Wick Polynomials under Microlocal Conditions

Wick powers exist because Hadamard subtraction is completed before coincidence. For a free real scalar field, normal ordering in a quasifree Hadamard state produces operator-valued distributions on a common dense invariant domain, and any two Hadamard choices are related by a canonical algebra isomorphism with smooth coefficients. A non-Hadamard reference can satisfy the field equation, positivity, and the canonical commutator while still leaving an ultraviolet singularity that has no canonical diagonal restriction.

Required background. Wavefront-set products, pullbacks, and pushforwards supplies the diagonal criterion; Hadamard states and the wavefront-set characterization supplies the universal singularity and the site’s commutator convention.

Helpful background. Local and microcausal functionals with Peierls brackets gives the functional language and its topological qualification; higher-point microlocal spectrum conditions controls the Wightman distributions of products of the new fields.

The chapter’s dependency map, hypothesis–conclusion table, and failure map keep diagonal existence, algebraic products, and the later local-covariance classification under their distinct hypotheses.

For a direct route, follow the diagonal obstruction, the operator construction, Wick products, the two microlocal domains, smooth state changes, and the explicit failure test.

Why subtraction must precede coincidence

Take a smooth, oriented and time-oriented, four-dimensional globally hyperbolic spacetime without boundary. For a concrete benchmark, let the field be massive,

$$P\phi=(\Box_g+m^2+\xi R)\phi=0, \qquad m>0,$$

although the microlocal construction also applies at $m=0$ when an appropriate Hadamard state is available. Let $E=G_{\mathrm{ret}}-G_{\mathrm{adv}}$. The site’s convention is

$$[\phi(f),\phi(h)]=-iE(f,h)\mathbf 1.$$

Fix a quasifree Hadamard state $\omega$ with two-point function $\omega_2$. In a geodesically convex neighborhood, let $H^+_\ell$ denote the oriented Wightman Hadamard parametrix used on the preceding page, so that

$$r_{\omega,\ell}:=\omega_2-H^+_\ell \in C^\infty(C\times C).$$

The superscript distinguishes this boundary value from the symmetric principal-value parametrix also denoted by $H$ in parts of the literature. If

$$H^{\mathrm s}_\ell =\frac12\left(H^+_\ell+H^{+\mathrm{op}}_\ell\right), \qquad \omega_{2,\mathrm s} =\frac12\left(\omega_2+\omega_2^{\mathrm{op}}\right),$$

then $\omega_{2,\mathrm s}-H^{\mathrm s}_\ell$ is smooth as well. An oriented subtraction must be paired with the ordered product $\phi(x)\phi(y)$; a symmetric subtraction must instead be paired with the anticommutator $\frac12\{\phi(x),\phi(y)\}$. Mixing those conventions leaves an erroneous antisymmetric term.

Now consider the diagonal embedding

$$\iota:M\longrightarrow M\times M, \qquad \iota(x)=(x,x).$$

Its normal set is

$$N_\iota=N^*\Delta\setminus0 =\{(x,k;x,-k):k\ne0\}.$$

At coincidence the Hadamard cone contains $(x,k;x,-k)$ for every nonzero future-directed null covector $k$. Therefore

$$N_\iota\cap\operatorname{WF}(\omega_2)\ne\varnothing, \qquad N_\iota\cap\operatorname{WF}(H^+_\ell)\ne\varnothing.$$

The pullback theorem consequently does not license either $\iota^*\omega_2$ or $\iota^*H^+_\ell$ separately. The legal order is to subtract on $C\times C$ first and only then restrict:

$$\iota^*(\omega_2-H^+_\ell)(x) =r_{\omega,\ell}(x,x).$$

This pullback exists because a smooth kernel has empty wavefront set. Failure of the wavefront criterion alone would only say that the general theorem supplies no canonical pullback; the explicit non-Hadamard example below also exhibits an actual quadratic cutoff divergence.

