On the global Hadamard parametrix in QFT and the signed squared geodesic distance defined in domains larger than convex normal neighbourhoods

We consider the global Hadamard condition and the notion of Hadamard parametrix whose use is pervasive in algebraic QFT in curved spacetime (see refences in the main text). We point out the existence of a technical problem in the literature concerning well-definedness of the global Hadamard parametrix in normal neighbourhoods of Cauchy surfaces. We discuss in particular the definition of the (signed) geodesic distance $\sigma$ and related structures in an open neighbourhood of the diagonal of $M\times M$ larger than $U\times U$, for a normal convex neighborhood $U$, where $(M,g)$ is a Riemannian or Lorentzian (smooth Hausdorff paracompact) manifold. We eventually propose a quite natural solution which slightly changes the original definition by B.S. Kay and R.M. Wald and relies upon some non-trivial consequences of the paracompactness property. The proposed re-formulation is in agreement with M.J. Radzikowski's microlocal version of the Hadamard condition.


Introduction
The use of Hadamard states is nowadays pervasive in algebraic QFT (aQFT) in curved spacetime (see, e.g., [BFK96, BF00, HW01, HW02, Mo03, Mo08, DMP09, DMP11, DFP08, Sa15, FS08, FV03,FV13], and [BDFY15] for a recent survey on aQFT). The rigorous definition of Hadamard state in terms of short-distance behaviour of the two-point function was stated in the celebrated paper [KW91] by B.S. Kay and R.M. Wald for the first time. Some years later, that technically complex definition was translated into the language of microlocal analysis within a pair of nice papers by M.J. Radzikowski [Rad96a,Rad96b].
The original geometric definition of [KW91] of a global Hadamard parametrix has been exploited for instance to deal with rigorous intepretations of the Hawking radiation, see [MP12] and the recent interesting paper [KPV21]. Just to mention some other applications of the Hadamard parametrix in aQFT (the following list of examples is by no means exhaustive) we can say that it plays a crucial role in the definition of locally-covariant Wick powers [HW01,KM15], including the definition of the stress-energy tensor operator [Mo03]. The Hadamard parametrix has been also employed in the study of quantum energy inequalities [FS08]. It has been also used in semiclassical approaches to the quantum gravity and cosmological applications [DFP08,MPS21].
Locally (and a bit roughly) speaking, in a globally-hyperbolic four-dimensional spacetime (M, g), an algebraic state ω of a real scalar Klein-Gordon quantum field is of Hadamard type if its two-point function Λ ω has the Hadamard short-distance singularity, Λ ω (x, y) = 1 (2π) 2 ∆(x, y) 1/2 σ(x, y) + v(x, y) ln σ(x, y) + H ω (x, y) when viewed as an integral kernel (see Section 3 for some technical details here disregarded). H ω is a smooth function depending on the state ω, whereas ∆, v, σ are universal geometric objects constructed out of the local geometry only. In particular, σ(x, y) is the so-called signed squared geodesic distance of x, y ∈ M . It is defined as the squared length -with the appropriate sign -of the geodesic segment joining x and y. The so-called Hadamard parametrix is the singular universal part Λ ω (x, y) − H(x, y).
Since there are many geodesics, in principle, joining x and y, a standard possibility is to assume that the identity above is true in a normal convex neighbourhood 1 (see Section 2). This is an open set U such that every pair of points x, y ∈ U can be joined by a unique geodesic segment γ : [0, 1] → M that belongs to the set: γ([0, 1]) ⊂ U . This elementary precaution is not enough however in the global definition discussed in [KW91]. It is because (1) is assumed to be valid for pairs (x, y) contained in many normal convex neighbourhoods. In principle this gives rise to a cumbersome many-valued function σ. This is one of the most difficult technical issues tackled in [KW91].
