Conservation of the stress tensor in perturbative interacting quantum field theory in curved spacetimes Stefan Hollands and Robert M. Wald ∗ † Enrico Fermi Institute and Department of Physics 5 0 University of Chicago 0 5640 S. Ellis Avenue, Chicago, IL 60637, USA 2 n a February 7, 2008 J 8 1 2 v Contents 4 7 0 1 Introduction 3 4 0 4 2 The nature and properties of time-ordered products 6 0 2.1 The construction of the free quantum field algebra and the natureof time-ordered products / c 2.2 Properties of time ordered products: Axioms T1-T9 . . . . . . . . . . . . . 13 q - r g 3 The Leibniz rule 18 v: 3.1 Formulation of the Leibniz rule, T10, and proof of consistency withaxioms T1-T9 18 i 3.2 Anomalies with respect to the equations of motion . . . . . . . . . . . . . . 26 X r a 4 Quadratic interaction Lagrangians and retarded response 30 4.1 Formulation of the general condition T11 . . . . . . . . . . . . . . . . . . . 30 4.2 External source variation: Axiom T11a . . . . . . . . . . . . . . . . . . . . 34 4.3 Metric variation: Axiom T11b . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.4 External potential variation . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5 Some key consequences of our new requirements 40 5.1 Consequences for the free field . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.2 Consequences for interacting fields . . . . . . . . . . . . . . . . . . . . . . . 46 ∗[email protected] †[email protected] 1 6 Proof that there exists a prescription for time-ordered productssatisfying T11a and 6.1 Proof that T11a can be satisfied . . . . . . . . . . . . . . . . . . . . . . . . 54 6.2 Proof that T11b can be satisfied when D > 2 . . . . . . . . . . . . . . . . 55 6.2.1 Proof that D is supported on the total diagonal . . . . . . . . . . 58 n 6.2.2 Proof that D is a c-number . . . . . . . . . . . . . . . . . . . . . . 59 n 6.2.3 Proof that D is local and covariant and scales almosthomogeneously 63 n 6.2.4 Proof that D = 0 when one of the Φ is equal to ϕ . . . . . . . . . 63 n i 6.2.5 Proof that D satisfies a wave front set condition and dependssmoothly and analytically n 6.2.6 Proof that D is symmetric when Φ = T . . . . . . . . . . . . . . 74 n 1 ab 6.2.7 Proof that D can be absorbed in a redefinition of thetime-ordered products 77 n 7 Outlook 79 A Infinitesimal retarded variations 82 B Functional derivatives 84 Abstract We propose additional conditions (beyond those considered in our previous pa- pers) that should be imposed on Wick products and time-ordered products of a free quantum scalar field in curved spacetime. These conditions arise from a sim- ple “Principle of Perturbative Agreement”: For interaction Lagrangians L that are 1 suchthattheinteractingfieldtheorycanbeconstructedexactly—as occurswhenL 1 is a “pure divergence” or when L is at most quadratic in the field and contains no 1 more than two derivatives—then time-ordered products mustbedefined so that the perturbative solution for interacting fields obtained from the Bogoliubov formula agrees with the exact solution. The conditions derived from this principle include a version of the Leibniz rule (or “action Ward identity”) and a condition on time- ordered products that contain a factor of the free field ϕ or the free stress-energy tensor T . The main results of our paper are (1) a proof that in spacetime dimen- ab sions greater than 2, our new conditions can be consistently imposed in addition to our previously considered conditions and (2) a proof that, if they are imposed, then for any polynomial interaction Lagrangian L (with no restriction on the number of 1 derivatives appearing in L ), the stress-energy tensor Θ of the interacting theory 1 ab will be conserved. Our work thereby establishes (in the context of perturbation theory) the conservation of stress-energy for an arbitrary interacting scalar field in curved spacetimes of dimension greater than 2. Our approach requires us to view time-ordered products as maps taking classical field expressions into the quantum fieldalgebra rather thanas mapstaking Wick polynomials of thequantum fieldinto the quantum field algebra. 