Weak Adiabatic Limit in Quantum Field Theories with Massless Particles

We construct the Wightman and Green functions in a large class of models of perturbative QFT in the four-dimensional Minkowski space in the Epstein–Glaser framework. To this end we prove the existence of the weak adiabatic limit, generalizing the results due to Blanchard and Seneor. Our proof is valid under the assumption that the time-ordered products satisfy certain normalization condition. We show that this normalization condition may be imposed in all models with interaction vertices of canonical dimension 4 as well as in all models with interaction vertices of canonical dimension 3 provided each of them contains at least one massive field. Moreover, we prove that it is compatible with all the standard normalization conditions which are usually imposed on the time-ordered products. The result applies, for example, to quantum electrodynamics and non-abelian Yang–Mills theories.


Introduction
Relativistically covariant perturbative quantum field theory (QFT) in the Minkowski space is one of the most successful modern physical theories. Its models, especially the standard model, have been repeatedly tested in experiments and provide a very accurate description of almost all phenomena in particle physics. The fundamental objects of interest in QFT in the flat spacetime are the vacuum expectation values (VEVs) of the products and the timeordered products of the interacting fields called the Wightman and Green functions, respectively. Conceptually, the Wightman functions are of more fundamental character. They are used for examining the basic physical structure of a theory. Their properties: Poincaré covariance, spectrum condition, hermiticity, locality, positivity and clustering, reflect the basic assumptions of relativistic QFT. Moreover, by the Wightman reconstruction theorem [47] their knowledge allows to obtain the operator formulation of the theory. On the other hand, the Green functions are of more practical importance and it is usually easier to obtain them in the perturbation theory. Furthermore, by the LSZ reduction formula [38] they are directly related to the S-matrix elements. Because of this they have been studied in the physical literature more extensively than the Wightman functions.
The perturbative construction of the Green functions in a large class of models was given by Lowenstein [41], who generalized the BPHZ method [4,32,52], and by Breitenlohner and Maison [7,8], who used the dimensional regularization [6,49]. Their results apply to all models with interaction vertices of infrared dimension equal at least to 4. The infrared dimension is the additive number calculated by assigning dimension 2 to massive fields, dimension 3 2 to massless Dirac fields and dimension 1 to massless boson fields and each derivative. Since the infrared dimension is not lower than the canonical dimension, the Green functions exist in particular in all models with interaction vertices of the canonical dimension equal to 4. Let us recall that the canonical dimension is the additive number which is used in the classification of models of QFT into super-renormalizable, renormalizable and non-renormalizable models. It is calculated by assigning dimension 1 to each derivative as well as scalar and vector fields 1 and dimension 3 2 to Dirac fields. The Wightman functions have not been defined with the use of the momentum space methods. However, both the Wightman and Green functions can be constructed in the Epstein-Glaser (EG) approach [22]. The basic objects in this approach are the time-ordered products of free fields, which satisfy a set of axioms formulated in position representation. The EG method is based on the observation that the time-ordered product of n + 1 local fields at points x 1 , . . . , x n+1 is determined uniquely by the time-ordered products of at most n fields for all test functions with support away from the main diagonal x 1 = · · · = x n+1 . The solution of the ultraviolet (UV) problem in the EG approach amounts to finding an extension of the above distribution to the space of all test functions which is compatible with the axioms. In order to define an interacting model one specifies its interaction vertices L 1 , . . . , L q , which are polynomials in free fields and their derivatives. To each interaction vertex L l we associate a coupling constant e l and a switching function g l ∈ S(R 4 ). To avoid the infrared (IR) problem in the definition of the scattering matrix and the interacting fields one multiplies every interaction term by a switching function belonging to the Schwartz class. In order to define the physical scattering matrix and the physical interacting fields one has to remove this IR regularization by taking the so-called adiabatic limit.
Let g denote the list of the switching functions g 1 , . . . , g q . The IRregularized scattering matrix is given by Vol. 19 (2018) Weak Adiabatic Limit in QFTs with Massless Particles 877 The IR-regularized interacting fields C int (g; x) and their time-ordered products T(C 1,int (g; x 1 ), . . . , C m,int (g; x m )) are defined using the Bogoliubov formulas (cf. Sect. 2.7). The physical scattering matrix S is obtained in the following way  By definition g = (g 1, , . . . , g q, ) and g l, (x) := g l ( x), where g l is a Schwartz function such that g l (0) = 1 for all l ∈ {1, . . . , q}. Note that in the limit 0 the interaction is adiabatically turned on and off. We say that the strong adiabatic limit exists if in each order of the perturbation theory the limit (1) exists for any Ψ from the chosen domain and its value is independent of the choice of the Schwartz functions g 1 , . . . , g q . Similarly, the weak adiabatic limit exists if in each order of the perturbation theory the limits (1) exist in the sense of Schwartz distributions and their values are independent of the choice of the Schwartz functions g 1 , . . . , g q .
The existence of the weak adiabatic limit in purely massive theories was shown by Epstein and Glaser in [22]. This result was subsequently extended to the case of quantum electrodynamics (QED) and the massless ϕ 4 theory by Blanchard and Seneor [1]. The proof of the existence of the weak adiabatic limit in a more general class of models has not been given in the literature. It is the aim of the present paper to fill this gap. For purely massive theories the existence of the strong adiabatic limit was shown by Epstein and Glaser in [23]. Because of the infrared problem this limit does not exist in most theories with massless particles, e.g., in QED. Adiabatic limit of the inclusive cross-sections in low orders of the perturbation theory of QED was considered in [43] (see also [20,21]). Finally, the existence of the expectation values of the products of the interacting fields in thermal states has been recently proved by Fredenhagen and Lindner in the EG framework with the use of the time-slice axiom and the time-averaged Hamiltonian [25] (see also [12,39]). Let us also note that the method of Epstein and Glaser has been reformulated and generalized in a number of ways. It is the basis of QFT in curved spacetime which is used to construct interacting quantum fields propagating on an arbitrary globally hyperbolic spacetime [10,34,35]. The ideas of Epstein and Glaser combined with the formalism of deformation quantization and functional approach led also to the foundation of the perturbative algebraic quantum filed theory-the formulation of perturbative QFT in terms of abstract local algebras [26,42].
The main result of the paper is the proof of the existence of the weak adiabatic limit in the four-dimensional Minkowski space in models with the interaction vertices L 1 , . . . , L q satisfying one of the following conditions: (1) ∀ l∈{1,...,q} dim(L l ) = 4 or (2) ∀ l∈{1,...,q} dim(L l ) = 3 and L l contains at least one massive field, where dim(B) is the canonical dimension of the polynomial B. (The case of models with the interaction vertices of mixed dimensions is briefly discussed in "Appendix B.") It is a generalization of the results due to Blanchard and Seneor [1] mentioned in the previous paragraph. Our result provides a method for the construction of the Wightman and Green functions in the class of models given above. To our knowledge this is the first construction of the Wightman functions in non-abelian Yang-Mills theories.
The general structure of our proof of the existence of the weak adiabatic limit resembles that of [1]. The most important distinguishing feature, which allows us to obtain a significantly more general result, is the use of the notion of regularity of a distribution expressed in terms of the generalized big O notationt(q, q ) = O dist (|q| δ ) (cf. Sect. 3.3). Like the proof in [1], our proof needs specific normalization of some of the time-ordered products corresponding to Feynman diagrams with only massless external lines. The required normalization condition N.wAL (the Normalization condition sufficient for the existence of the weak Adiabatic Limit) is formulated in Sect. 3.6. We prove that this condition is compatible with all the standard normalization conditions like, for example, Poincaré covariance. In particular, in the case of QED the condition N.wAL is compatible with all the normalization conditions introduced in [16]. In purely massless models the condition N.wAL is a consequence of almost homogeneous scaling of the VEVs of the time-ordered products. This implies the compatibility of N.wAL with all the normalization conditions used in [33] in the definition of non-abelian Yang-Mills theories without matter (in the special case of the flat spacetime). We also prove that the following condition has to be satisfied for the weak adiabatic limit to exist: The self-energies of the massless fields (which are used in the definition of a given model) have to be normalized such that the physical masses of these fields vanish. We call this condition the correct mass normalization of massless fields. Note that it follows from the condition N.wAL. The Wightman and Green functions cannot be defined by means of the weak adiabatic limit in models in which the correct mass normalization of massless fields is not possible. An example of such model is the massless ϕ 3 theory.
Using the technique of the proof of the existence of the weak adiabatic limit we also show that it is possible to significantly restrict the freedom in the definition of the time-ordered products by imposing the condition which we call the central normalization condition. In the case of QED the time-ordered product defined in this way satisfy all the standard normalization conditions, including the Ward identities. The corresponding advanced and retarded products constitute the so-called central solution of the splitting problem [22,43], whose existence up to now was only established in purely massive models [22].
In the framework of perturbative algebraic quantum field theory [26,42] the existence of the weak adiabatic limit allows to define a real, normalized and Poincaré-invariant functional on the algebra of interacting fields obtained by means of the algebraic adiabatic limit. In the case of models which are constructed on the Fock space with a positive-definite covariant inner product this functional is positive and can be interpreted as an interacting vacuum state.
The plan of the paper is as follows: In Sect. 2 we describe the EG approach to the perturbation theory, present some auxiliary results and introduce the notation that is used throughout the paper. Section 3 contains the main results, including the proof of the existence of the weak adiabatic limit, which is outlined in Sect. 3.2 and given in Sect. 3.6. In Sect. 4 we study the compatibility of the condition N.wAL, needed in our proof of the existence of the weak adiabatic limit, with the standard normalization conditions usually imposed on the time-ordered products. In Sect. 4.2 we formulate the central normalization condition. Section 4.4 contains the list of the properties of the Wightman and Green functions constructed in the paper. In Sect. 5 we construct a Poincaréinvariant functional on the algebra of interacting fields. In "Appendix A" we show that the correct mass normalization of massless fields is necessary for the existence of the weak adiabatic limit in the second order of perturbation theory. In "Appendix B" we consider the case of models with the interaction vertices of different dimensions. "Appendix C" contains the comparison of the Green functions defined in the Epstein-Glaser framework and the Green functions obtained with the use of the Gell-Mann and Low formula.
We use the (+ − −−) signature of the Minkowski metric. By V + we denote the closed cone of the future-directed timelike and lightlike vectors. Our convention for the Fourier transform of Schwartz distributions is