At operator level, introduce the symmetric point-split field

$$B_{H,2}(x,y) :=\frac12\{\phi(x),\phi(y)\} -H^{\mathrm s}_\ell(x,y)\mathbf1.$$

Using the common commutator of $\phi$ and $\omega_2$, this can be rewritten as

$$B_{H,2}(x,y) =:\!\phi(x)\phi(y)\!:_\omega +\bigl(\omega_{2,\mathrm s}-H^{\mathrm s}_\ell\bigr)(x,y)\mathbf1.$$

The first term has the improved wavefront behavior supplied by Hadamard normal ordering, while the second is smooth. The Wick square is the diagonal pullback, smeared from the outset:

$$:\!\phi^2\!:_H(f) :=\left\langle\iota^*B_{H,2},f\right\rangle, \qquad f\in C_0^\infty(C).$$

For a general compactly supported $f$, cover its support by finitely many convex normal neighborhoods and patch the local constructions with a partition of unity; the later local-covariance axioms control the allowed smooth differences between such prescriptions.

In the state $\omega$ its expectation is the smooth function

$$\omega\!\left(:\!\phi^2\!:_H(x)\right) =\bigl[\omega_2-H^+_\ell\bigr](x,x).$$

Thus the familiar notation $\lim_{y\to x}[\omega_2-H^+_\ell](x,y)$ is an ordinary limit only after smoothness has been proved. Before that proof, coincidence means a distributional pullback, not evaluation of two divergent functions.

Wick powers are operator-valued distributions

Reference-state normal ordering is compactly encoded by the formal generating identity

$$:e^{i\phi(f)}:_\omega =\exp\!\left(\frac12\omega_2(f,f)\right)e^{i\phi(f)}.$$

The exponentials need not belong to the polynomial field algebra; the equation means the collection of identities obtained by differentiating in $f$. Equivalently, one recursively subtracts every internal contraction from $\phi(x_1)\cdots\phi(x_n)$. The resulting jointly normal-ordered kernel

$$:\!\phi^{\otimes n}\!:_\omega(x_1,\ldots,x_n)$$

is symmetric and is an operator-valued distribution on $M^n$. Symmetry follows because the antisymmetric part of the ordered field product is exactly canceled by the antisymmetric part of $\omega_2$.

Coincidence is imposed by the compactly supported diagonal distribution

$$\left\langle f\,\delta_{\Delta_n},t\right\rangle =\int_M f(x)t(x,\ldots,x)\,d\mathrm{vol}_g(x),$$

whose wavefront set is

$$\operatorname{WF}(\delta_{\Delta_n}) =\left\{(x,k_1;\ldots;x,k_n): \sum_{j=1}^n k_j=0,\ \text{not all }k_j=0\right\}.$$

One chooses smooth kernels $t_\epsilon$ converging to $f\delta_{\Delta_n}$ inside a fixed admissible wavefront cone and defines

$$:\!\phi^n\!:_\omega(f) =\lim_{\epsilon\downarrow0} :\!\phi^{\otimes n}\!:_\omega(t_\epsilon).$$

The limit is not an operator-norm limit. Matrix elements on the common microlocal domain converge as distributions in the Hörmander topology, and sequential continuity makes the answer independent of the admissible approximating family. Brunetti, Fredenhagen, and Köhler first prove this convergence for products of auxiliary Wick monomials, identify the original Gelfand–Naimark–Segal (GNS) domain as a core, reconstruct Wightman fields on a common dense invariant domain generated by finitely many smeared Wick monomials, and establish the microlocal spectrum condition for the resulting hierarchy Brunetti, Fredenhagen, and Köhler 1996, §5, Proposition 5.3 and Theorems 5.4, 5.7–5.8, preprint pp. 15–18 (Open PDF).

Three statements should remain separate:

• $:\!\phi^n\!:_\omega(f)$ is an operator on a specified common invariant dense domain in the GNS representation of $\omega$;
• its matrix elements are distributions obeying microlocal spectrum bounds, so suitable products and limits can be tested;
• $:\!\phi^n\!:_\omega(x)$ is distributional notation, not a Hilbert-space operator obtained by evaluating the field at a point.

No essential self-adjointness or boundedness claim follows from this construction.

Wick’s theorem and the product domain

Curvature changes the two-point kernel, not the contraction combinatorics. As an identity of operator-valued distributions on $M\times M$, Wick’s theorem gives

$$\begin{aligned} :\!\phi^n\!:_\omega(x) :\!\phi^m\!:_\omega(y) =\sum_{k=0}^{\min(n,m)}& \binom{n}{k}\binom{m}{k}k!\, \omega_2(x,y)^k\\ &\times :\!\phi^{n-k}(x)\phi^{m-k}(y)\!:_\omega. \end{aligned}$$

Every contraction runs between the two normal-ordered factors; internal contractions have already been removed. Powers $\omega_2(x,y)^k$ are defined on $M\times M$: every nonzero first-slot covector has the same future orientation, so no sum of covectors from the factors can vanish. The higher-point graph cone controls the remaining normal-ordered kernel and the products of several Wick fields.