2 1 Introduction In [13] and [14], we took an axiomatic approach toward defining Wick powers and time- ordered products of a quantum scalar field, ϕ, in curved spacetime. We provided a list of axioms that these quantities are required to satisfy (see conditions T1-T9 of [14] or section 2 below) and then succeeded in proving both their uniqueness (up to specified renormalization ambiguities) [13] and their existence [13, 14]. Our previous analysis restricted attention to the case where the Wick powers and the factors appearing in the time-ordered products do not contain derivatives of the scalar field ϕ. In fact, however, as we already noted in [13, 14], our uniqueness and existence results extend straightforwardly to the case where the Wick powers and the factors ap- pearing in the time-ordered products are arbitrary polynomial expressions in ϕ and its derivatives1. We excluded the explicit consideration of expressions containing derivatives partly for simplicity but also because it was clear to us that additional axioms should be imposed on these quantities—and, consequently, stronger uniqueness and existence theo- rems should be proven—but it was not clear to us precisely what form these additional axioms should take. The main purpose of this paper is to provide these additional axioms, to investigate some of their consequences—most notably, conservation of the stress-energy of theinteracting field—and toprove the desired stronger existence anduniqueness results for our new strengthened set of axioms. Some simple examples should serve to illustrate the issues involved in determining what additional conditions should be imposed. One obvious possible requirement is the “Leibniz rule”. Consider, for example, the Wick monomials ϕ2 and ϕ ϕ in D = 4 a ∇ spacetimedimensions. Theuniqueness theoremof[13]appliestobothoftheseexpressions. It establishes that the first is unique up to the addition of c R11, where c is an arbitrary 1 1 constant and R denotes the scalar curvature. Similarly, the second is unique up to the addition of c R11, where c is an independent arbitrary constant. However, it would 2 a 2 ∇ be natural to require that ϕ2 = 2ϕ ϕ (1) a a ∇ ∇ where the left side denotes the distributional derivative of ϕ2. If we wished to impose eq. (1), then we would need to strengthen our previous existence theorem to show that eq. (1) can be imposed in addition to our previous axioms. (This is easily done.) Our above uniqueness result would then be strengthened in that we would have c = 2c , i.e, 1 2 c and c would no longer be independent. Note that the Leibniz rule eq. (1) has an 1 2 obvious generalization to arbitrary Wick polynomials, but it is not so obvious, a priori, what form the Leibniz rule should take on factors occurring in time-ordered products. A second “obvious” requirement that one might attempt to impose on Wick polyno- mials and time-ordered products is that they respect the equations of motion of the free 1AxiomT9wasexplicitlystatedin[14]onlyforthecaseofexpressionsthatdonotcontainderivatives. Its generalization to expressions with derivatives is given in section 2 below. 3 field ϕ. Consider the case of a massless Klein-Gordon field, so that a ϕ = 0. Then it a ∇ ∇ would seem natural to require the vanishing of any Wick monomial containing a factor of a ϕ—such as the Wick monomials ϕ a ϕ and ( ϕ)( a ϕ). Similarly, it would a a b a ∇ ∇ ∇ ∇ ∇ ∇ ∇ be natural to require the vanishing of any time-ordered product with the property that any of its arguments contains a factor of this form. However, it turns out that—as we will explicitly prove in section 3 below—it is not possible to impose this “wave equation” requirement together with the Leibniz rule requirement of the previous paragraph. Should one impose the Leibniz rule or the free equations of motion (or neither of them) on Wick polynomials or time-ordered products? If the Leibniz rule is imposed, what form should it take for time-ordered products? Should any conditions be imposed in addition to the Leibniz rule or, alternatively, to the free equations of motion? In this paper, we will take the view that these andother similar questions should notbe answered by attempting to make aesthetic arguments concerning properties of Wick polynomials and time-ordered products for the free field theory defined by the free Lagrangian L . 0 Rather, we will consider the properties of the interacting quantum field theory defined by adding to L an interaction Lagrangian density L , which may contain an arbitrary (but 0 1 finite) number of powers of ϕ and its derivatives. As discussed in detail e.g., in section 3 of [15] (see also subsection 4.1 below), an arbitrary interacting quantum field Φ (with Φ L1 denoting an arbitrary polynomial in ϕ and its derivatives) is defined perturbatively by the Bogoliubov formula, which expresses Φ in terms of the free-field time-ordered products L1 with factors composed of Φ and L . The main basic idea of this paper is to invoke 1 the following simple principle, which we will refer to as the “Principle of Perturbative Agreement”: If the interaction Lagrangian L is such that the quantum field theory defined 1 by the full Lagrangian L +L can be solved exactly, then the perturbative construction of 0 1 the quantum field theory must agree with the exact construction. There are two separate cases in which this principle yields nontrivial conditions. The first is where the interaction Lagrangian corresponds to a pure “boundary term”, i.e., in differential forms notation, the interaction Lagrangian is of the form dB, where B is a smooth (D 1)-formof compact support depending polynomially onϕ and its derivatives. − Such an “interaction” produces an identically vanishing contribution to the action, and the interacting quantum field theory is therefore identical to the free theory. As we shall show in subsection 3.1, the imposition of the requirement that all perturbative corrections vanish for any interaction Lagrangian of the form dB precisely yields the Leibniz rule for Wick polynomials and yields a generalization of the Leibniz rule for time-ordered products. This generalization states that, in effect, derivatives can be freely commuted throughthe“timeordering”. Wewillrefertothisconditionasthegeneralized Leibnizrule and will label it as “T10”. Our condition T10 corresponds to the “action Ward identity” proposed in [18, 9] and proven recently in the context of flat spacetime theories in [10]. In order for condition T10 to be mathematically consistent, it is necessary that we adopt the viewpoint of [2] and [8]—which we already adopted in [15] for other reasons—that time-ordered products are maps from classical field expressions (on which the classical 4 equations of motion are not imposed) into the quantum algebra of observables. This viewpoint and the reasons that necessitate its adoption are explained in detail in section 2. A proof that condition T10 can be consistently imposed in addition to conditions T1-T9 is given in subsection 3.1. The second case where the above principle yields nontrivial conditions is where the interaction Lagrangian is at most quadratic in the field and contains a total of at most two derivatives. This includes interaction Lagrangians consisting of terms of the form Jϕ, Vϕ2, and hab ϕ ϕ, corresponding to the presence of an external classical source, a b ∇ ∇ a spacetime variation of the mass, and a variation of the spacetime metric. In all of these cases, the exact quantum field algebra of the theory with Lagrangian L + L can be 0 1 constructed directly, in a manner similar to the theory with Lagrangian L . Our demand 0 that perturbation theory reproduce this construction yields new, nontrivial conditions on time-ordered products (which are most conveniently formulated in terms of retarded products). The general form of this requirement, which we label as “T11”, is formulated insubsection 4.1. Ausefulinfinitesimal version of thisconditionforthecaseofanexternal current interaction—which we label as condition T11a—is derived in subsection 4.2, and a corresponding infinitesimal version for the case of a metric variation—which we label as condition T11b—is derived in subsection 4.3. The consequences of our additional conditions are investigated in section 5. The main results proven there—which also constitute some of the main results of this paper—are that our conditions imply the following: (i) The free stress-energy tensor, T , in the ab free quantum theory must be conserved. (ii) For an arbitrary polynomial interaction Lagrangian, L , (a) the interacting quantum field ϕ always satisfies the interacting 1 L1 equations of motion and (b) the interacting stress-energy tensor, Θab, of the interacting L1 theory always is conserved. This is rather remarkable in that, a priori, one might have expected properties (i) and (ii) to be entirely independent of conditions T1-T11. Indeed, one might have expected that if one required that (i) and (ii) be satisfied in perturba- tion theory, one would obtain a further set of requirements on Wick polynomials and time-ordered products. The fact that no additional conditions are actually needed pro- vides confirmation that T10 and T11 are the appropriate conditions that are needed to supplement our original conditions T1-T9. In effect, the analysis of section 5 shows the following: Suppose that the definition of time-ordered products satisfies T1-T10. Then, if the definition of time-ordered products is further adjusted, if necessary, so that in per- turbation theory the quantum field satisfies the correct field equation in the presence of an arbitrary classical