Epstein-Glaser Approach
In this section we outline the EG approach to perturbative QFT in the fourdimensional Minkowski space. We follow closely the exposition from [22] and treat fields as operator-valued distributions defined on a suitable domain of the Fock space. For details we refer the reader to [22,43,44].

Algebra F of Symbolic Fields
In perturbative QFT the interacting models are built with the use of the free fields. Let be the set of symbols denoting types of free fields needed for the definition of a model under consideration. The elements of this set are called the basic generators. All components of vector or spinor fields are included in this set as separate symbols (if fields of these types are present in the model). We assume P. Duch Ann. Henri Poincaré that an involution denoted by * is defined in G 0 . It means that if a charged field A i belongs to G 0 , then also its hermitian conjugation denoted by A * i belongs to it, i.e., A * i = A i for some i ∈ {1, . . . , p}. For example, in the case of QED there are 12 basic generators: the four components of the vector potential A μ = A * μ , and the four components of the spinor field ψ a and its hermitian conjugate ψ * a = (ψγ 0 ) a . The basic generators (2.1) supplemented with the symbols corresponding to their derivatives form the set of the generators To every generator we associate the following quantum numbers: where q ∈ Z is the electric charge number. Following [3] we define the algebra of symbolic fields denoted by F. It is a free unital graded-commutative * -algebra over C generated by the elements of the set G. The adjoint is defined uniquely by the following conditions: the antilinearity and the identity ( The graded commutativity means that for any B 1 , B 2 ∈ F which are monomials in the generators it holds The definition of f (B) is extended to all monomials B ∈ F by f (cB) = f (B) for c ∈ C and the additivity f (B 1 B 2 ) = f (B 1 ) + f (B 2 ) for any monomials B 1 , B 2 ∈ F. The same holds for the other quantum numbers. We say that B ∈ F is a homogeneous polynomial if it is a linear combination of monomials with the same quantum numbers. The set of homogeneous polynomials is denoted by F hom . The definitions of the quantum numbers are naturally extended to F hom .
By definition a monomial is an element of F which is a product of generators. The monomials are labeled by super-quadri-indices. The monomial A r ∈ F labeled by the super-quadri-index r is given by the formula supported on a finite subset of G. In addition, if A i is a fermionic field, then we demand that r(i, α) ∈ {0, 1} for all α ∈ N 4 0 . We say that the super-quadri-index r involves only the field A i (involves only the massless fields) if r(i , α) = 0 Vol. 19 (2018) Weak Adiabatic Limit in QFTs with Massless Particles 881 for i = i (for all i such that A i is a massive field). We write r ≥ s iff r(i, α) ≥ s(i, α) for all (i, α) ∈ G. If r ≥ s, then the super-quadri-index u = r−s is given by u(i, α) = r(i, α) − s(i, α) for all (i, α) ∈ G. Moreover, we set Note that if fermionic fields are present then the order of factors in product (2.2) matters. To remove the ambiguity in the definition of A r , we fix some linear ordering in the set G {1, . . . , p} × N 4 0 (its precise form is irrelevant for our purposes) and always multiply the generators in this order. All monomials in F are proportional to A r for some super-quadri-index r. The set {A r : r is a super-quadri-index} is a linear basis of the algebra F. For example, in the case of QED the electric current j μ = ψγ μ ψ = ψ * a γ 0 ab γ μ bc ψ c ∈ F (we use the Einstein summation convention) is a combination of the monomials ψ * a ψ c . The derivative of B ∈ F with respect to the generator ∂ α A i is defined as a graded derivation Let s be a super-quadri-index. We define the linear map F B → B (s) ∈ F given by For example, if B = A r , then B (s) is proportional to A r−s for r ≥ s and vanishes otherwise. The polynomial C ∈ F is a sub-polynomial of the polynomial B ∈ F iff C equals B (s) up to a multiplicative constant for some super-quadriindex s. All sub-polynomials of a monomial A r are of the form A s , where a super-quadri-index s is such that r ≥ s.
The representation of the SL(2, C) group, which is the covering group of the Lorentz group, acts on the algebra of symbolic fields F in a standard way, and the action is called ρ.

Wick Polynomials
Let D be a linear space over C equipped with a non-degenerate sesquilinear inner product (·|·), and let L(D) be the space of linear maps D → D. By an operator-valued Schwartz distribution on D we mean a map T : S(R N ) → L(D) such that for any Ψ, Ψ ∈ D 882 P. Duch Ann. Henri Poincaré , by the nuclear theorem.
To every symbolic field B ∈ F we associate the Wick polynomial :B(x): which is an operator-valued Schwartz distribution [47,51] on a suitable domain D 0 in the Fock Hilbert space to be specified below. For example, A i (x) denotes one of the basic free fields and (∂ α A i )(x) = ∂ α A i (x)-its derivative. All Wick monomials at point x ∈ R 4 are up to a multiplicative constant of the form for some super-quadri-index r. Observe that :B(x): = 0 does not imply B = 0. For example, we have ( ϕ)(x) = ϕ(x) = 0 if ϕ is free field fulfilling the wave equation, whereas the symbol ϕ ∈ F is by definition a nonzero generator.
The definition of all basic free fields A i (x) and the Fock spaces on which they act is standard [11,44,50]. In the case of the vector field we use the Gupta-Bleuler approach [2,30] which is nicely summarized in the monographs [44,48]. Let us only mention that in this approach the vector field is defined on the Krein space with two inner products. The field is hermitian only with respect to the covariant inner product which is not positive definite.
The Hilbert space H on which the model is defined is the tensor product of the Fock spaces on which the fields A 1 (x), . . . , A p (x) corresponding to basic generators act. The vacuum state in the Fock space H is denoted by Ω. We introduce the following dense subspace in H From now on, all the operators we consider are elements of L(D 0 ). As shown in [51] Wick polynomials are well defined as operator-valued Schwartz distributions on D 0 . The fact that D 0 is embedded in the Hilbert space H plays no role. The space D 0 is equipped with a Poincaré covariant and non-degenerate inner product, which is denoted by (·|·). The product is positive definite unless the vector fields are present in the model. The hermitian conjugation and the notion of unitarity are defined with respect to this product. The unitary representation of the inhomogeneous SL(2, C) group, which is the covering group of the Poincaré group, denoted by U (a, Λ), where a ∈ R 4 and Λ is a Lorentz transformation, is defined on D 0 in the standard way. In the case of pure translations we write U (a) ≡ U (a, 1).

