The Star Product in Interacting Quantum Field Theory

We provide a direct combinatorial formula for the interacting star product in perturbative Algebraic Quantum Field Theory (pAQFT). Our expression is non-perturbative in the coupling constant and is well defined on regular observables if the interaction is also regular. Under the same assumptions, we also show that the quantum M{\o}ller operator exists non-perturbatively (in the coupling constant) provided that the classical M{\o}ller operator can be constructed exactly. This provides a first step towards understanding how pAQFT can be formulated such that the only formal parameter is $\hbar$, while the coupling constant can be treated as a number. In the introductory part of the paper, apart from reviewing the framework, we make precise several statements present in the pAQFT literature and recast these in the language of (formal) deformation quantization. Finally, we use our formalism to streamline the proof of perturbative agreement provided by Drago, Hack, and Pinamonti and to generalize some of the results obtained in that work to the case of a non-linear interaction.


Introduction
Constructing interacting quantum field theory (QFT) models in 4 dimensions is one of the most important challenges facing modern theoretical physics. Even though there is no final consensus on how the actual axiomatic framework underlying QFT should be formulated, most attempts at construction of models try to fit into one of the established axiomatic systems: Wightman-Gårding [41], Haag-Kastler [25,22] or Osterwalder-Schrader [34]. On the other hand, most computations in QFT are done using less rigorous methods and often rely on a perturbation theory expansion organized in terms of Feynman graphs.
A new approach that combines the advantages of a mathematically sound axiomatic framework with perturbative methods has emerged in the last two decades; it is called perturbative algebraic quantum field theory (pAQFT). The foundations were laid in [10,11,12,13,9,3] and further results concerning fermionic fields and gauge theory were obtained in [39,18,17]. For a review see [40].
One of the ingredients of pAQFT is formal deformation quantization -the construction of a (quantum) star product for the algebra of observables. Recently, this star product was constructed in [17] by an indirect method (when restricted to regular functionals, i.e., linear or non-linear, but also non-local observables). There, the theory was formulated in terms of formal power series in and the coupling constant. Ultimately, the aim of pAQFT is to go beyond formal deformation quantization and to obtain exact results. The first step towards this goal is to find a formulation that avoids formal expansion in the coupling constant. It is desirable to do so, since in many situations the split into free and interacting theory is unnatural and the physical results should not depend on this split. This is the case, for example, in quantum gravity, as indicated in [7].
Motivated by these considerations, in this paper we provide a direct formula for the interacting star product of regular observables, which is a non-perturbative expression in the coupling constant (although still formal in ). We also show how to non-perturbatively construct the quantum Møller operator, assuming the classical Møller operator can be constructed. Our main results are Theorems 5. 10 and 5.13. This is the first result on convergence in the coupling constant obtained in the framework of pAQFT in 4 dimensions (a construction of Sine Gordon model in 2D was provided recently in [8]) and it is a major step towards understanding the geometrical and combinatorial structures underlying this framework. We hope that this will eventually lead to more general non-perturbative results. Moreover, our result is of interest in its own right, as an alternative to the Kontsevich formula [29]. The detailed comparison of our approach with that of Kontsevich is made in Section 5.3.2.
The paper is organized as follows. In Sections 2-3 we review the construction of classical field theory in the pAQFT framework and provide a more rigorous proof of the result of [11,2] that the retarded Møller map intertwines between the Peierls brackets of the free and interacting theories. In Section 4 we discuss deformation quantization. We introduce several natural quantization maps, useful in pAQFT and prove a theorem that completely characterizes the ambiguity in constructing the time-ordering operator on regular functionals. This result mimics the Main Theorem of Renormalization proven for local functionals in [3]. In Section 5 we introduce the formal S-matrix and the quantum Møller operator and we show how the latter can be constructed non-perturbatively on regular functionals, provided the classical Møller operator is known. Next we prove a direct formula for the interacting star product and show that it makes sense non-perturbatively in the coupling constant. In this section we also discuss the relation to the formula of Kontsevich, provide some useful formulea for the quantum Møller operator, and finally we discuss the principle of perturbative agreement.

Kinematical structure
In the framework of perturbative algebraic quantum field theory (pAQFT) we start with the classical theory, which is subsequently quantized. We work in the Lagrangian framework, but there are some modifications that we need to make to deal with the infinite dimensional character of field theory. In this section we give an overview of mathematical structures that will be needed later on to construct models of classical and quantum field theories. Since we do not fix the dynamics yet, the content of this section describes the kinematical structure of our model.

Functionals on the configuration space
We model classical and quantum observables as smooth (in the sense of [1,23,31,33]) functionals on E.
By definition F (1) (ϕ), if it exists, is an element of the complexified dual space E ′ C .
= E ′ ⊗ C. More generally, F (n) (ϕ) induces a continuous map on the completed projective tensor product E⊗ π k ∼ = Γ(E ⊠n → M n ), where ⊠ is the exterior tensor product of vector bundles. Here we denote the map on E⊗ π k by the same symbol as the original differential, i.e., F (n) (ϕ). Definition 2.3. The spacetime support of a map, F , from E to some set is defined by exists an open neighbourhood V in E and k ∈ N such that for all ϕ ∈ V we have where j k x (ϕ) is the k'th jet prolongation of ϕ and α is a density-valued function on the jet bundle. We denote the space of local functionals by F loc .
We equip the space F loc of local functionals on the configuration space with the pointwise product using the prescription where ϕ ∈ E. F loc is not closed under this product, but we can consider instead the space F of multilocal functionals, which is defined as the algebraic closure of F loc under the product (2). We can also introduce the involution operator * on F using complex conjugation, i.e., F * (ϕ) . = F (ϕ) .
In this way we obtain a commutative * -algebra. Functional derivatives of smooth functionals on E are compactly supported distributions. We can distinguish certain important classes of functionals by analyzing the wavefront (WF) set properties of their derivatives.
Local and multilocal functionals satisfy some important regularity properties. Firstly, for local functionals the wavefront set of F (n) (ϕ) is orthogonal to T D n , the tangent bundle of the thin diagonal. In particular, F (1) (ϕ) has empty wavefront set, and so is smooth for each fixed ϕ ∈ E. The latter is true also for multilocal functionals, i.e., F ∈ F. Note that using the metric volume form µ g we can therefore identify F (1) (ϕ) with an element of E * C .
For a regular functional, F (n) (ϕ) can be identified with an element of Γ c ((E * ) ⊠n → M n ) C .

