The Quantum Sine Gordon model in perturbative AQFT

We study the Sine-Gordon model with Minkowski signature in the framework of perturbative algebraic quantum field theory. We calculate the vertex operator algebra braiding property. We prove that in the finite regime of the model, the expectation value - with respect to the vacuum or a Hadamard state - of the Epstein Glaser S-matrix and the interacting current or the field respectively, both given as formal power series, converge.


Introduction
Perturbative algebraic quantum field theory (pAQFT) is an approach to perturbation theory in quantum field theory that follows the paradigm of local quantum physics proposed by Haag and Kastler [HK64,Haa93]. The important feature of this framework is that one separates the construction of the algebra of observables (local aspects of the theory) from the choice of a state (global features). This is of particular importance for generalizing the framework to quantum field theory on a large class of Lorentzian manifolds [BF00, BFV03,HW02]. The pAQFT framework has been since then applied to a wide class of physical problems including quantization of a bosonic string [BRZ14,Zah16] and effective quantum gravity [BFR13]. However, up to now it has not been tested on an interacting model, for which non-perturbative results exists. The present work is meant to bridge this gap and establish convergence results in the massless Sine Gordon model on R 2 in the ultraviolet-finite regime. This answers in particular the question raised in [Sum12], whether there exists an exact interacting model in pAQFT. As it turns out, we can reduce our proofs of summability to the proof from [Frö76], despite the fact that, contrary to that paper, we work in the hyperbolic setting. This way we not only test the robustness of the pAQFT framework, but also provide the first construction of the formal S-matrix in the massless Sine Gordon model on R 2 (in the ultraviolet-finite regime) that is performed directly in the Lorentzian signature. We also construct the interacting currents.
The construction of the local algebras is solely based on the fundamental solutions and solutions of the underlying linear (hyperbolic) partial differential equation. In this approach, no Fock space is needed to calculate e.g. the S-matrix, which turns out to be a formal power series over a certain space of functionals. Moreover, it is not necessary to pass to a Wick rotated Euclidean version of the theory with an underlying elliptic PDE. Renormalization is formulated as an inductive procedure of extending the time ordering map from local functionals to microcausal functionals, in the spirit of [EG73] (i.e. in particular in finite volume). The infrared cutoff is given by a compactly supported test function cutting off the interaction term, and it is usually kept throughout the calculations. The idea behind this is that local measurements do not depend on the particular choice of this test function. To calculate expectation values at some point a state is chosen, but it is not necessary to take this choice -which on general manifolds is not canonical -as the starting point of the construction.
In future work, one should extend these findings to the superrenormalizable regime of the model, in the spirit of [DH93]. We expect that it is straightforward to directly extend our results to globally hyperbolic 2-dimensional spacetimes if the (fundamental) solutions are explicitly known.

Vertex operators in the framework of pAQFT
In this section we rephrase vertex operators as they arise for normal ordered exponentials of the massless scalar field in 2 dimensions in perturbative Algebraic Quantum Field Theory (pAQFT). The basic idea is to rewrite the constructions from perturbative QFT and understand the combinatorics of e.g. the Wick theorem algebraically, in terms of formal power series.
Let M be the D-dimensional Minkowski spacetime, i.e. R D with the diagonal metric diag(1, −1, . . . , −1) and corresponding inner product denoted by x · x. The starting point in the pAQFT construction of models is the classical configuration space E that specifies the type of objects we want to describe. In general it is the space of smooth section of some vector bundle over M. For the scalar field theory, we have E . = C ∞ (M, R). We equip E with its standard Frechét topology. Next, we consider the space of smooth functional on E (smoothness understood in the sense of [Bas64,Ham82,Mil84,Nee06]). Among these, there are some important classes of functionals that are important for the construction of models in pAQFT. Firstly, we introduce the notion of local functionals.
where j k x (ϕ) is the k-th jet prolongation of ϕ and α is a density-valued function on the jet bundle.
The spacetime localisation of a functional is provided by the notion of spacetime support The notion of smoothness that we use implies that functional derivatives of a smooth functional F ∈ C ∞ (E, C) can be seen as compactly supported distributions, i.e. F (n) (ϕ) ∈ Γ ′ (M n , R) C ≡ E ′ (M n ) C , where the superscript C indicates complexification. One can require a stronger condition, i.e. consider functionals whose derivatives are smooth. This motivates the following definition.
