Model-independent comparison between factorization algebras and algebraic quantum field theory on Lorentzian manifolds

This paper investigates the relationship between algebraic quantum field theories and factorization algebras on globally hyperbolic Lorentzian manifolds. Functorial constructions that map between these two types of theories in both directions are developed under certain natural hypotheses, including suitable variants of the local constancy and descent axioms. The main result is an equivalence theorem between (Cauchy constant and additive) algebraic quantum field theories and (Cauchy constant, additive and time-orderable) prefactorization algebras.


Introduction and summary
Factorization algebras and algebraic quantum field theory are two mathematical frameworks to axiomatize the algebraic structure of observables in a quantum field theory. While from a superficial point of view these two approaches look similar, there are subtle differences. A prefactorization algebra F assigns to each spacetime M a vector space F(M ) of observables and to each tuple f = (f 1 : M 1 → N, . . . , f n : M n → N ) of pairwise disjoint spacetime embeddings a factorization product F(f ) : n i=1 F(M i ) → F(N ) satisfying suitable properties, cf. [CG17] and Section 2.2. On the other hand, an algebraic quantum field theory A assigns to each spacetime M an associative and unital algebra A(M ) of observables and to each spacetime embedding f : M → N an algebra morphism A(f ) : A(M ) → A(N ) such that suitable axioms hold true, cf. [BFV03,FV12,BDFY15,BSW17] and Section 2.3. The main difference is that, in contrast to an algebraic quantum field theory A, a prefactorization algebra F does not in general come endowed with a multiplication of observables in In this paper we shall develop functorial constructions (cf. Theorems 3.11 and 4.7) that allow us to relate prefactorization algebras and algebraic quantum field theories, provided that we assume certain natural hypotheses on both sides. We shall focus on the case where spacetimes are described by oriented and time-oriented globally hyperbolic Lorentzian manifolds, i.e. on the case of relativistic quantum field theory. Our main result is an equivalence theorem between (Cauchy constant and additive) algebraic quantum field theories and (Cauchy constant, additive and time-orderable) prefactorization algebras, cf. Theorem 5.1. Our equivalence theorem is considerably stronger and more general than the earlier comparison result by Gwilliam and Rejzner [GR17]: (1) We work in a model-independent setup, supplemented by natural additional hypotheses such as Cauchy constancy, additivity and time-orderability, while [GR17] only studies the example of a free Klein-Gordon field. (2) We investigate in detail uniqueness, associativity, naturality and Einstein causality of the multiplications µ M : F(M ) ⊗ F(M ) → F(M ) determined by a Cauchy constant additive prefactorization algebra F, which requires rather sophisticated arguments from Lorentzian geometry. These questions were not addressed in [GR17]. (3) Our equivalence theorem admits an interpretation in terms of operad theory (cf. Remark 5.2), which provides a suitable starting point for generalizations to higher categorical quantum field theories [CG17, BSS15, BS17, BSW19b, BS19] such as gauge theories. (The present paper does not study this generalization and will focus on the case of 1-categorical quantum field theories.) Let us now explain in more detail our constructions and results while outlining the content of the present paper: In Section 2 we recall the necessary preliminaries from Lorentzian geometry, factorization algebras and algebraic quantum field theory. All prefactorization algebras and algebraic quantum field theories will be defined on the usual category Loc of oriented and timeoriented globally hyperbolic Lorentzian manifolds. We introduce an additivity axiom for both prefactorization algebras and algebraic quantum field theories, which roughly speaking demands that the observables in a spacetime M are generated by the observables in the relatively compact and causally convex open subsets U ⊆ M . It is shown that factorization algebras, i.e. prefactorization algebras satisfying