Dynamics
Dynamics is introduced in the Lagrangian framework. We begin with recalling some crucial definitions after [3]. Here, D(M) denotes the space of smooth, compactly supported, real functions (test functions).
iii) Let Iso(M) be the oriented and time oriented isometry group of the spacetime Definition 3.2. An action is an equivalence class of Lagrangians under the equivalence relation [3] The physical meaning of (3) is to identify Lagrangians that "differ by a total divergence".
An action is something like a local functional with (possibly) noncompact support. It is used just as L(1) would be if it existed.
where h ∈ E c and f ∈ D(M) is chosen in such a way that f = 1 on supp h.
Since L(f ) is a local functional, L ′ doesn't depend on the choice of f . Note that two Lagrangians equivalent under the relation (3) induce the same Euler-Lagrange derivative, so dynamics is a structure coming from actions rather than Lagrangians. = L ′ for any Lagrangian L ∈ S.
We are now ready to introduce the equations of motion (eom's).
understood as a condition on ϕ ∈ E.
Remark 3.1. The space of solutions of (4) may be pathological. Instead, in this algebraic setting, we should work with the quotient of F or F reg by the ideal generated by S ′ . This plays the role of the algebra of functionals on the space of solutions. However, in this paper we are concerned with "off shell" constructions, i.e., the quotient is not taken.
Definition 3.6. The second variational derivative S ′′ of the action S is defined by where L ∈ S and f = 1 on supp h 1 ∪ supp h 2 .
By definition, S ′′ (ϕ) : E c × E c → C is a bilinear map, and it induces a continuous linear operator P S (ϕ) : E C → E * C . Note that if S is quadratic then P S . = P S (ϕ) is the same for all ϕ and S ′ (ϕ) = P S ϕ. This is the case for the free scalar field, where The crucial assumption in the pAQFT approach is that P S (ϕ) is a normally hyperbolic operator. For such operators there exist unique retarded and advanced Green's functions (fundamental solutions) and the support properties where f ∈ E * C c . Note that, by the Schwarz kernel theorem, these operators can be written in terms of their integral kernels, which then satisfy appropriate support properties and The causal propagator is Due to (5) the causal propagator is antisymmetric, i.e., its integral kernel satisfies We equip the algebra of functionals with a Poisson bracket called the Peierls bracket [35].
where ∆ S (ϕ) is used as a complex bidistribution in Γ ′ c ((E * ) ⊠2 → M 2 ) C (this can be understood as the complexified dual of the appropriate completion of E * c ⊗ E * c ).
Remark 3.2. This is not limited to the multi-local functionals, F. It works just as well on the algebra of microcausal functionals, F µc , defined below. For a quadratic action, the Peierls bracket gives a Poisson bracket on the algebra of regular functionals, F reg .
In this paper, we will mainly consider an "action" of the form where S 0 is a quadratic (in ϕ) action and V ∈ F reg . This doesn't satisfy the above definition of an action, but it can be used in much the same way. For example, the Euler-Lagrange derivative should be understood as S ′ = S ′ 0 + λV (1) . Remark 3.3. The λV is used as a regularized interaction term. The fact that V has compact support is an infrared (IR) regularization, and the fact that it is a regular (rather than local) functional is an ultraviolet (UV) regularization. This type of IR regularization is the usual technique used in Epstein Glaser renormalization [15].
With this, the definitions of the propagators must be modified slightly (this is equivalent to the definition given in Lemma 1 in [2]): If λ is a formal parameter, it is easy to see that satisfies this definition, and there is an analogous formula for ∆ R S . Remark 3.4. To reduce clutter, we will often omit the • symbol when composing linear operators, as in eq. (8).

Classical Møller maps off-shell
To avoid functional analytic difficulties we define all the structures only for regular functionals F reg or local functionals F loc . We mostly work perturbatively, i.e., we use formal power series in λ, which plays the role of a coupling constant.
From now on, we fix a free (quadratic) action S 0 and consider "actions" of the form S = S 0 + λV with V ∈ F reg . For two such actions, the retarded and advanced Møller maps r S 1 ,S 2 , a S 1 ,S 2 : E S 2 → E S 1 are defined by the requirements that for any ϕ ∈ E S 2 , ϕ − r S 1 ,S 2 (ϕ) has past-compact support and ϕ − a S 1 ,S 2 (ϕ) has futurecompact support. We will be using the retarded maps, but everything we say adapts easily to the advanced maps.
Following [11], the off-shell retarded Møller map is defined by the conditions and These retarded Møller maps are perturbatively well defined, and restriction to solutions gives the on-shell Møller maps [11]. To simplify the notation we abbreviate r λV . = r S 0 +λV,S 0 . Møller maps act on functionals by pullback.
Lemma 3.1. If λ is a formal parameter, then the retarded Møller map satisfies the Yang-Feldmann equation Conversely, a map satisfying the Yang-Feldmann equation (12) must be the retarded Møller map.
Apply ∆ R S 0 (which is independent of ϕ).
Rearranging gives Since r 0 = id, integrating gives (12). The Yang-Feldman equation is equivalent to saying that the inverse of the Møller map is This gives r −1 λV as a formal power series in λ. The constant term is just the identity map. Therefore this is invertible as a formal power series. Its inverse is unique, so the Møller map is that unique inverse.
Nonperturbatively, the Yang-Feldmann equation may be taken as a definition of the Møller map. There is certainly no problem with defining r −1 λV by eq. (13), but its inverse r λV may not actually exist. An idea how to use the Nash-Moser inverse function theorem on locally convex topological vector spaces to tackle this problem has been proposed in [6].
We will now provide an alternative proof of the result of [11] stating that r λV intertwines between the free and interacting Poisson brackets. The advantage of our proof is that it is explicitly performed to all orders and can be generalized to the non-perturbative setting, while the argument in [11] is essentially a proof of the infinitesimal version of the statement.
The transpose ρ T : E * → E * is defined by reversing the arguments in the integral kernel for ρ.
Lemma 3.2. The derivative of the inverse Møller map is and Proof. Equation (14) follows immediately from eq. (13). The image of ∆ R S 0 : E * c → E is the set of ψ ∈ E such that supp ψ is pastcompact and supp P S 0 ψ is compact. Because P S (ϕ) − P S 0 = λV (2) (ϕ) has compact support, the image of Im ∆ R S (ϕ) = Im ∆ R S 0 . Also note that With this in mind -and hiding the ϕ arguments -we have Composing this with the transpose ρ T = id E * c + λV (2)  Proof. Note that the last term of eq. (15) is symmetric, so subtracting the transpose of this equation gives simply It is simpler to prove the equivalent property with r −1 λV . First note that the derivative of r −1 This gives

Deformation quantization
Starting with the Poisson algebra (F * , ⌊·, ·⌋ S 0 ) (where F * = F reg or another appropriate space of functionals, e.g., F µc , which will be introduced in Definition 4.5) formal deformation quantization [5] means constructing an associative algebra where the product ⋆ is given by a power series in which each B n is a bidifferential operator (in the sense of calculus on E) and in particular

