the rest of this introduction for F . However, all of this generalizes to F loc . Kasia Rejzner Time-ordered products and factorization algebras 13 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Dynamics Dynamics is introduced by a generalized action S , a natural transformation S : D → F loc , 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Dynamics Dynamics is introduced by a generalized action S , a natural transformation S : D → F loc , where D ( M ) = C ∞ c ( M , R ) and D acts on morphisms by pushforward. For example for the free scalar field: � 2 ϕ 2 + 1 � � 2 ∇ µ ϕ ∇ µ ϕ 1 S M ( f )[ ϕ ] = fd µ . M Kasia Rejzner Time-ordered products and factorization algebras 14 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Dynamics Dynamics is introduced by a generalized action S , a natural transformation S : D → F loc , where D ( M ) = C ∞ c ( M , R ) and D acts on morphisms by pushforward. For example for the free scalar field: � 2 ϕ 2 + 1 � � 2 ∇ µ ϕ ∇ µ ϕ 1 S M ( f )[ ϕ ] = fd µ . M The Euler-Lagrange derivative of S is denoted by dS and defined � � S M ( f ) ( 1 ) [ ϕ ] , ψ by � dS M ( ϕ ) , ψ � = , where f ≡ 1 on supp ψ , ψ ∈ D ( M ) . The field equation is: dS M ( ϕ ) = 0, so geometrically, the solution space is the zero locus of dS M (seen as a 1-form supp ( f ) on E ( M ) ). M supp ( ψ ) f ≡ 1 Kasia Rejzner Time-ordered products and factorization algebras 14 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Dynamics Dynamics is introduced by a generalized action S , a natural transformation S : D → F loc , where D ( M ) = C ∞ c ( M , R ) and D acts on morphisms by pushforward. For example for the free scalar field: � 2 ϕ 2 + 1 � � 2 ∇ µ ϕ ∇ µ ϕ 1 S M ( f )[ ϕ ] = fd µ . M The Euler-Lagrange derivative of S is denoted by dS and defined � � S M ( f ) ( 1 ) [ ϕ ] , ψ by � dS M ( ϕ ) , ψ � = , where f ≡ 1 on supp ψ , ψ ∈ D ( M ) . The field equation is: dS M ( ϕ ) = 0, so geometrically, the solution space is the zero locus of dS M (seen as a 1-form supp ( f ) on E ( M ) ). M supp ( ψ ) Again, we drop M from notation f ≡ 1 when no confusion arises. Kasia Rejzner Time-ordered products and factorization algebras 14 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Symmetries We use the BV framework, where symmetries are identified with vector fields (directions) on E . C ϕ F E ( M ) Kasia Rejzner Time-ordered products and factorization algebras 15 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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. C ϕ F E ( M ) Kasia Rejzner Time-ordered products and factorization algebras 15 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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: ∂ X F ( ϕ ) := � F ( 1 ) ( ϕ ) , X ( ϕ ) � C ϕ F E ( M ) Kasia Rejzner Time-ordered products and factorization algebras 15 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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: ∂ X F ( 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 ) � =: δ S ( X )( ϕ ) . C ϕ F E ( M ) Kasia Rejzner Time-ordered products and factorization algebras 15 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Free scalar field (classical) E = C ∞ ( M , R ) and the equation of motion is dS ( ϕ ) = P ϕ = 0, where P = − ( ✷ + m 2 ) . Kasia Rejzner Time-ordered products and factorization algebras 16 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Free scalar field (classical) E = C ∞ ( M , R ) and the equation of motion is dS ( ϕ ) = P ϕ = 0, where P = − ( ✷ + m 2 ) . Space of solutions: E S ⊂ E . Denote functionals that vanish on E S by F 0 . They are of the form: δ S ( X ) for some X ∈ V . Kasia Rejzner Time-ordered products and factorization algebras 16 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Free scalar field (classical) E = C ∞ ( M , R ) and the equation of motion is dS ( ϕ ) = P ϕ = 0, where P = − ( ✷ + m 2 ) . Space of solutions: E S ⊂ E . Denote functionals that vanish on E S by F 0 . They are of the form: δ S ( X ) for some X ∈ V . The space of on-shell observables (i.e. functionals on E S ) F S is the quotient F S = F / F 0 . Kasia Rejzner Time-ordered products and factorization algebras 16 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Free scalar field (classical) E = C ∞ ( M , R ) and the equation of motion is dS ( ϕ ) = P ϕ = 0, where P = − ( ✷ + m 2 ) . Space of solutions: E S ⊂ E . Denote functionals that vanish on E S by F 0 . They are of the form: δ S ( X ) for some X ∈ V . The space of on-shell observables (i.e. functionals on E S ) F S is the quotient F S = F / F 0 . δ S is called the Koszul differential. Symmetries constitute its kernel. Kasia Rejzner Time-ordered products and factorization algebras 16 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Free scalar field (classical) E = C ∞ ( M , R ) and the equation of motion is dS ( ϕ ) = P ϕ = 0, where P = − ( ✷ + m 2 ) . Space of solutions: E S ⊂ E . Denote functionals that vanish on E S by F 0 . They are of the form: δ S ( X ) for some X ∈ V . The space of on-shell observables (i.e. functionals on E S ) F S is the quotient F S = F / F 0 . δ S is called the Koszul differential. Symmetries constitute its kernel. δ S We obtain a sequence: 0 → Sym ֒ → V − → F → 0. Kasia Rejzner Time-ordered products and factorization algebras 16 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Free scalar field (classical) E = C ∞ ( M , R ) and the equation of motion is dS ( ϕ ) = P ϕ = 0, where P = − ( ✷ + m 2 ) . Space of solutions: E S ⊂ E . Denote functionals that vanish on E S by F 0 . They are of the form: δ S ( X ) for some X ∈ V . The space of on-shell observables (i.e. functionals on E S ) F S is the quotient F S = F / F 0 . δ S is called the Koszul differential. Symmetries constitute its kernel. δ S We obtain a sequence: 0 → Sym ֒ → V − → F → 0. In this talk, I discuss only the case where there are no non-trivial (not vanishing on E S ) local symmetries, Kasia Rejzner Time-ordered products and factorization algebras 16 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Free scalar field (classical) E = C ∞ ( M , R ) and the equation of motion is dS ( ϕ ) = P ϕ = 0, where P = − ( ✷ + m 2 ) . Space of solutions: E S ⊂ E . Denote functionals that vanish on E S by F 0 . They are of the form: δ S ( X ) for some X ∈ V . The space of on-shell observables (i.e. functionals on E S ) F S is the quotient F S = F / F 0 . δ S is called the Koszul differential. Symmetries constitute its kernel. δ S We obtain a sequence: 0 → Sym ֒ → V − → F → 0. In this talk, I discuss only the case where there are no non-trivial (not vanishing on E S ) local symmetries, Introduce the BV complex: BV . = ( PV , δ S ) . Then the space of classical on-shell observables is given by F S = H 0 ( BV ) and higher cohomology groups vanish. Kasia Rejzner Time-ordered products and factorization algebras 16 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 } = ∂ X F , F ∈ F , X ∈ V , Kasia Rejzner Time-ordered products and factorization algebras 17 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 } = ∂ X F , 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 } = ∂ X F , F ∈ F , X ∈ V , It satisfies the graded Leibniz rule: { X , Y ∧ Z } = { X , Y } ∧ Z + ( − 1 ) | Y | ( | X | + 1 ) Y ∧ { X , Z } . Derivation δ S is not inner with respect to { ., . } , but locally it can be written as δ S X = { X , S ( f ) } for f ≡ 1 on supp X , X ∈ V . Kasia Rejzner Time-ordered products and factorization algebras 17 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison dgA(Q)FT: Classical For convenience, we will from now on restrict our category Loc n 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison dgA(Q)FT: Classical For convenience, we will from now on restrict our category Loc n 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 of cohomology. Kasia Rejzner Time-ordered products and factorization algebras 19 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 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 A ( N ) → A ( O ) is a quasi-isomorphism. Kasia Rejzner Time-ordered products and factorization algebras 19 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Peierls bracket For M globally hyperbolic, P possesses unique retarded and advanced Green’s functions ∆ R , ∆ A . supp ∆ R ( f ) Their difference is the Pauli-Jordan function ∆ . = ∆ R − ∆ A . The Poisson bracket (Peierls bracket) of the supp f free theory is ⌊ F , G ⌋ . � F ( 1 ) , ∆ G ( 1 ) � = , supp ∆ A ( f ) for F , G local functions on E ( M ) . Kasia Rejzner