Weiss descent, are in particular additive prefactorization algebras. We also introduce a Cauchy constancy (or time-slice) axiom for both kinds of theories, which formalizes a concept of time evolution in a globally hyperbolic Lorentzian manifold. In Section 3 we construct a functor A (−) : PFA add,c → AQFT add,c that assigns a Cauchy constant additive algebraic quantum field theory A F to each Cauchy constant additive prefactorization algebra F, see Theorem 3.11 for the main result. The crucial step is to define canonical multiplications µ M : F(M ) ⊗ F(M ) → F(M ) for such F (cf. (3.1)), which is done by using Cauchy constancy. Proving naturality and Einstein causality of these multiplications requires the additivity axiom, cf. Propositions 3.7 and 3.10. In Section 4 we construct a functor F (−) : AQFT → tPFA that assigns a time-orderable prefactorization algebra F A to each algebraic quantum field the-ory A, see Theorem 4.7 for the main result. The difference between time-orderable and ordinary prefactorization algebras on Loc is that the former just encode factorization products F(f ) : n i=1 F(M i ) → N for tuples of pairwise disjoint morphisms f that are in a suitable sense time-orderable, see Definition 4.1. There is a natural forgetful functor PFA → tPFA from ordinary to time-orderable prefactorization algebras, which is however not full, see Remarks 4.2 and 4.5. Our results suggest that the concept of time-orderable prefactorization algebras from Section 4 is better suited to the category of Lorentzian spacetimes Loc than the more naive concept from Section 2.2 that allows also for factorization products for non-time-orderable tuples of pairwise disjoint morphisms. In Section 5 we explain that the construction A (−) : PFA add,c → AQFT add,c from Section 3 factors through the forgetful functor PFA add,c → tPFA add,c , thereby defining a functor A (−) : tPFA add,c → AQFT add,c that assigns a Cauchy constant additive algebraic quantum field theory to each Cauchy constant additive time-orderable prefactorization algebra. Our main Equivalence Theorem 5.1 proves that this functor admits an inverse that is given by the restriction F (−) : AQFT add,c → tPFA add,c of the functor from Section 4 to Cauchy constant and additive theories. Hence, Cauchy constant additive algebraic quantum field theories are naturally identified with Cauchy constant additive time-orderable prefactorization algebras. In Section 5.2, we apply our general results to the simple example given by the free Klein-Gordon field A KG ∈ AQFT add,c . We observe as in [GR17] that the corresponding time-orderable prefactorization algebra F KG ∈ tPFA add,c describes the time-ordered products from perturbative algebraic quantum field theory, cf. [FR13,Rej16].

Lorentzian geometry
In order to fix our notations, we shall briefly recall some basic definitions and properties of Lorentzian manifolds. We refer to [BGP07] for a concise introduction.
A Lorentzian manifold is a manifold M together with a metric g of signature (− + · · · +). A non-zero tangent vector 0 In what follows we always consider time-oriented Lorentzian manifolds, denoted collectively by symbols like M , suppressing the metric g and time-orientation t from our notation. A time-like or causal curve γ : I → M is called future directed if g(t,γ) < 0 and past directed if g(t,γ) > 0. The chronological future/past of a point x ∈ M is the subset I ± M (x) ⊆ M of all points that can be reached from x by future/past directed time-like curves. The causal future/past of a point x ∈ M is the subset J ± M (x) ⊆ M of all points that can be reached from x by future/past directed causal curves and x itself. Given any subset S ⊆ M , we define I ± M (S) : Definition 2.5. We denote by Loc the category whose objects are all oriented and timeoriented globally hyperbolic Lorentzian manifolds M and morphisms are all orientation and time-orientation preserving isometric embeddings f : M → N with causally convex and open image f (M ) ⊆ N .
We introduce the following terminology to specify important (tuples of) Loc-morphisms that enter the definitions of algebraic quantum field theories and factorization algebras.
We shall write f 1 ⊥ f 2 for causally disjoint morphisms.

