Relating Nets and Factorization Algebras of Observables: Free Field Theories
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Abstract
In this paper we relate two mathematical frameworks that make perturbative quantum field theory rigorous: perturbative algebraic quantum field theory (pAQFT) and the factorization algebras framework developed by Costello and Gwilliam. To make the comparison as explicit as possible, we use the free scalar field as our running example, while giving proofs that apply to any field theory whose equations of motion are Green-hyperbolic (which includes, for instance, free fermions). The main claim is that for such free theories, there is a natural transformation intertwining the two constructions. In fact, both approaches encode equivalent information if one assumes the time-slice axiom. The key technical ingredient is to use time-ordered products as an intermediate step between a net of associative algebras and a factorization algebra.
Recently there have appeared two, rather elaborate formalisms for constructing the observables of a quantum field theory via a combination of the Batalin–Vilkovisky framework with renormalization methods. One [FR12b], later referred to as FR, works on Lorentzian manifolds and weaves together (a modest modification of) algebraic quantum field theory (AQFT) with the Epstein–Glaser machinery for renormalization. The other [CG17a, CG17b], later referred to as CG, works with elliptic complexes (i.e., “with Euclidean theories”) and constructs factorization algebras using renormalization machinery developed in [Cos11]. To practitioners of either formalism, the parallels are obvious, in motivation and techniques and goals. It is thus compelling (and hopefully eventually useful!) to provide a systematic comparison of these formalisms, with hopes that a basic dictionary will lead in time to effortless translation.
The primary goal in this paper is to examine in detail the case of free field theories, where renormalization plays no role and we can focus on comparing the local-to-global descriptions of observables. In other words, in the context of this free theory, we show how to relate the key structural features of AQFT and factorization algebras. In the future we hope to compare interacting field theories, which demands an examination of renormalization’s role and deepens the comparison by touching on more technical features.
The key to our comparison result is that while the approach of [Cos11, CG17a, CG17b] constructs the space of quantum observables by deforming the differential on the classical observables, one can equivalently leave the differential unchanged and deform the factorization product instead. This deformation of the factorization product corresponds, in the formalism of [FR12b, FR12a], to the passage from the pointwise product to the time-ordered product by means of the time-ordering operator \({\mathcal {T}}\). Hence one can either work with the pointwise product \(\cdot \) and the differential \({\hat{s}}\doteq {\mathcal {T}}^{-1}\circ s\circ {\mathcal {T}}\), or with the product \(\cdot _{{}^{\mathcal {T}}}\) and the differential s. In both cases the differential is a derivation with respect to the corresponding product, but only for arguments with disjoint supports, and in fact one obtains a prefactorization structure valued in Beilinson-Drinfeld algebras.
In the Lorentzian framework of [FR12b, FR12a], there is, in addition to the commutative time-ordered product, a non-commutative product \(\star \), identified as the operator product of quantum observables. The products \(\cdot _{{}^{\mathcal {T}}}\) and \(\star \) are related by time-ordering and we show in Sect. 6.3 how to reconstruct \(\star \) from \(\cdot _{{}^{\mathcal {T}}}\) for the algebra of free fields.
A secondary goal of this paper is to facilitate communication between communities, by providing a succinct treatment of this key example in each formalism. We expect that interesting results—and questions!—can be translated back and forth.
Indeed, one consequence of this effort at comparison is that it spurred a modest enhancement of each formalism. On the FR side, we introduce a differential graded (dg) version of the usual axioms for the net of algebras. Prior work fits nicely into this definition, and in the future we hope to examine its utility in gauge theories. On the CG side, we show that the free field construction applies to Lorentzian manifolds as well as Euclidean manifolds. (The case of interacting theories in the CG formalism does not port over so simply, as it exploits features of elliptic complexes in its renormalization machinery.)
As an overview of the paper, we begin by raising key questions about how the formalisms agree and differ. To sharpen these questions, we give precise descriptions of the outputs generated by each formalism, namely the kinds of structure possessed by observables. On the FR side, one has a net of algebras; on the CG side, a factorization algebra of cochain complexes. With these definitions in hand, we can state our main results precisely. As a brief, imprecise gloss, our main result is that the FR and CG constructions agree where they overlap: if one restricts the CG factorization algebra of observables to the opens on which the FR net is defined (and takes the zeroth cohomology), then the factorization algebra and net determine the same functor to vector spaces. We also explain how one can recover as well the algebraic structures on the nets (Poisson for the classical theory, associative for the quantum) from the constructions. Next, we turn to carefully describing the constructions in each formalism, so that we can prove the comparison results. We recall in detail how each formalism constructs the observables for the free theory given by a Green-hyperbolic operator, producing on the one hand, a net of algebras on a globally hyperbolic Lorentzian manifold, and on the other, a factorization algebra. With the constructions in hand, the proof of the comparison results is straightforward. Finally, we draw some lessons from the comparison and point out natural directions of future inquiry.
1 A Preview of the Key Ideas
Before delving into the constructions, we discuss field theory from a very high altitude, ignoring all but the broadest features, and explain how each formalism approaches observables. With this knowledge in hand, it is possible to raise natural questions about how the formalisms differ. The rest of the paper can be seen as an attempt to answer these questions.
1.1 Classical theories
- (1)
a smooth manifold M (the “spacetime”),
- (2)
a smooth fiber bundle over the manifold \(\pi : E \rightarrow M\) whose smooth sections \(\Gamma (M,E)\) are the “fields,”
- (3)
and a system of partial differential equations on the fields (the “equations of motion” or “Euler-Lagrange equations”) that are variational in nature.
In this paper, the focus is on free fields and we will write the equations as \(P(\phi ) = 0\) where \(\phi \) is a field and P denotes the equations of motion operator. (There are many variations and refinements on this loose description, of course, but most theories fit into this framework.)
Here the manifold M is equipped with a metric g, and an important difference is that the FR formalism requires g to have Lorentzian signature while the CG formalism requires g to be Riemannian. We use the notation \({\mathcal {M}}\equiv (M,g)\).
In this paper we focus on the Lorentzian case and we will assume that P is a Green-hyperbolic operator, i.e. it has unique retarded and advanced Green functions (see [Bär15] for a lucid and extensive discussion of this notion). Note that this class of operators allows one to treat the free scalar field and the free Dirac fermion as special cases.
Any measurement of the system should then be some function of Sol(M), the set of global solutions. In other words, the algebra of functions \({\mathcal {O}}(Sol(M))\) constitutes an idealized description of all potential measuring devices for the system. (An important issue later in the text will be what kind of functions we allow, but we postpone that challenge for now, simply remarking that solutions often form a kind of “manifold,” possibly singular and infinite-dimensional, so that \({\mathcal {O}}\) is not merely set-theoretic.) Even better, we obtain a covariant functor \({\mathcal {O}}(Sol(-)): \mathbf {Open}(M) \rightarrow \mathbf {CAlg}\) to the category \(\mathbf {CAlg}\) of commutative algebras. As Sol is a sheaf, \({\mathcal {O}}(Sol(-))\) should be a cosheaf, meaning that it satisfies a gluing axiom so that the global observables are assembled from the local observables.
Nothing about this general story depends on the signature of the metric, and each formalism gives a detailed construction of a cosheaf of commutative algebras for a classical field theory (although some technical choices differ, e.g., with respect to functional analysis). It is with quantum field theories that the formalisms diverge.
1.2 Quantization
The CG formalism provides a functor \(Obs^q: \mathbf {Open}(M) \rightarrow \mathbf {Ch}\), which assigns a cochain complex (or differential graded (dg) vector space) of observables to each open set. This cochain complex is a deformation of a commutative dg algebra \(Obs^{cl}\), where \(H^0(Obs^{cl}(U)) = {\mathcal {O}}(Sol(U))\).
The FR formalism provides a functor \({\mathfrak {A}}: \mathbf {Caus}({\mathcal {M}}) \rightarrow \mathbf {Alg}^*\), which assigns a unital \(*\)-algebra to each “causally convex” open set (so that \(\mathbf {Caus}({\mathcal {M}})\) is a special subcategory of \(\mathbf {Open}(M)\) depending on the global hyperbolic structure of \({\mathcal {M}}\)). The algebra \({\mathfrak {A}}(U)\) is, in practice, a deformation quantization of the Poisson algebra \({\mathcal {O}}(Sol(U))\).
- (1)
Why does the FR formalism (and AQFT more generally) restrict to a special class of opens but the CG formalism does not? And what should the FR formalism assign to a general open?
- (2)
Why does the FR formalism (and AQFT more generally) assign a \(*\)-algebra but the CG formalism assigns only a vector space? And can the CG approach recover the algebra structure as well?
2 Nets Versus Factorization Algebras
This section sets the table for this paper. We begin with some background notation (which is mostly self-explanatory, so we suggest the reader only refer to it if puzzled) before reviewing quickly the key definitions about nets and factorization algebras. We made an effort to make the definitions accessible to those from the complementary community.
2.1 Notations
2.1.1 Geometry
We fix throughout a smooth vector bundle over the manifold \(\pi : E \rightarrow M\). As we work throughout with manifolds equipped with a metric, we use the associated volume form of (M, g) to identify smooth functions with densities. We also assume that E is equipped with a nondegenerate bilinear pairing on the fibers, so as to identify sections of E with sections of the dual bundle \(E^*\).
2.1.2 Functional analysis
\({\mathcal {E}}(M) \doteq \Gamma (M,E)\), \({\mathcal {E}}^*(M) \doteq \Gamma (M,E^*)\) with their natural Fréchet topologies,
\({\mathcal {E}}'(M)\) for the strong topological dual (i.e., the space of continuous linear \({\mathbb {R}}\)-valued functions on a given topological space), which consists of compactly supported distributions,
\({\mathcal {D}}(M)\doteq \Gamma _c(M,E)\), \({\mathcal {D}}^*(M)\doteq \Gamma _c(M,E^*)\) with their natural inductive limit topologies, and
\({\mathcal {D}}'(M)\) for the strong topological dual (i.e., the space of continuous linear \({\mathbb {R}}\)-valued functions on a given topological space), which consists of non-compactly supported distributions.
We will also work with certain natural completions of tensor products, which arise by geometric constructions.
\({\mathcal {E}}_n(M)\doteq \Gamma (M^n,E^{\boxtimes n})\), which is equal to the completed projective tensor product \({\mathcal {E}}(M)^{{\widehat{\otimes }}n}\),
\({\mathcal {E}}_n'(M)\doteq \Gamma (M^n,E^{\boxtimes n})'\), with the strong topology.
\({\mathcal {D}}_{n}(M)\doteq \Gamma _c(M^n,E^{\boxtimes n})\),
\({\mathcal {D}}_{n}'(M) \doteq \Gamma _c(M^n,E^{\boxtimes n})'\), with the strong topology.
Since we fixed an explicit isomorphism \(E \cong E^*\) of vector bundles, we have preferred inclusions \({\mathcal {E}}(M) \cong {\mathcal {E}}_n^*(M)\hookrightarrow {\mathcal {D}}_n'(M)\) and \({\mathcal {D}}_n(M)\cong {\mathcal {D}}_n^*(M) \hookrightarrow {\mathcal {E}}_n'(M)\).
Remark 2.1
We note that these conventions differ from those in [CG17a], where \({\mathcal {E}}_c(M)\) denotes the compactly supported smooth sections, \({\overline{{\mathcal {E}}}}(M)\) the distributional sections, and \({\overline{{\mathcal {E}}}}_c(M)\) the compactly supported distributional sections.
We indicate the complexification of a real vector space V by a superscript \(V^{\scriptscriptstyle {{\mathbb {C}}}}\).
2.1.3 Categories
Myriad categories will appear throughout this work, and so we introduce some of the key ones, as well as establish notations for generating new ones. Categories will be indicated in bold.
We start with a central player. Let \(\mathbf {Nuc}\) denote the category of nuclear, topological locally convex vector spaces, which is a subcategory of the category of topological locally convex spaces \(\mathrm {\mathbf {TVec}}\). It is equipped with a natural symmetric monoidal structure via the completed projective tensor product \({\widehat{\otimes }}\) (although we could equally well say ‘injective’ as the spaces are nuclear). A reason to work with nuclear spaces is the fact that injective and projective tensor products are isomorphic for such spaces, hence it is sufficient to work with just one monoidal structure. Nuclearity is preserved under taking strong duals, a direct sum, an inductive limit of a countable family of nuclear spaces, a product and a projective limit of any family of nuclear spaces. Moreover, spaces of smooth sections and their strong duals introduced in Sect. 2.1.2 are nuclear. (See [Tre67] for an accessible treatment of nuclear spaces.)
We emphasize that we make this choice as it suits our purposes. Many of our ideas and constructions work with other categories, such as \(\mathrm {\mathbf {TVec}}\), but require one to be more attentive to which monoidal structures are in play.
Remark 2.2
Given the spaces appearing in our construction, it is often worthwhile to work instead with convenient vector spaces [KM97], but we will not discuss that machinery here, pointing the interested reader to [CG17a, Rej16].
If we wish to discuss the category of unital associative algebras of such vector spaces, we write \(\mathbf {Alg}(\mathbf {Nuc})\). Here the morphisms are continuous linear maps that are also algebra morphims. Similarly, we write \(\mathbf {CAlg}(\mathbf {Nuc})\) for unital commutative algebras in \(\mathbf {Nuc}\) and \(\mathbf {PAlg}(\mathbf {Nuc})\) for unital Poisson algebras therein. We will typically want \(*-\)structures (i.e., an involution compatible with the multiplication), and we use \(\mathbf {Alg}^*(\mathbf {Nuc})\), \(\mathbf {CAlg}^*(\mathbf {Nuc})\), and \(\mathbf {PAlg}^*(\mathbf {Nuc})\), respectively.
More generally, for \(\mathrm {{\mathbf {C}}}\) a category with symmetric monoidal structure \(\otimes \), we write \(\mathbf {Alg}(\mathrm {{\mathbf {C}}}, {\otimes })\) for the unital algebra objects in that category. Often we will write simply \(\mathbf {Alg}(\mathrm {{\mathbf {C}}})\), if there is no potential confusion about which symmetric monoidal structure we mean.