Exponential star products
The Peierls bracket of a quadratic action such as S 0 is a constant bidifferential operator, in the sense that it does not depend on ϕ. Some of the ⋆-products that we need to consider are simple in a similar way.
Let F * denote the space of functionals satisfying an appropriate tower of WF set conditions for the functional derivatives (for example take * = µc or * = reg). More precisely, F ∈ F * if WF(F (n) (ϕ)) ⊂ Λ n , for all ϕ ∈ E, n ∈ N, where the cone Λ n has to be chosen appropriately (e.g. for * = µc it is given by Ξ n and for * = reg it is the empty set).
This has an obvious generalization to functionals of two variables.
Let m : F * ⊗ F * → F * denote the pointwise product. In this notation, F · G = m(F ⊗ G). Note that F * ⊗ F * ⊂ F 2 * , and m is pullback by the diagonal. It is clear from definition 4.1 that m extends to a map m : Remark 4.1. Note the nontrivial compatibility condition implicit here between the choice of F * (i.e. the choice of Λ n ) and the singularity structure of K.
The involution * (given by complex conjugation) extends to F * [[ ]] if we just let * = . However, this is an antiautomorphism of ⋆ if and only if K(y, x) = K(x, y).
In terms of eq. (16), if we expand (18) in powers of , then the first term is just B 0 = m, and the second term is B 1 = m • D, i.e., From this, we can see that ⋆ gives a deformation quantization with respect to the Peierls bracket induced by S 0 if and only if So, only the antisymmetric part of K is relevant to compatibility with the Peierls bracket.
is an isomorphism.
Proof. Firstly, the hypothesis that F * is in the domain of all powers of D Y means that α Y : Because of the symmetry of Y , applying D Y to a product gives More concisely, In other words, m intertwines those two operators. This implies that it intertwines their powers and their exponentials, therefore Since the various differential operators commute. Composing this identity on the right with e D 1 gives This means that is a homomorphism. Finally, α Y is a formal power series with leading term the identity map, therefore it is invertible and hence an isomorphism.
Remark 4.2. In the finite-dimensional setting, a proof of this result was given in [32], although it may have been known well before then. In the context of quantum field theory, it was discussed in [3].
This shows that only the antisymmetric part of K really matters, which suggests that the simplest choice is to take K to be antisymmetric. In that case, eq. (19) Unfortunately, this ⋆ 0 does not extend to a larger space of functionals than F reg . This is the fault of ∆ S 0 , whose wavefront set is where ∼ means that there exists a null geodesic strip that both (x, k) and (x ′ , k ′ ) belong to. Recall that a null geodesic strip is a curve in T * M of the form (γ(λ), k(λ)), λ ∈ I ⊂ R,where γ(λ) is a null geodesic parametrized by λ and k(λ) is given by The problem is that the tensor powers of ∆ S 0 have a WF set that at singular points contains the whole cotangent bundle [9,3]. This means that the second order term of ⋆ 0 is only well defined on regular functionals. We can obtain a better behaved star product by a choice of K that has a smaller WF set. Specifically, we want it to be the half of WF(∆ + S 0 ) with k future-pointing, i.e., k ∈ (V + ) x . This is better, because sums of future-pointing vectors are always future-pointing, and do not give the entire cotangent space. This leaves room for F (n) and G (n) to have nontrivial WF sets but still give a well-defined star product of F and G.
With this in mind, as shown in [38], there exists a real, symmetric, distributional bisolution to the field equation, H, such that has WF set and ∆ + S 0 is called a Hadamard distribution. This motivates the following definition: Definition 4.6. Given a Hadamard distribution, the Wick product (denoted ⋆ H ) is the exponential product on F µc [[ ]] given by ∆ + S 0 . This gives a deformation quantization of F µc with respect to the Peierls bracket [3] with * as an involution. By Prop. 4.2, is an isomorphism.
The choice of H is far from unique. It is only determined up to the addition of any smooth real symmetric function. Prop. 4.2 shows that the products given by any two choices are equivalent.

Time-ordered products
In this section we discussed the time-ordered product. One uses this structure in pAQFT to construct the S-matrix and the interacting fields (see e.g., [3]). We will come back to these in Section 5. Here, we want to review some basic properties of this product, emphasizing the importance of the time-ordering map T that establishes the equivalence between the time-ordered product and the pointwise product.
We first consider time-ordered products of regular functionals, to understand the algebraic structure. We want the time-ordered product ·T to be a binary operation on F reg [[ ]] that satisfies the condition where the relation "≺" means "not later than" i.e., supp G ≺ supp F means that there exists a Cauchy surface to the future of supp G and to the past of supp F . We will assume that ·T is defined as is a differential operator, • TF = F for F linear, • and supp TF is contained in the causal completion of supp F . 1 A natural question to ask is what is the freedom in choosing T? A similar problem arises in Epstein-Glaser renormalisation [15], where one constructs n-fold timeordered products recursively (as multilinear maps on local functionals). There, the non-uniqueness of these maps is characterized by the Main Theorem of Renormalization (for various versions of this result see [37,36,26,12,3]). The following theorem provides a solution to this problem for T restricted to regular functionals.
Theorem 4.3. Any two operators T andT satisfying these conditions (and defining products satisfying eq. (25)) are related bỹ with a n (ϕ) a formal power series with coefficients in symmetric distributions supported on the thin diagonal of M n and depending at most linearly on ϕ, and a n (ϕ) is a multiple of .
Proof. First, define X = log(T −1 •T). Because T andT are differential, X must be differential. This means that it can be written in the form where a n is valued in formal power series of symmetric distributions. If F is any linear functional, then TF =TF = F , so XF = 0, therefore a 0 = a 1 = 0.
1 A more elegant way to formulate such a condition is to require T to be natural in the following sense. Consider the category of bounded causally convex subsets of M with inclusions as morphisms. Let F reg be the functor from this category to the category of vector spaces, assigning to O ⊂ M the space of regular functionals supported in O. We can require T to be a natural transformation from F reg to itself. This condition implies in particular the support property.
The ratio T −1 •T = e X is almost a homomorphism of the pointwise product in the sense that if F and G have causally separated support (supp Consequently, X is almost a derivation in the sense that if F and G have causally separated support, then Any finite set of distinct points of M can be placed in an order consistent with the causal partial order, and there is a time slicing of M (into Cauchy surfaces) consistent with this order. Let x 1 , . . . , x n ∈ M be such a list of points. There exist neighborhoods of these points that are causally separated by Cauchy surfaces of this time slicing. Consider any p 1 , . . . , p n ∈ E * c supported on these respective neighborhoods, and any ϕ 0 ∈ E. Define linear functionals For any numbers m 1 , . . . , m n ∈ N, apply X to the product F m 1 1 . . . F mn n and use eq. (26). Evaluating at ϕ 0 , this shows that Because the section p i can take any values around x i , this shows that a m 1 +···+mn (ϕ 0 ) is not supported at Up to symmetry, this is any point outside the thin diagonal. Finally, consider a point x ∈ M. If F is supported on a causally complete neighborhood U ∋ x, then a n (ϕ), F (n) (ϕ) must be supported on U. Take F homogeneous of degree n, so that Since TF has to be supported in U, we conclude that a n (ϕ) − a n (ϕ + ψ), f = 0 .
We can now use this fact to conclude that the derivative a (1) n (ϕ), seen as a distribution on M n+1 , has to be supported on the diagonal. Since X maps regular functionals to regular functionals, this implies that a n (ϕ) can depend at most linearly on ϕ.
There is a natural choice of T given in terms of the Dirac propagator, ∆ D where Proposition 4.4. The time-ordered product ·T defined in terms of T is the exponential product in the sense of definition 4.2 defined by K = i∆ D S 0 .
Proof. The Leibniz rule for differentiation implies that which is in fact a co-product structure. Hence Because both ⋆ 0 and ·T are exponential products, they are related by an exponential factor. With obvious notation, , and therefore the relation is From this relation, it is easy to see why ·T satisfies eq. (25). Consider the case Given a choice of Hadamard distribution, H, the Feynman propagator is defined as ∆ F S 0 = i∆ D S 0 + H. There is another time-ordered product, ·T H , which is the exponential product on F reg [[ ]] given by ∆ F S 0 . Again, this is commutative and equivalent to the pointwise product by the isomorphism Note that this is exactly the same as the relation between ⋆ 0 and ·T ; consequently, by identical reasoning, they are also related by eq. (25) . Renormalization in pAQFT is a matter of extending these consistently to local functionals. There is freedom (renormalization ambiguity) in this [3]. Table 1: Some of the exponential products, and the distributions used to construct them.