More generally, one can impose different, less restrictive conditions on the regularity structure of functionals derivatives of functionals, seen as distributions. To describe the singularity structure, it is convenient to use Hörmander's wavefront (WF) set [Hör03], a refined notion of the singular support. It is a subset of the cotagent bundle whose projection onto the base is the singular support of a distribution (and whose covariables give the so-called high frequency cone). It yields the following very simple sufficient criterion for the existence of products of distributions: if no two covariables (at the same base point) from the two respective wavefront sets can bee added to give 0, the product existst as a distribution. Later on, we will see that the following class of functionals is a good choice for building models of pAQFT's; for other choices see [DB14,Dab14a,Dab14b] Definition 3. A functional F ∈ C ∞ (E, R) is called microcausal if it is compactly supported and satisfies WF(F (n) (ϕ)) ⊂ Ξ n , ∀n ∈ N, ∀ϕ ∈ E , where Ξ n is an open cone defined as where (V ± ) x is the closed future/past lightcone understood as a conic subset of T * x M.
The construction of models in pAQFT starts with the free theory with the equation of motion of the form where P is a normally hyperbolic operator. For such operators there exist retarded (forward) and advanced (backward) fundamental solutions ∆ R , ∆ A respectively. One defines the commutator function (also called the causal propagator) as ∆ = ∆ R − ∆ A . As a distribution, ∆ has WF set of the form where the equivalence relation ∼ means that there exists a null geodesic strip such that both (x, k) and (x ′ , k ′ ) belong to it. One can then split [Rad96] ∆ as a sum of two distributions in such a way that H is symmetric and the WF set of W (interpreted physically as the 2-point function) is The latter condition allows us to introduce the following non-commutative product for microcausal functionals F, G ∈ F µc , Here, the formula for D H−H ′ is meant as a way to write a distribution using formal integral kernels. Note that the algebraic structure presented here looks the same for arbitrary dimension D of M, but the concrete form of W , H and ∆ would be different.
Let us now discuss how the star product ⋆ H allows one to formulate the algebraic version of Wick's theorem and define Wick ordered quantities without using a concrete Hilbert space representation. We follow the construction introduced in [DF01b,BDF09]. Given a regular functional F ∈ F reg , the formal power series formula for the Wick ordered expression is Now approximate F ∈ F µc by regular functionals F = lim F n where F n ∈ F reg and define the corresponding normally ordered quantity as . On regular functionals we can introduce a star product that is independent of H: where δG δϕ(y) dxdy for kernel K and where µ denotes the pullback along ϕ → ϕ ⊗ ϕ. This works also after the limiting procedure since where the down arrows are embeddings hence injective. The relation between ⋆ H and ⋆ encodes the combinatorics of the Wick theorem. While the star product ⋆ is the "standard" product of the quantum theory, one can trade such products of Wick ordered quantities :F : and :G: for ⋆ H -products of ordinary functionals F and G, using formula (2.10). This is a big advantage of the pAQFT framework, since it allows us to write down concrete expressions and discuss convergence of formal power series without going to a Hilbert space (or Krein space as in [Pie88]) representation. The only input is the 2-point function W .