Time-ordered products and factorization algebras 20 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Peierls bracket For M globally hyperbolic, P possesses unique retarded and advanced Green’s functions ∆ R , ∆ A . supp ∆ R ( f ) Their difference is the Pauli-Jordan function ∆ . = ∆ R − ∆ A . The Poisson bracket (Peierls bracket) of the supp f free theory is ⌊ F , G ⌋ . � F ( 1 ) , ∆ G ( 1 ) � = , supp ∆ A ( f ) for F , G local functions on E ( M ) . This structure extends to BV and for O ∈ Caus ( M ) we define P ( O ) := ( BV ( O ) , ⌊ ., . ⌋ O ) as the dg classical filed theory model on M . The on-shell classical theory is obtained as ( H 0 ( BV ( O )) , ⌊ ., . ⌋ O , · ) , where · is the pointwise product. Kasia Rejzner Time-ordered products and factorization algebras 20 / 36
(p)AQFT Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Free scalar field quantization We define a ⋆ -product (deformation quantization of the classical Poisson algebra): ∞ � n ( F ⋆ G )( ϕ ) . � � � F ( n ) ( ϕ ) , (∆ + ) ⊗ n G ( n ) ( ϕ ) = , n ! n = 0 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 Algebraic quantum field theory and its generalizations Factorization algebras pAQFT Comparison Free scalar field quantization We define a ⋆ -product (deformation quantization of the classical Poisson algebra): ∞ � n ( F ⋆ G )( ϕ ) . � � � F ( n ) ( ϕ ) , (∆ + ) ⊗ n G ( n ) ( ϕ ) = , n ! n = 0 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 (p)AQFT 1 Algebraic quantum field theory and its generalizations pAQFT Factorization algebras 2 Comparison 3 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 U 1 , . . . , U n , with n > 0, and an open V containing every U i , a morphism A ( { U i } ; V ) : A ( U 1 ) ⊗ · · · ⊗ A ( U n ) → 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 � � � A ( T ij ) A ( U i ) i j i commutes for A ( V ) any collection { U i } , as above, contained in V and for any collections { T ij } j where for each i , the opens { T ij } j are pairwise disjoint and each contained in U i , 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 � � � A ( T ij ) A ( U i ) i j i commutes for A ( V ) any collection { U i } , as above, contained in V and for any collections { T ij } j where for each i , the opens { T ij } j are pairwise disjoint and each contained in U i , the morphisms A ( { U i } ; 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 opens (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 opens (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 opens (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 P 0 ) 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 P 0 ) 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 P 0 ) 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 P 0 ) 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 ) a a · 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 P 0 ) 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 ) a a · 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 Statement of the main results Factorization algebras Time-ordered products Comparison (p)AQFT 1 Algebraic quantum field theory and its generalizations pAQFT Factorization algebras 2 Comparison 3 Statement of the main results Time-ordered products Kasia Rejzner Time-ordered products and factorization algebras 27 / 36
(p)AQFT Statement of the main results Factorization algebras Time-ordered products Comparison 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 Statement of the main results Factorization algebras Time-ordered products Comparison Comparison I Comparing nets and factorization algebras of observables: the free scalar field , O. Gwilliam, KR, CMP 2020 . There is a natural transformation ι cl : c ◦ P| Caus ( M ) ⇒ c ◦ P of functors to commutative dg algebras CAlg ( Ch ( Nuc )) . Kasia Rejzner Time-ordered products and factorization algebras 28 / 36
(p)AQFT Statement of the main results Factorization algebras Time-ordered products Comparison Comparison I Comparing nets and factorization algebras of observables: the free scalar field , O. Gwilliam, KR, CMP 2020 . There is a natural transformation ι cl : c ◦ P| Caus ( M ) ⇒ c ◦ P of 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
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