Quantization maps
In this section we discuss quantization maps and formalize constructions known from [3] and [10]. If we consider two choices of Hadamard distribution, H and H ′ , then Proposition 4.2 gives an isomorphism α H ′ −H between the deformation quantizations. These are coherent in the sense that This means that we can think of just one abstract algebra 2 , A. The different star products come from different ways of identifying the underlying vector space of Here are some key features of A: • There is a surjective homomorphism P : A → F µc (evaluation at the classical limit).
The compatibility with P ensures that the star product reduces to the pointwise product modulo . In physics terms, a quantization map is a choice of operator ordering.
Remark 4.3. In a less formal version, Q would be a map from an algebra of bounded classical observables to the algebra of sections of a continuous field of C * -algebras. This can still induce a star product, but eq. (29) needs to become an asymptotic expansion.
In analogy to (25) we define the time-ordered version of the non-commutative product on A such that The exponential product ⋆ H is also induced by a quantization map, This can also be done for the dense subalgebra F reg ⊂ F µc and a subalgebra A reg ⊂ A. In this way, there is a quantization map, Q Weyl : F reg → A reg , that produces the Moyal-Weyl product, ⋆ 0 . The quantization maps are related by Because these quantization maps are bijective, Q Weyl induces a commutative "pointwise" product on A reg , and Q H induces one on A.
If ∆ + S 0 is the 2-point function of some choice of "vacuum" state, then Q H is the corresponding normal-ordering map. The commutative product induced by Q H is the normal-ordered (Wick) product. Of course, this depends upon a choice of H.
The time-ordered product is a more natural commutative product. The time ordered product on A reg is induced by the quantization map The choice of T fixes the freedom in defining the product T as We denote by ⋆ T the star product on F reg [[ ]] induced by Q T ; this is given by the multiplication map The disadvantages of Q T are that it does not extend naturally to F µc and it is not * -linear. The relations between ⋆ T and the other star products are:

The formal S-matrix and Møller operators
Consider a theory with action S = S 0 + λV , where V ∈ F reg and λ is the coupling constant, treated from now on as a formal parameter (similarly to ).
Remark 5.1. Here V plays the role of an interaction term where some infrared (IR) and ultraviolet (UV) regularisations have been implemented. The former guarantees compact support and the latter regularity. An example regular functional is ϕ → f (x 1 , . . . , x 3 )ϕ(x 1 ) . . . ϕ(x 3 ), where f is a compactly supported density on M 3 and ϕ ∈ C ∞ (M, R). The physical interaction is recovered in the limit Instead of taking this limit directly, in pAQFT one proceeds in two steps. First the map S is extended to a larger subset of A, to deal with the potential UV divergences (see for example [3] and [40] for a review). Next, one takes the algebraic adiabatic limit to deal with the IR problem. For more details see [19,40] and [24] for an alternative formulation.
We wish to construct a formal deformation quantization of (F * , ⌊·, ·⌋ S ). Proposition 3.3 shows that r λV intertwines the Peierls brackets for S and S 0 . If ⋆ is a formal deformation quantization for S 0 , then this gives an obvious formal deformation quantization for S. Simply define ⋆ r by This is simple, but it is not a good choice. We will explain the reasons for this at the end of this section, but before we do that, we need to introduce some further definitions. A central object in pAQFT for constructing interacting theories is a formal S-matrix.
The physical interpretation is that S becomes the scattering matrix of the theory, in the adiabatic limit (i.e., the cutoff function becomes constant), if it exists. One uses S to construct interacting field using the quantum Moller operator given by the formula of Bogoliubov (see e.g., [3] and [40] for a review): It appears from this formula that R A : The interacting product • int on A reg is then defined by For the purpose of practical computations, it is easier to work with functionals rather than the abstract algebra A reg , so we should identify A reg with F reg [[ ]] by using a quantization map, but there are several to choose from.
The most obvious way to work is based on the quantization map Q Weyl . With this identification, the formal S-matrix becomes S 0 .
If we instead use Q H , then the formula for R H,λV is the same, but with ⋆ H and T H .
These formulas are difficult to work with, because they each use two different products, neither of which is the natural pointwise product on the space of functionals.
Instead, the most convenient quantization map for computations is actually Q T . With this identification, the formal S-matrix becomes simply The Møller operator becomes R T,λV .
The inverse of this is particularly simple: Note that any formula for R T,λV can be converted to a formula for R H,λV (and vice versa) by R H,λV • T H = T H • R T,λV .
Next, note that the symmetric tensor algebra SE is the universal enveloping algebra of the abelian Lie algebra E. It acts by differential operators on functionals, and we denote this action as ⊲. This extends to formal power series in λ and . We make use of this action in the following proposition to prove a more explicit and concise formula for the Møller operator.
for w ∈ E * c .
Proof. First, for some arbitrary functional F , consider F ⋆ T as an operator acting on G: For F = e iλV / , this becomes . The original proof of the next result is due to [9] and was presented for the case of local functionals. Here we are working only with regular functionals, which allows a simpler proof.
That is, the Møller operator contains no negative powers of .
Proof. In the expression the numerator is of order , so this cancels the in the denominator, thus J does not contain any negative powers of . This means that ]. The fact that J ≈ 1 to 0'th order in λ means that R −1 T,λV is a formal power series with the identity map in leading order. As a formal power series, it can thus be inverted, and R T,λV exists. Finally, the other forms of the Møller operator are equivalent to this one.
Let us now discuss the classical limit. Recall that P : A → F µc , A reg → F reg is the evaluation at the classical limit; it corresponds to setting = 0. Proposition 5.3. The classical limit of the quantum Møller operator is the classical Møller operator: Equivalently, for F ∈ F reg , Proof. Again, it is easiest to prove the equivalent statement for R T,λV . To 0'th order in , This givesr This is just a Taylor series expansion, sõ In other words,r −1 λV is the pullback by the operator that acts on E[[λ]] as Rearranging this gives an equation satisfied by the inverse map, which is the Yang-Feldmann equation, and so by Lemma 3.1,r λV = r λV .
Denote the ⋆ H,int -commutator by It is now easy to see that the theory defined by ⋆ H,int is indeed a quantization of the classical theory defined by the Poisson bracket ⌊ · , · ⌋ S 0 +λV given in Definition 3.7. Proof. This is a straightforward consequence of Propositions 3.3 and 5.3. The quantum Møller operator intertwines ⋆ H with ⋆ H,int (by definition), so it intertwines their commutators. To first order in , Remark 5.2. The equivalent, more abstract, statement is that for any A, B ∈ A reg , Next we show that the quantum Møller operator can be constructed nonperturbatively, provided the classical Møller operator is known exactly. To this end, we will extract the "classical part" of the quantum Møller operator and see what remains. It will then become clear that, at a fixed order in , the remaining "purely quantum part" contains only finitely many terms in its coupling constant expansion.
Proposition 5.5. Define ⊲ r , J 1 , and Υ λV by The inverse quantum Møller operator can be computed as Proof. This J 1 is chosen so that J = J 0 J 1 . Proposition 5.1 gives This last expression contains ϕ 3 times. Note that the second ϕ is just a constant as far as J 0 (ϕ)⊲ is concerned.
Note that the fraction in the definition of J 1 is of order at least , so that every occurrence of λ is accompanied by a factor of . If J 1 is expanded in powers of , then each coefficient contains only finitely many powers of λ, so λ no longer needs to be treated as a formal parameter. This gives a non-perturbative definition of the quantum Møller operator, provided that the classical Møller operator exists non-perturbatively.
This shows the relationship between the interacting product and the naive product constructed using the classical Møller operator. The naive product ⋆ T,r is defined by . The interacting product ⋆ T,int in the identification given by Q T is defined by Equation (36) shows that Note that Υ λV is the identity map modulo . This means that Υ λV , the "purely quantum" part of the quantum Møller operator, is the "gauge" equivalence [29] relating the naive product ⋆ T,r to the preferred choice ⋆ T,int .
Remark 5.3. Rearranging (36) results in r −1 λV R T,λV = Υ −1 λV , so Υ −1 λV can be seen an the "purely quantum" part of the Møller operator. Such a map has been introduced in [14] for quadratic interactions (where it is called β and in our notation we have β . = r −1 λV • R 0,λV ), so our current discussion is a natural generalization of that result. The only missing step in that comparison is to transform the quantum Møller map from the Q T -identification to the Q Weyl or Q H -identification (i.e., to go from R T,λV to R 0,λV or R H,λV ). We will come back to this in Section 5.5. Now we are ready to come back to the problem of comparing ⋆ T,r with ⋆ T,int . The reason for ⋆ T,int to be a better choice is related to locality and the adiabatic limit. Ultimately the interacting star product will be used to construct local nets of algebras. It was shown in [4,19] (in a slightly different setting) that if we use ⋆ T,int to construct local algebras assigned to bounded causally convex regions O ⊂ M, then perturbing the interaction by a compactly supported functional, supported outside O changes the local algebra of O only by an inner automorphism. This means that the abstract net of algebras depends only locally on the interaction. We can therefore first introduce a cutoff for the interaction and then remove it using the algebraic adiabatic limit construction [4,16]. Such construction would not be possible for ⋆ T,r or any naive product defined from the classical Møller operator by eq. (31).
Another problem with the naive construction is this. The interacting product can be constructed equivalently in different identifications using the appropriate versions of the quantum Møller operator. For example, ⋆ H,int and ⋆ T,int are equivalent through T H . The naive interacting products, constructed with the classical Møller operator are not equivalent in this natural way.