In this paper, we treat the non-perturbative case, so we need to generalize the setting above to the situation where is not a formal parameter, but a number. We proceed in a similar way. We equip the space of smooth functionals C ∞ (E, C) with the topology τ of pointwise convergence of all the derivatives. The n-th functional derivative of F ∈ C ∞ (E, C) at ϕ ∈ D is treated as an element of E(M n ) C , equipped with the standard weak topology. We define A similarly to

The Sine-Gordon model
We start by applying normal ordering to vertex operators. Denote x), and where the Wick product is defined by (2.8). These are the vertex operators. The smeared vertex operators are then defined by evaluating in a test function f In the construction of the Sine Gordon model the starting point is the free theory given by (minus) the massless Klein Gordon (i.e wave) operator P = −✷ on two-dimensional Minkowski . Let ∆ R and ∆ A denote the retarded and advanced (forward and backward) fundamental solutions 1 of P , The 2-point function of the free massless field ϕ in 2 dimensions is [Pie88] W with constants d and b. It is well known that this 2-point function is not positive. Therefore, if one wants to study representations of the abstract algebra from section 2, one needs to apply the Krein space construction [MPS90]. However, the algebraic structure of the theory (e.g. the vertex operators algebra) can be studied independently of the representation and this is the approach we take in the present paper. We first write the 2-point function in terms of a symmetric (H) and antisymmetric (∆) contribution, W = i 2 ∆ + H. The antisymmetric contribution is the causal propagator and a short calculation (see the appendix A) then shows The wavefront sets of the distributions ∆, ∆ A/R are such that Hörmander's sufficient criterion for the existence of products of distributions does not apply, essentially because the wavefront set of the Heaviside function is that of the δ-distribution. However, we will use that the characteristic function χ [0,∞) squares to itself and hence we set θ 2 = θ (as distributions). Similarly, we set the product of θ and θ • j where j(x) = −x, to 0, which is justified since the product of the locally integrable functions χ [0,∞) and χ (−∞,0] is 0 in L 1 loc . Explicitly, we find for n ∈ N, 1 To see that these are indeed fundamental solutions, observe that − 1 Note that the powers of the 2-point function W are well-defined as usual, but contrary to the behaviour of massless fields in higher dimensions, also powers of the commutator function ∆ are. Therefore, we can study the exponential series of these distributions in the following sense: Let u ∈ D ′ (R k ) be such that arbitrary powers u n , n ∈ N, are again distributions. Then one can investigate if the series converges (in R) for any g ∈ D(R k ). In terms of formal integral kernels, we write Observe that the last equality is simply short hand notation (not an interchanging of integration and taking sums). However, if u is smooth and e u(x) converges pointwise, then since g is compactly supported, e u(x) g(x) converges uniformly (on supp g) and in this case, the integral and the summation can indeed be interchanged. In the same spirit we regard the identity Let us now calculate the commutation relation of two vertex operators :V a (f ): from (3.11). Here, we face the additional complication that the distributions depend on ϕ ∈ E. We will treat this dependence pointwise as discussed at the end of section 2.
In terms of formal integral kernels and using the formula for products of normally ordered functionals (2.10), we find W (x, y) n a n a ′n i 2n exp(iaϕ(x))f (x) exp(ia ′ ϕ(y))g(y) . (3.16) In the spirit of the comments above we interpret the sum as the distribution e −aa ′ W (x,y) , so Reversing the order of the vertex operators (i.e. interchanging a and a ′ and f and g, or the roles of x and y) we find and deduce that in terms of formal integral kernels, the commutation relations are .
we see a posteriori that the exponential series is well defined in this case. We get, in particular, for |x − y| = 0 and t > t ′ , (3.17) Choosing ϕ = 0 in the above is motivated by the fact that ω 0,H is then exactly the vacuum expectation value in the state whose 2-point function is given by W = i 2 ∆ + H (see for example the discussion around formula (67) in [FR15], where the more complicated case of curved spacetimes is treated). As explained in section 2, instead of working in ( , ⋆ H ) and motivated by the discussion above, we introduce the notation For the vacuum expectation value of the product of two vertex operators (cp. [Frö76, Lemma 2.2]) we then find The motivation behind the pAQFT approach is to make precise the Dyson formula for the scattering matrix and interacting fields. Recall that heuristically, the Dyson formula for the interacting time evolution operator U I (t, s) is where λ is the coupling constant, T denotes time-ordering and the interaction Lagrangian :L I : is an operator-valued "function". This formula suffers from both the UV and IR divergences. A way to give it mathematical sense is to use the framework of Epstein and Glaser [EG73]. Here, the IR problem is solved by systematically treating the interaction Lagrangian as an operator valued distribution, and evaluating the n-fold time ordered product T (:L I : ⊗ · · · ⊗ :L I :) in g ⊗n , where g is a compactly supported test function. The UV divergences are controlled after carefully defining the time-ordered products.