For example,

$$\begin{aligned} :\!\phi^2\!:_\omega(x) :\!\phi^2\!:_\omega(y) ={}&:\!\phi^2(x)\phi^2(y)\!:_\omega\\ &+4\omega_2(x,y):\!\phi(x)\phi(y)\!:_\omega +2\omega_2(x,y)^2\mathbf1. \end{aligned}$$

Smearing $x$ and $y$ independently with smooth compactly supported tests gives a valid operator product on the common domain. Setting $y=x$ is a new operation: it asks for a diagonal pullback of the entire right-hand side. The cross-contraction terms retain Hadamard singularities and meet $N^*\Delta$, so an ordinary same-point product of two Wick fields is not automatically defined. A separately constructed Wick power such as $:\!\phi^4\!:$ is not the naïve pointwise product $(:\!\phi^2\!:)^2$. Time-ordered products require a further extension onto diagonals and are treated later in this chapter.

Extended Wick algebra and functional derivative cones

The operator construction can be enlarged beyond smooth smearings. Let $\overline V_\pm$ be the closed future and past causal covector cones and define

$$\overline V_\pm^{\,n} :=\left\{(x_1,k_1;\ldots;x_n,k_n): k_j\in\overline V_\pm\ \text{for every }j\right\}.$$

For a symmetric compactly supported distribution $t\in\mathcal E'(M^n)$, require

$$\operatorname{WF}(t) \cap\left(\overline V_+^{\,n}\cup \overline V_-^{\,n}\right)=\varnothing.$$

This excludes a wavefront tuple whose covectors are all future causal or all past causal. The diagonal kernel $f\delta_{\Delta_n}$ passes: nonzero covectors that sum to zero cannot all lie in the same closed causal cone. One may therefore smear the normal product to obtain generators $W_n(t)$. Their products have the form

$$W_n(t)W_m(s) =\sum_{k=0}^{\min(n,m)} W_{n+m-2k}\!\left(t\otimes_k^\omega s\right),$$

where $t\otimes_k^\omega s$ contracts $k$ pairs with $\omega_2$ and includes the Wick combinatorial factors. Wavefront composition shows that every contracted kernel is again admissible. This is the extended Wick algebra of Hollands and Wald 2001, §2.2, Definition 2.1 and Theorem 2.1, preprint pp. 9–12 (Open PDF).

The same cone appears in the functional formulation, but it describes a different object. A Bastiani-smooth functional $F:C^\infty(M)\to\mathbb C$—smooth in the standard locally convex differential calculus—is called microcausal when every compactly supported distributional derivative satisfies

$$\operatorname{WF}\!\left(F^{(n)}[\varphi]\right) \cap\left(\overline V_+^{\,n}\cup \overline V_-^{\,n}\right)=\varnothing.$$

For the local polynomial $F_f(\varphi)=\int f\varphi^r\,d\mathrm{vol}_g$, these derivatives are supported on the thin diagonal and have conormal covectors summing to zero, so the condition holds. In a binary Peierls bracket or star product, the cone restriction licenses the termwise cross-contractions between derivatives of the two microcausal functionals and the causal or Hadamard kernel. It does not license an internal self-contraction such as pairing $f\delta_\Delta$ directly with $H(x,y)$, which would simply reintroduce the forbidden coincidence operation.

Two domains must not be conflated. The microlocal domain of smoothness above is a dense set of Hilbert-space vectors on which vector-valued Wick distributions may be smeared. A microcausal functional is a nonlinear map on classical configuration space whose derivatives obey cone conditions. Moreover, the pointwise microcausal condition alone does not provide enough uniform control to make the full class closed as smooth functionals under every Peierls or star product. Equicausal functionals add an equicontinuity condition over configuration space; local functionals and Wick polynomials lie in that refined class, which is closed under the relevant products Hawkins, Rejzner, and Visser 2026, §5 and §7, Theorems 5.8–5.9 and 7.4, preprint pp. 25–36 (Open PDF).