current source J (as required by T11a), then the interacting field also will satisfy the correct field equation for an arbitrary self-interaction. Furthermore, if, in perturbation theory, the stress-energy tensor remains conserved in the presence of an arbitrary metric variation (as is a consequence of T11b), it also will remain conserved in the presence of an arbitrary self-interaction. Finally, in section 6, we prove that condition T11a and—in spacetimes of dimen- sion D > 2—condition T11b can be consistently imposed, in addition to conditions T1- 5 T10. The proof that condition T11a can be consistently imposed is relatively straight- forward, and is presented in subsection 6.1. The proof that condition T11b also can be imposed when D > 2 is much more complex technically, and is presented in the seven sub- subsections of 6.2. Despite its complexity, the proof is logically straightforward except for a significant subtlety that is treated in sub-subsection 6.2.6. Here we find that a potential obstruction to satisfying T11b arises from the requirement that time-ordered-products containing more than one factor of the stress-energy tensor be symmetric in these factors. We show that this potential obstruction does not actually occur for the theory of a scalar field, as treated here. However, this need not be the case for other fields, and, indeed, it presumably is the underlying cause of the inability to impose stress-energy conservation in certain parity violating theories in curved spacetimes of dimension D = 4k + 2, as found in [1]. For scalar fields, we are thereby able to show that condition T11b can be consistently imposed in curved spacetimes of dimension D > 2. However, for D = 2 a further difficulty arises from the simple fact that the freedom to modify the definition of ϕ ϕ by the addition of an arbitrary local curvature term does not give rise to a simi- a b ∇ ∇ lar freedom to modify the definition of T , and we find that, as a consequence, condition ab T11b cannot be satisfied for a scalar field in D = 2 dimensions. It is our view that conditions T1-T11 provide the complete characterization of Wick polynomials and time-ordered products of a quantum scalar field in curved spacetime. Notation and Conventions. Our notation and conventions generally follow those of our previous papers [13]-[15]. The spacetime dimension is denoted as D, and (M,g) always denotes an oriented, globally hyperbolic spacetime. We denote by ǫ = √ g dx0 − ∧ dxD−1 the volume element (viewed as a D-form, or density of weight 1) associated ···∧ with g. Abstract index notation is used wherever it does not result in exceedingly many indices. However, abstract index notation is generally not used for g = g and ǫ = ǫ . ab ab...c 2 The nature and properties of time-ordered prod- ucts 2.1 The construction of the free quantum field algebra and the nature of time-ordered products Consider a scalar field ϕ on an arbitrary globally hyperbolic spacetime, (M,g), with classical action 1 S = L = (gab ϕ ϕ+m2ϕ2 +ξRϕ2)ǫ. (2) 0 0 a b −2 ∇ ∇ Z Z 6 The equations of motion derived from this action have unique fundamental advanced and retarded solutions ∆adv/ret(x,y) satisfying ( a m2 ξR)∆adv/ret = δ, (3) a ∇ ∇ − − together with the support property supp∆adv/ret (x,y) M M x J−/+(y) , (4) ⊂ { ∈ × | ∈ } where J−/+(S) is the causal past/future of a set S in spacetime. Here we view the distribution kernel of ∆adv/ret as undensitized, i.e., acting on test densities rather than scalar test functions2, i.e., we view ∆adv/ret as a linear map from compactly supported, smooth densities to smooth scalar functions. The quantum theory of the field ϕ is defined by constructing a suitable *-algebra of observables as follows: We start with the free *-algebra with identity 11 generated by the formal expressions ϕ(f) and ϕ(h)∗ where f,h are smooth compactly supported densities on M. Now factor this free *-algebra by the following relations: (i) ϕ(α f +α f ) = α ϕ(f )+α ϕ(f ), with α ,α C; 1 1 2 2 1 1 2 2 1 2 ∈ (ii) ϕ(f)∗ = ϕ(f¯); (iii) ϕ(( a m2 ξR)f) = 0; and a ∇ ∇ − − (iv) ϕ(f )ϕ(f ) ϕ(f )ϕ(f ) = i∆(f ,f )11, where ∆ denotes the causal propagator for 1 2 2 1 1 2 − the Klein-Gordon operator, ∆ = ∆adv ∆ret. (5) − We refer to the algebra, (M,g), defined by relations (i)–(iv) as the CCR-algebra (for A “canonical commutation relations”). Quantum states on the CCR-algebra are simply A linear maps ω from into C that are normalized in the sense that ω(11) = 1 and that A are positive in the sense that ω(a∗a) is non-negative for any a . This algebraic notion ∈ A of a quantum state