Graphical computations
To simplify and organize the computations, we will represent our structures in terms of graphs. Definition 5.3. G(n) is the set of isomorphism classes of directed graphs with n vertices labelled 1, . . . , n (and possibly unlabelled vertices with valency ≥ 1).
We make G a category by defining a morphism u : α → β to be a function that • maps vertices of α to vertices of β and edges of α to edges or vertices of β, • respects sources and targets of edges, • maps labelled vertices to labelled vertices, • and preserves the order of labelled vertices.
For γ ∈ G: e(γ) is the number of edges; v(γ) is the number of unlabelled vertices; Aut(γ) is the group of automorphisms. It is chosen to give the concept of extension that will be useful below. The meaning of isomorphism is the same.
Definition 5.4. A graph γ ∈ G(n) determines an n-ary multidifferential operator, γ, on functionals as follows: • An edge represents ∆ A S 0 (x, y) with the direction from y to x -i.e., such that this is only nonvanishing when the edge points from the future to the past; • if the labelled vertex j has valency r, this represents the order r derivative of the j'th argument; • likewise, an unlabelled vertex of valency r represents V (r) .
Definition 5.5. In diagrams, ∆ A S 0 will be denoted by a dashed line with an arrow, so all graphs in G will be drawn with dashed lines for the edges.

Non-interacting product
Definition 5.6. G 1 (2) ⊂ G(2) is the subset of graphs with no unlabelled vertices in which all edges go from 2 to 1.
In terms of these, the non-interacting product can be expressed as Example 5.1. The 3 term is given by the graph 1 2 which is the unique graph in G 1 (2) with 3 edges. Its automorphism group is S 3 , which has order 6 and gives the correct coefficient, 1 6 (−i ) 3 .

Inverse Møller operator
Definition 5.7. G 2 (1) ⊂ G(1) is the set of graphs such that every edge goes from 1 to an unlabelled vertex.
Proof. Taking a Taylor expansion of V about ϕ in eq. (34) shows that The term V (n) (ϕ), (∆ A S 0 w) ⊗n gives the same differential operator as the graph with n edges directed from the vertex 1 to a single unlabelled vertex. We have here a sum over all possible products of such terms. This corresponds to all possible graphs in G 2 (1).
The automorphisms of a graph γ ∈ G 2 (1) do not permute the unlabelled vertices with different valencies, thus the automorphism group is a Cartesian product of the group of automorphisms for each of those subgraphs.
If γ has k unlabelled vertices of valency n, then an automorphism can permute each of those vertices, and for each such vertex, it can permute the n edges leading to it. That subgroup of automorphisms is thus a semidirect product of S k and (S n ) k . This has order k!(n!) k .

Composition of operators
The following is analogous to the concept of an extension of groups.
Definition 5.8. For α ∈ G(n) and γ ∈ G(m), an extension of γ by α at j ∈ γ is a pair of an injective and a surjective morphism , and the restriction of v to the compliment of u(α) is injective.
Two extensions are equivalent if there exists a commutative diagram between them with the identity on α and γ, and an automorphism on β. which maps the first graph to the bottom edge of the second graph, and then collapses that subgraph to the vertex 1 of the third graph.
Definition 5.9. The partial composition of multilinear maps is denoted by • j and means the composition of one map into the j'th argument of another.
Lemma 5.7. Let α ∈ G(n) and γ ∈ G(m) and j = 1, . . . , m. The partial composition at j is where the sum is over equivalence classes of extensions of γ by α at j.
where the sum is over the set of β ∈ G(n + m − 1) such that there exists such an extension of γ by α at j, and d β is the number of subgraphs of β isomorphic to α such that the quotient is isomorphic to γ with the subgraph mapped to j.
Proof. The composition γ • j α can be thought of as inserting α in place of j ∈ γ. If r is the valency of j ∈ γ, then this is taking an order r derivative of α. The edges represent ∆ A S 0 , which is constant, so the product rule tells us that this derivative is given by a sum over all possible ways of attaching the r edges to vertices of α (instead of to j ∈ γ). Any such attachment gives a graph β with a subgraph identified with α and the quotient identified with γ. This gives precisely the set of equivalence classes of extensions as defined above.
For the second expression, we need to write this as a sum over the graphs β that can appear as extensions, but one graph can appear in inequivalent extensions, so we need to understand the number of extensions with a given β.
An extension α ֒→ β ։ γ certainly determines a subgraph of β that is isomorphic to α. If two extensions determine the same subgraph, then it is easy to construct an automorphism of β that gives an equivalence between the extensions. (All of the edges of β are either in the subgraph or map to edges of γ.) So, having the same subgraph is a weaker condition than equivalence of extensions.
By definition, d β is the number of possible images of α in extensions. Any two extensions with the same image are related by an automorphism of α and an automorphism of γ, therefore the number of extensions is d β |Aut α||Aut γ| .
An equivalence of extensions is always given by an automorphism of β, so an equivalence class consists of |Aut β| extensions, therefore the number of equivalence classes of extensions is d β |Aut α||Aut γ| |Aut β| .
Each of these graphs has trivial automorphism group, except for the last, for which it has order 2. However, there are also 2 ways of mapping 1 into it to give the quotient 2 1 . This is why the coefficient of that term is also 1.