It is often useful to forget extra structure. We use \({\mathfrak {v}}: \mathbf {PAlg}^*(\mathbf {Nuc}) \rightarrow \mathbf {Nuc}\) and \({\mathfrak {v}}: \mathbf {Alg}^*(\mathbf {Nuc}) \rightarrow \mathbf {Nuc}\) to denote forgetful functors to vector spaces. We use \({\mathfrak {c}}: \mathbf {PAlg}^*(\mathbf {Nuc}) \rightarrow \mathbf {CAlg}^*(\mathbf {Nuc})\) to denote the forgetful functor to commutative algebras.
In a similar manner, if \(\mathrm {{\mathbf {C}}}\) is an additive category, we write \(\mathbf {Ch}(\mathrm {{\mathbf {C}}})\) to denote the category of cochain complexes and cochain maps in \(\mathrm {{\mathbf {C}}}\). Thus \(\mathbf {Ch}(\mathbf {Nuc})\) denotes the category of cochain complexes in \(\mathbf {Nuc}\) (which, unfortunately, is not a particular nice place to do homological algebra). We note that we allow unbounded complexes, but in practice our constructions here produce complexes bounded on one side. (If we treated gauge theories, we would have complexes unbounded in both directions.)
Remark 2.3
This category \(\mathbf {Ch}(\mathbf {Nuc})\) admits a natural notion of weak equivalence: a cochain map is a weak equivalence if it induces an isomorphism on cohomology. Thus it is a relative category and presents an \((\infty ,1)\)-category, although we will not need such notions here.
There is another important variant to bear in mind. In the original axiomatic framework of Haag and Kastler, the notion of subsystems is encoded in the injectivity requirement for algebra morphisms. We use the superscript “\(\mathrm{inj}\)”, if we want to impose this condition on morphisms, for a given category. Hence \(\mathbf {Alg}^*(\mathbf {Nuc})^\mathrm{inj}\) consists of the category whose objects are nuclear, topological locally convex unital \(*\)-algebras but whose morphisms are injective continuous algebra morphisms.
2.1.4 Dealing with \(\hbar \)
In perturbative field theory, one works with \(\hbar \) as a formal variable. In our situation, since we restrict to free fields, this is overkill: one can actually set \(\hbar = 1\) throughout, and all the constructions are well-defined. But \(\hbar \) serves as a helpful mnemonic for what we are deforming and as preparation for the interacting case.
We thus introduce categories involving \(\hbar \) that emphasize its algebraic role and minimize any topological issues. As a gesture at the topological issues, note that the ring \({\mathbb {C}}[[\hbar ]]\) is equipped with an adic topology, and so one might want to work with topological vector spaces that are modules over \({\mathbb {C}}[[\hbar ]]\) in a continuous way. There are then some nontrivial compatibilities to discuss. Instead, we will restrict our attention to a special class of objects where we can avoid such discussions, as follows.
The category \(\mathbf {Alg}({\mathbf {Nuc}}_{\mathbf {\hbar }})\) then consists of algebra objects in that symmetric monoidal category, \(\mathbf {Ch}({\mathbf {Nuc}}_{\mathbf {\hbar }})\) denotes cochain complexes therein, and \(\mathbf {Alg}(\mathbf {Ch}({\mathbf {Nuc}}_{\mathbf {\hbar }}))\) denotes dg algebras therein. We use again the notations \({\mathfrak {v}}\) and \({\mathfrak {c}}\) as forgetful functors, hopefully without producing confusion.
2.2 Overview of the pAQFT setting
The framework of AQFT formalizes rigorously the core ideas of Lorentzian field theory, building on the lessons of rigorous quantum mechanics, but the standard calculational toolkit for interacting QFT does not fit into the framework. Perturbative AQFT is a natural modification of the framework within which one often can realize a version of the usual calculations, while preserving the structural insights of AQFT.
2.2.1 .
Let \({\mathcal {M}}=(M,g)\) be an n-dimensional spacetime, i.e., a smooth n-dimensional manifold with the metric g of signature \((+,-,\dots ,-)\). We assume \({\mathcal {M}}\) to be oriented, time-oriented and globally hyperbolic (i.e. it admits foliation with Cauchy hypersurfaces). To make this concept clear, let us recall a few important definitions in Lorentzian geometry.
Definition 2.4
timelike, if \(g({\dot{\gamma }},{\dot{\gamma }})> 0\),
spacelike, if \(g({\dot{\gamma }},{\dot{\gamma }})< 0\),
lightlike (or null), if \(g({\dot{\gamma }},{\dot{\gamma }})= 0\),
causal, if \(g({\dot{\gamma }},{\dot{\gamma }})\ge 0\).
The classification of curves defined above is the causal structure of \({\mathcal {M}}\).
Definition 2.5
A set \({\mathcal {O}}\subset {\mathcal {M}}\) is causally convex if for any causal curve \(\gamma :[a,b]\rightarrow {\mathcal {M}}\) whose endpoints \(\gamma (a),\gamma (b)\) lie in \({\mathcal {O}}\), then every interior point \(\gamma (t)\), for \(t\in [a,b]\), also lies in \({\mathcal {O}}\) for every \(t\in [a,b]\).
With these definitions in hand, we can define the category of open subsets on which we specify algebras of observables.
Definition 2.6
Let \(\mathbf {Caus}({\mathcal {M}})\) be the collection of relatively compact, connected, contractible, causally convex subsets \({\mathcal {O}}\subset {\mathcal {M}}\). Note that the inclusion relation \(\subset \) is a partial order on \(\mathbf {Caus}({\mathcal {M}})\), so \((\mathbf {Caus}({\mathcal {M}}),{\subset })\) is a poset (and hence a category).
2.2.2 .
To formulate a classical theory, we start with making precise what we mean by the model for the space of classical fields.
Definition 2.7
Note how this definition formalizes the sketch of classical field theory in Sect. 1: we have a category of open sets—here, \(\mathbf {Caus}({\mathcal {M}})\)—and a functor to a category of Poisson algebras, since the observables of a classical system should form such a Poisson algebra.
In the AQFT community, the underlying commutative algebra of \({\mathfrak {P}}\) is known as the space of classical fields. In more formal language, we introduce a forgetful functor \({\mathfrak {c}}: \mathbf {PAlg}^*(\mathbf {Nuc}) \rightarrow \mathbf {CAlg}(\mathbf {Nuc})\) and state the following.
Definition 2.8
The space of classical fields is the functor \({\mathfrak {c}}\circ {\mathfrak {P}}\).
Remark 2.9
Note that there is a conflict here with terminology in the CG framework (and with some other communities working in physics), where a field is an element of \({\mathcal {E}}\), i.e., a configuration in the AQFT sense. Thus the space of fields in the CG sense corresponds to the configuration space in the AQFT sense. In the CG framework, \({\mathfrak {c}}\circ {\mathfrak {P}}\) is the commutative algebra of observables on the classical fields, aka functions on the configuration space.
It is useful to introduce a further axiom that articulates more precisely how the dynamics of a classical theory should behave. Here, a time orientation plays an important role.
Definition 2.10
Given the global timelike vector field u (the time orientation) on M, a causal curve \(\gamma \) is called future-directed if \(g(u,{\dot{\gamma }}) > 0\) all along \(\gamma \). It is past-directed if \(g(u,{\dot{\gamma }}) < 0\).
Definition 2.11
A causal curve \(\gamma :(a,b) \rightarrow M\) is future inextendible if \(\lim _{t \rightarrow b} \gamma (t)\) does not exist in M.
Definition 2.12
A Cauchy hypersurface in \({\mathcal {M}}\) is a smooth subspace of \({\mathcal {M}}\) such that every inextendible causal curve intersects it exactly once.
Remark 2.13
The significance of Cauchy hypersurfaces lies in the fact that one can use them to formulate the initial value problem for partial differential equations, and for normally hyperbolic equations this problem has a unique solution.
With this notion in hand, we have a language for enforcing equations of motion at an algebraic level.
Definition 2.14
A model is said to be on-shell if in addition it satisfies the time-slice axiom: for any \({\mathcal {N}}\in \mathbf {Caus}({\mathcal {M}})\) a neighborhood of a Cauchy surface in the region \({\mathcal {O}}\in \mathbf {Caus}({\mathcal {M}})\), then \({\mathfrak {P}}\) sends the inclusion \({\mathcal {N}}\subset {\mathcal {O}}\) to an isomorphism \({\mathfrak {P}}({\mathcal {N}})\cong {\mathfrak {P}}({\mathcal {O}})\). Otherwise the model is called off-shell.
Remark 2.15
Note that being on-shell codifies the idea that the set of solutions is specified by the initial value problem on a Cauchy hypersurface.
2.2.3 .
We now turn to the quantum setting.
Definition 2.16
Definition 2.17
A QFT model is said to be on-shell if in addition it satisfies the time-slice axiom (where one simply replaces \({\mathfrak {P}}\) by \({\mathfrak {A}}\) in the definition above). Otherwise, it is off-shell.
Often a quantum model arises from a classical one by means of quantization. In order to formalize this, we need some notation. Given a functor \({\mathfrak {F}}\), let \({\mathfrak {F}}[[\hbar ]]\) denote the functor sending \({\mathcal {O}}\) to \({\mathfrak {F}}({\mathcal {O}}) {\widehat{\otimes }} {\mathbb {C}}[[\hbar ]]\).
Definition 2.18
- (1)
\({\mathfrak {v}}\circ {\mathfrak {A}}\cong {\mathfrak {v}}\circ {\mathfrak {P}}[[\hbar ]]\),
- (2)
\({\mathfrak {c}}\circ {\mathfrak {P}}\cong {\mathfrak {A}}/(\hbar )\), and
- (3)
the brackets \(-\frac{i}{\hbar }[.,.]_{\mathcal {O}}\) coincides with \(\{.,.\}_{\mathcal {O}}\) modulo \(\hbar \),
a grading-preserving associative multiplication \(\star \) so that \(a \star b \in A^{m+n}\) if \(a \in A^m\) and \(b \in A^n\), and
- a differential \(\mathrm{d}: A \rightarrow A\) that increases degree by one, satisfies \(\mathrm{d}^2 = 0\), and is a derivation, so thatfor homogeneous elements \(a, b \in A\).$$\begin{aligned} \mathrm{d}(a \star b) = \mathrm{d}a \star b + (-1)^{|a|} a \star \mathrm{d}b \end{aligned}$$
2.3 A dg version of pAQFT
We articulate here a very minimal generalization of the usual AQFT axioms that allows dg algebras, rather than plain algebras, as the target category. It will be apparent that free field theories fits these axioms, and we intend to show that the perturbative construction of gauge theories does as well. We forewarn the reader that we do not impose certain conditions (notably isotony) because we do not yet know an appropriate dg generalization.
Remark 2.19
Others have suggested modifications of AQFT in a dg direction, particularly [BDHS13, BSS17a, BS17], who explore the case of abelian gauge theories in depth and even examine some nonperturbative facets. A generalization to non-abelian gauge theories has been obtained on the classical level in [BSS17b]. We expect, based on explicit models constructed in [FR12a], that our minimal, perturbative definitions apply verbatim to gauge theories like Yang-Mills theories and can be seen as the infinitesimal version of the axioms of homotopy AQFT proposed by Benini and Schenkel [BS17].
2.3.1 .
Before we get to our definition, let us sketch the big picture. The basic principle is to replace ordinary categories and functors by higher categorical analogues. Hence we want to articulate a version of a QFT model as a functor between \(\infty \)-categories, and so the first step is to determine the source and target \(\infty \)-categories.
Consider the category \(\mathbf {Ch}({\mathbf {Nuc}}_{\mathbf {\hbar }})\) as a relative category with quasi-isomorphism as the notion of weak equivalence; this determines an \(\infty \)-category we momentarily denote \({{\mathcal {D}}}({\mathbf {Nuc}}_{\mathbf {\hbar }})\). Then take the \(\infty \)-category of \(*\)-algebras \({{\mathcal {A}}}\mathrm{lg}^*({{\mathcal {D}}}({\mathbf {Nuc}}_{\mathbf {\hbar }}))\).
Consider the category \(\mathbf {Alg}^*(\mathbf {Ch}({\mathbf {Nuc}}_{\mathbf {\hbar }}))\) as a relative category with quasi-isomorphism as the notion of weak equivalence. This determines another \(\infty \)-category.
Once one has fixed a target \(\infty \)-category \({\mathcal {C}}\), then one can view a functor of \(\infty \)-categories \({\mathfrak {A}}: \mathrm{N}(\mathbf {Caus}({\mathcal {M}})) \rightarrow {\mathcal {C}}\) as a higher version of the data of a QFT model: in a homotopy-coherent fashion, it assigns a \(*\)-algebra to each \({\mathcal {O}}\) in \(\mathbf {Caus}({\mathcal {M}})\).
Remark 2.20
It is an interesting question—particularly from the perspective of examples and applications—to determine when such a functor of \(\infty \)-categories \({\mathfrak {A}}\) can be represented by a strict functor between explicit (relative) categories, such as \({\widetilde{{\mathfrak {A}}}}: \mathbf {Caus}({\mathcal {M}}) \rightarrow \mathbf {Alg}^*(\mathbf {Ch}({\mathbf {Nuc}}_{\mathbf {\hbar }}))\). We do not address that question here, although we expect it admits a clean answer. We feel, however, that it is a question distinct from the issue of formulating a good, abstract definition.
To obtain a higher version of a QFT model, however, we need to articulate versions of Einstein causality and the time-slice axiom. Again there are several approaches. As yet we do not feel it is clear which approach is most natural or compelling, so we encourage interested readers to explore and advocate the approach that appeals to them.
Thankfully, as we will see, the constructions from the FR and CG formalisms yield ordinary functors that ought to determine functors of higher categories in almost any imaginable approach, as will be manifest to those familiar with higher categories. To flag the provisional nature of the definitions we provide below, we include the adjective “semistrict,” since we mix lower and higher categorical approaches.
2.3.2 .
Recall that \(\mathbf {Ch}(\mathbf {Nuc})\) denotes the category whose objects are cochain complexes in \(\mathbf {Nuc}\) and whose morphisms are continuous cochain maps. We equip it with the completed projective tensor product \({\widehat{\otimes }}\) to make it symmetric monoidal. So far we have only specified an ordinary category, but we can view it as presenting an \(\infty \)-category by making it a relative category: a morphism is a weak equivalence if it is a quasi-isomorphism.