Factorization algebras
Factorization algebras are typically considered in the context of topological, complex or Riemannian manifolds, see [CG17] for a detailed study. In order to obtain a meaningful comparison to algebraic quantum field theory, which is typically considered in the context of globally hyperbolic Lorentzian manifolds, we shall introduce below a variant of factorization algebras on the category Loc from Definition 2.5. A similar concept of factorization algebras on Loc appeared before in [GR17]. For what follows let us fix any cocomplete closed symmetric monoidal category (C, ⊗, I, τ ), e.g. the category of vector spaces Vec K over a field K.
A prefactorization algebra F on Loc with values in C is given by the following data: , with the convention that to the empty tuple ∅ → N is assigned a morphism I → F(N ) from the monoidal unit.
A morphism ζ : F → G of prefactorization algebras is a family ζ M : F(M ) → G(M ) of Cmorphisms, for all M ∈ Loc, that is compatible with the factorization products, i.e. for all in C commutes.
Definition 2.8. We denote by PFA the category of prefactorization algebras on Loc.
Factorization algebras are prefactorization algebras that satisfy a suitable descent condition with respect to Weiss covers [CG17]. For proving our results in this paper, it is sufficient to assume a weaker descent condition that we shall call additivity in reference to a similar property in algebraic quantum field theory [Few13]. As explained below, this includes in particular all factorization algebras on Loc. Before we can formalize the additivity property, we have to introduce some further terminology and notations. We may restrict the orientation, time-orientation and metric on M to the causally convex open subsets U ∈ RC M and thereby define objects U ∈ Loc. Every inclusion U ⊆ V in RC M then defines a Loc-morphism ι V U : U → V . Hence, we can regard RC M ⊆ Loc as a subcategory, for every M ∈ Loc, and restrict any prefactorization algebra F ∈ PFA to a functor F| M : RC M → C.
Definition 2.12. A prefactorization algebra F ∈ PFA is called additive if, for every M ∈ Loc, the canonical morphism is an isomorphism in C. We denote by PFA add ⊆ PFA the full subcategory of additive prefactorization algebras.
Remark 2.13. The additivity condition formalizes the idea that F(M ) is "generated" by the images of the maps F(U ) → F(M ), for all relatively compact and causally convex open subsets U ⊆ M . Interpreting F(M ) as a collection of observables for a quantum field theory, this means that all observables described by F(M ) arise from relatively compact regions U ⊆ M . △ Proposition 2.14. Every factorization algebra F on Loc is an additive prefactorization algebra.
Proof. Suppose that F is a factorization algebra [CG17], i.e. it satisfies a cosheaf condition with respect to all Weiss covers of every M ∈ Loc. For every M ∈ Loc, the cover defined by RC M is a Weiss cover. Indeed, given finitely many points The property of being a factorization algebra then implies that the canonical diagram is a coequalizer in C. Our claim then follows by observing that the cocones of (2.4) are canonically identified with the cocones of (2.5). Indeed, any cocone {α U : F(U ) → Z} of (2.4) defines a cocone of (2.5) because U ∩ V ∈ RC M (whenever nonempty) and hence the diagram in C commutes. Vice versa, any cocone {α U : F(U ) → Z} of (2.5) defines a cocone of (2.4) because U ∩ V = U , for all U ⊆ V , and hence the diagram As a last definition, we would like to introduce a suitable local constancy property that is adapted to the category Loc. This property will play a crucial role in establishing our comparison results. Recall from Definition 2.6 the concept of Cauchy morphisms.
is an isomorphism in C, for every Cauchy morphism f : M c → N . We denote by PFA c ⊆ PFA the full subcategory of Cauchy constant prefactorization algebras. The full subcategory PFA add,c ⊆ PFA add of Cauchy constant additive prefactorization algebras is defined analogously.

Algebraic quantum field theories
Let C be a cocomplete closed symmetric monoidal category as in the previous subsection. We briefly review the basic definitions for C-valued algebraic quantum field theories on Loc following [BSW17]. See also [BFV03,FV12,BDFY15] for a broader introduction to algebraic quantum field theories and their applications to physics.