F -Products
In this section we introduce the notion of the F-product. The time-ordered products and all other products which are used in the EG approach are examples of the F-products. We say that a multi-linear map is an F-product iff the following conditions hold for any B 1 , . . . , B n ∈ F hom : A.1 Translational covariance: A.3 Wick expansion: The product F (B 1 (x 1 ), . . . , B n (x n )) is uniquely determined by the VEVs of the product F of the sub-polynomials of B 1 , . . . , B n : The above properties of F-products are modeled on the ordinary product of normally ordered operators  for all a ∈ R 4 . For any translationally invariant t ∈ S (R 4(n+1) ), n ∈ N 0 , we can define the associated distribution t ∈ S (R 4n ) by t(x 1 , . . . , x n ) ≡ t(x 1 , . . . , x n , 0) := d 4 y t(x 1 + y, . . . , x n + y, y) h(y), (2.4) where h ∈ S(R 4 ), d 4 y h(y) = 1. The RHS of the above equation is independent of h. The Fourier transforms of t and t are related bỹ t(q 1 , . . . , q n+1 ) = (2π) 4 δ(q 1 + · · · + q n+1 )t(q 1 , . . . , q n ).
Let us define the product and the graded commutator of two F-products: The product of F and F is by definition the following multi-linear map which, as it turns out, is again an F-product. The only non-trivial part in the proof of this fact is the verification that the product of F and F fulfills condition (2.3), which follows from the properties of the Wick products (cf. Section 4 in [22]). The graded commutator of F and F is defined by where the sum is carried out over l ∈ N 0 and functionsū,ū ,ī,ī ,ᾱ,ᾱ such that: ( where ∝ indicates the equality up to a possible factor (−1). Given a list of functionsū,ū ,ī,ī ,ᾱ,ᾱ of the above type we define the following maps characterizing this list e,d : where G 0 is the set of the basic generators (2.1). We set is the total super-quadri-index of the list s. It follows from (2.7) that the lists s = (s 1 , . . . , s n ), s = (s 1 , . . . , s n ) and functionsū,ū ,ī,ī ,ᾱ,ᾱ which appear in representation (2.6) satisfy the following constraints The term in the sum on the RHS of (2.6) for which l = 0 (i.e., s = s = 0 and The terms in the sum on the RHS of (2.6) corresponding to functionsū,ū , i,ī ,ᾱ,ᾱ , which satisfy the conditionē(A i ) = e s (A i ) = e s (A i ) = 0 for all massive fields A i ∈ G 0 (i.e., the super-quadri-indices from the lists s, s involve only massless fields) are called the massless contributions. We close this section with a lemma which further restricts the form of terms which may appear in the sum on the RHS of (2.6).

Time-Ordered Product
The scattering matrix and the interacting fields in perturbative QFT are given in terms of the time-ordered products. By definition the time-ordered products form a family of F-products indexed by n ∈ N 0 which satisfies the following axioms (besides the conditions A.1-A.3 stated in the previous section, which we also call the axioms): A.4 T(∅) = 1, T(B(x)) = :B(x):, where 1 on the LHS of the above equality is the unity in F. A.5 Graded symmetry: For any B 1 , . . . , B n ∈ F hom it holds where π is any permutation of the set {1, . . . , n} and f (π) ∈ Z/2Z is the number of transpositions in π that involve a pair of fields with odd fermion number. In particular, the time-ordered product of polynomials (dim(B j ) + c), (2.12) where sd(·) is the Steinmann scaling degree of a distribution and the constant c ∈ {0, 1} is fixed. [45]). Let t ∈ S (R N ). The scaling degree of t with respect to the origin is given by

Definition 2.4 (Steinmann scaling degree
where by definition inf ∅ = +∞. For the proof of the existence of the time-ordered products satisfying the above axioms we refer the reader to [10,22]. We stress that the timeordered products, like any other F-products, are operator-valued Schwartz distributions on D 0 indexed by the symbolic fields B ∈ F, and not the Wick polynomials :B(x):∈ S (R 4 , L(D 0 )). This is the feature of the so-called offshell formalism [9,18,46]. Observe that, for example, the time-ordered product T( ϕ(x 1 ), ϕ(x 2 )) does not have to vanish even if ϕ is a free field satisfying the wave equation.
Setting c = 0 in condition (2.12) is the standard choice [10,22,26]. The case c > 0 corresponds in the BPHZ approach to the special form of the socalled over-subtractions [52]. Its compatibility with the rest of axioms may be proved using Lemma 6.6 of [10] as in the standard case. We always assume that c = 0 in the case of models with at least one interaction vertex of dimension 4. However, in the case of models with interaction vertices of dimension 3 it is also possible to set c = 1. The choice c = 1 makes the UV behavior of the theory worse, but at the same time allows more freedom in the construction of time-ordered products. This freedom may be used to improve its IR properties.

Normalization Freedom
Assume that all the time-ordered products with at most n arguments are given. In this section we will characterize the ambiguity in defining the time-ordered products with n + 1 arguments. To this end, by axiom A.3 it is enough to consider the family of numerical distributions (2.13) We claim that any two possible definitions of the above family differ by a family of distributions for all B 1 , . . . , B n+1 ∈ F hom and all permutations π ∈ P n+1 , where f (π) ∈ Z/2Z is the number of transpositions in π that involve a pair of fields with odd fermion number, for some constants c γ ∈ C indexed by multi-indices γ, |γ| ≤ ω, where The bound |γ| ≤ ω in sum (2.15) is a consequence of axiom A.7. It follows that v(B 1 (x 1 ), . . . , B n+1 (x n+1 )) = 0 if ω < 0 and the VEV of the corresponding time-ordered product is determined uniquely by the time-ordered products with at most n arguments.

Generating Functional and T, A, R, D Products
has odd fermion parity, i.e., f (B k ) is odd, then g k is a Schwartz function valued in the Grassmann algebra with odd Grassmann parity) and g = m j=1 g j ⊗ B j ∈ S(R 4 ) ⊗ F. The generating functional of the time-ordered products is given by . . , Bj n (xn)).

(2.17)
It is a map that sends elements of S(R 4 ) ⊗ F into formal power series in g 1 , . . . , g m with coefficients in L(D 0 ). The argument of Texp should be treated symbolically. Since S(0) = 1 the formal power series S(g) is invertible. A nontrivial but very important consequence of the axiom of causality A.6 is the so-called causal factorization property [5,22] which holds whenever supp g + V + ∩ supp g = ∅, i.e., the support of g does not intersect the causal future of the support of g . The support of g may be arbitrary. Now we will introduce the abbreviated notation which will be used throughout the paper.
generates the time-ordered products in the following sense where we used the identities . The anti-time-ordered product T(I) can be expressed as a combination of products of the time-ordered products T (I 1 ) . . . T(I n ), where the concatenation of the sequences I 1 , . . . , I n is some permutation of the sequence I.
The advanced and retarded products are given by Let us list the properties of the products introduced above. It is evident that they are F-products. Moreover, they are graded-symmetric separately in arguments collectively denoted by I and J. A closer inspection reveals that D and A products with n arguments can be expressed by products of the timeordered products with at most n − 1 arguments. Using the causal factorization property (2.18) one proves that the advanced, retarded and causal products have the following support properties: The condition in the definition of the sets Γ ± n,m means that each of the points y 1 , . . . , y n is in the causal future/past of some of the points x 1 , . . . , x m . In particular, in the case when J contains only one element, using translational invariance of the VEVs of the F-products we obtain:

Definition of Wightman and Green Functions
In order to define a model of interacting QFT one has to specify its interaction vertices which are distinguished homogeneous elements of the algebra of symbolic fields F such that for all l ∈ {1, . . . , q}: (1) L l is a Lorentz scalar, where c is the constant which appears in axiom A.7. The first condition is crucial for the Lorentz covariance of the model. The second one is needed for quantum mechanical consistency of the model, e.g., for the unitarity of the scattering matrix. The third reflects the fact that we consider only renormalizable interactions. The last condition guarantees that the scattering matrix commutes with the fermion number operator. To prove the existence of the Wightman and Green function in the case of models with massless particles we impose some additional conditions on the interaction vertices which are formulated in Sect. 3.6 as Assumption 3.1.
To each interaction vertex L l we associate a parameter e l ∈ R also called a coupling constant. The scattering matrix and the interacting fields are defined as formal power series in the independent parameters e 1 , . . . , e q . In some models the physical coupling constants accompanying different interaction terms are related. In this case we assume that there exists a set of independent parameters such that all e 1 , . . . , e q are polynomial functions of them (for example, it may hold e 1 = e, e 2 = e 2 for some e ∈ R). In our proof of the existence of the weak adiabatic limit it is possible to assume that the parameters e 1 , . . . , e q are independent. The relation between e 1 , . . . , e q of the above-mentioned form may be always imposed in the final expression for the Wightman and Green functions by reorganizing the resulting formal power series.
Set g = q l=1 e l g l ⊗ L l and h = h ⊗ C, where g 1 , . . . , g q , h ∈ S(R 4 ) and C ∈ F hom . The advanced and retarded interacting fields with IR regularization are defined by the Bogoliubov formulas [5,22] Vol. 19 (2018) Weak Adiabatic Limit in QFTs with Massless Particles 891 The functions g 1 , . . . , g q ∈ S(R 4n ) are called the switching functions. They switch off the interaction as |x| → ∞ and play the role of the IR regularization.
. The time-ordered product of advanced fields with IR regularization is given by [22] T(C 1,adv (g; where J := (C 1 (x 1 ), . . . , C m (x m )), and we used notation (2.19). The timeordered product of the retarded fields is given by a similar formula with the advanced product replaced by the retarded product. The time-ordered products of the advanced or retarded fields are graded-symmetric and satisfy the axiom of causality. However, they are not translationally covariant. The Wightman function of the advanced/retarded fields with IR regularization are the VEVs of the product of the advanced/retarded fields whereas the Green functions of the advanced/retarded fields with IR regularization are defined by (Ω| T(C 1,adv/ret (g; x 1 ), . . . , C m,adv/ret (g; x m ))Ω). (2.26) They are formal power series in coupling constants e 1 , . . . , e q with coefficients in S (R 4m ). In order to obtain the physical Wightman and Green functions one has to get rid of the switching functions. To this end, one takes the adiabatic limit (1.2) of each term in the expansion of (2.25) and (2.26) in powers of the coupling constants.