corresponds to the usual notion of a state as a normalized vector in a Hilbert space as follows: Given a representation, π, of on a Hilbert space, , (so A H that each a is represented as a linear operator π(a) on ), then any normalized ∈ A H vector state ψ defines a state ω in the above sense via taking expectation values, | i ∈ H ω(a) = ψ π(a) ψ . Conversely, given a state, ω, the GNS construction establishes that h | | i one always can find a Hilbert space, , a representation, π of on , and a vector H A H ψ such that ω(a) = ψ π(a) ψ . | i ∈ H h | | i By construction, the only observables contained in are the correlation functions of A the quantum field ϕ. Even if we were only interested in considering the free quantum 2Consequently, the delta-distribution in eq. (3) is also undensitized. 7 field defined by the action eq. (2), there are observables of interest that are not contained in , such as the stress-energy tensor of the quantum field A δL 1 1 Tab = 2ǫ−1 0 = aϕ bϕ gab cϕ ϕ gabm2ϕ2 c δg ∇ ∇ − 2 ∇ ∇ − 2 ab +ξ[Gabϕ2 2 a(ϕ bϕ)+2gab c(ϕ ϕ)]. (6) c − ∇ ∇ ∇ ∇ We will refer to any polynomial expression, Φ, in ϕ and its derivatives as a “Wick polyno- mial”. All Wick polynomials, such as T , that involve quadratic or higher order powers ab of ϕ are intrinsically ill defined on account of the distributional character of ϕ. It is natural, however, to try to interpret Wick polynomials as arising from “unsmeared” ele- ments of that are then made well defined via some sort of “regularization” procedure. A In Minkowski spacetime, a suitable regularization is accomplished by “normal ordering”, which can be interpreted in terms of a subtraction of expectation values in the Minkowski vacuum state. However, in curved spacetime, regularization via “vacuum subtraction” is, in general, neither available (since there will, in general, not exist a unique, preferred “vacuum state”) nor appropriate (since the resulting Wick polynomials will fail to be local, covariant fields [13]). The necessity of going beyond observables in becomes even more clear if one at- A temptstoconstructthetheoryofaself-interactingfield(withapolynomialself-interaction) in terms of a perturbation expansion off of a free field theory. First, the interaction La- grangian, L , itself will be a Wick polynomial and thereby corresponds to an observable 1 that does not lie in . Second, the nth order perturbative corrections to ϕ—or, more gen- A erally, thenthorderperturbativecorrectionstoanyWickmonomialΦ—areformallygiven by the Bogoliubov formula (see eq. (91) below), which expresses the Wick monomial Φ , L1 for the interacting field as a sum of Φ and correction terms involving the “time-ordered products” of expressions containing one factor of Φ and n factors of L . For the case of 1 two Wick monomials, Φ and Φ , the time-ordered product is formally given by 1 2 T (Φ (x )Φ (x )) = ϑ(x0 x0)Φ (x )Φ (x )+ϑ(x0 x0)Φ (x )Φ (x ) (7) 1 1 2 2 1 − 2 1 1 2 2 2 − 1 2 2 1 1 where ϑ denotes the step function. (The formal generalization of eq. (7) to time-ordered products with n-arguments is straightforward.) However, even if Wick monomials have been suitably defined, the time-ordered product (7) is not well defined since the Wick monomials also have a distributional character, and taking their product with a step function is, in general, ill defined. Nevertheless, in Minkowski spacetime, time-ordered products can be defined by well known renormalization procedures. Thus, the perturbative construction of the quantum field theory of an interacting field requires the definition of Wick polynomials and time-ordered products, both of which necessitate enlarging the algebra of observables beyond the original CCR-algebra, . A These steps were successfully carried out in [13, 14], based upon prior results obtained in [4, 5]. The first key step is to construct an algebra of observables, (M,g), which W 8 is large enough to contain all Wick polynomials and time-ordered products. To do so, consider the following expressions in (M,g): A n W (u) = u(x ,...,x ) : ϕ(x ) : n 1 n i ω Z i Y δn ≡ u(x1,...,xn)inδf(x ) δf(x )eiϕ(f)+12ω2(f,f) , u ∈ C0∞ (8) 1 n (cid:12) Z ··· (cid:12)f=0 (cid:12) (cid:12) where ω2 is the two-point function of an arbitrarily chosen Hadama(cid:12)rd state. Thus, ω2 is a distribution on M M with antisymmetric part equal to (i/2)∆, satisfying the spectrum × condition given in eq. (31) and satisfying the Klein-Gordon equation in each entry, i.e., (P 1)ω = 0 = (1 P)ω where P is the Klein-Gordon operator associated with L , 2 2 0 ⊗ ⊗ P = a m2 ξR. (9) a ∇ ∇ − − It follows from the above relations (i)–(iv) in the CCR-algebra that W (u)∗ = W (u¯), and n n that W (u) W (u′) = W (u u′), (10) n m n+m−2k k · ⊗ 2k≤m+n X where the “k-times contracted tensor product” is defined by k ⊗ n!m! (u u′)(x ,...,x ) d=ef S u(y ,...,y ,x ,...,x ) k 1 n+m−2k 1 k 1 n−k ⊗ (n k)!