Let us denote by Alg := Alg As (C) the category of associative and unital algebras in C. An algebraic quantum field theory A on Loc with values in C is a functor A : Loc → Alg that satisfies the Einstein causality axiom: for every pair of causally disjoint morphisms (f 1 :

denotes the (opposite) multiplication on A(N ). A morphism κ :
A → B of algebraic quantum field theories is a natural transformation between the underlying functors.
Definition 2.16. We denote by AQFT the category of algebraic quantum field theories on Loc.
For proving some of the results of this paper, we require a relatively mild variant of an additivity property in the sense of [Few13]. Recall from Definition 2.9 the category RC M of relatively compact and causally convex open subsets of M ∈ Loc.
Definition 2.17. An algebraic quantum field theory A ∈ AQFT is called additive if, for every M ∈ Loc, the canonical morphism is an isomorphism in Alg. We denote by AQFT add ⊆ AQFT the full subcategory of additive algebraic quantum field theories.
Remark 2.18. Because RC M is a directed set by Lemma 2.11, the colimit in Definition 2.17 can be computed in the underlying category C, see e.g. [Fre17, Proposition 1.3.6]. Hence, additivity of A can be tested by studying its underlying functor A : Loc → C to the category C. △ Furthermore, we introduce a suitable local constancy property that is also known in the literature as the time-slice axiom.
We denote by AQFT c ⊆ AQFT the full subcategory of Cauchy constant algebraic quantum field theories. The full subcategory AQFT add,c ⊆ AQFT add of Cauchy constant additive algebraic quantum field theories is defined analogously.
Our construction consists of three steps, which will be carried out in detail in individual subsections below.
Step (1) consists of proving that, for each M ∈ Loc, the object F(M ) ∈ C carries canonically the structure of an associative and unital algebra in C. This step relies on Cauchy constancy, while it does not require that the additivity property holds true.
Step (2) consists of proving that these algebra structures are compatible with the maps F(f ) : F(M ) → F(N ) induced by Loc-morphisms f : M → N . Here our additivity property turns out to be crucial. Finally, in step (3) we show that the resulting functor Loc → Alg satisfies the properties of a Cauchy constant additive algebraic quantum field theory, cf. Section 2.3.

Object-wise algebra structure
All results of this subsection do not use the additivity property from Definition 2.12. Hence, we let F ∈ PFA c be any Cauchy constant prefactorization algebra.
where the upward-left pointing arrow is an isomorphism because F is by hypothesis Cauchy constant. A priori, it is not clear whether different choices of such ι M U : U → M lead to the same multiplication map in (3.1). The possible choices are recorded in the following category. There exists a unique morphism (ι M U : Lemma 3.2. For every M ∈ Loc, the category P M is non-empty and connected.
Proof. Non-empty: Choose any Cauchy surface Σ of M and define Σ ± : would provide a zig-zag that proves connectedness of P M .

It remains to show that
and claim that Σ is a Cauchy surface of M . This can be easily confirmed by checking that every inextensible time-like curve γ : I → M meets Σ exactly once.
where one also uses the composition properties (2.1) of prefactorization algebras.
To obtain a unit for F(M ), we recall that there exists a unique empty tuple of disjoint morphisms ∅ → M to which the prefactorization algebra assigns a C-morphism that we shall denote by η M : I → F(M ). The main result of this subsection is as follows. Proof. To prove that the multiplication µ M is associative, we consider two Cauchy surfaces Σ 0 , Σ 1 of M such that Σ 1 ⊂ I + M (Σ 0 ), i.e. Σ 1 is in the future of Σ 0 . Using the independence result from Corollary 3.3 and the composition properties of prefactorization algebras from Section 2.2, one easily confirms that µ M (id⊗µ M ) is the upper path and µ M (µ M ⊗id) the lower path from F(M ) ⊗3 to F(M ) in the commutative diagram where as before we denote by Σ ± := I ± M (Σ) ⊆ M the chronological future/past of a Cauchy surface Σ of M . Unitality of the product follows immediately from the fact that there exists a unique empty tuple ∅ → N for each N ∈ Loc and the composition properties (2.1) of prefactorization algebras.