Definition 2.21
A semistrict dg classical field theory model on a spacetime \({\mathcal {M}}\) is a functor \({\mathfrak {P}} : \mathbf {Caus}({\mathcal {M}}) \rightarrow \mathbf {PAlg}^*(\mathbf {Ch}(\mathbf {Nuc}))\), so that each \({\mathfrak {P}}({\mathcal {O}})\) is a locally convex dg Poisson \(*\)-algebra satisfying Einstein causality: spacelike-separated observables Poisson-commute at the level of cohomology. That is, for \({\mathcal {O}}_1,{\mathcal {O}}_2\in \mathbf {Caus}({\mathcal {M}})\) that are spacelike to each other, the bracket \(\left\lfloor {\mathfrak {P}}({\mathcal {O}}_1),{\mathfrak {P}}({\mathcal {O}}_2)\right\rfloor \) is exact (and so vanishes at the level of cohomology) in \({\mathfrak {P}}({\mathcal {O}}')\) for any \({\mathcal {O}}' \in \mathbf {Caus}({\mathcal {M}})\) that contains both \({\mathcal {O}}_1\) and \({\mathcal {O}}_2\).
It satisfies the time-slice axiom if for any \({\mathcal {N}}\in \mathbf {Caus}({\mathcal {M}})\) a neighborhood of a Cauchy surface in the region \({\mathcal {O}}\in \mathbf {Caus}({\mathcal {M}})\), then the map \({\mathfrak {P}}({\mathcal {N}}) \rightarrow {\mathfrak {P}}({\mathcal {O}})\) is a quasi-isomorphism.
Remark 2.22
Isotony holds at the cochain level for the constructions and example with which we are familiar, but it may fail at the level of cohomology, as it does in the setting of gauge theory. (Consider, as a toy model, how ordinary cohomology can be viewed as arising from sheaf cohomology of a locally constant sheaf. Locally, the sheaf is simple but its cohomological behavior depends on the topology of each open.) One might guess that isotony holds at the level of cohomology for inclusions \({\mathcal {O}}\rightarrow {\mathcal {O}}'\) between contractible opens, but we hesitate to impose that condition until we have explored more examples.
One can further loosen the definition, if one wishes, by asking for associativity of morphisms only up to homotopy coherence. This is a formal change to implement and not relevant to our focus in this paper. We will introduce, however, the appropriate notion of weak equivalence of models, so that we have a relative category implicitly presenting an \(\infty \)-category.
Definition 2.23
A natural transformation \(\eta : {\mathfrak {P}}\Rightarrow {\mathfrak {P}}'\) between two semistrict dg classical field theory models is a weak equivalence if the map \(\eta _{{\mathcal {O}}}: {\mathfrak {P}}({\mathcal {O}}) \rightarrow {\mathfrak {P}}'({\mathcal {O}})\) is a quasi-isomorphism for every \({\mathcal {O}}\in \mathbf {Caus}({\mathcal {M}})\).
We now turn to the quantum setting.
Definition 2.24
A semistrict dg QFT model on a spacetime \({\mathcal {M}}\) is a functor \({\mathfrak {A}}: \mathbf {Caus}({\mathcal {M}}) \rightarrow \mathbf {Alg}^*(\mathbf {Ch}({\mathbf {Nuc}}_{\mathbf {\hbar }}))\), so that each \({\mathfrak {A}}({\mathcal {O}})\) is a locally convex unital \(*\)-dg algebra satisfying Einstein causality: spacelike-separated observables commute at the level of cohomology. That is, for \({\mathcal {O}}_1,{\mathcal {O}}_2\in \mathbf {Caus}({\mathcal {M}})\) that are spacelike to each other, the bracket \([{\mathfrak {A}}({\mathcal {O}}_1),{\mathfrak {A}}({\mathcal {O}}_2)]\) is exact in \({\mathfrak {A}}({\mathcal {O}}')\) for any \({\mathcal {O}}' \in \mathbf {Caus}({\mathcal {M}})\) that contains both \({\mathcal {O}}_1\) and \({\mathcal {O}}_2\).
It satisfies the time-slice axiom if for any \({\mathcal {N}}\in \mathbf {Caus}({\mathcal {M}})\) a neighborhood of a Cauchy surface in the region \({\mathcal {O}}\in \mathbf {Caus}({\mathcal {M}})\), then the map \({\mathfrak {A}}({\mathcal {N}}) \rightarrow {\mathfrak {A}}({\mathcal {O}})\) is a quasi-isomorphism.
Again, we introduce a notion of weak equivalence.
Definition 2.25
A natural transformation \(\eta : {\mathfrak {A}}\Rightarrow {\mathfrak {A}}'\) between two semistrict dg classical field theory models is a weak equivalence if the map \(\eta _{{\mathcal {O}}}: {\mathfrak {A}}({\mathcal {O}}) \rightarrow {\mathfrak {A}}'({\mathcal {O}})\) is a quasi-isomorphism for every \({\mathcal {O}}\in \mathbf {Caus}({\mathcal {M}})\).
2.4 Overview of factorization algebras
In their work on chiral conformal field theory, Beilinson and Drinfeld introduced factorization algebras in an algebro-geometric setting. These definitions also encompass important objects in geometric representation theory, playing a key role in the geometric Langlands program. Subsequently, Francis, Gaitsgory, and Lurie identified natural analogous definitions in the setting of manifolds, which provide novel approaches in, e.g., homotopical algebra and configuration spaces. Below we describe a version of factorization algebras, developed in [CG17a], that is well-suited to field theory.
As this brief history indicates, factorization algebras do not attempt to axiomatize the observables of a field theory. Instead, they include examples from outside physics, such as from topology and representation theory, and permit the transport of intuitions and ideas among these fields. We will explain below further structure on a factorization algebra that makes it behave like the observables of a field theory in the Batalin–Vilkovisky formalism.
2.4.1 The core definitions
Let M be a smooth manifold. Let \(\mathbf {Open}(M)\) denote the poset category whose objects are opens in M and where a morphism is an inclusion. A factorization algebra will be a functor from \(\mathbf {Open}(M)\) to a symmetric monoidal category \(\mathrm {{\mathbf {C}}}\) with tensor product \(\otimes \) equipped with further data and satisfying further conditions. We will explain this extra information in stages. (Note that almost all the definitions below apply to an arbitrary topological space, or even site with an initial object, and not just smooth manifolds.)
Definition 2.26
for each open \(U \subset M\), an object \({\mathcal {F}}(U) \in \mathrm {{\mathbf {C}}}\),
- for each finite collection of pairwise disjoint opens \(U_1,\ldots ,U_n\), with \(n > 0\), and an open V containing every \(U_i\), a morphism$$\begin{aligned} {\mathcal {F}}(\{U_i\}; V): {\mathcal {F}}(U_1) \otimes \cdots \otimes {\mathcal {F}}(U_n) \rightarrow {\mathcal {F}}(V), \end{aligned}$$
- composition is associative, so that the triangle commutes for 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 \({\mathcal {F}}(\{U_i\}; V)\) are equivariant under permutation of labels, so that the triangle commutes for any \(\sigma \in S_n\).
Note that if one restricts to collections that are singletons (i.e., some \(U \subset V\)), then one obtains simply a precosheaf \({\mathcal {F}}: \mathbf {Open}(M) \rightarrow \mathrm {{\mathbf {C}}}\). By working with collections, we are specifying a way to “multiply” elements living on disjoint opens to obtain an element on a bigger open. In other words, the topology of M determines the algebraic structure. (One can use the language of colored operads to formalize this interpretation, but we refer the reader to [CG17a] for a discussion of that perspective. Moreover, one can loosen the conditions to be homotopy-coherent rather than on-the-nose.)
A factorization algebra is a prefactorization algebra for which the value on bigger opens is determined by the values on smaller opens, just as a sheaf is a presheaf that is local-to-global in nature. A key difference here is that we need to be able to reconstruct the “multiplication maps” from the local data, and so we need to modify our notion of cover accordingly.
Definition 2.27
A Weiss cover\(\{U_i\}_{\{i \in I\}}\) of an open subset \(U \subset M\) is a collection of opens \(U_i \subset U\) such that for any finite set of points \(S= \{x_1,\ldots ,x_n\} \subset U\), there is some \(i \in I\) such that \(S \subset U_i\).
Remark 2.28
Note that a Weiss cover is also a cover, simply by considering singletons. Typically, however, an ordinary cover is not a Weiss cover. Consider, for instance, the case where \(U = V \sqcup V'\), with \(V,V'\) disjoint opens. Then \(\{V,V'\}\) is an ordinary cover by not a Weiss cover, since neither V nor \(V'\) contains any two element set \(\{x,x'\}\) with \(x \in V\) and \(x' \in V'\). Nonetheless, Weiss covers are easy to construct. For instance, a Weiss cover of an n-manifold M is given by the collection of open subsets that are each homeomorphic to a finite union of copies of \({\mathbb {R}}^n\).
This notion of cover determines a Grothendieck topology on M; concretely, this means it determines a notion of cover for each open of M that behaves nicely with respect to intersection of opens and refinements of covers. In particular, we can talk about (co)sheaves relative to this Weiss topology on M.
Definition 2.29
Typically, our target category \((\mathrm {{\mathbf {C}}}, \otimes )\) is vector spaces of some kind (such as topological vector spaces), in which case the coproducts \(\coprod \) denote direct sums \(\oplus \) and the coequalizer simply means that \({\mathcal {F}}(U)\) is the cokernel of the difference of the maps for the inclusions \(U_i \cap U_j \subset U_i\) and \(U_i \cap U_j\subset U_j\). Note that we have implicitly assumed that \(\mathrm {{\mathbf {C}}}\) possesses enough colimits, and we will assume that henceforward.
Remark 2.30
The prefactorization algebras we construct in this paper use spaces of smooth or distributional sections, and hence live in nuclear spaces. In Chapter 6, Section 5 of [CG17a], it is checked directly that the relevant colimits exist for these functors in the closely related category of differentiable vector spaces. In short, it is proved there that our main constructions form factorization algebras. (The arguments mimic the proofs that smooth functions form a sheaf—partitions of unity play a role—but exploit the Weiss condition at one key point.) We do not examine here the colimit condition in nuclear spaces.
Remark 2.31
In fact, our target category is usually cochain complexes of vector spaces, and we want to view cochain complexes as (weakly) equivalent if they are quasi-isomorphic. Hence, we want to work in an \(\infty \)-categorical setting. In such a setting, the cosheaf condition becomes higher categorical too: we replace the diagram above by a full simplicial diagram over the Čech nerve of the cover and we require \({\mathcal {F}}(U)\) to be the homotopy colimit over this simplicial diagram. For exposition of these issues, see [CG17a].
In practice, another condition often holds, and it’s certainly natural from the perspective of field theory.
Definition 2.32
In brief, if \({\mathcal {F}}\) is a multiplicative factorization algebra, one can reconstruct \({\mathcal {F}}\) if one knows how it behaves on a collection of small opens. For instance, suppose M is a Riemannian manifold and one knows \({\mathcal {F}}\) on all balls of radius \(\le 1\), then one can reconstruct \({\mathcal {F}}\) on every open of M. (See Chapter 7 of [CG17a] for how to reconstruct from a Weiss basis.) Our examples are often multiplicative, or at least satisfy the weaker condition that the map is a dense inclusion.
Remark 2.33
It is natural to wonder if there is a functor adjoint to the forgetful (aka inclusion) functor of factorization algebras into prefactorization algebras, by analogy to the sheafification functor from presheaves to sheaves. We do not know the answer to this question. There exists a cosheafification functor from precosheaves to Weiss cosheaves, but the underlying precosheaf of a prefactorization algebra does not know about structure maps involving multiple disjoint opens, so it seems unlikely that Weiss cosheafification is sufficient (by itself) to produce a factorization algebra.
2.4.2 Relationship with field theory
By now, the reader may have noticed that there has been no discussion of fields or Poisson algebras or so on. Indeed, the definitions here are more general and less involved than for the AQFT setting because they aim to apply outside the context of field theory (e.g., there are interesting examples of factorization algebras arising from geometric representation theory and algebraic topology) and because there is no causality structure to track. By contrast, AQFT aims to formalize precisely the structure possessed by observables of a field theory on Lorentzian manifolds, and hence must take into account both causality and other characterizing features of field theories (e.g., Poisson structures at the classical level).
Let us briefly indicate how to articulate observables of field theory in this setting, suppressing important issues of homological algebra and functional analysis, which are discussed below in the context of the free scalar field and in [CG17a, CG17b] in a broader context. The necessary extra ingredient is that on each open U, the object \({\mathcal {F}}(U)\) has an algebraic structure.
Definition 2.34
In other words, the category of prefactorization algebras \(\mathbf {PFA}(M,(\mathrm {{\mathbf {C}}},\otimes ))\) is itself symmetric monoidal. In many cases the full subcategory \(\mathbf {FA}(M, (\mathrm {{\mathbf {C}}},\otimes ))\) is closed under this symmetric monoidal product. In particular, if the tensor product \(\otimes \) in \(\mathrm {{\mathbf {C}}}\) preserves colimits separately in each variable (or at least geometric realizations), then \({\mathcal {F}}\otimes {\mathcal {G}}\) is a factorization algebra when \({\mathcal {F}}, {\mathcal {G}}\) are.
Thus, if \(\mathrm {{\mathbf {C}}}\) is some category of vector spaces, one can talk about, e.g., a commutative algebra in \(\mathbf {PFA}(M,(\mathrm {{\mathbf {C}}},\otimes ))\). That means \({\mathcal {F}}\) is equipped with a map of prefactorization algebras \(\cdot : {\mathcal {F}}\otimes {\mathcal {F}}\rightarrow {\mathcal {F}}\) satisfying all the conditions of a commutative algebra. Similarly, one can talk about Poisson or \(*\)-algebras.