Generalized A, R, D Products with a Partition
. Fix a strictly increasing sequence of natural numbers P = (p 0 , . . . , p k ), such that p 0 = 0 and p k = m, and define a partition J 1 , . . . , J k of J such that J j is a contiguous subsequence of J starting at the position p j + 1 and ending at p j+1 . The generalized advanced product with the partition P is given by Similarly, the generalized retarded product with the partition P is given by If I is the list of the interaction vertices where T adv (g; J) is given by (2.24). Since T adv/ret (g; C(x)) = C adv/ret (g; Note that the Green functions are expressed by the ordinary advanced product A(I; J). This is not true for the Wightman functions. In formula (2.31) the arguments of A(I; J; P ) collectively denoted by I are always of form (2.29) and identity (2.30) applies. However, in the inductive proof of the existence of the weak adiabatic limit presented in Sect. 3 we will have to consider also the case when I is the list of sub-polynomials of the interaction vertices ln (y n )). It follows from the definitions of the generalized A, R and D products that The above formula implies that D(I; J; P ) may be expressed as a combination of terms of the form or a combination of terms of the form where the concatenation of the sequences I 1 , I 2 , I 3 is some permutation of the sequence I and the number of elements of I 3 is strictly less than the number of elements of I. Moreover, as a consequence of the causal factorization property the generalized A, R and D products have the same support properties (2.22) as the standard A, R and D products.

Existence of Weak Adiabatic Limit
In this section we present our main results. The objective is to prove the existence of the weak adiabatic limit in a large class of models including all models with interaction vertices of dimension four. To this end, we have to prove that limits (1.2) exist, where the interacting fields are the advanced or retarded fields defined in Sect. 2.7. In view of the discussion of Sect. 2.8 it is enough to show that the limits: and the analogous limit with the generalized advanced product A(I; J; P ) replaced by the generalized retarded product R(I; J; P ) (1) exist for all n, m ∈ N 0 , l 1 , . . . , l n ∈ {1, . . . , q}, C 1 , . . . , C m ∈ F, f ∈ S(R 4m ) and an arbitrary sequence P of the form which is considered in Sect. 2.8, (2) are independent of g ∈ S(R 4n ) such that g(0, . . . , 0) = 1 and (3) have the same values in the retarded and advanced case.
Ann. Henri Poincaré By definition g (y 1 , . . . , y n ) := g( y 1 , . . . , y n ). We consider first theories with only massive particles. In this case the existence of limit (3.1) was shown by Epstein and Glaser in [22]. In Sect. 3.1 we give a slightly modified proof of this fact. It shall be regarded as the preparation for the more involved proof for theories with massless particles which is outlined in Sect. 3.2 and presented in full detail in Sect. 3.6. Sections 3.3, 3.4 and 3.5 contain intermediate results.

Proof for massive theories
The proof of the existence of the weak adiabatic limit for theories with only massive particles relies on the presence of the mass gap in the energymomentum spectrum of these theories. This property, which holds only in the case of purely massive models, means that the vacuum state is separated from the rest of the spectrum. The presence of the mass gap implies that there is restricted to test functions supported in O is a smooth function. Before proving this let us observe that with the use of the above distribution limit (3.1) can be rewritten in the form . . , q n ). Assuming thatã is smooth in O, the existence of the above limit is an immediate consequence of the following obvious lemma.
Let us show that (3.2) is indeed a smooth function in some neighborhood of the origin. To this end, for each n ∈ N + we define a function Θ n ∈ C ∞ (R 4n ), called a UV regular splitting function, such that it vanishes in some neighborhood of the origin and Θ n (y 1 , . . . , y n ) := ρ 3n(y 0 1 + · · · + y 0 n ) |(y 1 , . . . , y n )| for |(y 1 , . . . , y n )| ≥ , where is an arbitrary positive constant of the dimension of length, |(y 1 , . . . , y n )| = (|y 1 | 2 + · · · + |y n | 2 ) 1/2 and ρ ∈ C ∞ (R) is a real function having the following properties: The precise form of the functions Θ n for |(y 1 , . . . , y n )| < is irrelevant for our purposes-we only assume that Θ n is supported outside the origin. For |(y 1 , . . . , y n )| ≥ it holds: Moreover,Θ n -the Fourier transform of Θ n -is a smooth function outside the origin andΘ n (k) vanishes at infinity faster than any power of 1/|k|.  We will show that the contribution to (3.2) from each of the three terms of the RHS of Eq. (3.3) is a smooth function in some neighborhood of zero. For the first two terms this follows from the lemma below with q 1 = · · · = q m = 0, B 1 = L l1 , . . . , B n = L ln . The lemma will be also used in the proof of the existence of the weak adiabatic limit for theories with massless particles. Its proof uses only the support properties of the functions Θ n , (1 − Θ n ) and of the generalized advanced and retarded products.
Ann. Henri Poincaré is a smooth function. The same holds for Proof. Because of the support property of the generalized retarded distribution (cf. Sect. 2.8) and the presence of the function Θ n the integrand in (3.4) for |(y 1 , . . . , y n )| ≥ may be nonzero only in the region where the cone Γ − n,m is given by (2.23). To prove the above inclusion we first note that for any (y 1 , . . . , y n ; Combining the above inequality with the first bound in (3.6) we get After summing both sides of the second bound in (3.6) and of bound (3.7) over j from 1 to n we obtain | y 1 | + · · · + | y n | ≤ 2n |(x 1 , . . . , x m )| + |y 0 1 | + · · · + |y 0 n |, where the concatenation of the sequences I 1 , I 2 , I 3 is some permutation of a sequence (L l1 (y 1 ), . . . , L ln (y n )). The above distributions vanish in some neighborhood of the origin as a consequence of both parts of Lemma 2.3 and the assumption that there are no massless fields. Note that the validity of the last statement follows ultimately from the presence of the mass gap in the energy-momentum spectrum. This is the only place in the proof where we use the assumption that all fields are massive. The contribution to (3.2) from the last term on the RHS of Eq. (3.3) is of the form Because of support properties of (3.8) and the smoothness ofΘ n outside the origin the above distribution is indeed a smooth function in some neighborhood of the origin. The above result implies the existence of limit (3.1). Since D(I; J; P ) = A(I; J; P ) − R(I; J; P ) and the distributiond vanishes in some neighborhood of zero limit (3.1) with the A product replaced by R also exists and has the same value as (3.1). This shows the existence of the weak adiabatic limit in massive theories.