Naturality of algebra structures
The aim of this subsection is to investigate compatibility between the algebra structures from Proposition 3.4 and the maps F(f ) : F(M ) → F(N ) induced by Loc-morphisms. For our main statement to be true it will be crucial to assume that F ∈ PFA add,c is a Cauchy constant additive prefactorization algebra in the sense of Definitions 2.12 and 2.15. As a first partial result, we prove the following general statement. commutes.
Remark 3.6. We would like to emphasize that our assumption that the image f (M ) ⊆ N is relatively compact was crucial for the proof of Lemma 3.5. In fact, if one does not assume that the image of the Loc-morphism f : M → N is relatively compact, then it is not true that the image f (Σ) ⊂ N of a Cauchy surface Σ of M can be extended to a Cauchy surface Σ of N . A simple example that demonstrates this feature is given by the subset inclusion ι V U : U → V of the following two diamond regions in 2-dimensional Minkowski spacetime (note that U is not relatively compact as a subset of V ): Proof. We already observed in the proof of Lemma 3.5 that F(f ) preserves the units.
For the multiplications we have to prove that Because F is by hypothesis additive (cf. Definition 2.12) and the monoidal product ⊗ in a cocomplete closed symmetric monoidal category preserves colimits in both entries, it follows that where in the last step we also used that RC M is directed by Lemma 2.11. For every U ∈ RC M , consider the diagram The top and bottom squares of this diagram commute because of Lemma 3.5 and the fact that both U ⊆ M and f (U ) ⊆ N are relatively compact subsets. The two triangles commute by direct inspection. By universality of the colimit in (3.8), this implies that the front square in (3.9) commutes, proving our claim.
commutes by the compatibility properties (2.3) of prefactorization algebra morphisms.

Algebraic quantum field theory axioms
The goal of this subsection is to show that the construction above assigns to each Cauchy constant additive prefactorization algebra a Cauchy constant additive algebraic quantum field theory. More precisely, we shall prove that the functor A (−) : PFA add,c → Alg Loc established in Corollary 3.8 factors through the full subcategory AQFT add,c ⊆ Alg Loc of Cauchy constant additive algebraic quantum field theories.
The main result of this subsection is as follows.
Proposition 3.10. Let F ∈ PFA add,c be any Cauchy constant additive prefactorization algebra.

denotes the (opposite) multiplication on F(N ) from Proposition 3.4.
Proof. Because F is by hypothesis additive (cf. Definition 2.12) and the monoidal product ⊗ in a cocomplete closed symmetric monoidal category preserves colimits in both entries, it follows that The two triangles coincide and commute by direct inspection. Furthermore, for every (U 1 , U 2 ) ∈ RC M 1 ×RC M 2 , the outer square commutes as a consequence of Lemma 3.9 applied to the causally disjoint pair (f 1 : U 1 → N ) ⊥ (f 2 : U 2 → N ), whose images f 1 (U 1 ), f 2 (U 2 ) ⊆ N are relatively compact subsets. Hence, by universality of the colimit in (3.15), the inner square commutes as well, which is our claim.
Proposition 3.10 leads to the following refinement of Corollary 3.8.

From AQFT to PFA
In this section we show that every algebraic quantum field theory A ∈ AQFT defines a variant of a prefactorization algebra on Loc where the factorization products are defined only for those tuples of pairwise disjoint morphisms f : M → N that are in a suitable sense time-orderable. We shall call this type of prefactorization algebras time-orderable and denote the corresponding category by tPFA. Our construction defines a functor F (−) : AQFT → tPFA to the category of time-orderable prefactorization algebras. Cauchy constancy and additivity do not play a role in this section, however we shall prove that these properties are preserved by our functor.