It is equivalent to say that \({\mathcal {F}}\) is in \(\mathbf {CAlg}(\mathbf {PFA}(M,(\mathrm {{\mathbf {C}}},\otimes )))\) or to say it is a prefactorization algebra with values in \(\mathbf {CAlg}(\mathrm {{\mathbf {C}}},\otimes )\), the category of commutative algebras in \((\mathrm {{\mathbf {C}}},\otimes )\). This equivalence does not apply, however, to factorization algebras, due to the local-to-global condition: a colimit of commutative algebras does not typically agree with the underlying colimit of vector spaces. For instance, in the category of ordinary commutative algebras \(\mathbf {CAlg}(\mathbf {Vec},\otimes )\), the coproduct is \(A \otimes B\), but in the category of vector spaces \(\mathbf {Vec}\), it is the direct sum \(A \oplus B\). (This issue is very general: for an operad \({\mathcal {O}}\), the category \({\mathcal {O}}\text {-alg}(\mathrm {{\mathbf {C}}},\otimes )\) of \({\mathcal {O}}\)-algebras has a forgetful functor to \(\mathrm {{\mathbf {C}}}\) that always preserves limits but rarely colimits.) Thus, a commutative algebra in factorization algebras \({\mathcal {F}}\in \mathbf {CAlg}(\mathbf {PFA}(M,(\mathrm {{\mathbf {C}}},\otimes )))\) assigns a commutative algebra to every open U and a commutative algebra map to every inclusion of disjoint opens \(U_1,\ldots , U_n \subset V\), but it satisfies the coequalizer condition in \(\mathrm {{\mathbf {C}}}\), not in \(\mathbf {CAlg}(\mathrm {{\mathbf {C}}},\otimes )\).
This terminology lets us swiftly articulate a deformation-theoretic view of the Batalin–Vilkovisky framework.
Definition 2.35
A classical field theory model is a 1-shifted Poisson (aka\(P_0\)) algebra \({\mathcal {P}}\) in factorization algebras \(\mathbf {FA}(M, \mathbf {Ch}(\mathbf {Nuc}))\). That is, to each open \(U \subset M\), the cochain complex \({\mathcal {P}}(U)\) is equipped with a commutative product \(\cdot \) and a degree 1 Poisson bracket \(\{-,-\}\); moreover, each structure map is a map of shifted Poisson algebras.
Note that we always work with the completed projective tensor product \({\widehat{\otimes }}\) with nuclear spaces, so we will suppress it from the notation. In other words, we simply write \(\mathbf {Ch}(\mathbf {Nuc})\) instead of \((\mathbf {Ch}(\mathbf {Nuc}), {\widehat{\otimes }})\).
In parallel, we have the following.
Definition 2.36
an \(\hbar \)-linear commutative product \(\cdot \),
an \(\hbar \)-linear, degree 1 Poisson bracket \(\{-,-\}\), and
- a differential such that$$\begin{aligned} \mathrm{d}(a \cdot b) = \mathrm{d}(a) \cdot b + (-1)^a a \cdot \mathrm{d}(b) + \hbar \{a,b\}. \end{aligned}$$
Remark 2.37
We include the condition of flatness to ensure that tensoring need not be derived. All of our examples will be free in the appropriate sense.
Remark 2.38
The condition on the differential is an abstract version of a property possessed by the divergence operator for a volume form on a finite-dimensional manifold. Thus, the differential of a BD algebra behaves like a “divergence operator,” as explained in Chapter 2 of [CG17a], and hence encodes (some of) the kind of information that a path integral would.
2.5 A variant definition: locally covariant field theories
Above, we have worked on a fixed manifold, but most field theories are well-defined on some large class of manifolds. For instance, the free scalar field theory makes sense on any manifold equipped with a metric of some kind. Similarly, (classical) pure Yang-Mills theory makes sense on any 4-manifold equipped with a conformal class of metric and a principal G-bundle. One can thus replace \(\mathbf {Open}(M)\) by a more sophisticated category whose objects are “manifolds with some structure” and whose maps are “structure-preserving embeddings.” (In the scalar field case, think of manifolds-with-metric and isometric embeddings.) In a field theory, the fields restrict along embeddings and the equations of motion are local (but depend on the local structure), so that solutions to the equations Sol forms a contravariant functor out of this category. Likewise, one can generalize the models of classical or quantum field theory to this kind of setting, as we now do.
Remark 2.39
This discussion is not necessary for what happens elsewhere in the paper, so the reader primarily interested in our comparison results should feel free to skip ahead.
2.5.1 The Lorentzian case
We begin by replacing the fixed spacetime \({\mathcal {M}}\) by a coherent system of all such spacetimes.
Definition 2.40
Let \(\mathrm {\mathbf {Loc}}_n\) be the category where an object is a connected, (time-)oriented globally hyperbolic spacetime of dimension n and where a morphism \(\chi :{\mathcal {M}}\rightarrow {\mathcal {N}}\) is an isometric embedding that preserves orientations and causal structure. The latter means that for any causal curve \(\gamma : [a,b]\rightarrow N\), if \(\gamma (a),\gamma (b)\in \chi (M)\), then for all \(t \in ]a,b[\), we have \(\gamma (t)\in \chi (M)\). (That is, \(\chi \) cannot create new causal links.)
We can extend \(\mathrm {\mathbf {Loc}}_n\) to a symmetric monoidal category \(\mathrm {\mathbf {Loc}}_n^{\otimes }\) by allowing for objects that are disjoint unions of objects in \(\mathrm {\mathbf {Loc}}_n\). The relevant symmetric monoidal structure is the disjoint union \(\sqcup \). Note that a morphism in \(\mathrm {\mathbf {Loc}}_n^\otimes \) must send disjoint components to spacelike-separated regions.
We are now ready to state what is meant by a locally covariant field theory in our setting, following the definition proposed in [BFV03]. We use here a very minimal version of the axioms for the locally covariant field theory functor. From the physical viewpoint, it might be necessary to require some further properties, e.g. dynamical locality (for more details see [FV12a, FV12b]).
Note that isotony is implicit in the requirement that morphisms in \(\mathbf {Alg}^*(\mathbf {Nuc})^\mathrm{inj}\) are injective. It is likewise implicit in the following definitions.
Definition 2.41
Definition 2.42
Given two isometric embeddings \(\chi _1:{\mathcal {M}}_1\rightarrow {\mathcal {M}}\) and \(\chi _1:{\mathcal {M}}_1\rightarrow {\mathcal {M}}\) whose images \(\chi _1({\mathcal {M}}_1)\) and \(\chi _2({\mathcal {M}}_2)\) are spacelike-separated, the subalgebrascommute, i.e., we have$$\begin{aligned} {\mathfrak {A}}\chi _1({\mathfrak {A}}({\mathcal {M}}_1)) \subset {\mathfrak {A}}({\mathcal {M}}) \supset {\mathfrak {A}}\chi _2({\mathfrak {A}}({\mathcal {M}}_2)) \end{aligned}$$for any \(a_1 \in {\mathfrak {A}}({\mathcal {M}}_1)\) and \(a_2 \in {\mathfrak {A}}({\mathcal {M}}_2)\).$$\begin{aligned} {[}{\mathfrak {A}}\chi _1(a_1),{\mathfrak {A}}\chi _2(a_2)]=\{0\}, \end{aligned}$$
Definition 2.43
A model \({\mathfrak {P}}\) (\({\mathfrak {A}}\)) is called on-shell if it satisfies in addition the time-slice axiom: If \(\chi :{\mathcal {M}}\rightarrow {\mathcal {N}}\) contains a neighborhood of a Cauchy surface \(\Sigma \subset {\mathcal {N}}\), then the map \({\mathfrak {P}}\chi : {\mathfrak {P}}({\mathcal {M}}) \rightarrow {\mathfrak {P}}({\mathcal {N}})\) (respectively, \({\mathfrak {A}}\chi : {\mathfrak {A}}({\mathcal {M}}) \rightarrow {\mathfrak {A}}({\mathcal {N}})\)) is an isomorphism.
Remark 2.44
The category \(\mathbf {Alg}^*(\mathbf {Nuc})^\mathrm{inj}\) has a natural symmetric monoidal structure via the completed tensor product \({\widehat{\otimes }}\). Then Einstein causality can be rephrased as the condition that \({\mathfrak {A}}\) is a symmetric monoidal functor from \(\mathrm {\mathbf {Loc}}_n^{\otimes }\) to \(\mathbf {Alg}^*(\mathbf {Nuc})^{\mathrm{inj},{\widehat{\otimes }}}\), as discussed in [BFIR14].
2.5.2 The factorization algebra version
Let us begin with the simplest version.
Definition 2.45
Let \(\mathbf {Emb}_n\) denote the category whose objects are smooth n-manifolds and whose morphisms are open embeddings. It possesses a symmetric monoidal structure under disjoint union.
Then we introduce the following variant of the notion of a prefactorization algebra. Below, we will explain the appropriate local-to-global axiom.
Definition 2.46
A prefactorization algebra onn-manifolds with values in a symmetric monoidal category \((\mathrm {{\mathbf {C}}}, \otimes )\) is a symmetric monoidal functor from \(\mathbf {Emb}_n\) to \(\mathrm {{\mathbf {C}}}\).
This kind of construction works very generally. For instance, if we want to focus on Riemannian manifolds, we could work in the following setting.
Definition 2.47
Let \(\mathbf {Riem}_n\) denote the category where an object is Riemannian n-manifold (M, g) and a morphism is open isometric embedding. It possesses a symmetric monoidal structure under disjoint union.
Definition 2.48
A prefactorization algebra on Riemanniann-manifolds with values in a symmetric monoidal category \((\mathrm {{\mathbf {C}}}, \otimes )\) is a symmetric monoidal functor from \(\mathbf {Riem}_n\) to \(\mathrm {{\mathbf {C}}}\).
Remark 2.49
In these definitions, the morphisms in \(\mathbf {Riem}_n\) form a set, but one can also consider an enrichment so that the morphisms form a space, perhaps a topological space or even some kind of infinite-dimensional manifold. This kind of modification can be quite useful. For instance, this would allow to view isometries (i.e., isometric isomorphisms) as a Lie group, rather than as a discrete group.
In general, let \({\mathcal {G}}\) denote some kind of local structure for n-manifolds, such as a Riemannian metric or complex structure or orientation. In other words, \({\mathcal {G}}\) is a sheaf on \(\mathbf {Emb}_n\). A \({\mathcal {G}}\)-structure on an n-manifold M is then a section \(G \in {\mathcal {G}}(M)\). There is a category \(\mathbf {Emb}_{\mathcal {G}}\) whose objects are n-manifolds with \({\mathcal {G}}\)-structure \((M,G_M)\) and whose morphisms are \({\mathcal {G}}\)-structure-preserving embeddings, i.e., embeddings \(f : M \hookrightarrow N\) such that \(f^* G_N = G_M\). This category is fibered over \(\mathbf {Emb}_{\mathcal {G}}\). One can then talk about prefactorization algebras on \({\mathcal {G}}\)-manifolds.
We now turn to the local-to-global axiom in this context.
Definition 2.50
A Weiss cover of a \({\mathcal {G}}\)-manifold M is a collection of \({\mathcal {G}}\)-embeddings \(\{ \phi _i : U_i \rightarrow M\}_{i \in I}\) such that for any finite set of points \(x_1,\ldots ,x_n \in M\), there is some i such that \(\{x_1,\ldots ,x_n\} \subset \phi _i(U_i)\).
With this definition in hand, we can formulate the natural generalization of our earlier definition.
Definition 2.51
A factorization algebra on\({\mathcal {G}}\)-manifolds is a symmetric monoidal functor \({\mathcal {F}}: \mathbf {Emb}_{{\mathcal {G}}} \rightarrow \mathrm {{\mathbf {C}}}\) that is a cosheaf in the Weiss topology.
One can mimic the definitions of models for field theories in this setting.
Definition 2.52
A \({\mathcal {G}}\)-covariant classical field theory is a 1-shifted (aka\(P_0\)) algebra \({\mathcal {P}}\) in factorization algebras \(\mathbf {FA}(\mathbf {Emb}_{{\mathcal {G}}},\mathbf {Ch}(\mathbf {Nuc}))\).
Definition 2.53
A \({\mathcal {G}}\)-covariant quantum field theory is a Beilinson-Drinfeld (BD) algebra \({\mathcal {A}}\) in factorization algebras \(\mathbf {FA}(\mathbf {Emb}_{{\mathcal {G}}}, \mathbf {Ch}({\mathbf {Nuc}}_{\mathbf {\hbar }}))\).
3 Comparing the Definitions
- (1)
In the CG formalism a model for a field theory defines a functor on the poset \(\mathbf {Open}(M)\) of all open subsets. By contrast, the FR formalism a model defines a functor on the subcategory \(\mathbf {Caus}({\mathcal {M}})\). Why this restriction? How should one extend an FR model to a functor on the larger category of all opens? Is it a factorization algebra?
- (2)
In the FR formalism, a model assigns a Poisson algebra (or \(*\)-algebra) to each open in \(\mathbf {Caus}({\mathcal {M}})\), whereas in the CG formalism, a model assigns a shifted Poisson algebra (or BD algebra) to every open. Are these rather different kinds of algebraic structures related?
3.1 Free field theory models
We now turn to stating our main result, which is a comparison of the FR and CG procedures. First, we need to state what each formalism accomplishes with the free field. In the following sections, we spell out in detail how to construct the models asserted and prove the propositions.
We remark that these statements are likely hard to understand at this point; the point we emphasize here is just that we get models in both the FR and CG senses.
Proposition 3.1
The space of fields \({\mathfrak {F}}({\mathcal {O}})\) is the space generated (as a commutative algebra) by continuous linear functionals on distributional solutions of \(P\phi = 0\) on \({\mathcal {O}}\);
the commutative product \(\cdot \) is the obvious pointwise product of the space of functionals on the solution space of \({\mathcal {O}}\);
the Poisson bracket is the Peierls bracket \(\left\lfloor .,.\right\rfloor \) (see [Pei52] and the remark below).
Remark 3.2
In Proposition 3.1 we mention the Peierls bracket, which is a Poisson bracket introduced by Peierls in [Pei52]. It is defined using the Lagrangian formalism (in contrast to the usual canonical bracket introduced in the Hamiltonian framework), in a fully covariant way, as a bracket on the algebra of functions on the space of solutions to the equations of motion. A key feature is that it has a well-defined off-shell extension to a Poisson bracket on the space of all functionals on the configuration space (see [DF03]). We come back to this structure in Sect. 6.4.
Remark 3.3
Note that allowing for distributional solutions enforces a restriction on the dual, so that \({\mathfrak {F}}\) is generated by functionals of the form \(\phi \mapsto \int \phi f\), where f is a compactly supported test density on M, modulo the ideal generated by functionals of the form \(\phi \mapsto \int P\phi f\).
Analogously, the CG approach to free theories applies to Lorentzian manifolds, as we show below, and we obtain the following.