Changing the Hadamard reference

Let $\omega$ and $\omega'$ be quasifree Hadamard states for the same operator and commutator, and set

$$d(x,y)=\omega'_2(x,y)-\omega_2(x,y).$$

The kernel $d$ is real, symmetric, smooth, and a bisolution. Smoothness can be proved without comparing local series term by term. The wavefront sum rule gives $\operatorname{WF}(d)\subset\mathcal C^+$. Cancellation of the common commutator gives $d=d^{\mathrm{op}}$, so transposition also gives $\operatorname{WF}(d)\subset\mathcal C^-$. Since $\mathcal C^+\cap\mathcal C^-=\varnothing$, $\operatorname{WF}(d)=\varnothing$.

The change of normal ordering is therefore finite and local in the spacetime variables:

$$\alpha_{\omega'\to\omega} \!\left(:\!\phi^n\!:_\omega'(x)\right) =\sum_{j=0}^{\lfloor n/2\rfloor} \frac{n!}{(n-2j)!j!2^j} [-d(x,x)]^j :\!\phi^{n-2j}\!:_\omega(x).$$

The sign is fixed already at $n=2$:

$$\alpha_{\omega'\to\omega} \!\left(:\!\phi^2\!:_\omega'\right) =:\!\phi^2\!:_\omega-d(x,x)\mathbf1.$$

For general distributional generators, the same map contracts pairs of arguments with $-d$ before lowering the degree. It is a canonical $*$-isomorphism between the two extended Wick algebras even when their GNS representations are not unitarily equivalent Hollands and Wald 2001, Lemma 2.1 and Eqs. (20)–(24), preprint pp. 13–15 (Open PDF).

Now hold the geometric $H^+_\ell$ prescription fixed and compare its Wick-square expectation in the two states. All state-independent terms cancel:

$$\begin{aligned} \omega\!\left(:\!\phi^2\!:_H(x)\right) -\omega'\!\left(:\!\phi^2\!:_H(x)\right) &=\bigl[\omega_2-\omega'_2\bigr](x,x)\\ &=-d(x,x). \end{aligned}$$

This is a smooth function because the diagonal pullback of $d$ is ordinary evaluation. It is also independent of the common subtraction scale and any common state-independent finite term. The curved-spacetime application develops the physical point-splitting interpretation.

The commutator supplies an independent sign and normalization check. Differentiate the normal-ordering generating identity, or argue recursively with Wick’s rule. Commuting $\phi(y)$ through each unspecialized field factor produces $-iE(x,y)$, while the contraction coefficients reorganize the remaining polynomial into $n:\!\phi^{n-1}\!:(x)$. Hence

$$\boxed{ [ :\!\phi^n\!:(x),\phi(y) ] =-inE(x,y):\!\phi^{n-1}\!:(x) }$$

as an identity of operator-valued distributions. The minus sign follows from $[\phi(x),\phi(y)]=-iE(x,y)\mathbf1$; writing $+inE(x,y)$ would instead correspond to the reversed propagator convention or to $+inE(y,x)$.

A positive non-Hadamard subtraction fails

The obstruction can be made explicit without appealing to an unspecified singular kernel. Let $\omega_{0,2}$ be Hadamard and choose $\alpha>0$. Define

$$w_\alpha =(1+\alpha)\omega_{0,2} +\alpha\omega_{0,2}^{\mathrm{op}}.$$

Both summands are bisolutions. The transpose term is of positive type, and the coefficients preserve positivity. The antisymmetric part is unchanged:

$$w_\alpha-w_\alpha^{\mathrm{op}} =\omega_{0,2}-\omega_{0,2}^{\mathrm{op}} =-iE.$$

Thus $w_\alpha$ is the two-point function of a legitimate quasifree state. It is nevertheless non-Hadamard because

$$\operatorname{WF}(w_\alpha) =\mathcal C^+\cup\mathcal C^-.$$

Choose $H^+_\ell$ matching the singular part of $\omega_{0,2}$. Then

$$w_\alpha-H^+_\ell =\bigl(\omega_{0,2}-H^+_\ell\bigr) +\alpha\bigl(\omega_{0,2}+\omega_{0,2}^{\mathrm{op}}\bigr).$$

The first term is smooth, but the second retains both frequency orientations. At coincidence its wavefront set contains $(x,k;x,-k)\in N_\iota$, so the canonical diagonal-pullback theorem fails.