Let A ∈ AQFT be an algebraic quantum field theory. Our aim is to construct from this data factorization products in C. This is however problematic in view of the equivariance property (2.2) of prefactorization algebras. In fact, if we would use (4.1) for all pairs of disjoint morphisms f = (f 1 , f 2 ) : M → N , then (2.2) would be satisfied if and only if the diagram in (2.7) commutes, which is in general not the case unless f 1 ⊥ f 2 are causally disjoint. By closer inspection of (3.1), one observes that (4.1) is not supposed to be the correct definition for all pairs of disjoint morphisms, but only for those pairs f = (f 1 , f 2 ) : M → N where f 1 (M 1 ) ⊆ N is "later" (in a suitable sense) than f 2 (M 2 ) ⊆ N . This would solve the problem concerning the equivariance property discussed above. The following definition formalizes a concept of time-ordering that allows us to prove our desired statements. In this picture the left and right boundaries are identified as indicated, thereby producing the Lorentzian cylinder N = (R × S 1 , g = −dt 2 + dφ 2 , t = ∂ ∂t ). △ The following technical lemma is the crucial ingredient for our proofs below. (ii): Since it is sufficient to prove that the composition of time-ordered tuples of pairwise disjoint morphisms is time-ordered. Therefore, assuming that f and g i , for i = 1, . . . , n, are time-ordered, we have to show that (f 1 g 11 , . . . , f n g nkn ) is time-ordered, i.e. J + N (f i g ii ′ (L ii ′ )) ∩ f j g jj ′ (L jj ′ ) = ∅ for the following two cases: Case 1 is i < j and arbitrary i ′ = 1, . . . , k i and j ′ = 1, . . . , k j . Case 2 is i = j and j < j ′ . Case 1 follows immediately from the hypothesis that f is time-ordered, i.e. J + N (f i (M i )) ∩ f j (M j ) = ∅ for all i < j. For case 2 we use that g i is time-ordered, i.e. J + M i (g ii ′ (L ii ′ )) ∩ g ij ′ (L ij ′ ) = ∅ for all j < j ′ , and hence by the properties of Loc-morphisms This proves that (f 1 g 11 , . . . , f n g nkn ) is time-ordered.
(iii): Suppose that ρ −1 ρ ′ : f ρ → f ρ ′ reverses the time-ordering between f k and f ℓ , i.e. ρ(i) = k = ρ ′ (i ′ ) and ρ(j) = ℓ = ρ ′ (j ′ ) with i < j and j ′ < i ′ or vice versa with j < i and i ′ < j ′ . Let us consider the case i < j and j ′ < i ′ , the other one being similar. It follows that which implies that f k ⊥ f ℓ are causally disjoint. This observation implies the existence of a factorization ρ −1 ρ ′ = τ 1 · · · τ N : f ρ → f ρ ′ , where τ l : f ρτ 1 · · · τ l−1 → f ρτ 1 · · · τ l is a transposition of adjacent causally disjoint pairs of morphisms, for all l = 1, . . . , N . These data are required to satisfy the analogs of the prefactorization algebra axioms from Section 2.2 for time-orderable tuples. A morphism ζ : F → G of time-orderable prefactorization algebras is a family ζ M : F(M ) → G(M ) of C-morphisms, for all M ∈ Loc, that is compatible with the time-ordered products as in (2.3).
Definition 4.4. We denote by tPFA the category of time-orderable prefactorization algebras on Loc. In analogy to Definitions 2.12 and 2.15, we introduce the full subcategories tPFA add , tPFA c , tPFA add,c ⊆ tPFA of additive, Cauchy constant and Cauchy constant additive time-orderable prefactorization algebras.