Remark 1: If the spacetime points x and y belong to many convex normal neighbourhoods, different choices of such a neighbourhood may not only lead to different values for σ(x, y) around (x, y), but also to different singularity structures as determined by the Hadamard parametrix (1). Therefore, in principle, the introduced issue not only affects the construction of the Hadamard parametrix, but also the singularity structure that the Hadamard condition is assumed to describe. In particular, Hadamard states may not satisfy (1) in every convex normal neighbourhood. (See also the end of Example 2.) If x and y are causally related, a natural choice of U for the given x, y exists which solves the problem of the definition of σ(x, y) and it was adopted in [KW91] (see Remark 13 below for more details). U , if it exists, is a normal convex neighbourhood that simultaneously includes x, y and their causal double cone J(x, y) (see (7)). Obviously, every other normal convex neighbourhood U ′ which both includes x, y and J(x, y) must also contain the causal geodesic γ joining x and y in U . As U ′ is convex, γ is also its own geodesic γ ′ joining x and y in U ′ . There is, in fact, only one geodesic segment (parametrized in [0, 1]) joining a pair of causally related points x and y in common for the subfamily of the said normal convex sets. For these pairs (x, y), σ(x, y) can be therefore unambiguously defined.
Physics is properly reflected by the family of causal geodesics, but a mathematically coherent definition of the Hadamard parametrix needs to consider also non-causal geodesics: the noncausal ones "arbitrarily close" to the causal ones. For technical reasons, in [KW91] σ was therefore also required to be smooth and well-defined in a neigbourhood O of that special family of causally related pairs (x, y). We stress that the neighbourhood O must also contain noncausally connected pairs. The argument used to give a non-ambiguous definition of σ cannot be used for those pairs. The existence of O with a non-ambiguous extension of the definition of σ was assumed in [KW91] and also in [Rad96a,Rad96b] without a proof. In this author's view, it remains a gap in the whole construction. This work is devoted to that gap.
Referring to Remark 1, one eventually sees that the problem of the definition of σ for noncausally connected pairs actually affects the definition of the Hadamard parametrix in (1), but not its singularity structure (see Remark 20).
We shall not try to directly prove the existence of that O. Our solution relies on a thin refinement of the definition of Hadamard parametrix which is possible thanks to a consequence of the paracompactness property. The final new definition of Hadamard state, which is a quite slight modification of the original definition in [KW91], though it is based on a non-trvial topological result, turns out to be in agreement with the microlocal version of the Hadamard condition.
To achieve our final goal, in the first part of the paper, we shall focus on the more abstract and mathematically-minded problem of a well-posed definition of σ (and related geometric objects) in a neighbourhood of the diagonal of M × M . This issue is the core of the problem with the Hadamard parametrix, but it may have other applications in mathematical physics, so that it deserves a separate study.
Example 2: A concrete elementary illustration of the problems one faces when trying to define σ(x, x ′ ) in a non-trivial spacetime is the following one 2 . Consider the spacetime (M, g) constructed out of the 1 + 1 Minkowski spacetime periodically identified under (t, x) → (t, x + 2L) (c = 1). Differently from the Minkowski space, M is not normal convex in its own right. To define σ(x, x ′ ), one is therefore forced to make a choice of a normal convex open set containing x and x ′ . Let x = (0, L/2) and x ′ = (L/2, L) ≡ (L/2, −L). These points are causally related and J(x, x ′ ) (a null line segment) can be thickened up to become a normal convex neighbourhood U . We wish to define the function σ near (x, x ′ ). Let us first consider nearby points that are still causally connected. In particular, let y = x and y ′ = (L/2, L − ǫ) = (L/2, −L − ǫ) with 0 < ǫ ≪ L. These are causally related and near to (x, x ′ ). In the considered case, we can also assume, enlarging U if necessary, that U ⊃ J(y, y ′ ). We can create further normal convex neighbourhoods which include (y, y ′ ) by (i) thickening the line segment (in R 2 ) between (0, L/2) and (L/2, L − ǫ) which is a timelike geodesic between y and y ′ . It produces a convex neighbourhood U ′ which we can assume to satisfy again U ′ ⊃ J(y, y ′ ). The line segment between (0, L/2) and (L/2, L − ǫ) belongs to both U and U ′ . Consequently, σ(y, y ′ ) referred to U must coincide with σ(y, y ′ ) referred to U ′ .
(ii) thickening the line segment between (0, L/2) and (L/2, −L−ǫ) which is a spacelike geodesic between y and y ′ . In this case we obtain a value for σ(y, y ′ ) different from the one computed in U .