Proposition 3.4
FR | CG | |
---|---|---|
Classical | \({\mathfrak {P}}\) | \({\mathcal {P}}\) |
Quantum | \({\mathfrak {A}}\) | \({\mathcal {A}}\) |
We remark that these propositions might seem distinct on the surface, since the CG result involves cochain complexes while the FR result does not. This distinction disappears when one examines the actual constructions: both use a BV framework, and hence the FR construction actually builds a cochain-level functor as well. We formalize a dg version of pAQFT in Sect. 2.3 below, which makes the comparison even more obvious.
3.2 The comparison results
With these models in hand, a clean comparison result can be stated. Before making the formal statement, we first explain it loosely.
We then want to compare the functors \(H^0({\mathcal {P}}/{\mathcal {A}})|_{\mathbf {Caus}({\mathcal {M}})}\) to the corresponding FR functors. The targets of these functors, however, are different. For instance, \({\mathcal {P}}\) takes values in 1-shifted Poisson algebras and hence so does \(H^0 {\mathcal {P}}\) (although the bracket must then be trivial for degree reason). By contrast, \({\mathfrak {P}}_\mathrm {pol}\) takes values in Poisson \(*\)-algebras. (The subscript \(\mathrm {pol}\) indicates that we will use polynomial algebras for the comparison. See Remark 6.1 for a discussion of natural variant constructions, notably with regular functions.) Hence we apply forgetful functors to land in the same target category. We now state our comparison result for the classical level.
Theorem 3.5
This identification is not surprising, as both approaches end up looking at (a class of) functions on solutions to the equations of motion.
We can extend to the quantum level, but here we need the forgetful functor \({\mathfrak {v}}: \mathbf {Alg}^*({\mathbf {Nuc}}_{\mathbf {\hbar }}) \rightarrow {\mathbf {Nuc}}_{\mathbf {\hbar }}\), since \(H^0 {\mathcal {A}}\) is a priori just a vector space.
Theorem 3.6
The second part of the quantum comparison theorem is likely cryptic at the moment, as it involves the notations \(\alpha _{\partial _{G^{\mathrm{D}}}}\) and \(\star _{G^{\mathrm C}}\) and the terminology “time-ordered products” that we have not yet introduced. We will explain these in the next section, as they are the key to understanding how the two approaches to QFT relate. We wish to clarify now, however, the main thrust of the theorem.
To paraphrase the theorem, the factorization algebra \({\mathcal {A}}\) knows information equivalent to the QFT model \({\mathfrak {A}}\). Conversely, one can recover from \({\mathfrak {A}}\), the precosheaf structure of \({\mathcal {A}}\) restricted to \(\mathbf {Caus}({\mathcal {M}})\). (This assertion is true when one uses the cochain-level refinement of \({\mathfrak {A}}\), as we will see below when reviewing the explicit FR construction.)
What is even more important is that there is a natural way to identify the algebra structures on either side. We will show that one can read off the FR deformation quantization \({\mathfrak {A}}\) from the CG factorization algebra \({\mathcal {A}}\) and conversely.
3.3 Key ingredients of the argument
In this section we recall the relevant background about quantum field theories, notably the notions appearing in the theorems above. We explain, in particular, how the associative algebra structure appears in \({\mathfrak {A}}\), why it is important to the physics, and how it relates to constructions in the CG formalism.
3.3.1 Time-ordered products and why they are important
In [FR12a] it was shown that the maps \({\mathcal {T}}_n\) arise from a commutative, associative product \(\cdot _{{}^{\mathcal {T}}}\) defined on a certain domain of \(C^\infty ({\mathcal {E}},{\mathbb {C}})[[\hbar ]]\). Here, to avoid problems related to renormalization, we will consider \(\cdot _{{}^{\mathcal {T}}}\) on the subset \({\mathfrak {F}}_{\mathrm {reg}}[[\hbar ]]\) of \(C^\infty ({\mathcal {E}},{\mathbb {C}})[[\hbar ]]\). (See Definition 3.12 for its description.)
More abstractly, we introduce the following notion.
Definition 3.7
\({\mathfrak {P}}_0\subset {\mathfrak {P}}\), a subfunctor of the classical theory functor that characterizes the domain of definition of the time-ordered product,
- a functorwhich gives the time-ordered product as a commutative product,$$\begin{aligned} {\mathfrak {A}}_T :\mathbf {Caus}({\mathcal {M}}) \rightarrow \mathbf {CAlg}^*(\mathbf {Nuc}_\hbar ), \end{aligned}$$
- a natural embeddingwhich identifies \({\mathfrak {A}}_T\) as a subspace of \({\mathfrak {A}}\), but only as a vector space,$$\begin{aligned} \xi :{\mathfrak {v}}\circ {\mathfrak {A}}_T \Rightarrow {\mathfrak {v}}\circ {\mathfrak {A}}, \end{aligned}$$
- and a natural isomorphism of commutative algebras$$\begin{aligned} {\mathcal {T}}: {\mathfrak {c}}\circ {\mathfrak {P}}_0[[\hbar ]] \Rightarrow {\mathfrak {A}}_T, \end{aligned}$$
Remark 3.8
Note that the existence of the natural isomorphism \({\mathcal {T}}\) and the fact that \({\mathfrak {A}}\) is a quantization of \({\mathfrak {P}}\) imply that there is a natural embedding \({\mathfrak {v}}\circ {\mathfrak {A}}_T \Rightarrow {\mathfrak {v}}\circ {\mathfrak {A}}\), but \(\xi \) does not have to coincide with this embedding. However, one can choose \(\xi \) to be the identity map and choose the quantization map in definition (2.18) as \(\xi \circ {\mathcal {T}}\). Such choice has been used in [HR16] and it greatly simplifies the construction of the interacting star product.
Remark 3.9
The way in which we phrased definition 3.7 is general enough to cover also the situation where renormalization is needed. For the purpose of this paper (where we work only with regular functionals, so no renormalization is needed), we can take \({\mathfrak {P}}_0\) to be just \({\mathfrak {P}}\) and \(\xi \) to be a natural isomorphism.
This definition intertwines the product on classical and quantum observables in a nontrivial way, and as mentioned in Theorem 3.6, it is the key to relating the algebraic structures on \({\mathcal {A}}\) and \({\mathfrak {A}}\). Hence our goal is to construct this time-ordered product on free fields and show how it appears in the comparison map \(\iota ^q\). We explain that in the next few subsections, which are thus somewhat technical. The main ingredient is various propagators, or Green’s functions, for the equation of motion.^{3}
3.3.2 Propagators
Symbol | Meaning |
---|---|
\(G^{\mathrm {A}}\) | Advanced propagator |
\(G^{\mathrm {R}}\) | Retarded propagator |
\(G^{\mathrm {C}}\doteq G^{\mathrm {R}}-G^{\mathrm {A}}\) | Causal propagator |
\(G^{\mathrm {D}}\doteq \frac{1}{2}\left( G^{\mathrm {R}}+G^{\mathrm {A}}\right) \) | Dirac propagator |
The causal propagator \(G^{\mathrm C} \) is related to another important type of bi-solution of P, namely the Hadamard function.
Definition 3.10
- (1)
\(G^+\) is a distributional bi-solution for P.
- (2)
\(2\,\mathrm {Im}\,G^+=G^{\mathrm C}\)
- (3)\(G^+\) fulfills the microlocal spectrum condition: its wavefront set^{4} iswhere \((x,k)\sim (x',k')\) means that there exists a null geodesic connecting x and \(x'\) and \(k'\) is the parallel transport of k along this geodesic, \({\dot{T}}\) denotes the tangent bundle with the zero section removed and \(({\overline{V}}_+)_x\) is the closure of the cone of positive, future-pointing vectors in \(T_x^*M\).$$\begin{aligned} \mathrm {WF}(G^+)=\{(x,k;x',-k')\in {\dot{T}}M^2|(x,k)\sim (x',k'), k\in ({\overline{V}}_+)_x\}, \end{aligned}$$
- (4)
\(G^+\) is of positive type, i.e. \(\left<G^+,f\otimes {\bar{f}}\right>\ge 0\), for all non-zero \(f\in {\mathcal {D}}(M)\otimes {\mathbb {C}}\). The bracket denotes the dual pairing between distributions and test functions.
Symbol | Meaning |
---|---|
\(G^+\doteq \frac{i}{2}G^{\mathrm {C}}+H\) | Hadamard function |
\(G^{\mathrm {F}}\doteq iG^{\mathrm {D}}+H\) | Feynman propagator for \(G^+\) |
- (1)
new products on the observables and
- (2)
automorphisms of the (underlying vector spaces of) observables.
3.3.3 Smooth maps between locally convex vector spaces
In this work, we model observables as \({\mathbb {C}}[[\hbar ]]\)-valued functions on the space of solutions to some linear differential equations (elliptic or hyperbolic). On various stages of the comparison between the CG and FR approaches, we also consider functions between arbitrary locally convex topological vector spaces. For such functions one can introduce the notion of smoothness, which we are going to use later. We start by introducing smooth functions on \({\mathcal {E}}(M)\). For future convenience, we state here the general definition of a functional derivative of a function between two Hausdorff locally convex spaces.
Definition 3.11
This definition applies in particular to functions from \({\mathcal {E}}(M)\) to \({\mathbb {C}}\). Iterating it n times we define \(C^n\)-functionals of \({\mathcal {E}}(M)\). If a functional is \(C^n\) for all \(n\in {\mathbb {N}}\), we call it (Bastiani) smooth and write \(F\in C^\infty ({\mathcal {E}}(M),{\mathbb {C}})\). Detailed properties of such functionals have been investigated in [BDLGR17].
Among all smooth functionals, a special role is played by the regular ones. Regularity properties of a smooth functional are formulated in terms of the wavefront (WF) sets of its derivatives, since \(F^{(n)}(\phi )\in {{\mathcal {E}}_n'}^{\!\scriptscriptstyle {{\mathbb {C}}}}(M)\). (Recall from Sect. 2.1 that this notation means compactly supported distributional sections on \(M^n\), and the superscript \({\mathbb {C}}\) denotes the complexification.) See [BDLGR17, section 3.4] for a proof.
Definition 3.12
Definition 3.13
Clearly, \({\mathfrak {F}}_\mathrm {pol}(M)\subset {\mathfrak {F}}_\mathrm {reg}(M)\).
3.3.4 Exponential products
A propagator G is an element of \({\mathcal {D}}_2'(M)\) and as such, can be viewed as a bi-vector field on \({\mathcal {E}}(M)\). To make this precise, we first need to make sense of the tangent bundle \(T{\mathcal {E}}(M)\). To this end, we have to equip \({\mathcal {E}}(M)\) with an infinite dimensional manifold structure. One obvious choice is to use the Fréchet topology of \({\mathcal {E}}(M)\).^{5} With this choice, since \({\mathcal {E}}(M)\) is a vector space, we find that the tangent bundle \(T{\mathcal {E}}(M)\) can be identified with \({\mathcal {E}}(M)\times {\mathcal {E}}(M)\). Similarly, the total space of the bundle arising from tensoring the tangent bundle with itself (as a vector bundle over the base manifold \({\mathcal {E}}(M)\)) can be identified with \({\mathcal {E}}(M)\times {\mathcal {E}}(M)^{\otimes 2}\) and hence admits a natural completion to the vector bundle \({\mathcal {E}}(M)\times {\mathcal {D}}'_2(M) \rightarrow {\mathcal {E}}(M)\). The propagator G is a constant section of this completed bundle.
Definition 3.14
Direct computation shows that \(\star _G\) is associative. As we will see later, the product structure of \({\mathfrak {A}}({\mathcal {O}})\) comes from \(\star _{G^{\mathrm{C}}}\).
3.4 The time-slice axiom and the algebra structures
A dissatisfying aspect of the comparison results is that they involve forgetful functors: it seems like we ignore the crucial Poisson, respectively associative, algebra structures, although the constructions (e.g., with propagators) certainly involved them. As discussed, these algebraic structures play a crucial role in physics and hence appear in the axioms of AQFT, but they are not built into the CG construction. It is natural to ask how to resolve this tension.
We provide two perspectives that we feel clarify substantially this issue, one rooted in a key maneuver of the FR work and another using results in higher algebra in conjunction with the CG perspective. Both depend on a prominent and useful feature of these examples: they satisfy the time-slice axiom. That is, if \(\Sigma \) is a Cauchy hypersurface for nested opens \({\mathcal {O}}\subset {\mathcal {O}}'\) in \(\mathbf {Caus}({\mathcal {M}})\), then \({\mathfrak {A}}({\mathcal {O}}) \rightarrow {\mathfrak {A}}({\mathcal {O}}')\) is an isomorphism. The factorization algebra satisfies a cochain-level analog of this axiom: the map \({\mathcal {A}}({\mathcal {O}}) \rightarrow {\mathcal {A}}({\mathcal {O}}')\) is a quasi-isomorphism.
The time-slice property suggests formulating a version of \({\mathfrak {A}}\) and \({\mathcal {A}}\) living just on a Cauchy hypersurface itself. We will state a natural comparison result before explaining the idea why one should exist from the CG perspective.
3.4.1 The result on comparison of algebraic structures
We now turn to formulating a precise framework for describing how the algebraic structures intertwine.
Let \(\Sigma \) be a Cauchy hypersurface of \({\mathcal {M}}\), which inherits a canonical Riemannian metric. Consider the collection of open balls in \(\Sigma \) such that each ball B contains a point \(x \in B\) such that the closure of B is contained inside the injectivity radius of x. This definition plays nicely with inclusion, so let \(\mathbf {CBall}(\Sigma )\) denote the full subcategory of \(\mathbf {Open}(\Sigma )\) for these opens.^{6} Each ball has an associated diamond\(D_{\mathcal {M}}(B) \in \mathbf {Caus}({\mathcal {M}})\), which is the union \(D^+_{\mathcal {M}}(B) \cup D^-_{\mathcal {M}}(B)\), where \(D^{\pm }_{\mathcal {M}}(B)\) consists of every point p in the future/past of B such that every inextendible timelike past/future curve through p passes through B. This construction determines a functor \(D_{\mathcal {M}}: \mathbf {CBall}(\Sigma ) \rightarrow \mathbf {Caus}({\mathcal {M}})\), and hence we obtain the following construction.
Definition 3.15
Let \({\mathfrak {A}}|_\Sigma : \mathbf {CBall}(\Sigma ) \rightarrow \mathbf {Alg}(\mathbf {Nuc})\) denote the composite functor \({\mathfrak {A}}\circ D_{\mathcal {M}}\).