Idea of Proof for Theories with Massless Particles
The proof of the existence of the Wightman and Green functions in massive models was based on the fact that distribution (3.8) vanishes in some neighborhood of the origin. This in turn follows from the existence of the mass gap in the energy-momentum spectrum of massive theories and is no longer true when massless particles are present. In fact, in theories with massless particles distribution (3.2) is usually not a continuous function in any neighborhood of zero. Note that by Part (B) of Lemma 3.1 it is enough to control the behavior of this distribution in the vicinity of the origin. To quantify the regularity of distributions t ∈ S (R N ) near 0 ∈ R N we shall introduce a distributional condition t(q) = O dist (|q| δ ), where δ ∈ R. It generalizes the condition f (q) = O(|q| δ ), expressed in terms of the standard big O notation, which applies when f is a function. We will prove that in a large class of models with massless particles (cf. Assumption 3.1 stated in Sect. 3.6 for the precise specification of this class) it is possible to normalize the time-ordered products such that distribution (3.2) is of the form a(q 1 , . . . , q n ) = c + O dist (|q 1 , . . . , q n | 1−ε ) for some c ∈ C and any ε > 0 (3.9) and distribution (3.8) is of the form d(q 1 , . . . , q n ) = O dist (|q 1 , . . . , q n | 1−ε ) for any ε > 0. (3.10) As we will see the existence of the weak adiabatic limit follows immediately from the above conditions. Conditions (3.9) and (3.10) are satisfied if the time-ordered products fulfill the normalization condition N.wAL, which is equivalent to N.wAL'. (Both conditions are formulated in Sect. 3.6.) The latter condition says that for any 898 P. Duch Ann. Henri Poincaré F-product F which is a product of the time-ordered products and all lists u = (u 1 , . . . , u k ) of super-quadri-indices which involve only massless fields 2 (3.11) The functions e u (·), d u (·) are given by (2.10), and p is the number of basic generators. According to Theorem 3.19, stated in Sect. 3.6, this normalization condition may be imposed in all models satisfying Assumption 3.1. We prove this theorem by induction. We assume that the time-ordered products with at most n arguments satisfy the condition N.wAL' and show that it is possible to define time-ordered products with n + 1 arguments such that this conditions holds. We first use the following fact which is an immediate consequence of the statement (1') of Theorem 3.16 stated in Sect. 3.4 entitled Product. Let F and F be two F-products. Assume that their VEVs satisfy condition (3.11). Then the VEV of their product (2.5) also satisfies this condition. Consequently, the VEVs of the D and A products with n + 1 arguments fulfill condition (3.11). The second part of the proof of the inductive step is based on Theorem 3.17 from Sect. 3.5 entitled Splitting. Using this theorem we show that if the VEV of the D product with n + 1 arguments satisfies condition (3.11), then it is possible to define the advanced product with n + 1 arguments such that its VEV also satisfies condition (3.11). The existence of the weak adiabatic limit in the class of models satisfying our assumptions is stated as Theorem 3.20. Let us explain the intuitive content of this theorem. Consider the distributioñ a u1,...,u k+m (q 1 , . . . , q k ; q 1 , . . . , q m ) := d 4 y 1 . . .
Note that after smearing it with a test function in q 1 , . . . , q n we get a continuous function of q 1 , . . . , q m . For q 1 = · · · = q m = 0 and u 1 = . . . u k+m = 0 the above distribution coincides with (3.2). We will prove that: (A) for any list u = (u 1 , . . . , u k+m ) of super-quadri-indices involving only massless fields such that at least one of them is nonzero it holds a u1,...,u k+m (q 1 , . . . , q k ; q 1 , . . . , q m ) = O dist (|q 1 , . . . , q k | d−ε ) for any ε > 0, (3.14) where the above distribution is defined by (3.12) with the A product replaced by the D product. Because of the presence of the graded commutator in (2.32) it is possible to prove (3.14) for all u 1 , . . . , u n+m which involve only massless fields using the validity of (3.13) for all u 1 , . . . , u n+m which involve only massless fields such that at least one of them is nonzero. Condition (3.14) implies (3.10). Using (3.14) and (3.10) as well as Theorem 3.18 from Sect. 3.5 entitled Splitting we obtain (3.13) and (3.9) with k = n.

Mathematical Preliminaries
where q ∈ R N and q ∈ R M iff there exist a neighborhood O of the origin in R N × R M and a family of functions t α ∈ C(O) indexed by multi-indices α such that (1) t α ≡ 0 for all but finite number of multi-indices α, t(q, q ) = α ∂ α q t α (q, q ) for (q, q ) ∈ O. Note that the differential operator ∂ α q and the factor |q| δ+|α| above involve only the variable q ∈ R N . If N = 0 we write t(q ) = O dist (| · | δ ). By definition for any Let us make a couple of remarks about the above definition: 1. In our applications the exponent δ which appears in Definition 3.3 will never be an integer. Usually, we set δ = d − ε, where d ∈ Z and ε ∈ (0, 1). 2. The condition t(q, q ) = O dist (|q| δ ) controls the behavior of the distribution t only near the origin. In particular if a distribution t ∈ S (R N ×R M ) vanishes in a neighborhood of 0 then t(q, q ) = O dist (|q| δ ) for arbitrarily is a continuous function for q ∈ O 2 . In particular, the distribution t(q, 0) is well defined in O 1 and it holds t(q, 0) = O dist (|q| δ ). 4. If t ∈ C(R N ×R M ) is such that |t(q, q )| ≤ const |q| δ in some neighborhood of 0 then t(q, q ) = O dist (|q| δ ). There are, however, t ∈ C(R N × R M ) such that t(q, q ) = O dist (|q| δ ) and the bound |t(q, q )| ≤ const |q| δ is violated in every neighborhood of 0. An example of such function in the case N = 1, M = 0 and δ = 2 is t(q) = q sin(1/q) + 3q 2 cos(1/q) = ∂ q [q 3 cos(1/q)]. 5. Finally, let us remark that a very similar characterization of the regularity of distributions near the origin in one dimension was introduced by Estrada in [24] for the investigation of the existence of an extension t ∈ D (R) of a distribution t 0 ∈ D ((0, ∞)).

Definition 3.4.
We say that a distribution t ∈ S (R N ) has a value t(0) ∈ C at zero in the sense of Lojasiewicz [40] iff the limit below exists for any g ∈ S(R N ) such that . The value t(0) is usually called in the physical literature the adiabatic limit of the distribution t at 0. The distribution t ∈ S (R N ) has zero of order ω ∈ N + at the origin in the sense of Lojasiewicz iff ∂ γ q t(q) q=0 = 0 for all multi-indices γ such that |γ| < ω, where ∂ γ q t(q) q=0 is defined in the sense of Lojasiewicz.
We have t(0) = c in the sense of Lojasiewicz.

then t has zero of order ω at the origin in the sense of Lojasiewicz.
Proof. Let us begin with the proof of part (A). It follows from Definition 3.3 that there exists χ ∈ D(R N ), χ ≡ 1 on some neighborhood of 0, such that for all q ∈ R N . For arbitrarily large ρ > 0 it holds and consequently, On the other hand, since |t α (q)| ≤ const |q| |α|+δ . This finishes the proof of Part (A). Part (B) follows immediately from Part (A). Part (C) is a consequence of Part (B) and the fact that Definition 3.6. The function Θ : R N → R is called a UV regular splitting function in R N iff (1) Θ is smooth, An example of a UV regular splitting function with N = 4n is the function Θ n defined in Sect. 3.1. Note that in comparison with [1,22,23] the splitting functions which we will use are homogeneous only outside certain neighborhood of the origin. Because they are smooth everywhere we call them UV regular splitting functions. The UV-regularized splitting of a Schwartz distribution t ∈ S (R N × R M ) is defined in the position space by t Θ (y, x) := Θ(y)t(y, x). (3.15) The result of the splitting t Θ is again a Schwartz distribution. Equivalently, in momentum space the splitting of a distribution t ∈ S (R N × R M ) is given by for any g ∈ S(R N ). The splitting (3.15), which was introduced above, will be applied to distributions t of the form If M = 0, the above condition means that the distribution t is an arbitrary element of S (R N ).

Lemma 3.7. Let N ≥ 2 and Θ be a UV regular splitting function in
where (1)Θ hom β is smooth on R N \{0} and homogeneous of degree −N + 1, Ann. Henri Poincaré

(3)Θ β (k) vanishes at infinity faster than any power of |k|.
It follows thatΘ β (k) andΘ(k) are smooth outside the origin and vanish at infinity faster than any power of |k|. Moreover,Θ β is absolutely integrable.
Proof. Let Θ β (y) := (−i)Θ(y)y β /|y| 2 , Θ hom β be the unique homogeneous distribution of degree −1 such that Θ hom β (y) = Θ β (y) for |y| > and Θ rest β (y) = Θ β (y) − Θ hom β (y). The first equality in (3.18) follows from It is evident thatΘ rest for every g ∈ S(R N ) such that 0 / ∈ supp g, where Δ is the Laplacian on R N . Due to the fact that Θ β is smooth everywhere and homogeneous of degree −1 outside some neighborhood of zero it holds Because we can choose m at will the functionΘ β (k) is smooth outside the origin and vanishes at infinity faster than any power of |k|. Since Θ hom β is homogeneous of degree −1,Θ hom β is homogeneous of degree −N + 1.
where Θ is an arbitrary UV regular splitting function.
Proof. Suppose that 0 / ∈ suppt. It follows from definition (3.16) of the splitting of a distribution, the fact thatΘ is smooth outside the origin and the assumed form (3.17) of the distribution t that in some neighborhood of the origin t Θ (q, q ) is a smooth function of q and q . This concludes the proof in this case. Thus, because of the assumptiont(q, q ) = O dist (|q| d−ε ) it is enough to consider distributionst which are globally of the form for any g ∈ S(R N ). Using Lemma 3.7 we obtain where the integrand on the RHS of the above equation is absolutely integrable. The theorem follows from Lemma 3.9 with D = d + |α| ≥ 1.