Remark 4.5. Each ordinary prefactorization algebra on Loc defines a time-orderable one by restriction to time-orderable tuples of pairwise disjoint morphisms. This defines a functor PFA → tPFA, which is faithful, but not necessarily full due to the fact that not all pairwise disjoint tuples f : M → N are time-orderable, cf. Remark 4.2. This functor clearly preserves both additivity and Cauchy constancy. △ With these preparations we can now carry out our envisaged construction of a time-orderable prefactorization algebra F A ∈ tPFA from a given algebraic quantum field theory A ∈ AQFT.
In particular, we can now complete our attempt from the beginning of this section to define the time-ordered factorization products. Let f = (f 1 , . . . , f n ) : M → N be a time-orderable tuple of pairwise disjoint morphisms with time-ordering permutation ρ ∈ Σ n . We define the corresponding time-ordered product in C, where µ (n) N denotes the n-ary multiplication in the associative and unital algebra A(N ) in the given order, i.e. µ (n) N (a 1 ⊗ · · · ⊗ a n ) = a 1 · · · a n with juxtaposition denoting multiplication in A(N ). As before, for n = 0 we assign to the empty tuple ∅ → N the C-morphism η N : I → A(N ) corresponding to the unit of A(N ). Proof. Consider time-ordering permutations ρ, ρ ′ ∈ Σ n for f . Recalling Lemma 4.3 (iii), the right permutation ρ −1 ρ ′ : f ρ → f ρ ′ is generated by transpositions of adjacent causally disjoint pairs of morphisms. Hence, the claim follows from the Einstein causality axiom (2.7) of the algebraic quantum field theory A ∈ AQFT.
Theorem 4.7. Let A ∈ AQFT be an algebraic quantum field theory. Then the following data defines a time-orderable prefactorization algebra F A ∈ tPFA: Proof. Lemma 4.3 immediately implies that F A satisfies the axioms of time-orderable prefactorization algebras. More explicitly, Lemma 4.3 (i) implies the equivariance axiom (2.2) for all timeorderable tuples and Lemma 4.3 (ii) implies the composition axiom (2.1) for all time-orderable tuples. By definition, we also have that F A (id M ) = id F A (M ) , for all M ∈ Loc.
Concerning functoriality of the assignment A → F A , we have to show that every AQFTmorphism κ : A → B canonically defines a tPFA-morphism F A → F B . Observe that, for every time-orderable tuple f : M → N with time-ordering permutation ρ ∈ Σ n , the diagram Proof. The only non-trivial check to confirm that A (−) • F (−) = id AQFT add,c amounts to show that, for every A ∈ AQFT add,c , the multiplications on A F A (M ) and on A(M ) coincide, for all M ∈ Loc. By (3.1) and (4.7), the multiplication on A F A (M ) is given by Conversely, to show that F (−) • A (−) = id tPFA add,c , we have to confirm that the time-ordered products of F A F ∈ tPFA add,c coincide with the original time-ordered products of F ∈ tPFA add,c . In arity n = 0 and n = 1 this is obvious. For n ≥ 2, this is more complicated and requires some preparations. Using equivariance under permutation actions, it is sufficient to compare the time-ordered products for time-ordered (in contrast to time-orderable) tuples f = (f 1 , . . . , f n ) : M → N . Because of additivity, we can further restrict to the case where f : M → N has relatively compact images, i.e. f i (M i ) ⊆ N is relatively compact, for all i = 1, . . . , n. We shall now show that, due to Cauchy constancy, we can further restrict our attention to time-ordered tuples h = (h 1 , . . . , h n ) : L → N with relatively compact images for which there exists a Cauchy surface Σ of N such that By direct inspection one observes that Σ fulfills (5.2).
Using (5.2), we obtain a factorization where on the right-hand side we regard h i : L i → Σ + as morphisms to Σ + , for i = 1, . . . , n−1, and h n : L n → Σ − as a morphism to Σ − . Iterating this construction, we observe that it is sufficient to prove that where N ∈ Loc and the Cauchy surface Σ of N is arbitrary. Using (4.7) and (3.1), we obtain that which clearly coincides with the original time-ordered product F(ι N Σ ) : F(Σ + ) ⊗ F(Σ − ) → F(N ). This concludes our proof.