Likewise, we provide a version of \({\mathcal {A}}|_\Sigma \). One could use a limit construction, but we prefer the concrete approach.
Definition 3.16
Let \({\mathcal {A}}|_\Sigma : \mathbf {CBall}(\Sigma ) \rightarrow \mathbf {Ch}({\mathbf {Nuc}}_{\mathbf {\hbar }})\) denote the functor that assigns to B, the BD algebra \({\mathcal {A}}(D_{\mathcal {M}}(B))\).
We thus obtain a nice comparison statement.
Theorem 3.17
Remark 3.18
Note that we use the superscript Alg to indicate that we have a new functor that factors through algebras. In particular, we have \({\mathfrak {v}}\circ H^0({\mathcal {A}}|_\Sigma )^{Alg} = H^0({\mathcal {A}}|_\Sigma )\).
We prove this result in Sect. 6, after we spell out the explicit constructions of our models. The argument does something more refined: we show that the factorization product agrees with the star product up to exact terms. In other words, we implicitly lift \({\mathcal {A}}|_\Sigma \) to a homotopy associative algebra object in \(\mathbf {FA}(\Sigma , \mathbf {Ch}({\mathbf {Nuc}}_{\mathbf {\hbar }}))\). We refrain, however, from spelling out a full homotopy-coherent algebra structure (e.g., \(A_\infty \) structure).
There is an obvious classical analogue to this result. The associative algebra \(H^0({\mathcal {A}}|_\Sigma )^{Alg}\) is naturally filtered by powers of \(\hbar \), and its associated graded algebra is isomorphic to the commutative algebra \(H^0({\mathcal {P}}|_\Sigma )[[\hbar ]]\). Hence, the commutative algebra \({\mathfrak {c}}\circ H^0({\mathcal {P}}|_\Sigma )\) acquires an unshifted Poisson bracket, by taking the \(\hbar \)-component of the commutator of the associative algebra.
Corollary 3.19
A version of this statement at the cochain-level, for \({\mathcal {P}}\), would also be appealing. We now turn to explaining a version that relies on homotopical algebra, but in Sect. 6.4 we use formulas to explain how the Peierls bracket follows from the BV bracket.
3.4.2 The argument via higher abstract nonsense
We wish to explain why \({\mathcal {P}}\) and \({\mathcal {A}}\), when restricted to a Cauchy hypersurface, obtain Poisson and associative structures, respectively. A priori they have a shifted Poisson and BD structure. How could this transmutation of algebraic structure occur?
The key is a pair of interesting results from higher algebra that will relate certain factorization algebras to associative and Poisson algebras. We state the results before extracting the consequence relevant to us.
The second result explains how to relate different kinds of shifted Poisson algebras. Let \(P_n\) denote the operad encoding \((1-n)\)-shifted Poisson brackets, so that \(P_1\) algebras are the usual Poisson algebras (in a homotopy-coherent sense).
Theorem 3.20
For \(n = 0\), these results combine to say that a locally constant factorization algebra with a 1-shifted Poisson structure determines a homotopy-coherent version of an 0-shifted Poisson algebra. Now consider the map \(q: {\mathcal {M}}\rightarrow {\mathbb {R}}\) by taking the leaf space with respect to the foliation by Cauchy surfaces. The pushforward factorization algebra \(q_* {\mathcal {P}}\) has a 1-shifted Poisson structure but it is also locally constant, since the solutions to the equation of motion is a locally constant sheaf in terms of the “time” parameter \({\mathbb {R}}\). Hence, by general principles, we know that \(q_* {\mathcal {P}}\) determines a 0-shifted Poisson algebra.
In this case, the homotopy-Poisson algebra must be strict at the level of cohomology, since the cohomology \(H^*{\mathcal {P}}\) is concentrated in degree 0. This strict Poisson structure agrees with the Poisson structure on \({\mathfrak {P}}\), as we will see.
At the quantum level, things are analogous but simpler. The pushforward factorization algebra \(q_* {\mathcal {A}}\) is also locally constant and hence determines a homotopy-associative algebra. Since the cohomology \(H^*{\mathcal {A}}\) is concentrated in degree 0, it equips \(H^0 {\mathcal {A}}\) with a strict associative structure. One can see it agrees with canonical quantization by a modest modification of arguments from Section 4.4 of [CG17a]. Thus, it agrees the associative structure on \({\mathfrak {A}}\). Hence, by keeping track of the \(\hbar \)-filtration, we deduce that we obtain a correspondence between the Poisson algebra structures.
Our proofs of the comparison theorems take a different tack. Following Section 4.6 of [CG17a], we exhibit natural Poisson and associative algebra structures by explicit formulas involving the propagators. These match on the nose with the time-ordered product, which gives us a direct relation with the star product of \({\mathfrak {A}}\). Hence, in the quantum case, we see directly that these agree with the associative algebra structures coming from the abstract machinery described above. By keeping track of the \(\hbar \)-filtration, we deduce that we obtain a correspondence between the Poisson algebra structures.
Remark 3.21
At the core of these identifications is a relationship between the standard deformation quantization of symplectic vector spaces and the standard BV quantization of free theories, which we exhibited here via explicit formulas. Work-in-progress of the first author with Rune Haugseng suggests a general explanation via higher abstract nonsense. In [GH16], they constructed a functor of linear BV quantization on dg vector spaces with a 1-shifted, linear Poisson bracket. Loosely speaking, one finds that additivity intertwines this linear BV quantization with the usual Weyl quantization of ordinary Poisson vector spaces: namely, taking \(E_1\) algebras on the domain and codomain of linear BV quantizations yields the dg version of standard deformation quantization.
4 Constructing the CG Model for the Free Scalar Field
After all that formalism, we turn in a concrete direction and sketch the construction of free field theories in the CG formalism. We give a brief treatment here as this example is treated at length in Sections 4.2 and 4.3 of [CG17a] for the case of a Riemannian manifolds. As we shall see, the constructions apply verbatim to Lorentzian manifolds.
Let \({\mathcal {M}}= (M,g)\) denote a Lorentzian manifold. Lazily, we write \(\mathrm{d}x\) for the associated volume form on M. We will consider the case \(({\mathbb {R}}, dx)\) as a running example.
4.1 The classical model
To start, consider the classical theory. The equation of motion is \(P\phi = 0\). The running example is the free scalar field, with \(\Box \phi + m^2 \phi = 0\) and \(\phi \) a smooth function on M. The space of distributional solutions \(V\subset {\mathcal {D}}'(M)\) consists of “waves”, and let \(V^*\) denote the continuous linear dual. The natural algebra of observables—of a purely algebraic flavor—is \(\mathrm {Sym}_{alg}(V^*)\), the polynomial functions on V. (Such functions should be contained in more sophisticated choices of observables, and indeed are often a dense subalgebra.) In the BV framework, one replaces this commutative algebra by a commutative dg algebra that resolves it and that also remembers the larger space of fields.
Example 1
For the free scalar field on the real line, the space of solutions is a two-dimensional vector space V spanned by \(\{e^{\pm imx}\}\). Here \(\mathrm {Sym}(V^*) \cong {\mathbb {C}}[p,q]\), a polynomial algebra with two generators. These generators can be identified with “position” and “momentum” at \(x = 0\), since the value of a function and its derivative at one point determine a solution of the equation.
In constructing this resolution, one eventually has to make some choices about functional analysis. We will begin by avoiding any analysis and construct a purely algebraic version, in order to exhibit the structure of the BV approach, but then we will turn to a functional-analytic completion convenient for free theories. (See Section 3.5 or Appendix B of [CG17a] for a seemingly interminable discussion of such functional analysis issues.)
There is manifestly a surjection \(\mathrm {Sym}_{alg}({\mathcal {D}}(M)) \rightarrow \mathrm {Sym}_{alg}(V^*)\) by restricting a function on all fields to a function on fields that satisfy the equation of motion. We now extend this surjection to a resolution \({\widetilde{PV}}\). (It might help some readers to know that we are going to write down the Koszul resolution for a linear equation, which in this case are the equations of motion.)
Remark 4.1
It is an illuminating exercise to show that \(({\widetilde{PV}},\mathrm{d})\) provides a cochain complex resolving the polynomial functions \(\mathrm {Sym}(V^*)\) on the space of solutions V. (It helps to bear in mind that we have written down a Koszul resolution for a linear equation, albeit on an infinite-dimensional vector space.) This resolution has the special property that polynomial functions on all scalar fields is given by the truncation consisting of the degree 0 component. Hence, the commutative dg algebra also remembers, in this way, the ambient space of scalar fields.
Example 2
This completion encodes a flavor of polynomial functions on distributional solutions to the equation of motion.
Theorem 4.2
The proof involves a long detour into functional analysis, so we banish it to the appendix, where we introduce some arcane terminology that leads to a sharper version of the theorem as well as a proof. (Notably we improve the dense inclusion to an isomorphism, but using a symmetric algebra built by completing the bornological tensor product.)
Example 3
In particular, for the free scalar field on \(M = {\mathbb {R}}\), we have \(H^*(PV({\mathbb {R}}),\mathrm{d}) \cong {\mathbb {C}}[p,q]\), where p, q are two variables.
Remark 4.3
Note that this bilinear pairing is ill-defined if one replaces compactly supported smooth functions by distributions. This issue is a key problem in setting up the BV formalism and begets many of the divergences in perturbation theory.
To summarize, we give the following definition.
Definition 4.4
It is simply a completion of the functor \({\mathcal {F}}_{theory}\) defined earlier. We believe it is actually a factorization algebra, but verifying this belief requires understanding homotopy colimits in \(\mathbf {Ch}(\mathbf {Nuc})\), in particular whether the usual formulas for homotopy colimits of cochain complexes hold in this setting (see Appendix C.5 of [CG17a]).
Remark 4.5
Two variations on this approach are needed when dealing with interacting theories. First, one replaces polynomial functions by formal power series, i.e., \(\oplus _{n \ge 0}\) becomes \(\prod _{n \ge 0}\). Second, one cannot restrict to smoothed observables but should allow distributional observables, i.e., \({\mathcal {D}}\) is replaced by \({\mathcal {E}}'\) (the space of compactly supported distributions). In the setting of elliptic differential operators (or elliptic complexes, more generally), the commutative dg algebras with smoothed or distributional algebras are (continuously) quasi-isomorphic. Moreover, the differential is still determined by the equations of motion but is more complicated as it has terms changing the homogeneity of observables. In particular, the smoothed and distributional algebras cease to be quasi-isomorphic in the interacting case.
4.2 The quantum model
We now turn to BV quantization, which modifies the differential by adding the BV Laplacian. This extra term is related to a shifted Poisson structure on PV(U).
Definition 4.6
We now examine the following useful result.
Lemma 4.7
In other words, the cohomology of the classical and quantum observables agree up to adjoining \(\hbar \) to the classical ones. Note that this isomorphism does not respect the commutative algebra structure on the cohomology of the classical observables. Indeed, the differential of a BD algebra is not a derivation with respect to the commutative product, and hence the commutative product does not descend to the cohomology.
Proof
The filtration by powers of \(\hbar \) determines a spectral sequence that computes the cohomology of the quantum observables. The first page is just the cohomology of the classical observables. Since that is totally concentrated in degree 0, the spectral sequence collapses. \(\quad \square \)
5 Constructing the pAQFT Model for a Free Field Theory
In this section we describe the pAQFT construction for the classical and quantum models for a free field theory and prove Proposition 3.1. It is a succinct review of a more extensive treatment available [FR15, Rej16].
The construction itself explicitly produces dg algebras; to recover algebras, one takes the cohomology, which happens to be concentrated in degree zero. Thus, before going into the details, we proffer a dg version of AQFT, as defined in Sect. 2.3.
5.1 Constructing the dg models
In this section we spell out the construction of a semistrict dg model of a free field theory. This is mainly a review of [FR12a], but with more detail and recast in notation compatible with the CG framework. The goal is to provide a kind of Koszul resolution of the algebra of functions on the space of solutions to the equations of motion. We need to pin down some functional analytic choices, along with the homological algebra, before we articulate the central construction, given in Definition 5.10.
5.1.1 Functionals
Regular functionals on the configuration space \({\mathcal {E}}\) were defined in Definition 3.12. We will use these to model classical observables.
Definition 5.1
We will always consider \({\mathfrak {F}}_{\mathrm {reg}}\) together with this topology. It was shown in [FR12b] [Appendix A] that this topology is nuclear. The idea behind the proof (following [BDF09]) is to use the fact that all the spaces \({\mathcal {D}}_n^{\scriptscriptstyle {{\mathbb {C}}}}(M)\), \(n\in {\mathbb {N}}\) are nuclear and \(\tau \) is the initial topology with respect to the evaluation maps \(F\mapsto F^{(n)}(\phi )\in {\mathcal {D}}_n^{\scriptscriptstyle {{\mathbb {C}}}}(M)\), \(\phi \in {\mathcal {E}}(M)\), \(n\in {\mathbb {N}}\). As nuclearity is preserved under projective limits, the result follows.
5.1.2 Polyvector fields
Our goal is to construct a cohomological resolution of \(\mathrm {Sym}(V')\), and we will produce a Koszul-type resolution. Since dS is a kind of 1-form, this resolution is built using the algebra of regular polyvector fields on \({\mathcal {E}}(M)\), with the differential determined by the equation of motion. Our focus at the moment is on the algebra; we postpone discussion of the differential until the paragraphs around Eq. (8).
For a finite-dimensional manifold M, such a Koszul resolution can be understood as arising from the shifted cotangent manifold \(T^*[-1]M\). When M is a vector space V, \(T^*[-1]M\) corresponds to the graded vector space \(V \oplus V^*[-1]\). Its ring of functions then looks like a graded-symmetric algebra \(\mathrm {Sym}(V^* \oplus V[1])\), whose degree \(-n\) component is then \(\mathrm {Sym}(V^*) \otimes \Lambda ^n(V)\). In our setting we thus work with the following.
Definition 5.2
Remark 5.3
Polyvector fields are elements of \({\mathscr {O}}({\mathcal {E}}(M) \oplus {\mathcal {E}}(M)[-1])\). In order to define regular polyvector fields, we need to analyze the WF sets of derivatives of \(F\in {\mathscr {O}}({\mathcal {E}}(M) \oplus {\mathcal {E}}(M)[-1])\).