In Minkowski spacetime the same failure is visible without microlocal notation. Put $\omega_{\mathbf p}=\sqrt{|\mathbf p|^2+m^2}$. With a sharp spatial-momentum cutoff,

$$\begin{aligned} 2\alpha\int_{|\mathbf p|\leq\Lambda} \frac{d^3\mathbf p}{(2\pi)^3\,2\omega_{\mathbf p}} &=\frac{\alpha}{2\pi^2} \int_0^\Lambda \frac{p^2\,dp}{\sqrt{p^2+m^2}}\\ &=\frac{\alpha}{4\pi^2} \left[ \Lambda\sqrt{\Lambda^2+m^2} -m^2\operatorname{arsinh}\!\left(\frac{\Lambda}{m}\right) \right]\\ &=\frac{\alpha\Lambda^2}{4\pi^2} +O\!\left(m^2\log\frac{\Lambda}{m}\right). \end{aligned}$$

The residual coincidence value therefore diverges quadratically. The residual bidistribution itself is still well defined and may be paired with any compactly supported smooth test on $M^2$, including one whose support meets the diagonal; what fails is its pullback to the diagonal, hence the coincidence value needed for a Wick square. Its restriction to separated points is also well defined. One could impose an additional extension or subtraction prescription, but the resulting object would use new renormalization data; it would not be normal ordering justified by the original non-Hadamard reference.

What the construction does not yet prove

The distinctions can be summarized compactly.

Construction	Licensed conclusion	Additional input still required
Normal ordering relative to $\omega$	Wick fields and their admissibly smeared products in one Hadamard GNS representation	No preferred state and no natural assignment across spacetimes
Subtraction by $H^+_\ell$ or $H^{\mathrm s}_\ell$	State-independent removal of the universal local singularity before diagonal pullback	A consistent scale, smooth completion, and finite local prescription
Extended Wick algebra	Products after each kernel contraction passes its wavefront test	Same-point products and time ordering need their own diagonal construction
Microcausal functional condition	Termwise cross-contractions in binary Peierls or star products are distributionally meaningful	Uniform equicausal control for closure as smooth functionals; separate renormalization for internal coincidence
Local covariant Wick field	Not established on this page	Naturality, scaling, smooth or analytic background dependence, and ambiguity classification

In particular, reference-state normal ordering is generally nonlocal because the chosen state contains global information. The next page, local covariant Wick powers and operator products, imposes the missing naturality and scaling axioms and classifies the finite curvature-polynomial freedom. Time-ordered products and the renormalized stress tensor then treats genuine extension onto diagonals and conservation.

Common pitfalls

Taking coincidence term by term. Neither $\omega_2(x,x)$ nor $H^+_\ell(x,x)$ is defined by the canonical pullback theorem. Form the smooth difference—or the contraction-subtracted operator-valued distribution—before restricting.

Mixing oriented and symmetric parametrices. Subtract $H^+_\ell$ from an ordered two-point kernel, or subtract $H^{\mathrm s}_\ell$ from the anticommutator. Crossing the two conventions spoils the commutator cancellation.

Treating an operator-valued distribution as a point operator. The symbol $:\!\phi^n\!:(x)$ acquires meaning only after smearing, or as a kernel inside a separately licensed distributional operation.

Multiplying Wick fields again at the same point. Normal ordering removes contractions internal to each factor. Cross-contractions remain singular, so the product of two already normal-ordered fields still needs a new coincidence analysis.

Equating a reference-state prescription with local covariance. Smooth state changes give isomorphic algebras, but the state used for normal ordering can depend on the entire spacetime. Naturality across embeddings is a stronger requirement.

Claiming nonexistence from a failed sufficient test. An intersection with the conormal bundle says the standard pullback theorem does not define the operation. To prove a stronger failure, add an explicit calculation such as the quadratic divergence of $w_\alpha$.

Exercises

1. Locate the diagonal obstruction. Compute the normal set of $\iota(x)=(x,x)$ and show that it intersects the coincident Hadamard cone.