For any UV regular splitting function Θ and any
Proof. We have is the Taylor approximation of degree D of the function q →Θ β (q − k). We will show below that expressions (3.20) and (3.21) are well defined and have the required properties. Using Lemma 3.9 we get The second bound is a consequence of the Taylor theorem. Because of the assumptions about t the functions c γ given by (3.20) are well defined and continuous. Now let us turn to the function t defined by (3.21). Upon introducing new integration variables u = k/|q|, we find 904 P. Duch Ann. Henri Poincaré (3.23) Since |λq − |q|u| ≥ 1 2 |q||u| for |u| ≥ 2 and λ ∈ [0, 1], it follows from (3.22) and |t(q, q )| ≤ const |q| D−ε that for |u| ≥ 2 the integrand above is bounded by const |u| −N −ε . For |u| < 2 we use the inequalities: Consequently, the integral in (3.23) exists and is a bounded function of q and q . The continuity of t is evident because the LHS of (3.19) is a continuous function of q and q and c γ ∈ C(R M ). . . . , x n , 0). If n = 0 and d ≤ 0, then by definition the distribution t is an arbitrary constant. If n = 0 and d > 0, then t = 0. Observe that if t ∈ S (R 4(n+1) ) vanishes in some neighborhood of 0 in momentum space, then the IR-index of t is arbitrary large. If t has IR-index d ∈ Z, then it also has IR-index d ∈ Z for any d ≤ d.

Product Definition 3.10 (IR-index). The translationally invariant distribution
The following lemma is an immediate consequence of Part (C) of Theorem 3.5. Lemma 3.11. Let t ∈ S (R 4(n+1) ) be translationally invariant distribution which has IR-index d. Then the distributiont has zero of order d at the origin in the sense of Lojasiewicz.

Definition 3.12 (IR-index). The distribution
has IR-index d ∈ Z, with respect to the variables x 1 , . . . , x n iff for any ε > 0 and every f ∈ S(R 4m ) it holds The LHS of the above equation is the Fourier transform of the distribution . . . , x m ) t(x 1 , . . . , x n ; x 1 , . . . , x m ).
By definition, if n = 0 and d ≤ 0, then the above condition is equivalent to saying that the distribution t is a continuous function. If n = 0 and d > 0, Vol. 19 (2018) Weak Adiabatic Limit in QFTs with Massless Particles 905 then t = 0. Observe that if t ∈ S (R 4n × R 4m ) vanishes in some neighborhood of 0 × R 4m in momentum space, then the IR-index of t with respect to the first n variables is arbitrarily large. If t has IR-index d ∈ Z, then it also has IR-index d ∈ Z for any d ≤ d.
Proof. The variables q 1 , . . . , q n will be denoted collectively by q, the variables q 1 , . . . , q k by q and similarly for the primed variables. By Definition 3.3 the distributions t and t admit (in some neighborhood of the origin) the following representations: where the functions t α and t α are continuous and such that |t α (q)| ≤ const |q| |α|+δ and |t α (q )| ≤ const |q | |α |+δ . (3.30) The distribution t defined by (3.29) is expressed in some neighborhood of 0 by the following sum and s is a continuous function given in Lemma 3.14. To prove the above statement we observe that as a result of the support properties of the function s and of the measure dμ 0 the integration region in (3.32) may be restricted to |k 1 |, . . . , |k l | ≤ |Q|. Thus, it follows from the linearity of maps (3.28) that (3.31) holds for Q, q, q in sufficiently small neighborhood of 0 in R 4n+4n +4 . The continuity of the functions t αα is evident. Thus, to prove the statement of the lemma it is enough to show that |t αα (Q, q, q )| ≤ const |Q, q, q | |α|+δ+|α |+δ +2l+|β|+2 .
Finally, we arrive at the main result of this section.  Proof. We will prove first the statements about product (3.35). The proof is based on formula (2.6). We will show that the statements (1')-(3') hold independently for each term of the sum on the RHS of this formula. First note that as a result of Definitions 3.10 and 3.12 of the IR-and IR-index and Part (A) of Lemma 2.3 this is true for all the terms for which at least one of the super- quadri-indices s 1 , . . . , s n , s 1 , . . . , s n involves a massive field. The statements in the cases (1')-(3') follow now directly from Theorem 3.13 and constraints (2.11).
In order to prove the claim about the graded commutator (3.36) we use Part (B) of Lemma 2.3 and the claim about product (3.35) which has just been proved.

Splitting
Theorem 3.17. Assume that the distribution Then there exist constants c α ∈ C for multi-indices α, |α| < d such that the distribution for any ε ∈ (0, 1) and some constants c γ ∈ C, where γ is a multi-index. For the last term it is an immediate consequence of Theorem 3.8 with M = 0 and Definition 3.10 of the IR-index. In the case of the Fourier transforms of the first and the second terms we prove, along the lines of Lemma 3.2, that they are smooth functions and apply the Taylor theorem. has IR-index min{d, 0} with respect to y 1 , . . . , y n . (B) If d = 1, then for every f ∈ S(R 4m ) and ε > 0 there exists c ∈ C such  for every ε > 0 and f ∈ S(R 4m ). We will prove that if t(y 1 , . . . , y n ; x 1 , . . . , x m ) is any of the three terms on the RHS of Eq. (3.37) then there exist c γ ∈ C(R 4m ) for multi-indices γ, |γ| < d such that For the first and the second terms this follows from Lemma 3.2 from Sect. 3.1 and the Taylor theorem. For the last term it is an immediate consequence of Theorem 3.8. Thus, Eq.  q 1 ), . . . ,B n (q n );C 1 (p 1 ), . . . ,C m (p m ); P )Ω) To show Part (B) we note that the last term on the RHS of the above equation is a constant for d = 1 and use (2.27) and (3.38).

Proof for Theories with Massless Particles
In this section we prove the existence of the weak adiabatic limit in theories containing massless particles. The idea of the proof was described in Sect. 3.2.
Our proof is valid only if the time-ordered products of the sub-polynomials of the interaction vertices satisfy certain condition which is formulated below as the normalization condition N.wAL. In Theorem 3.19, we prove that this condition may be imposed in all models satisfying Assumption 3.1 stated below and is equivalent to the normalization condition N.wAL'. The latter condition is used in Theorem 3.20, which states the existence of the weak adiabatic limit. The proof of the compatibility of N.wAL with other standard normalization conditions usually imposed on the time-ordered products, like, for example, the unitarity or Poincaré covariance, is postponed to Sect. 4.3.
Our proof of the existence of the Wightman and Green functions applies to the following class of models. Assumption 3.1. We assume that all interaction vertices L 1 , . . . , L q of the models under consideration have even fermion number and satisfy one of the following conditions: (1) ∀ l∈{1,...,q} dim(L l ) = 4, (2) ∀ l∈{1,...,q} dim(L l ) = 3 and L l contains at least one massive field. 3 Moreover, we assume throughout that axiom A.7 holds with c = 0 in the case (1) and c = 1 in the case (2), i.e., c = 4 − dim(L l ).
Other conditions which have to be fulfilled by the interaction vertices are listed in Sect. 2.7. They ensure the correct physical properties of the model, but do not play any role in the proof of the existence of the weak adiabatic limit presented in this section. We will, however, use them afterward to prove that the Wightman and Green functions have all the standard properties (cf. Sect. 4.4). Note that in the proof of the existence of the Wightman and Green functions in purely massive theories we only use the fact that the interaction vertices have even fermion number.
Let us introduce some notations which will be used throughout this section. For any list of super-quadri-indices u = (u 1 , . . . , u k ) we set (3.40)

P. Duch Ann. Henri Poincaré
The functions e u (·), d u (·) are given by (2.10), and p is the number of basic generators. Observe that since c = 4 − dim(L l ), by axiom A.7 we get for any B 1 , . . . , B k ∈ F hom . Note that ω defined above coincides for k = n + 1 with ω given by (2.16). By F L we denote the subspace of F spanned by L (u) l with l ∈ {1, . . . , q} and u being a super-quadri-index involving only massless fields. Let us recall that according to our definition the time-ordered products are any F-products which satisfy the axioms formulated in Sect. 2.4. The freedom in the definition of the time-ordered products is characterized in Sect. 2.5. Our proof of the existence of the weak adiabatic limit is applicable only if the time-ordered products are defined in such a way that they fulfill the following normalization condition.
Let us consider also the following normalization condition.