Remark 5.4
Consider the special case \(X_1=\Gamma (M,E_1)\), \(X_2=\Gamma (M,E_2)\), where \(E_1\), \(E_2\) are vector bundles over M. It was shown in [Rej16, BDLGR17] that \(X_2^{{\widehat{\otimes }} n}\cong \Gamma '(M^n,E_2^{\boxtimes n})_{S_n}\) and that the kth functional derivative of \(F\in {\mathscr {O}}^n(E_1\oplus E_2[1])\) at a given point in \(X_1\), is an element of \(\Gamma '(M^{k+n},E_1^{\boxtimes k}\boxtimes E_2^{\boxtimes n})_{S_k\times S_n}\), symmetric in the first k and antisymmetric in the last n entries.
Definition 5.5
Let \(F\in {\mathscr {O}}^n({\mathcal {E}}(M)\oplus {\mathcal {E}}(M)[-1])\) be a polyvector field. We say F is regular if \(F^{(k)}(\phi )\) has empty WF set (i.e. is smooth). We use \(\mathfrak {PV}_{\mathrm {reg}}({\mathcal {O}})\) to denote the space of all regular polyvector fields on \({\mathcal {E}}({\mathcal {O}})\) where \({\mathcal {O}}\subset {\mathcal {M}}\).
In particular, among all regular polyvector fields we can distinguish the polynomial ones (analogous to Definition 3.13), which we denote by \(\mathfrak {PV}_{\mathrm {pol}}({\mathcal {O}})\).
This construction gives a functor \(\mathfrak {PV}_{\mathrm {reg}}\) from \(\mathbf {Caus}({\mathcal {M}})\) to \(\mathbf {CAlg}(\mathbf {Ch}(\mathbf {Nuc}))\), where the action on morphisms is induced by the pullback. (We currently have the zero differential on the polyvector fields, but we will introduce a differential depending on dS below.)
Clearly, \(\mathfrak {PV}^0_{\mathrm {reg}}={\mathfrak {F}}_{\mathrm {reg}}\) and \(\mathfrak {PV}_\mathrm {reg}\) is a graded commutative algebra by the usual product on functions and the wedge product of polyvector fields.
Remark 5.6
5.1.3 Poisson structure
It is crucial that P is a normally hyperbolic operator, so on a globally hyperbolic spacetime it has retarded/advanced Green’s functions \(G^{\mathrm {R}}\)/\(G^{\mathrm {A}}\), respectively and other propagators introduced in Sect. 3.3.2.
Hence we can lift our notion to the cochain level. We assign dg Poisson algebras to \({\mathcal {O}}\in \mathbf {Caus}({\mathcal {M}})\) by keeping track of support by means of (3).
Definition 5.7
The following proposition shows that this indeed gives us a functor from \(\mathbf {Caus}({\mathcal {M}})\) to \(\mathbf {PAlg}^*(\mathbf {Ch}(\mathbf {Nuc}))\).
Proposition 5.8
Proof
The analogous result for \(\mathfrak {PV}_\mathrm {pol}\) follows by the same arguments.
Remark 5.9
Note that the statement about the existence and uniqueness of retarded and advanced Green functions (needed in the proof of the proposition) is true only on opens that are themselves globally hyperbolic spacetimes (when equipped with the induced metric). Therefore, it is crucial to restrict to \(\mathbf {Caus}({\mathcal {M}})\), rather than to consider arbitrary opens.
If we forget the Poisson algebra structure, we obtain the cohomological (derived) description of the space of classical observables as the functor \({\mathfrak {v}}\circ {\mathfrak {P}}_\mathrm {reg}\equiv {\mathfrak {v}}{\mathfrak {P}}_\mathrm {reg}\) (or \({\mathfrak {v}}{\mathfrak {P}}_\mathrm {pol}\) if we restrict to polynomials).
Definition 5.10
Going on-shell corresponds to taking the \(H^0\) of \({\mathfrak {P}}_\mathrm {reg}\) (or \({\mathfrak {v}}{\mathfrak {P}}_\mathrm {pol}\)). We obtain the following:
Definition 5.11
Proposition 5.12
The assignment \({\mathcal {O}}\mapsto H^0({\mathfrak {P}}_{\mathrm {reg}})({\mathcal {O}})\) defines a classical on-shell model in the sense of Definition 2.43.
This result is part of Proposition 3.1, namely the classical piece of the model.
Proof
It remains only to verify the time-slice axiom, which was done in [Dim80] and also in [Chi08, CF08]. Here, for completeness, we provide an argument.
In addition, pick two other Cauchy surfaces \(\Sigma _\pm \) in \({\mathcal {N}}\), such that \(\Sigma _-\) is in the past \(J^-(\Sigma )\) of \(\Sigma \) and \(\Sigma ^+\) is in the future \(J^+(\Sigma )\) of \(\Sigma \).
Finally, pick a smooth function \(\chi \) that is equal to 1 on \( J^-(\Sigma _-)\), and vanishes on \( J^+(\Sigma _+)\). We use it to construct a partition of unity subordinate to the cover by \(J^+(\Sigma _-)\) and \(J^-(\Sigma _+)\). This partition leads us to decompose f as the linear combination \(\chi f+(1-\chi )f\). The first term is supported in the past \(J^-(\Sigma _+)\) of \(\Sigma _+\), and the second term is supported in the future \(J^+(\Sigma _-)\) of \(\Sigma _-\).
5.1.4 Star product
Definition 5.13
It is straightforward to check that \({\mathfrak {A}}_{\mathrm {reg}}\) is a QFT model in the sense of Definition 2.24. Hence we obtain an on-shell model as follows.
Definition 5.14
With this definition, the functor \({\mathcal {O}}\mapsto H^0({\mathfrak {A}}_\mathrm {reg})({\mathcal {O}})\) (as well as \({\mathcal {O}}\mapsto H^0({\mathfrak {A}}_\mathrm {pol})({\mathcal {O}})\)) is an on-shell QFT model in the sense of Definitions 2.16 and 2.17 . Since causality holds by construction, the only non-trivial step is to prove the time-slice axiom. This is done exactly as in Proposition 5.12.
Remark 5.15
Those familiar with the CG approach, notably Section 4.6 of [CG17a], will recognize that this definition is precisely the factorization product on \({\mathcal {A}}\).
Note that the product \(\cdot _{{}^{\mathcal {T}}}\) is not compatible with \(\delta _S\) (since \(G^{\mathrm{D}}\) is a Green function, rather than a bi-solution), we introduce the off-shell models \({\mathfrak {P}}^{\mathrm{off}}_\mathrm {reg}=(\mathfrak {PV}_\mathrm {reg},\left\lfloor .,.\right\rfloor )\) and \({\mathfrak {A}}_\mathrm {reg}^{\mathrm{off}}=({\mathfrak {v}}\mathfrak {PV}_\mathrm {reg}[[\hbar ]],\star )\).
Definition 5.16
- a functordefined by \({\mathfrak {A}}_T=(\mathfrak {PV}_{\mathrm {reg}}[[\hbar ]],\cdot _{{}^{\mathcal {T}}})\), where \(\cdot _{{}^{\mathcal {T}}}\) is given by (3.7).$$\begin{aligned} {\mathfrak {A}}_T :\mathbf {Caus}(M) \rightarrow \mathbf {CAlg}^*(\mathbf {Nuc}_\hbar ), \end{aligned}$$
- the obvious embedding$$\begin{aligned} \xi :{\mathfrak {v}}\circ {\mathfrak {A}}_T \Rightarrow {\mathfrak {v}}\circ {\mathfrak {A}}_\mathrm {reg}^{\mathrm{off}}. \end{aligned}$$
Lemma 5.17
The time-ordering map \({\mathcal {T}}\) determines a natural transformation \({\mathcal {T}}: {\mathfrak {c}}\circ {\mathfrak {P}}_\mathrm {reg}^{\mathrm{off}}[[\hbar ]] \Rightarrow {\mathfrak {A}}_T\).
Proof
On each \({\mathcal {O}}\in \mathbf {Caus}({\mathcal {M}})\), we define \({\mathcal {T}}^{\mathcal {O}}\doteq \text {exp}\left( \frac{i\hbar }{2}\partial _{G^{\mathrm D}_{\mathcal {O}}}\right) \). This map is well-defined, since \({\mathcal {T}}^{\mathcal {O}}\) is support-preserving.
Definition 5.18
Remark 5.19
The name “quantum observables” used to describe \({\mathfrak {v}}{\mathfrak {A}}_\mathrm {reg}^{q}\) is justified, as this corresponds to what one would describe as such in the physics literature (e.g. [HT92]). Even though \({\mathfrak {v}}{\mathfrak {A}}_\mathrm {reg}^{q}\) is quasi-isomorphic to \({\mathfrak {v}}{\mathfrak {P}}_\mathrm {reg}[[\hbar ]]\) (by means of \({\mathcal {T}}\)), it is conventional to work with the former. One reason is that it forms a BD algebra when equipped with the usual graded pointwise product and the Schouten bracket, while \({\mathfrak {v}}{\mathfrak {P}}_\mathrm {reg}[[\hbar ]]\) forms a BD algebra with \(\cdot _{{}^{\mathcal {T}}}\) and the bracket twisted by \({\mathcal {T}}\) (this is explained in more detail in Sect. 7.1)
6 Proof of Comparison Theorems
Let us build up the natural transformations \(\iota ^{cl}\) and \(\iota ^q\) in stages.
6.1 The classical case
Remark 6.1
In the FR framework one usually works with \(\mathfrak {PV}_{\mathrm {reg}}\), rather than \(\mathfrak {PV}_\mathrm {pol}\), since it contains also infinite sums of polynomials (e.g. Weyl generators \(e^{{\mathcal {O}}_f}\)). Here we stress that the precise comparison between FR and CG frameworks is most naturally done for \(\mathfrak {PV}_\mathrm {pol}\) and one can pass to \(\mathfrak {PV}_\mathrm {reg}\) by appropriate completion (on both sides).
6.2 The quantum case
The quantum case is a bit subtler. The FR approach assigns a dg algebra \({\mathfrak {A}}_\mathrm {pol}=(\mathfrak {PV}_\mathrm {pol}[[\hbar ]], \delta _S, \star )\) whereas the CG approach assigns merely a cochain complex \((PV[[\hbar ]], \mathrm{d}+ \hbar \triangle )\). On the face of it, these look rather different. In particular, the differentials are different, so the embedding that works for the classical case does not extend.
Remark 6.2
As explained in chapter 4.6 of [CG17a], the map \({\mathcal {T}}\) in (14) is not a morphism of factorization algebras. The issue arises when considering structure maps involving disjoint opens containing into a larger open; such maps do not arise when restricted to \(\mathbf {Caus}({\mathcal {M}})\).
6.3 The associative structures
Finally, we come to the comparison of algebra structures, i.e. we prove Theorem 3.17.
In comparing the FR and CG frameworks, a crucial role is played by the time-ordered product. To understand this, observe that in trying to pass from a net to a factorization algebra, we need to construct a commutative product that gives rise to the factorization product structure. A natural commutative product in the pAQFT framework is \(\cdot _{{}^{\mathcal {T}}}\). But going back to to non-commutative product \(\star \) given the commutative one is also easy, as long as we keep track of the supports of observables.
6.3.1 .
To communicate the key idea, we present this conversion process in the 1-dimensional case, where the situation is simple.
In \({\mathbb {R}}\), any interval is a causally convex neighborhood of a Cauchy surface, which in this case is given by a point in the interval. Let \(I_0=(-a,a)\subset {\mathbb {R}}\) be an interval with \(a>0\). For \(I_0\), we fix the point 0 as the Cauchy surface. We can also consider \(I_t=(t-a,t+a)\), which is a translation of \(I_0\) by t.
For \(|t| > 2a\), the factorization product allows us to compute the \(\cdot _{{}^{\mathcal {T}}}\)-commutator As t gets smaller, however, the two intervals \(I_0\) and \(I_t\) begin to overlap, so that we cannot invoke the factorization product. It is possible to resolve this issue—to describe the \(\cdot _{{}^{\mathcal {T}}}\)-commutator in terms of the factorization product—at the level of cohomology. The key point is that for any smaller interval \(I'_0 \subset I_0\), the inclusion \({\mathfrak {A}}(I'_0) \rightarrow {\mathfrak {A}}(I_0)\) is a quasi-isomorphism. Any cocycle \(A \in {\mathfrak {F}}_{\mathrm {reg}}(I_0)\) can be replaced by a cohomologous element with support in the smaller interval \(I'_0 \subset ~I_0\). Hence, at the level of cohomology, we can make the width a of the interval arbitrarily small, and so the \(\cdot _{{}^{\mathcal {T}}}\)-commutator can always be computed using the factorization product. In short, at the level of cohomology, we can recover the \(\star \)-commutator from the factorization product.
6.3.2 .
The general case is also easy to understand, as there is already a factorization algebra structure on the Cauchy surface (i.e. spacelike separated regions are taken care of) and the relation between \(\star \) and the factorization product for time-like separated observables works exactly the same as in the one-dimensional case. We will show, in fact, something slightly more refined by working at the cochain level: we will show that the factorization product agrees with \(\star \) up to exact terms. Let us spell this out in detail now.
As discussed in Remark 6.2, the map \({\mathcal {T}}\) does not respect the factorization product, but this map is an isomorphism when restricted to each open. Hence one can use it to transfer the factorization product on the quantum observables to a new factorization product on the underlying cochain complex of the classical observables \({\mathcal {P}}[[\hbar ]]\). That is, one forgets the original structure maps and borrows them from the quantum side. Denote this new factorization algebra by \({\mathcal {A}}_T\).
As in Sect. 3.4 we fix a small tubular neighborhood \({\widetilde{\Sigma }}\) of a Cauchy surface \(\Sigma \) and construct \({\mathcal {A}}_T\big |_{\Sigma }\). Now we show how to obtain a homotopy-associative product on this restricted factorization algebra.
Consider a time-slice \({\mathcal {N}}_+\) in the future of \({\widetilde{\Sigma }}\) and disjoint from it, so \({\mathcal {N}}_+\cap {\widetilde{\Sigma }}=\emptyset \). Let \({\mathcal {N}}\) be a larger time-slice that contains both \({\mathcal {N}}_+\) and \({\widetilde{\Sigma }}\). By the time-slice axiom, we can make all these slices arbitrarily “thin” in the time direction.