Interacting product
Definition 5.10. G 2 (2) ⊂ G(2) is the subset of graphs such that every edge either goes from a labelled vertex to an unlabelled vertex or from 2 to 1.
Proof. This follows by inserting eq. (38) into eq. (39) and using the product rule.
Definition 5.11. G 3 (n) ⊂ G(n) is the set of graphs such that: • Every unlabelled vertex has at least one ingoing edge and one outgoing edge; • there are no directed cycles; • for 1 ≤ j < k ≤ n, there does not exist any directed path from j to k.
In particular, this implies that 1 is a sink (has only ingoing edges) and n is a source (has only outgoing edges).
We are now ready to write down the graphical expansion of the interacting star product. Note that the graphs we are using are constructed from the free propagator ∆ A S 0 and from the derivatives of the interaction term V . Later on, we will re-express things in terms of the propagator ∆ A S of the interacting theory and the derivatives of the full action S. Theorem 5.9.
Remark 5.5. The right hand side is the same as in eq. (40), except that G 3 has replaced G 2 and λ has become −λ.
Proof. For this proof, let F ⋆ ? G denote the right hand side of eq. (41), so that we need to prove ⋆ ? = ⋆ T,int . That is, we need to prove that , and eq. (40) has already computed the left side. By Lemma 5.7 and the definition of G 4 , R −1 T,λV (F ) ⋆ ? R −1 T,λV (G) can be computed as some sum over G 4 (2). We will show that cancellation reduces this to the sum over G 2 (2) in eq. (40).
We first need to check that G 2 (2) ⊆ G 4 (2), so suppose that γ ∈ G 2 (2). By definition, all edges of γ either go from 1 or 2 to an unlabelled vertex or from 2 to 1. Let α ⊂ γ be the subgraph of edges from 1 (to unlabelled vertices) and denote the quotient graph as γ/α. Let β ⊂ γ be the subgraph of edges going from 2 to (unlabelled) vertices not in α. All other edges of γ must go from 2 to vertices of α, therefore and so γ ∈ G 4 (2). Eqs. (41) and (39) give where e = e(α) + e(β) + e(δ) and v = v(α) + v(β) + v(δ). Applying Lemma 5.7, this becomes where the last sum is over α, β ⊂ γ, such that 1 ∈ α, 2 ∈ β, α, β ∈ G 2 (1), and (γ/α)/β ∈ G 3 (2). Note that α ⊂ γ is the subgraph of all edges outgoing from 1 ∈ γ, so it is uniquely determined by γ. On the other hand, although β must contain all edges from 2 to sinks that are not in α, it may contain any edge that goes from 2 to any other unlabelled vertex not in α (see Example 5.6). The sum over β is over the binary choices of including or not including each of these edges, thus if there are any such edges. This reduces the expression for R −1 T,λV (F ) ⋆ ? R −1 T,λV (G) to a sum over γ without any such edges.
We are therefore interested in graphs γ ∈ G 4 (2) that do not contain any such ambiguous edge. This means that any edge from 2 ∈ γ must go to a vertex of α ⊂ γ or to a sink. By the definition of G 4 , 1 ∈ γ/α is a sink, so any vertex of α ⊂ γ other than 1 is a sink. In short, any edge from 2 ∈ γ must go to 1 or a sink. This implies that all unlabelled vertices are sinks, and so any edge goes from 1 or 2 to an unlabelled vertex or from 2 to 1. In other words, γ ∈ G 2 (2). In that case, all vertices of γ are in α or β, so v(γ) = v(α) + v(β). Therefore, With Lemma 5.8, this shows that ⋆ ? satisfies eq. (37), which is the defining property of ⋆ T,int .
Example 5.6. Consider the graph The subgraph α consists of the edges labelled (a) here (and the adjacent vertices). The edge (b) must be in the subgraph β, but the edge (c) may or may not be in β. The other edges cannot be in β. The sum over the 2 possible choices of β with or without (c) gives 0, so that γ does not contribute to the formula for

Nonperturbative expression for an interacting product
In this section we will show that the interacting star product can be written in terms of the full propagator ∆ A S and derivatives of S and, provided that ∆ A S is known exactly, the result is a formal power series in , but the coupling constant λ can be treated as a number.
Definition 5.13. G 5 (n) is the set of isomorphism classes of directed graphs with labelled vertices 1, . . . , n such that: • Each unlabelled vertex has valency at least 3 and is neither a source nor a sink; • there exist no directed cycles; • for 1 ≤ j < k ≤ n, there does not exist a directed path from j to k.
Also define Remark 5.6. A crucial consequence of the first condition in the definition of G 5 (n) is that derivatives of V appearing in the graphical expansion are at least 3rd derivatives, so one can replace these with derivatives of S (S 0 is quadratic).
• The labelled vertex j represents a derivative of the j'th argument.
• An unlabelled vertex represents a variational derivative of S.
Definition 5.15. In our diagrams, ∆ A S will be denoted by a solid line with an arrow. For this reason, graphs in G 5 will be drawn with solid lines for the edges. (This helps to distinguish G 5 from G.) Remark 5.7. S ′′ is the linearized equation of motion operator, thus S ′′ ∆ A is the identity operator. This means that if there were a bivalent vertex in γ, then ։ γ would be the same as if that vertex were removed, i.e.,

= .
For this reason, G 5 (n) can (and should) be thought of as a quotient of G 3 (n). This leads to the appropriate definition of morphisms.
Definition 5.16. For α, β ∈ G 5 , a morphism u : α → β is a function that • maps vertices of α to vertices of β and edges of α to directed paths in β, • respects sources and targets of edges, • maps labelled vertices to labelled vertices, • and preserves the order of labelled vertices.
The concept of an extension in G 5 follows from this definition.
The following theorem is the main result of this section. It delivers an explicit formula for the interacting star product that is non-perturbative in the coupling constant.
Theorem 5.10. For S = S 0 + λV , Proof. For γ ∈ G 5 (n), the operator ։ γ can be expressed as a sum of operators given by graphs in G 3 (n), with the following dictionary: • Because any unlabeled vertex in γ has valency at least 3 (say, r), S (r) = λV (r) , so this vertex corresponds to a vertex in G 3 and a factor of λ.
• By eq. (8), an edge in γ corresponds to a sum over all possible chains of edges and bivalent vertices in G 3 , with a factor of −λ for every vertex, i.e., Adding bivalent vertices along edges of graphs in G 5 (n) will give all graphs in G 3 (n).
Remark 5.8. Note that after re-expressing everything in terms of full propagators, at a fixed order in there are only finitely many terms in the λ-expansion. Hence the result is exact in the coupling constant, provided that ∆ A S 0 can be constructed. Theorem 5.9 shows that eq. (42) gives the interacting product for a free action plus a regular perturbation (up to time ordering). This implies in particular that the product is associative. It is worth understanding why it is associative in greater generality.
Theorem 5.11. Suppose that S is an action and K(ϕ)(x, y) is any Green's function for the linearized equation of motion. If ⋆ is defined by the right hand side of eq. (42), with K in place of ∆ A S , then ⋆ is an associative product on the domain of definition of the associativity condition.
Proof. As usual, the dependence on ϕ will not be written explicitly.
The defining property of a Green's function is S ′′ K = id Ec . Differentiating this gives and multiplying on the left by K gives In this equation, KS ′′ is acting on the image of K, where it acts as the identity, so We would first like to compute (F 1 ⋆ F 2 ) ⋆ F 3 .
To compute this, first observe that for α, γ ∈ G 5 (2), where the sum is over equivalence classes of extensions at 1. These are precisely the ways of breaking up 1 ∈ γ and attaching the edges to α (possibly by adding new vertices) to form a graph β. The axioms of G 5 imply that there exists a partial order, , on the vertices of β ∈ G 5 (3) that is generated by the edges and 1 ≻ 2 ≻ 3.
This preimage of 1 ∈ γ is completely determined by the structure of β alone. It is the complete subgraph whose vertices satisfy 2. For this reason, d β = 1 in the sense of Lemma 5.7. The number of equivalence classes of extensions is thus |Aut α||Aut γ| |Aut β| .
This uniqueness shows that any β can only appear in one term of the expansion of (F 1 ⋆ F 2 ) ⋆ F 3 . Conversely, any β ∈ G 5 (3) does occur in this expansion.
Together, this shows that An essentially identical calculation (using the subgraph determined by 2) shows that F 1 ⋆ (F 2 ⋆ F 3 ) is given by the same formula, thus provided that both sides are defined.
Example 5.7. Consider the graph This occurs in a unique extension at 1, corresponding to a term in F 1 ⋆ (F 2 ⋆ F 3 ).
Remark 5.9. We have not proven that eq. (42) gives the correct interacting product for a local action. However, Theorem 5.11 makes this a plausible conjecture.