Let us now recall this construction in the framework of pAQFT, see e.g. [Rej16]. To avoid UV problems for the moment, we consider for now only regular functionals. Let F, G ∈ F reg , then the time ordered product of F and G is defined as where ∆ D is the Dirac propagator defined as More generally, we define n-fold time-ordered products via a map T n : These formulae are well-defined for regular functionals, but usually, physically relevant interaction terms are local and non-linear, hence not regular. Therefore, one needs to extend S from a map on where the sum runs over all 1 ≤ i < j ≤ n and where D ij F . = ∆ F , δ 2 δϕ i δϕ j and ∆ F is the Feynman propagator defined by Let us now apply this approach to a potential of the form V = 1 2 (V a +V −a ), for now assuming that there is no need to renormalize. Then according to formula (4.21), the S-matrix is where the second identity follows from the fact that the time-ordered product is commutative.
We will now explicitly compute the vacuum expectation value of the S-matrix for the Sine-Gordon model and show that the resulting series converges in A (as defined at the end of section 2).
To calculate the explicit form of T H n , we first determine the Feynman propagator or the Sine-Gordon model which can also be written, using the well-known ǫ-prescription as cf. again the appendix A. We now calculate the expectation value of the formal S-matrix in the coherent state ω ϕ,H . The first step is to show that the only nonzero contributions to are those where a i = 0, in particular, all contributions with odd n vanish. To do so we follow the literature, e.g. [Frö76, Lemma 4.2] and [Col75] in spirit, adapted to (and perhaps clarified in) our formalism.
The main idea is to introduce an auxiliary finite mass m (later to be taken to zero again) to define T H n . Starting point is the massive propagator of that mass, ∆ F m (z) is well-defined for any mass m ≥ 0 and indeed, the limit m → 0 yields our Feynman propagator up to an additive constant where γ is the Euler constant. Since h is smooth, we can use it to define an equivalence relation between the two time ordered products w.r.t. ∆ F m and ∆ F m . We take the vacuum expectation value according to (3.17) in the state given by with the corresponding massive advanced and retarded propagators. To realize the modification of the Feynman propagator, we change the prescription of normal ordering and define new normal ordered operators as In this way we reproduce the well-known formula (set µ = 1) where Φ x is the evaluation functional and z = √ −x 2 + iǫ, as above. We now consider expectation values (in the state ω 0,Hm ) of time ordered products (w.r.t. ∆ F m in the sense of (4.22)) of redefined vertex operators α −1 h (V a i ): Using the formulae for the time ordering (4.20) and (4.22) and the explicit form of ∆ F m , as well as the explicit form (4.25) of the equivalence map α −1 h acting on exponentials, we find and where the minus sign in the exponential involving the Feynman propagator K 0 is i 2 from the functional derivative acting on the vertex operators. Our auxiliary mass m being arbitrarily small and all arguments x i −x j being bounded by the support properties of our test functions f , we now use the expansion for K 0 for small argument, K 0 (mz) = −(ln(mz/2) + γ)I 0 (mz), I 0 (mz) = 1 + a power series in mz starting with a quadratic term, to rewrite the Feynman propagator ∆ F m . We then find, with z i and z ij as above, Since I 0 (mz) → 1 in the limit m → 0, only those terms where i a 2 i + 2 i<j a i a j = (a 1 + · · · + a n ) 2 = 0 survive. In particular, all the odd n contributions vanish.
The remaining terms are then of the form with n even, i a i = 0 and with our Feynman propagator ∆ F from (4.24). According to (4.23), we have contributions where the first n 2 parameters a i are equal to a and the remaining n parameters are equal to −a. Using the explicit form of the Feynman propagator (4.24), we then find with τ ij = t i − t j and ζ ij = x i − x j and with the abbreviation To see this, observe that (with where the second equality follows from the idempotency of the Heaviside function, e aθ = 1 + aθ + 1 2 a 2 θ + · · · = 1 − θ + θe a for any real a. Observe that for a i = −a j , the product a i a j = −a 2 , and the term |τ 2 ij − ζ 2 ij | a i a j /4π in (4.26) is singular in t i − t j = ±(x i − x j ). Using translation invariance we reduce this to a problem of a singularity along τ = ±ζ (for a distribution defined on R 2 ). This singularity is within the support of the Heaviside functions in (4.26). However, this is a homogeneous distribution which, as long as a 2 /4π ∈ N, has a unique extension according to [Hör03,Thm 3.2.3] and the prescription from [BF00], see also [NST13]. We will see below directly that these contributions are well-defined for a 2 /4π < 1.