(m k)!k! × − − ZM2k k u′(y ,...,y ,x ,...,x ) ω (y ,y )ǫ(y )ǫ(y ) (11) k+1 k+i n−k+1 n+m−2k 2 i k+i i k+i i=1 Y where S denotes symmetrization in x ,...,x . If either m < k or n < k, then 1 n+m−2k the contracted tensor product is defined to be zero. The above product formula can be recognizedasWick’stheoremfornormalorderedproducts. Theenlargedalgebra (M,g) W is now obtained by allowing not only compactly supported smooth functions u C∞ as ∈ 0 arguments of W (u) but more generally any distribution u in the space n (M,g) = u ′(Mn) WF(u) (V+)n = WF(u) (V−)n = . (12) n E { ∈ D | ∩ ∩ ∅} Here, V+/− T∗M is the union of all future resp. past lightcones in the cotangent space ⊂ over M, and WF(u) is the wave front [16] set of a distribution u. The key point is that Hadamard property of ω and the wave front set condition on the u and u′ imposed in 2 the definition of the spaces ′(M,g) is necessary and sufficient in order to show that En the distribution products appearing in the contracted tensor product are well-defined and give a distribution in the desired class ′ (M,g). Note that the definition of the Em+n−2k 9 algebra (M,g) a priori depends on the choice of ω . However, it can be shown [13] 2 W that different choices give rise to *-isomorphic algebras. Thus, as an abstract algebra, (M,g) is independent of the choice of ω . 2 W Although the algebra (M,g) is “large enough” to contain all Wick polynomials W and time-ordered products, the above construction does not determine which elements of (M,g) correspond to given Wick polynomials or time-ordered products. (In particular, W the normal-ordered quantities W , eq. (8), with u taken to be a smooth function of one n variable times a delta-function, clearly do not provide an acceptable definition of Wick powers, since they fail to define local, covariant fields [13].) In [13, 14], an axiomatic approach was then taken to determine which elements of correspond to given Wick W polynomials and time-ordered products. In other words, rather than attempting to define Wick polynomials and time-ordered products by the adoption of some particular regu- larization scheme, we provided a list of properties that these quantities should satisfy. We proved the existence of Wick polynomials and time-ordered products satisfying these properties and also proved their uniqueness up to expected renormalization ambiguities. Asalreadydiscussed intheprevious section, oneofthemainpurposes ofthepresent paper is to supplement this list of axioms with additional conditions applicable to Wick poly- nomials and time-ordered products containing derivatives, and to prove correspondingly stronger existence and uniqueness theorems. We will shortly review the axioms that we previously gave in [13, 14]. However, before doing so, we shall explain a subtle but important shift in our viewpoint on the nature of Wick polynomials and time-ordered products. A Wick polynomial is a distribution, valued in the quantum field algebra that W corresponds to a polynomial expression in the classical field ϕ and its derivatives. It is therefore natural to consider the classical algebra, , of real polynomial expressions class C in the (unsmeared) classical field ϕ(x) and its derivatives, where we impose all of the normal rules of algebra (such as the associative, commutative, and distributive laws) and tensor calculus (such as the Leibniz rule) to the expressions in , and, in addition, we class C impose the wave equation on ϕ, i.e., we set ( a m2 ξR)ϕ(x) = 0. It would then a ∇ ∇ − − be natural to view Wick polynomials as maps from into distributions with values class C in . However, this viewpoint on Wick polynomials is, in general, inconsistent because W of the existence of anomalies. Indeed, we already mentioned in the introduction that—as we will explicitly show in section 3.2 below—under our other assumptions, it will not be consistent to set to zero all Wick monomials containing a factor of ( a m2 ξR)ϕ(x), a ∇ ∇ − − even though elements of that contain such a factor vanish. class C This difficulty has a simple remedy: We can instead define a classical field algebra of polynomial expressions in the unsmeared field ϕ(x) and its derivatives where we no longer impose the wave equation. Moreprecisely, let denote the realvector space ofall clas- class V 10