N.wAL'
The distribution )Ω) has IR-index equal to ω given by (3.40) for all F-products F which are finite linear combinations of products of the time-ordered products, all k ∈ N + , l 1 , . . . , l k ∈ {1, . . . , q} and all list of super-quadri-indices u = (u 1 , . . . , u k ) such that each u j involves only massless fields (i.e., e u (A i ) = 0 unless A i is a massless field). Proof. We will prove Part (A) of the theorem by induction with respect to the number of arguments of the time-ordered products. First, we have to consider the distributions of the form (Ω|L (u) l (x)Ω), (3.44) where the super-quadri-index u involves only massless fields and l ∈ {1, . . . , q}.
By assumption L l contains at least one massive field or dim(L l ) = 4. First note that (3.44) is always a constant, which is nonzero iff L (u) l contains a nonzero constant term in the decomposition into monomials. The latter condition is never satisfied if L l contains a massive field. Thus, (3.44) is zero in this case. If dim(L l ) = 4 then L (u) l may contain a nonzero constant term in the decomposition into monomials only if dim(A u ) = 4 which implies that ω = 4 − dim(A u ) = 0 (set k = 1 in Eq. (3.40)). As a result, the normalization condition N.wAL' is satisfied in both cases.
We proceed to the proof of the inductive step. Fix n ∈ N + and assume that N.wAL' holds for all k ≤ n. (It is enough to assume that the normalization condition N.wAL' holds only for F being the time-ordered product.) We will demonstrate that it is possible to define the time-ordered products such that N.wAL' is satisfied for k = n + 1. Using the inductive hypothesis and the statement (1') of Theorem 3.16 we obtain that for all m ∈ N + , super-quadriindices u 1 , . . . , u m which involve only massless fields and F-products F which may be expressed as a linear combinations of products of the time-ordered products with at most n arguments, the distribution (Ω|F (L With the use of the above results, Theorem 3.17 and relation (2.21) between the products A and A we get that for all B 1 , . . . , B n+1 ∈ F L there exist constants c γ ∈ C indexed by multi-indices γ, |γ| < ω such that has IR-index ω given by (3.42) Using the normalization freedom described in Sect. 2.5 we will show that it is possible to redefine the time-ordered products such that the condition N.wAL' is fulfilled. To this end, we modify the definition of the VEV of time-ordered product with n + 1 arguments (2.13) by adding to it the graded-symmetric map where f (π) ∈ Z/2Z is the number of transpositions in π that involve a pair of fields with odd fermion number. Since the graded-symmetric part of (3.46) has IR-index ω given by (3.42) with k = n + 1 the same holds for the VEV of the redefined time-ordered product. This proves the condition N.wAL' for F = T. The generalization for arbitrary F considered in N.wAL' follows from the statement about distribution (3.45) made above. This ends the proof of Part (A).
To prove Part (B), we first observe that by Lemma 3.11 the condition N.wAL' with F = T implies the condition N.wAL. The reverse implication is also true since after imposing the condition N.wAL the freedom in defining the time-ordered products with n + 1 arguments given the time-ordered products with n arguments is the same as after imposing N.wAL': Two possible definitions of (2.13) differ by map (2.14) which satisfies the conditions stated in Sect. 2.5 and is such that for all B 1 , . . . , B n+1 ∈ F L the distribution v(B 1 (x 1 ), . . . , B n+1 (x n+1 )) is of form (2.15), where the sum over γ is restricted to |γ| = ω . Now we formulate our main theorem according to which the weak adiabatic limit exists in models satisfying Assumption 3.1.
This, by Part (B) of Theorem 3.5, implies the existence of the weak adiabatic limit (3.1).
Proof. We will prove the theorem by induction with respect to k. For k = 0 by Eq. (2.28) we have A(∅; J; P ) = R(∅; J; P ) = T(J; P ). It follows that for any f ∈ S(R 4m ) is a smooth function of q 1 , . . . , q m . This implies Part (A) for k = 0 as a result of Definition 3.12 of the IR-index. (Note that d ≤ 0 since at least one superquadri-index u 1 , . . . , u m is nonzero.) Part (B) is trivially true in this case. Now, let us assume that Part (A) holds for k ≤ n − 1, n ∈ N + . We shall prove that both Part (A) and (B) hold for k = n. We will first show that for k = n and any list u = (u 1 , . . . , u k+m ) of super-quadri-indices such that every u j involves only massless fields, including the case when all u j vanish, the distribution has IR-index d given by (3.48)

Compatibility of Normalization Conditions
In this section we show the compatibility of the condition N.wAL with the standard normalization conditions usually imposed on the time-ordered products. In the first subsection we show that in purely massless models the condition N.wAL is implied by almost homogeneous scaling of VEVs of timeordered products. In Sect. 4.2 we formulate the central normalization condition N.C which is stronger than N.wAL and, in particular, fixes uniquely all time-ordered products of the sub-polynomials of the interaction vertex in the

Almost Homogeneous Scaling
In this section we assume that all fields under consideration are massless and set c = 0 in axiom A.7. Following [34,35] we introduce the definition of the almost homogeneous scaling of a distribution.
and P is the minimal natural number with this property. If P = 0 the above condition states that the distribution t is homogeneous of degree −D.
Thus, t scales almost homogeneously with degree D and power P iff ρ D t(ρx) is a polynomial of log ρ with degree P . This implies that in particular sd(t) = D, where sd(·) is the Steinmann scaling degree (cf. Definition 2.4). Moreover, Hence, it holds sd(t) = N − D.
If axiom A.7 holds with c = 0 then the time-ordered products of polynomials of massless fields can be normalized such that the following condition holds.
N.aHS Almost homogeneous scaling: For all polynomials B 1 , . . . , B k ∈ F hom of massless fields and their derivatives the distribution scales almost homogeneously with degree The above normalization condition is a special case of the condition called scaling in [18], which was imposed also for the time-ordered products of massive fields as a substitute of A.7. It is clear that in the case of purely massless models N.aHS is stronger than A.7 since for any t ∈ S (R N ) which scales almost homogeneously with degree D it holds sd(t) = D. The condition similar to N.aHS was introduced for the first time in [34,35] in the context of QFT in curved spacetime. For massless fields this method of normalization was used, for example, in [28,29,37].
follows from the normalization condition N.aHS. As a result, we obtain where ω is given by (3.42) and t ∈ S (R 4(k−1) ) is such that Thus, for any g ∈ S(R 4(k−1) ) and multi-index γ, |γ| < ω it holds where g is defined in terms g as in Definition 3.4 of the value of a distribution at a point in the sense of Lojasiewicz. This implies the normalization condition N.wAL.

Central Normalization Condition
In this section we formulate the normalization condition for the time-ordered products which is a generalization of a condition introduced by Epstein and Glaser in [22] for purely massive models. Let us describe the content of the latter condition. It involves the VEVs of the advanced products of the form (Ω| A(B 1 (x 1 ), . . . , B n (x n ); B n+1 (0))Ω) and says that their Fourier transforms (which in the case of purely massive models is an analytic function in some neighborhood of the origin) have zero of order ω + 1 at the origin (the central and the most symmetric point), where ω is given by (2.16). The advanced products which fulfill the above condition are known in the literature as the central or symmetric solutions of the splitting problem, or sometimes the central or symmetric extensions. 4 This condition fixes uniquely all the time-ordered products and is compatible with the standard normalization conditions listed in the next section. So far, the above normalization condition has been formulated rigorously only for theories without massless fields [22]. However, an expectation that a condition of a similar type can also be imposed in the massive spinor QED has been expressed by some authors [14,19,36,43].
Using the method of the proof of Theorem 3.19 one can show [13] that it is possible to define the time-ordered products such that the following condition 918 P. Duch Ann. Henri Poincaré holds. It may be viewed as a generalization of the condition which was discussed above. N.C Let k ∈ N + and A r1 , . . . , A r k ∈ F be arbitrary monomials. Consider the following distribution (1) Distribution (4.2) has IR-index (2) If all of the super-quadri-indices r 1 , . . . , r k involve at least one massive field, then distribution (4.2) has IR-index (3) If all but one of the super-quadri-indices r 1 , . . . , r k involve at least one massive field, then distribution (4.2) has IR-index The above condition significantly restricts the normalization freedom of the time-ordered products. Its usefulness lies in the fact that it respects many symmetries, e.g., the Poincaré symmetry (cf. Definition 3.10 of the IR-index). Part (2) of N.C fixes uniquely the time-ordered products to which it refers.
Parts (1) and (2) of N.C imply the normalization condition N.wAL. In the case of monomials which are products of massless fields the condition N.C is equivalent to the normalization condition N.aHS. Let us consider the application of the condition N.C in QED. We recall that the interaction vertex in QED is L = ψ / Aψ, where ψ is a massive Dirac spinor field and A μ is a massless vector field. Since dim(L) = 4 we set c = 0 in axiom A.7. The time-ordered products satisfying the condition N.C have the following property. N.C QED The distribution has IR-index for any n ∈ N + and any sub-polynomials B 1 , . . . , B n of the interaction vertex L of QED such that d ≥ 0. By Lemma 3.11 the above normalization condition implies that the Fourier transform of distribution (4.3) is of the form (2π) 4 δ(q 1 + · · · + q n+1 ) t(q 1 , . . . , q n ), where t(q 1 , . . . , q n ) has zero of order d at q 1 = · · · = q n = 0 in the sense of Lojasiewicz. The Fourier transform of the VEV of the corresponding advanced product has the same form. Consequently, the advanced product defined in terms of the time-ordered products satisfying N.C QED may be interpreted as the central splitting solution. It follows from the results of Sect. 2.5 that N.C QED fixes uniquely all the time-ordered products of the sub-polynomials of the interaction vertex L. As we will show in the next section this condition implies all the standard normalization conditions: N.U, N.P, N.CPT, N.FE and N.W for the timeordered products of the sub-polynomials of the interaction vertex L of QED.