Remark 5.2. We would like to mention very briefly a more abstract operadic perspective on the Equivalence Theorem 5.1. Recall from [BSW17] that there exists a Set-valued colored operad O (Loc,⊥) whose category of C-valued algebras is the category of algebraic quantum field theories, i.e. AQFT = Alg O (Loc,⊥) (C). We can also define a Set-valued colored operad P Loc such that tPFA = Alg P Loc (C). Concretely, the colors of P Loc are the objects of Loc and the sets of operations are P Loc We expect that this operadic perspective will become important when considering the case where the target category C is a higher category or model category. This generalization is crucial for the description of quantum gauge theories in terms of factorization algebras [CG17] or algebraic quantum field theories [BSS15, BS17, BSW19b, BS19]. The adjunction (5.7) then becomes a Quillen adjunction between model categories, and a reasonable equivalence theorem would state that suitable restrictions to homotopy-invariant analogs of Cauchy constant and additive theories induce a Quillen equivalence. Proving such an equivalence theorem in a higher categorical context is technically complicated and will not be considered in the present paper. △ Remark 5.3. If the underlying category C carries an involutive structure, the category AQFT of algebraic quantum field theories carries an induced involutive structure, which is used to formalize * -objects, cf. [BSW19a]. Working in the standard involutive category Vec C of complex vector spaces, these * -object are the usual algebraic quantum field theories that assign * -algebras to spacetimes. Theorem 5.1 allows us to transfer the involutive structure on AQFT add,c to the category tPFA add,c of Cauchy constant additive time-ordered prefactorization algebras. The resulting involutive structure, however, involves Cauchy constancy explicitly. As a consequence, it seems to lack a geometric interpretation, which is why we refrain from describing it explicitly. △

Example: The free Klein-Gordon field
We apply our general Equivalence Theorem 5.1 to the simple example given by the free Klein-Gordon field and thereby recover the results from [GR17]. Let us briefly recall the algebraic quantum field theory description of the free Klein-Gordon field. Because C ∞ c : Loc → Vec R is a cosheaf for (causally convex) open covers and P : C ∞ c → C ∞ c is a natural transformation, it follows that V : Loc → Vec R is a cosheaf too. Consider the complexified symmetric algebra Sym C (V(M )) ∈ CAlg, which is a commutative algebra in the closed symmetric monoidal category (Vec C , ⊗, C, τ ) of complex vector spaces. This algebra is deformed to a noncommutative algebra by introducing a ⋆-product. For this we first define a (de Rham type) differential d : Sym C (V(M )) → Sym C (V(M )) ⊗ V(M ) by setting on monomials d ϕ 1 · · · ϕ n := n i=1 ϕ 1 · · · i ∨ . · · · ϕ n ⊗ ϕ i , J + N (f 1 (M 1 )) ∩ f 2 (M 2 ) = ∅, we obtain from (4.7), (5.11) and the support properties of G ± N that In the case where f = (f 1 , f 2 ) : M → N is anti-time-ordered, i.e. J + N (f 2 (M 2 )) ∩ f 1 (M 1 ) = ∅, we obtain Using again the support properties of G ± N , we observe that the the two cases in (5.12) can be combined into a single formula where G D N := 1 2 (G + N + G − N ) is the so-called Dirac propagator, that is valid for every time-orderable tuple (f 1 , f 2 ). In perturbative algebraic quantum field theory (see e.g. [FR13,Rej16]), the products · T N := · N • exp i G D N , d ⊗ d are called time-ordered products. Our observations in this subsection can thus be summarized as follows: The prefactorization algebra F KG ∈ tPFA add,c corresponding to the free Klein-Gordon theory A KG ∈ AQFT add,c encodes the usual time-ordered products obtained by the Dirac propagator. This agrees with the observations in [GR17].