Low order terms
Explicitly, the product ⋆ T,int is given up to order 3 as and

Is there a Kontsevich-type formula?
In his famous paper on deformation quantization [29], Kontsevich presented a formula for constructing a ⋆-product from an arbitrary Poisson structure on a finitedimensional vector space. Every term is a polynomial in the Poisson structure and its derivatives. In that construction, B 1 is antisymmetric and proportional to the Poisson structure, so this can be thought of as constructing a ⋆-product from its first order term.
The first order term of ⋆ T,int is ∆ A S , which is not antisymmetric. This suggests a question. In analogy with Kontsevich's formula, can ⋆ T,int be constructed from ∆ A S and its functional derivatives?
To address this, we first need some notation. Kontsevich's formula uses a sum over graphs in which vertices represent the Poisson structure. In his graphs, every unlabelled vertex has 2 outgoing edges. Because ∆ A S has no symmetry, in our generalization, it will be necessary to distinguish these as left and right edges.
Definition 5.17. A K-graph is a directed graph in which: • There are labelled vertices 1 and 2, and possibly unlabelled vertices; • every edge is labelled as "left" or "right"; • every unlabelled vertex has 2 outgoing edges, one left and one right.
As with other graphs that we have considered, a K-graph determines a bidifferential operator. The vertices 1 and 2 represent the arguments. The unlabelled vertices represent ∆ A S . The edges represent derivatives. In diagrams, these will be drawn with solid arrows on the right edges. Using the rules above applied to parts of graphs, one can easily treat an arbitrary K-graph. The second order term can indeed be constructed in this way: In general, the operator given by a K-graph can also be given by a sum of graphs in which edges represent ∆ A S and vertices represent derivatives of S, but not vice versa. A K-graph in which each unlabelled vertex has at most one ingoing edge will give a single term; otherwise, it will give several terms.
We can come fairly close to expressing B This leaves a discrepancy of Note that these two terms each have 4 unlabelled vertices. A K-graph at order 3 has precisely 3 unlabelled vertices, and when it is translated at least one term has 3 or fewer unlabelled vertices. In order to express (47) as a combination of K-graphs, we must in particular find combinations of K-graphs in which terms with 3 or fewer unlabelled vertices cancel. A tedious search shows that those combinations cannot reproduce (47), and thus the third order term of ⋆ T,int cannot be constructed from ∆ A S and its derivatives. This means that ⋆ T,int is not given by anything analogous to Kontsevich's formula.

Interacting product in the standard identification
These calculations have used the identification of A reg with F reg [[ ]] defined by the quantization map Q T . In order to compare results and to try to extend this to all of A, we need to use the identification defined by Q H .
An expression for the interacting product ⋆ H,int in that identification is obtained The next step is to extend the domain of definition of ⋆ H,int to local non-linear arguments. The potential problem with this is that the expansion of the product ⋆ T,int in terms of Feynman diagrams contains loops involving the advanced propagator, and light-cone divergences could be present.
Definition 5.18. G 6 (n) is the set of isomorphism classes of graphs with directed and undirected edges and labelled vertices 1, . . . , n such that: • Each unlabeled vertex has valency at least 3, including at least one ingoing and one outgoing edge; • there exist no directed cycles; • for 1 ≤ j < k ≤ n, there does not exist a directed path from j to k.
Definition 5.19. A graph γ ∈ G 6 (n) defines an n-ary multidifferential operator, ։ γ , as follows: • an undirected edge represents ∆ F S 0 ; • the vertex j represents a derivative of the j'th argument; • an unlabelled vertex represents a derivative of S.
Remark 5.10. Note that the Feynman propagator ∆ F S 0 = i 2 ∆ R S 0 + ∆ A S 0 + H is defined by the free action S 0 and the Hadamard distribution H. It is not a natural object from the point of view of the full action S. This suggests that there should be a better way to pass to the "standard identification", but at the moment we don't have a concrete proposal. A graph in G 6 (1) has no directed edges and no unlabelled vertices; it is just a bouquet of undirected loops. Consider the unique graph with e(γ) = m loops. Because the functional derivative of ∆ F S 0 is 0, this graph gives the operator Definition 5.20. G 7 (n) ⊂ G 6 (n) is the subset of graphs with no loops at labelled vertices (i.e., no edge begins and ends at the same labelled vertex). G 8 (n) ⊂ G 6 (n) is the subset of graphs with no loops.
where d(γ) is the number of directed edges. In particular, this is a finite sum at each order in .
Proof. First, observe that G) .
Replacing F and G with T −1 H F and T −1 H G serves to cancel out all terms with loops at 1 or 2.
To extend to local V , the usual procedure is to extend D F to a map D F that coincides with D F on regular functionals and vanishes on local ones (see e.g. [40, Section 6.2.1]). This implies that T H acts as identity on local functionals.
Proposition 5.14. If V ∈ F loc , then the sum in (5.13) can be taken over G 8 (2).
Proof. If V ∈ F loc , then D F (S) = 0, so any loop at an unlabelled vertex gives 0.
By direct inspection of the graphs, it is not clear whether the expression above can be renormalized or not. The problem is related to the presence of free Feynman propagators together with interacting advanced propagators. This is potentially an issue, since ∆ F S 0 − i∆ A S doesn't have the right WF set properites (in contrast to ∆ F S 0 − i∆ A S 0 ). As mentioned before, this is caused by the fact that in constructing the interacting product we left the time-ordered product unchanged, since the time-ordering of ⋆ H,int results again in the same commutative product ·T.
Although we began with a perturbative construction using a free action S 0 , it is only the time-ordered product that remembers S 0 and we would like this dependence to be completely removed in the interacting theory.
We hope that the results of this paper will allow us in the future to find a better way to construct the interacting star product in the standard identification, while keeping ⋆ T,int unchanged.