This equation implies that \(m_T({\mathrm {O}_{f}}_+,\mathrm {O}_{g})\) and \(\mathrm {O}_f\star \mathrm {O}_g\) are cohomologous. Thus, at the level of cohomology, the product \(m_T \circ \beta _+ \otimes \mathrm{id}\) agrees with \(\star \), the product on \({\mathfrak {A}}_{\mathrm {reg}}(U)\).
6.4 Shifted vs. unshifted Poisson structures
6.4.1 .
There is a nice interpretation of the map \(\sigma \) in terms of the hyperbolic complex.
Theorem 6.3
Clearly, the map \(\sigma \) is induced by the second to last mapping in this sequence, whose image is exactly the kernel of the equations of motion operator.
6.4.2 .
There is yet another perspective on the Peierls bracket, related to the one presented above, but placing more emphasis on the BD algebra structure.^{7} We now proceed to formulating a precise statement by introducing some assumptions and notation.
Suppose that \({\mathcal {M}}\) is foliated with compact Cauchy surfaces. We will call any small open neighborhood of some Cauchy surface a Cauchy slice. Now fix such a Cauchy slice \({\mathcal {N}}\subset {\mathcal {M}}\). Since solutions to the equations of motion are locally constant in the time direction (with respect to the foliation), we can translate observables forward or backward in time. In particular, for a linear observable \(\mathrm {O}_f\in {\mathcal {P}}({\mathcal {N}})\) localized in this slice, we can produce an observable \(\beta _+(\mathrm {O}_f)\), which is \(\mathrm {O}_f\) shifted to the future, and an observable \(\beta _-(\mathrm {O}_f)\), which is \(\mathrm {O}_f\) shifted to the past. (We will give an explicit formula for \(\beta _{\pm }\) in the proof below.)
Lemma 6.4
Proof
7 Interpretation of the Results
Now that we have precise statements and arguments in place, it may be useful to step back and articulate what they mean. Here we explain how our dialogue has modified our own perspective on these formalisms.
7.1 The main lesson
The map \({\mathcal {T}}\) used in the comparison theorems plays a double role: it is both a cochain isomorphism between classical and quantum observables and also an intertwiner between two products \(\cdot _{{}^{\mathcal {T}}}\) and \(\cdot \). The take-home message is that
We will now make this statement more precise. The approach to quantization taken in pAQFT relies on deformation of the product, while the observables are left unchanged. According to this philosophy, the free quantum theory is obtained by deforming \(\cdot \) to the non-commutative star product \(\star \). Since \(\delta _{S}\) is a derivation with respect to \(\star \), the vector space of observables is just \({\mathfrak {v}}{\mathfrak {P}}_\mathrm {pol}[[\hbar ]]\). Now let’s check if this is compatible with the time-ordered product \(\cdot _{{}^{\mathcal {T}}}\). This structure does not form a differential graded commutative algebra, since \(\delta _S\) is not a derivation with respect to \(\cdot _{{}^{\mathcal {T}}}\). In fact the following identity holds:Quantum observables are described either by deforming the product (from \(\cdot \) to \(\cdot _{{}^{\mathcal {T}}}\)) and keeping the differential as \(\delta _S\) or, equivalently, by deforming the differential (from s to \({\hat{s}}\)) and keeping the product.
7.2 Yet another perspective
Another important fact about the time-ordered product is that it essentially encodes the same combinatorics as the path integral. In Sect. 3.3.1, for instance, we discussed the Dyson series, which displays this encoding.
7.3 A summary by way of a dictionary
Dictionary between the FR and the CG approaches for the free scalar field
Fredenhagen–Rejzner | Costello–Gwilliam |
---|---|
\(M=({\mathbb {R}}^4,\eta )\), \(\eta =\mathrm {diag}(1,-1,-1,-1)\) | \(M=({\mathbb {R}}^4,\mathbb {1})\) |
The space of field configurations | |
\({\mathcal {E}}=C^\infty (M,{\mathbb {R}})\) | |
\(T{\mathcal {E}}={\mathcal {E}}\times {\mathcal {E}}_c\), if \({\mathcal {E}}\) is equipped | \(U\subset M\), \(T_c{\mathcal {E}}(U)={\mathcal {E}}(U)\times {\mathcal {E}}_c(U)\) |
with the Whitney topology; here \({\mathcal {E}}_c\doteq C^\infty _c(M,{\mathbb {R}})\) | |
\({\mathfrak {F}}_{\mathrm {pol}}\) | smooth/smeared observables \(\mathrm {Sym}({\mathcal {E}}_c^!)\) |
Solutions to field equations: zero locus of a 1-formdSon\({\mathcal {E}}\) | |
\(dS\in \Gamma (T^*{\mathcal {E}})\), where \(T^*{\mathcal {E}}={\mathcal {E}}\times {\mathcal {E}}_c'\) | \(dS\in \Gamma (T_c{\mathcal {E}})\) |
Free field equation: | Free field equation: |
\(dS(\phi )=(\Box +m^2)\phi =0\) | \(dS(\phi )=(\Delta +m^2)\phi =0\) |
Multilocal polyvector fields: \(\mathfrak {PV}_\mathrm {reg}(U)\) | PV(U) |
Classical observables | |
\({\mathfrak {v}}{\mathfrak {P}}_\mathrm {pol}({\mathcal {O}})=(\mathfrak {PV}_\mathrm {pol}({\mathcal {O}}),\delta _{S})\), | \({\mathcal {P}}(U)=PV(U)\) as vector spaces, |
where \(\delta _{S}\doteq -\iota _{dS}\) (insertion of the 1-form dS) | the differential is insertion of dS |
Feynman propagator satisfies: | G is a Green’s function for \(\Delta +m^2\) |
\(-(\Box +m^2)\circ G^{\mathrm {F}}=-G^{\mathrm {F}}\circ (\Box +m^2)=i\delta \) | \((\Delta +m^2)\circ G=\delta \) |
Wick (normal) ordering operator | |
\({\mathcal {T}}=e^{\frac{i\hbar }{2}{\mathcal {D}}_{\mathrm {F}}}\), where \({\mathcal {D}}_{\mathrm {F}}=\left\langle G^{\mathrm {F}},\frac{\delta ^2}{\delta \phi ^2}\right\rangle \) | \(W=e^{\hbar \partial _G}\), where \(\partial _G\) is contraction |
with the Green’s function G | |
Quantum observables | |
\({\mathfrak {v}}{\mathfrak {A}}_\mathrm {pol}^q({\mathcal {O}})\doteq (\mathfrak {PV}_\mathrm {pol}({\mathcal {O}})[[\hbar ]],\hat{s},\bigtriangleup )\) where | \({\mathcal {A}}(U) = (PV(U)[[\hbar ]],\mathrm{d}-i\hbar \triangle ,\{-,-\})\) |
\(\hat{s}=\delta _{S}-i\hbar \bigtriangleup \) | |
\({\mathfrak {v}}{\mathfrak {A}}_\mathrm {pol}^q\) can be equipped with a graded commutative | factorization product |
product \(\cdot \) | |
There is a map | There is a cochain isomorphism |
\({\mathcal {T}}:{\mathfrak {v}}{\mathfrak {A}}_\mathrm {pol}^q({\mathcal {O}})\rightarrow {\mathfrak {v}}{\mathfrak {P}}_\mathrm {pol}({\mathcal {O}})[[\hbar ]]\) | \(W_U:{\mathcal {P}}(U)[[\hbar ]]\rightarrow {\mathcal {A}}(U)\) |
that intertwines the differentials, | that deforms the factorization product |
and induces a new product on \({\mathfrak {v}}{\mathfrak {P}}_\mathrm {pol}[[\hbar ]]^\mathrm{a}\): | as follows\(^\mathrm{b}\): |
\(F\cdot _{{}^{\mathcal {T}}}G={\mathcal {T}}({\mathcal {T}}^{-1}F\cdot {\mathcal {T}}^{-1}G)\) | \(\alpha \circledast \beta =e^{-\hbar \partial _G}\left( e^{\hbar \partial _G}\alpha \cdot e^{\hbar \partial _G}\beta \right) \) |
\({\mathcal {T}}_n(\Phi (f_1),\dots ,\Phi (f_n))(0)\equiv \left<G_n,f_1\otimes \dots \otimes f_n)\right>\) | |
\(G^{(n)}\) is the vev of the time-ordered product of n fields, i.e. the n-point Green’s function. | Euclidean Green’s functions (Schwinger functions) |
8 Outlook and Next Steps
In this paper we treated non-renormalized scalar field, so the obvious next steps are to perform renormalization and to generalize to gauge theories. We also discuss the possibility of incorporating the Wick rotation into our framework.
8.1 Interacting field theories
Definition 8.1
In Lorentzian signature, a mathematically rigorous framework for renormalization was provided by Epstein and Glaser [EG73]. In [FR12a] this framework was combined with the BV formalism, allowing one to construct physically useful dg quantum models.
In light of the results of this paper, it is natural to ask whether one can produce a factorization algebra in Lorentzian setting. Note that classical observables form a factorization algebra even in the Lorentzian setting, with no extra work: solutions to the equations of motion form a sheaf—of possibly singular and infinite-dimensional manifolds, but a sheaf nonetheless—and so functions on solutions forms a factorization algebra. We hazard the following guess about the quantization of this situation.
Conjecture 1
Epstein–Glaser renormalization determines a factorization algebra deforming the classical observables. The restriction to \(\mathbf {Caus}({\mathcal {M}})\) determines the dg quantum model of [FR12a].
Remark 8.2
We hope to address the precise relation of that renormalization framework to Costello’s [Cos11] in our future work. This direction of research is potentially divergent from the conjecture above.
8.2 Lifting Wick rotation to the algebraic level
On the other hand, in [Cos11, CG17a] the factorization algebras of QFT allow one to reconstruct Schwinger n-point functions. In this paper we have seen that the CG approach can be also applied to the Lorentzian case directly. However, it is instructive to see how the two are connected on the level of n-point functions on flat spacetime.
The relation between the Euclidean and the Lorentzian framework is usually established via analytic continuation of Schwinger n-point functions using the Osterwalder-Schrader axioms [OS73]. The relation of Schwinger functions to time-ordered products has been discussed in [EE79]. We expect that one should be able to formulate the Wick rotation on the level of factorisation algebras (or nets). We want to address this issue in our future work.
8.3 Gauge theories
It is in the context of gauge and gravity theories that the BV formalism demonstrates its full capacities and qualities, and it would be natural to develop analogues of the results here in those contexts.
The case of abelian gauge theories—where are free theories, albeit cohomological in nature—can be treated by almost identical methods; renormalization is not needed. In [CG17a] there is extensive discussion of the case of pure abelian Chern-Simons theory and of its factorization algebra. Its AQFT counterpart has been constructed in [DMS17].
The work of [BSS17a] is also quite relevant in this context.
More generally, the BV quantization of Yang-Mills theories and effective gravity has been performed in [FR12b, FR12a, BFR16] (based on earlier results of [Hol08]), where the appropriate dg algebras are explicitly given and the need for dg models becomes truly manifest. In these cases, to tackle nonabelian gauge theories or to couple to matter fields requires renormalization, and so the methods, along the lines discussed above for interacting scalar theories, are necessary.
We expect that comparison results, analogous to the ones obtained in the present work, will be easy to prove, provided that the renormalization schemes are shown to be equivalent.
Footnotes
- 1.
If one wants to fix a particular model for \(\infty \)-categories, such as quasicategories, then there is always a standard way to promote an ordinary category to such a higher category. For instance, one can take the nerve \(\mathrm{N}(\mathbf {Caus}({\mathcal {M}}))\) to obtain a quasicategory.
- 2.
We are not even discussing here whether it would be better to work with some other class of functional-analytic spaces. It should be clear that one might reasonably replace \(\mathbf {Nuc}\) by \(\mathbf{TVS} \), or some other category.
- 3.
Indeed, this project began when we realized we were using the same tricks with propagators.
- 4.The wavefront set of a distribution \(u\in {\mathcal {D}}'({\mathbb {R}}^n)\) is a subset of \({\dot{T}}^*{\mathbb {R}}^n\) (the co-tangent bundle minus the zero section) characterizing singular points and singular directions of u (i.e.,directions in the cotangent space in which the Fourier transform does not decay rapidly). More precisely, the complement of \(\mathrm {WF}(u)\) in \({\dot{T}}^*{\mathbb {R}}^n\) is the set of points \((x,{k}) \in {\dot{T}}^*{\mathbb {R}}^n\) for which there exists a “bump function” \(f \in {\mathcal {D}}({\mathbb {R}}^n)\) with \(f(x)=1\) and an open conic neighborhood C of k, withThis notion easily generalizes to open subsets of \({\mathbb {R}}^n\) and to manifolds [Hör03]. Note that if a \(WF(u)=\varnothing \), then u is a smooth function.$$\begin{aligned} \sup _{{k}\in C}(1+|{k}|)^N|\widehat{f \cdot u}({k})|<\infty \qquad \forall N \in {\mathbb {N}}_0. \end{aligned}$$
- 5.
- 6.
Each ball is diffeomorphic to an ordinary ball in Euclidean space via the exponential map at x, and thus no funny topological business appears. The spirit of [FV12a] is at work here. Their slightly larger category of “Cauchy balls” would work equally well for our purposes.
- 7.
We thank Kevin Costello for suggesting this formulation.
Notes
Acknowledgements
This project grew out of interactions at the Mathematisches Forschungsinstitut Oberwolfach in May 2016, and we are grateful for the comfortable and stimulating atmosphere provided by the Workshop “Factorization Algebras and Functorial Field Theories” and our fellow participants. Subsequently we have benefited from the chance to interact in person at the Max Planck Institute for Mathematics, Bonn, and the Perimeter Institute. The authors would like to thank both institutions for their remarkable hospitality and financial support. This research is also supported by the EPSRC grant |EP/P021204/1|. On a more personal level, we have benefited from the insights, experience, suggestions, feedback, and questions of many people. Clearly, Klaus Fredenhagen and Kevin Costello play a crucial role in our perspectives on QFT. In our work on this particular paper, we have enjoyed discussions with Pavel Safronov, Theo Johnson-Freyd, Marco Benini, Alex Schenkel and participants at the conference “Modern Mathematics of Quantum Theory” in York.
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