We now prove our main estimate.
Proposition 4. Let β := a 2 /4π < 1. Let p > 1 such that βp < 1. Let g ∈ C ∞ c (R 2n ). Then there is a constant C = C(p, g) such that for all n, we have the following estimate on the n-th order contribution to the vacuum expectation value of the S-matrix of Sine-Gordon theory (4.23), for even n and S n (V ) = 0 for n odd.
Proof. We have already seen that for n odd, the contributions to the S-matrix vanish. Now consider the n-th contribution S n (V )(g) ϕ for even n = 2k. We use the explicit formula according to (4.26). We first estimate the functions θ a i a j (τ, ζ), which are given in terms of Heaviside functions and an oscillating factor, by 1. Without these functions, the formal integral kernel of with β = a 2 /4π > 0, and with the time variable differences τ ij = t i − t j and the space variable differences ζ ij = x i − x j . We rewrite this formula using |τ 2 − ζ 2 | = |τ − ζ| |τ + ζ| and consider the two factors separately, with the underlined variables denoting the collection of all the respective difference variables.
We will see that we can simply carry over the argument from [Frö76]. To that end, we introduce new variables, Then (for i < j) and exactly the same for +-combination, Therefore, and likewise for the +-combination, with the primed variables, w + n,m (z ′ , w ′ ). Note that for the negative powers we indeed get an unordered product (no relation i < j). This follows directly from the fact that in the second factor in (4.27), the two indices are independent.
Despite the fact that we started from a theory with hyperbolic signature, we can proceed in the same way as in [Frö76]. We first reproduce "The main estimate c)" before Thm 3.4 from [Frö76], see also [Col75], and apply Cauchy's lemma, for β = a 2 /4π > 0 and likewise for the +-combination w + n,m (z ′ , w ′ ). We then observe that since we are employing a cutoff function g ∈ C c (M 2n ), we do not have to control the behaviour of our integrand for large arguments z i − z j , z i − w j , w i − w j (infrared behaviour), but only for small ones. This means that we get an estimate on S n (V )(g) 0 for n = 2k as follows where F is g in the new coordinates.
Since the plus and the minus case are exactly the same, we only discuss w − and the integration over the unprimed variables. Without loss of generality, we take F to be a tensor product in the unprimed and primed variables, respectively, and by slight abuse of notation, we write F (z, w). Hence we consider From here on, we assume β . = a 2 /4π < 1. We can now directly follow the arguments from [Frö76], which however, we choose to reproduce for the convenience of the reader.
We choose p > 1 with βp < 1. Consider 1 < q . = p p−1 < ∞. Now view the compactly supported smooth function F as an element of L q (R n ). Applying the Hölder inequality F H 1 = |F H|dx ≤ F q H p for 1 q + 1 p = 1, this yields where K F denotes the support of F in R n .
As in [Frö76,(3.15)], we get an estimate on the integral over the determinant as follows, where S k denotes the group of permutations of k elements and with a constant C p,F that is independent of π. To see the well-definedness of each contribution to the sum, recall that βp < 1, hence the inverse powers are locally integrable.
Therefore, we find the following estimate on S n (V ) 0 for n = 2k, which proves the claim.
We are hence able to reproduce the well-known result: Corollary 5. The expectation value of the S-matrix in the coherent state ω ϕ,H , for every ϕ ∈ E, with IR-cutoff is summable for β < 1, i.e. in the UV-finite regime.
The above result guarantees the pointwise convergence of the S-matrix composed with α H as a functional on E. The pointwise convergence of all the functional derivatives follows from the fact that the l-th derivative of the vertex operator V a (f ) in the direction of ψ ∈ E is given by (ia) l V a (f ψ), so one can use the same estimates as in proposition 4 to obtain bounds on the expressions of the form and the convergence of the formal S-matrix in the topology discussed at the end of section 2 follows.