Formulae for the Møller operator
In this section we prove some combinatorial formulae for the quantum Møller operator, which can be used to streamline computations and might be the starting point for investigating renormalization in the future.
In our terms, a corolla is a graph γ ∈ G(1) such that there is a single edge from 1 to each unlabelled vertex.
In our terms, a tree is a connected graph γ ∈ G(1) such that 1 is a source, and each unlabelled vertex has precisely one ingoing edge.
Proof. Any extension of a corolla by a tree is also a tree, so the composition of r −1 λV with (49) can be computed as a sum over trees. Consider a tree, β. What will the coefficient of this term be? For every subset of leaves (valency 1 vertices) of β, there is an extension α ֒→ β ։ γ, where α is β without those leaves, and γ is the corolla made from those leaves. Note how the sign of the term depends upon the number of leaves that are removed. Lemma 5.7 shows that if β has m leaves, then the coefficient of this term is a multiple of m k=0 m k (−1) k = 0 unless m = 0. Therefore the only term in the composition of r λV with (49) is given by the unique tree with no leaves; the composition is the identity.
Definition 5.21. G 9 (1) is the set of trees such that no unlabelled vertex has precisely one outgoing edge.
Any tree can be obtained from such a graph by adding vertices along edges. As in Theorem 5.10, summing over these gives interacting advanced propagators. This gives the formula where edges represent ∆ A S and vertices represent derivatives of λV . In a similar way, inverting eq. (39) gives a graphical formula for R T,λV .
Definition 5.22. G 10 (1) ⊂ G(1) is the set of graphs such that • Every unlabelled vertex has at least one incoming edge; • 1 is a source (has no incoming edges); • there are no directed cycles.
With this, Next, this leads to a formula for R H,λV . If V is local (and we set T H (V ) = V ) then this amounts to replacing all products by time-ordered products. In terms of graphs, this is given by attaching undirected ∆ F S 0 -edges to graphs from G 10 (1).
Definition 5.23. G 11 (1) is the set of isomorphism classes of graphs with directed and undirected edges and a labelled vertex 1, such that • Every unlabelled vertex has at least one incoming edge; • 1 is a source; • there are no directed cycles; • there are no loops.
With this, for V local, where undirected edges represent ∆ F S 0 , and d(γ) is the number of directed edges. (1) is the subset of graphs such that that no unlabelled vertex has one incoming edge, one outgoing edge, and no unlabelled edge.
Any graph in G 11 (1) can be obtained by adding vertices along directed edges of a graph in G 12 (1). In this way, the formula can be reexpressed using the interacting advanced propagator, where directed edges represent ∆ A S , undirected edges represent ∆ F S 0 , and unlabelled vertices represent derivatives of λV .
Finally, consider the graphs in G 12 (1) without trees branching off of them. These are characterized by the lack of univalent vertices, i.e., leaves.
Definition 5.25. G 13 (1) is the set of isomorphism classes of graphs with directed and undirected edges and a labelled vertex 1, such that • Every unlabelled vertex has at least one incoming edge and one other edge; • no unlabelled vertex has only one incoming and one outgoing edge; • 1 is a source (has no incoming edges); • there are no directed cycles; • there are no loops.
We define an operator Note that any graph giving a term of order m has at most 4m vertices and 5m edges. There are finitely many such graphs, so λ does not need to be a formal parameter in this formula.
Finally, observe that any graph in G 12 (1) can be obtained as an extension of a tree in G 9 (1) by a graph in G 13 (1). Consequently, the composition r λV • Ω λV is a sum over G 12 (1) with the same coefficients. Therefore R H,λV = r λV • Ω λV which is a nonperturbative formula for R H,λV .
Unfortunately, in order to apply the standard methods of Epstein-Glaser renormalization [12], this formula needs to be expanded in λ again, to show cancellation of the lightcone divergences. However, we hope that the non-perturbative formula can nevertheless lead to a well defined object, if we use a different renormalization method (e.g. through some regularization scheme). This will be investigated in our future work.

Perturbative Agreement
Consider the case that V is quadratic, so that V (2) (ϕ) is independent of ϕ, and higher derivatives vanish. In such a situation one can treat the interacting theory exactly and a natural question to ask is how this compares with the perturbative treatment. This issue has been discussed in the literature [27,14] under the name perturbative agreement. One way to look at it is to compare the interacting star product obtained by means of quantum Møller operators with the star product constructed directly from the advanced Green function for the quadratic action S 0 + λV . In this case, Theorem 5.10 becomes: Corollary 5.17. If V ∈ F reg is quadratic, then ⋆ T,int is the exponential product defined by −i∆ A S 0 +λV . Proof. An unlabelled vertex of γ ∈ G 5 (2) has valency r ≥ 3 and represents the derivative S (r) , which vanishes because S is quadratic, therefore ։ γ = 0 unless γ has no unlabelled vertices. This means that eq. (42) simplifies to something like eq. (38): In [27,14] the principle of perturbative agreement (PPA) is expressed as a compatibility condition for time-ordered products corresponding to S 0 and S 0 + λV . In order to prove it in our current setting we need to pass from the Q T -identification to the Q Weyl -identification.
Remark 5.11. Equation (50) is very similar to eq. (36). The difference is that R T,λV is factored in a different order into classical and "purely quantum" part, since eq. (36) implies R T,λV = r λV • Υ −1 λV . Remark 5.12. Note that our result holds for a quadratic V that is local and compactly supported and removing the IR regularization is a non-trivial step.
After transforming the quantum Møller map to the Q Weyl -identification, (50) allows us to compute the map β . = r −1 λV R 0,λV of [14]. Note the close resemblance between β and Υ −1 λV , already pointed out in Remark 5.3. We are now ready to provide a streamlined version of the proof of Theorem 5.3 of [14].

Equation (15) simplifies to
Taking the transpose gives the same identity for the advanced propagators, and adding them shows that ρ • ∆ D S • ρ T = ∆ D S 0 − iλδ . The point of this is that Remembering that T = α i∆ D is the map that intertwines between the time-ordered product corresponding to S 0 and the one corresponding to S. Another way of expressing this relation is where T is the time-ordering map corresponding to S 0 and T S is the analogous map corresponding to S.

Conclusions
In this paper we have derived an explicit formula for the deformation quantization of a general class of infinite dimensional Poisson manifolds. We have also investigated the relation between our formula and the Kontsevich formula. Although the latter hasn't been generalized to the infinite dimensional setting (as for now), one can check if our formula could in principle be derived from one that involves only the propagator (in our case, the advanced propagator) and its derivatives. By direct inspection of the graphs that appear in the third order of our expansion of the interacting star product, we have shown that our expressions cannot be derived by only using the Kontsevich-type graphs. The extra information that we need (apart from the knowledge of the propagator and its derivatives) is the action S. This is, however, always provided in the models we are working with, since they arise from classical field theory formulated in a Lagrangian setting (the Poisson structure is constructed using the action S). It would be interesting to understand if it is even possible to construct a star product in the infinite dimensional setting without using some additional structure; otherwise, our formula may be the best analogue of Kontsevich's formula in this setting. We want to investigate this problem in our future work.
Our results hold for a restricted class of functions on the manifold in question and in order to generalize these results, one needs to perform renormalization. We want to investigate this in future research. There are two possible way forward. One is to use regularization and perform computations in concrete examples, to see how the divergences can be removed. This is expected to work for the quantum Møller operator itself (since it was constructed by perturbative methods, e.g. by [12]) and could also shed some light of the singularities of the interacting product. For the latter, another strategy is to try to modify the formula by changing the way one passes from the time-ordered identification to the standard identification.
We also want to see how the present results are compatible with the algebraic adiabatic limit [3,20] and the general framework proposed in [24].