Solution

The transpose differential sends a product covector $(k,q)$ to $k+q$. It vanishes precisely when $q=-k$, so

$$N_\iota=\{(x,k;x,-k):k\ne0\}.$$

At equal base points the Hadamard relation includes $(x,k;x,-k)$ for every nonzero future-null $k$. Hence the intersection is nonempty. This proves that the general pullback theorem does not license the unsubtracted restriction; it does not, by itself, rule out every separately prescribed extension.

2. Check a local functional. Let $F_f(\varphi)=\int f\varphi^r\,d\mathrm{vol}_g$. Show that every derivative $F_f^{(n)}[\varphi]$ satisfies the microcausal cone condition.

Solution

For $n\leq r$, the derivative is a smooth compact coefficient times the delta distribution on the thin diagonal; it vanishes for $n>r$. Every nonzero wavefront tuple therefore has all base points equal and covectors satisfying

$$k_1+\cdots+k_n=0.$$

A collection of future-causal covectors cannot sum to zero unless every covector vanishes, and the same is true in the past cone. The joint zero tuple is excluded from a wavefront set. Thus neither $\overline V_+^{\,n}$ nor $\overline V_-^{\,n}$ is met.

3. Multiply two Wick squares. Derive the coefficients $4$ and $2$ in the displayed product formula, and explain why the formula does not define $(:\!\phi^2\!:)^2(x)$.

Solution

With one cross-contraction, choose one of two fields at $x$ and one of two at $y$, giving $\binom21^2 1!=4$. With two cross-contractions, the bijections between the two pairs give $\binom22^2 2!=2$. At $y=x$, the terms containing $\omega_2(x,y)$ and $\omega_2(x,y)^2$ retain diagonal Hadamard singularities. Their pullback meets $N^*\Delta$, so the two-variable identity cannot simply be evaluated at coincidence.

4. Change the fourth power. Express $:\!\phi^4\!:_\omega'$ in the $\omega$ prescription under the canonical state-change isomorphism when $d=\omega'_2-\omega_2$.

Solution

The terms have zero, one, and two pair contractions:

$$\alpha_{\omega'\to\omega} \!\left(:\!\phi^4\!:_\omega'\right) =:\!\phi^4\!:_\omega -6d(x,x):\!\phi^2\!:_\omega +3d(x,x)^2\mathbf1.$$

The coefficients are $4!/(2!1!2)=6$ and $4!/(0!2!2^2)=3$. The first contraction carries a minus sign; the double contraction carries its square.

5. Fix the commutator sign. Derive the commutator of $:\!\phi^n\!:_\omega(x)$ with $\phi(y)$ using the site’s convention.

Solution

Apply $[AB,C]=A[B,C]+[A,C]B$ before taking coincidence. Each of the $n$ unspecialized field factors contributes the c-number commutator $-iE(x,y)\mathbf1$. The contraction coefficients in the Wick recursion then regroup the remaining terms into the normal-ordered polynomial of degree $n-1$, giving

$$[ :\!\phi^n\!:_\omega(x),\phi(y) ] =-inE(x,y):\!\phi^{n-1}\!:_\omega(x).$$

6. Verify the wrong-frequency divergence. Evaluate the leading large-$\Lambda$ behavior of the residual $w_\alpha-H^+_\ell$ in Minkowski spacetime.

Solution

Only the added symmetric term matters at leading order. At coincidence it contributes

$$\frac{\alpha}{2\pi^2} \int_0^\Lambda\frac{p^2\,dp}{\sqrt{p^2+m^2}}.$$

For large $p$, the integrand is $p+O(p^{-1})$. Integration gives

$$\frac{\alpha\Lambda^2}{4\pi^2} +O\!\left(m^2\log\frac{\Lambda}{m}\right),$$

confirming that the residual is not smooth and has no naïve finite coincidence value.

References

• Brunetti, Romeo, Klaus Fredenhagen, and Michael Köhler. “The Microlocal Spectrum Condition and Wick Polynomials of Free Fields on Curved Spacetimes.” Communications in Mathematical Physics 180 (1996): 633–652. DOI. Open PDF.
• Hawkins, Eli, Kasia Rejzner, and Berend Visser. “A Novel Class of Functionals for Perturbative Algebraic Quantum Field Theory.” arXiv:2312.15203v3 [math-ph] (2026). Open manuscript.
• Hollands, Stefan, and Robert M. Wald. “Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime.” Communications in Mathematical Physics 223 (2001): 289–326. DOI. Open PDF.