The interacting fields
In the framework of pAQFT interacting fields are constructed with the use of the Bogoliubov formula [BS59,DF01a] (see also [Rej16] for a review) where F is a classical observables (a functional in F µc ). This formula has to be understood as a formal power series in λ We will now show that we can rewrite the latter using the anti Feynman propagator, To do so, we use the following identity, which we prove in the appendix. Let ∆ D denote the Dirac propagator (4.18) which defines the time ordering. Then for n ≥ 0, we have where ∆ denotes the causal propagator (3.12), which is a sum of two Heaviside functions. It follows -as in the calculation leading to (3.16) -that the time ordered product of 2 vertex operators satisfies (in the sense of formal integral kernels) Contrary to the typical situation in higher dimensional models [Sch95], the products of distributions in the above formula are well defined and no renormalization is needed. This generalizes to n arguments α H (:V a 1 :(x 1 ) ·T . . . ·T :V an :(x n )) where S n is the group of permutations of n elements. The following well-known formula for the n-fold antichronological product α H (T n (:V a 1 :(x 1 ) ⊗ · · · ⊗ :V an :(x n ))) is verified using the argument recounted in [Bah04, Rem. 2.1]. Now, we apply again the identity (5.33) and see that for V = 1 2 (V a + V −a ) (which is real), where in the second step we used that H is real and where · T H is defined by the exponential formula using the anti Feynman propagator ∆ AF , hence In order to apply (5.31), first we need to compute the l + 1-fold time-ordered products of V ⊗l with ∂ µ Φ(f ). Using formula (4.22) we obtain where D ij F . = ∆ F , δ 2 δϕ i δϕ j . The contribution to (5.31) from the first term is easily computed using the lemma. Lemma 6. Let K be an integral kernel and let A, B, C be smooth functionals, with C linear, then where A (1) , C (1) are functional derivatives.
Proof. The result follows directly from the Leibniz rule.
The first contribution is just ∂ µ Φ(f ), since in our case A is the ⋆ H inverse of B. The second contribution can be expanded into time-ordered and antichronological products and using the explicit expression for the vertex operators we obtain a sum of terms of the form For the second term, we also use the explicit form of the vertex operators and obtain We see, in particular, that each contribution to R n in formula (5.31) is proportional to products of n vertex operators. Now we use the same approach as in section 4 -we introduce an auxiliary finite mass m, replace all the propagators in the formula for the interacting field by ∆ F m and W m respectively, and change the normal-ordering prescription by applying α −1 h to the vertex operators. The resulting formulea differ only by the occurence of factors ∂ µ ∆ F m and ∂ µ W m . These, however, have well-defined limits for m → 0 and therefore we can conclude, as in section 4, that only the terms with n i=1 a i contribute. Observe that this is possible for the current ∂ µ Φ x , but would not work for Φ x itself. This the consequence of the well known infrared problem of massless scalar fields in 2 dimensional Minkowski spacetime.
We can now proceed as in proposition 4, taking into account the extra factors of ∂ µ ∆ F * f and ∂ µ W * f . We know that ∂ µ ∆ F and ∂ µ W are tempered distirbutions, so by the standard result (see e.g. [Fol92, §9.2]), their convolutions with Schwarz functions are smooth. These are then multiplied with test functions, so the end result is again a test function.
Finally, note that ∆ AF , ∆ F , W only differ in the signs in front of the Heaviside functions. The latter did not enter in the estimate in Proposition 4, so we conclude that the same estimate as in Proposition 4 holds for (5.31). Again, we conclude that the formal power series ∂ µ Φ(f ) I is summable.

Conclusion
We have proven the convergence of the vacuum expectation value of the Epstein Glaser S-matrix and the interacting current in the finite regime of the Sine Gordon model. To our knowledge, this is the first such proof done directly in the Lorentzian signature.
The main input into the proof of the main estimate (Proposition 4) was the fact that the retarded propagator ∆ R is idempotent and bounded, and that the Hadamard function H is a logarithm so that exponentials of H simply give rational functions. Just like in the earlier investigations [Frö76,DH93], this ties the method of proof to 2-dimensional theories.
We expect that it should be possible to apply the above arguments directly to 2-dimensional curved spacetimes where the 2-point function is known explicitly.
It will be interesting to extend this analysis to the superrenormalizable case where 1 ≤ β < 2, extending the ideas of [DH93].

Acknowledgment
We thank David Bücher and John Imbrie for fruitful discussions in Göttingen and Oberwolfach.