Standard Normalization Conditions
In this section we list the standard normalization conditions which are usually imposed on the time-ordered products and argue that they are compatible with the normalization condition N.C.
N.U Unitarity: N.FE Let B 1 , . . . , B k ∈ F hom be sub-polynomials of the interaction vertex, and let A i ∈ G 0 be a basic generator. It holds: where the second sum is over generators C ∈ G. This condition says that the time-ordered products with a basic generator among its arguments are uniquely determined by the time-ordered products with less arguments. It also implies that the interacting fields satisfy the field equations.
where q(B) is the charge number (cf. Sect. 2.1) and j μ = ψγ μ ψ is the free electric current.
For the proof that N.U, N.P, N.CPT, N.FE, N.W (the last condition holds only in QED) can be simultaneously imposed, see, for example, [3,43]. The compatibility of N.U, N.P, N.CPT with N.C follows from the invariance of N.C under Poincaré and CPT transformations and the fact that the condition N.C implies the analogous condition for the VEVs of the anti-time-ordered products.
The compatibility of N.FE and N.C is less straightforward. We will outline its proof in the case of QED. Assume that the time-ordered products with at most n arguments satisfy N.FE and N.C. Define the time-ordered products with n + 1 arguments such that N.C holds. We have where D F 0 (x) is the massless Feynman propagator, c γ ∈ C are some constants, and Equation (4.4) implies that Assume that B 1 , . . . , B n are sub-polynomials of the interaction vertex and d ≥ 0. The condition N.C implies that the distribution in the last line of the above equation has IR-index d + 3. Since the distribution on the LHS of (4.4) has IR-index d + 1, the distribution on the LHS of (4.5) has IR-index d + 3. This implies that c γ = 0 and N.FE holds if the basic generator A i is the vector potential A μ . To prove that N.FE is satisfied when A i is the spinor field ψ a or ψ a it is enough to use the fact that the Feynman propagator of a massive particle is smooth in the vicinity of zero. Now let us suppose that the time-ordered products with at most n arguments satisfy N.W and N.C. Define the time-ordered products with n + 1 arguments such that N.C holds. We will show that N.W is satisfied for k = n. By the results of Appendix B of [16] it is enough to prove that vanishes if B 1 , . . . , B n are sub-polynomials of the interaction vertex of QED. We also know that the above distribution is of the form γ |γ|≤d+1 If d +1 ≥ 0, then the constants c γ must be zero since by N.C distribution (4.6) has IR-index d + 2.

Properties of Wightman and Green Functions
If the time-ordered products satisfy the normalization conditions formulated in Sect. 4.3, then the Wightman and Green functions have a number of important properties. In fact, the Wightman functions fulfill almost all the standard axioms, which are listed in [47], in the sense of formal power series in the coupling constants. It is only not clear whether the cluster decomposition property holds. Let C 1 , . . . , C m ∈ F be arbitrary polynomials. The following conditions are satisfied.
). The definition of the adjoint in F is given in Sect. 2.1. In the proof of the above property we use the identity C adv (g; h) * = C * adv (ḡ;h), which follows from the normalization condition N.U. (4) Local (anti-)commutativity: For C k , C k+1 ∈ F hom it holds if x k and x k+1 are spatially separated, i.e., (x k − x k+1 ) 2 ≤ 0. (5) Positive definiteness condition: Let j 0 ∈ N + , {1, . . . , j 0 } j → f j ∈ S(R 4nj ) and C j,1 , . . . , C j,nj ∈ F, where n j ∈ N + . Then the formal power series j,k∈N0 . . , C k,n k (y n k )) is nonnegative in the sense of Definition 5.2 stated in Sect. 5.2. The above condition is satisfied in models which can be defined on the Fock space with a positive-definite inner product. It follows from Theorem 5.4 formulated in Sect. 5.2. Similar equations are satisfied in other models. This condition is a direct consequence of the normalization condition N.FE. Note that it is not included in the list of the standard axioms [47]. Now let us state the properties of the Green functions.

Vacuum State
In this section we consider the framework of perturbative algebraic quantum field theory-the formulation of perturbative QFT in terms of local abstract algebras [26,42]. In the first subsection we introduce the notation and remind the reader the construction of the net of local abstract algebras of interacting fields [10], known in the literature under the name of the algebraic adiabatic limit. In Sect. 5.2 we define a Poincaré-invariant functional on the algebra of interacting fields. In the case of models which are constructed on the Fock space with a positive-definite covariant inner product this functional is positive. As a result, it is a state which can be interpreted as an interacting vacuum state.

Algebraic Adiabatic Limit
The construction of the local abstract algebras of interacting operators is based on the following observation made in [10]. For simplicity, we assume that there is only one interaction vertex L. The switching function is denoted by g, the coupling constant by e, and the interacting operators by C adv (g; h) = The action of the Poincaré transformations on F is given by the automorphisms defined in the following way on the generators of F α a,Λ (C adv (·; h)) = (ρ(Λ −1 )C) adv (·; h a,Λ ), where h a,Λ (x) = h(Λ −1 (x − a)) and ρ is the representation of the Lorentz group acting on F introduced in Sect. 2.1. As shown in [17,26] the net O → F(O) constructed above fulfills the axioms: (1) isotony, (2) Poincaré covariance and (3) Einstein causality (see [31] for their definition) in the sense of formal power series. The construction of the algebra F and the net O → F(O) outlined above is called in the literature the algebraic adiabatic limit.
In the case of theories with local symmetries the algebra F plays only an auxiliary role. From physical point of view more important is the algebra of observables, which is a certain quotient algebra consisting of equivalence classes of gauge-invariant fields. In the case of QED the construction of the algebra of observables was carried out by Dütsch and Fredenhagen in [16] and in the case of non-abelian Yang-Mills theories without matter by Hollands in [33].

Poincaré-Invariant State
Elements of F will be denoted by B(·). They are linear combinations of products of C adv (·; h) for some C ∈ F and h ∈ D(R 4 ).  It turns out that the existence of the weak adiabatic limit allows to define a real, normalized and Poincaré-invariant functional on the algebra of interacting fields F. exists. This, in particular, implies that the limit on the RHS of Eq. (5.2) exists and is independent of g. As a result, the functional σ is well defined. It is evident that σ is normalized. Hermiticity of the Wightman functions implies that the functional σ is real. Poincaré invariance of σ, σ( α a,Λ ( B(·) ) ) = σ( B(·) ) for all B ∈ F, is a consequence of Poincaré covariance of the Wightman functions. The above lemma follows immediately from the following fact. In the case of models with vector fields such as QED the covariant inner product on D 0 is indefinite and the functional σ given by (5.2) is not positive. Nevertheless, it is expected that it can be used to define a Poincaré-invariant state on the algebra of observables.

Summary and Outlook
We proved the existence of the weak adiabatic limit in the Epstein-Glaser approach to perturbative QFT in a large class of models. The result implies the existence of the Wightman and Green functions, which have most of the standard properties following from the Wightman axioms. It can also be used to construct a real, normalized and Poincaré-invariant functional on the algebra of interacting fields obtained by means of the algebraic adiabatic limit. In the case of models which are defined on the Fock space with a positive-definite covariant inner product we proved that this functional is positive in the sense of formal power series. Consequently, it is a state which we interpret as an interacting vacuum state.
In the construction of models containing vector fields one uses a covariant inner product which is not positive definite [44]. Consequently, in such models the above-mentioned functional is indefinite and in particular is not a state. However, in the case of models with vector fields besides the algebra of interacting fields one considers also the algebra of observables. In fact, only the latter algebra has a direct physical interpretation. It is defined as a quotient algebra consisting of equivalence classes of gauge-invariant fields. The algebras of observables in quantum electrodynamics and non-abelian Yang-Mills theories without matter fields were constructed in [16,33], respectively. Moreover, it was shown that each of the sub-algebras of these algebras consisting of observables localized in a bounded open subset of the Minkowski spacetime can be represented on a pre-Hilbert space with a positive-definite inner product. (In fact, in [33] this was proved even in the case of curved spacetime.) The definition of a state on the global algebra of observables, in particular the definition of a Poincaré-invariant state, is an open problem. An interesting generalization of our results would be a construction of a Poincaré-invariant functional on the algebra of observables in the case of quantum electrodynamics and non-abelian Yang-Mills theories without matter fields, and a proof that resulting functional is positive.
The techniques developed in the proof of the existence of the weak adiabatic limit were also used to establish that one can define the time-ordered products in such a way that they satisfy the central normalization condition. This result implies, in particular, the existence of the central splitting solution in quantum electrodynamics. The time-ordered products of sub-polynomials of the interaction vertex of quantum electrodynamics normalized in this way are fixed uniquely and satisfy the standard normalization conditions, including the Ward identities.