This illustrates why, when defining σ(y, y ′ ) in the causally connected case, the condition on the geodesically convex neighbourhood that it contains J(y, y ′ ) permits to select a common notion of distance. Next, consider the causally disconnected case. Let y = x = (0, L/2) and y ′ = (L/2, L + ǫ) = (L/2, −L + ǫ) with 0 < ǫ ≪ L. These are not causally connected. We can still create normal convex neighbourhoods containing y and y ′ by thickening the line segment between (0, L/2) and (L/2, L + ǫ) or the one between (L/2, −L + ǫ) and (0, L/2) giving spacelike geodesics of differing lengths and different values of σ(y, y ′ ). There are actually infinitely many other possibilities that can be obtained using other image points (L/2, (2n + 1)L + ǫ), n ∈ Z, throughout a very slim thickening of these segments, wrapping on the cylinder without any self-intersection.
An explicit illustration of the content of Remark 1 arises by referring to the example above and considering the timelike related points p = (0, 0) and p 0 = (2L, 0), which can also be connected by null geodesics. Thickening a timelike or a null geodesic from p to p 0 to a convex normal neighbourhood leads to distinct singularity structures.
2 Extension of the signed squared geodesic distance and related structures Smooth manifolds are hereafter assumed to be Hausdorff and paracompact 3 . We adopt the Lorentzian signature (−, +, · · · , +) and we follow [ON83] concerning basic defintions, notation and results in the theory of Lorentzian manifolds (see [Mi19] for an up-to-date general review).  Remark 4: (1) U as above is geodesically starshaped with respect to every p ∈ U : for every other q ∈ U there is only one geodesic segment γ (2) As U is geodesically starshaped with respect to every point p ∈ U , it is not difficult to prove that V (U ) p in Def. 3 is completely determined by U and p ∈ U .
(3) The set ∪ p∈U {p}×V Among other important constructions, the Whitehead theorem and the properties of exp allow one to define the so-called (signed) squared geodesic distance also known as Synge's world function. If U is an open normal convex set in (M, g), where the sign − appears only if g is Lorentzian and γ It is evident that σ U (p, q) strictly depends on the choice of the normal convex neighborhood containing the points p, q. If there were another normal convex neighborhood U ′ ∋ p, q, in general σ U (p, q) = σ U ′ (p, q) because the two sides refer to generally different geodesic segments: one stays in U and the other stays in U ′ , though both geodesics join p and q. This fact prevents one from defining σ as a global smooth function over M × M .

Assignment of geodesics around the diagonal of M × M and extension of σ thereon
A natural issue which pops out at this juncture is whether or not σ can be more globally defined, The root of the problem is that, generally speaking, there are many geodesics connecting a pair of points p, q and σ(p, q) depends on the choice of one of those curves. One restricts to work in a "small" neighbourhood A of the diagonal of M × M because it seems that the choice should be easier if p and q are close to each other. (There are however results concerning really global definitions of σ, on the whole M × M , when assuming suitable hypotheses on the topology of M [CM04].) To address the issue above, one may therefore wonder if it is possible to define a jointly smooth assignment of geodesic segments γ pq (t) = Γ(t, p, q) where t ∈ [0, 1], (p, q) varies in a neighborhood A of ∆ M and γ pq (0) = p, γ pq (t) = q. Indeed, equipped with such an assignment, σ can be defined on A by direct use of (2).
Remark 6: If (M, g) is Riemannian and its injectivity radius is positive, then other known ways exist to define a (smooth) notion of squared geodesic distance in a neighbourhood of the diagonal of M × M (see, e.g., [Sh91] for the case of a bounded geometry manifold in particular). However, we refer here to the general case where (M, g) may be Lorentzian, or Riemannian with zero injectivity radius.
An idea to construct Γ and σ in A (see also the discussion on p.131 of [ON83]) relies on the insight that sufficiently small normal convex neighbourhoods are expected to have intersections which are normal convex as well. In that case, if U ∩U ′ is convex and both p, q ∈ U , p, q ∈ U ′ , then the unique geodesic segment Γ U (t, p, q) := γ If a covering C of M exists made of normal convex open sets such that U, U ′ ∈ C implies that U ∩ U ′ is convex as well, then a jointly smooth assignment of geodesic segments Γ : [0, 1] × A → M joining the arguments (p, q) ∈ A (i.e., Γ(0, p, q) = p and Γ(1, p, q) = q) is well-defined and smooth on the open neighbourhood A of ∆ M . It suffices to define Indeed, if (x, y) ∈ A , then there must exist U ∈ C such that x, y ∈ U . Next, the right-hand side of (2) is well defined, since it does not depend on U if there are other elements in C containing x, y as pointed out above. Γ is also jointly smooth on A because it is locally jointly smooth. In this way, an associated signed squared geodesic distance σ : A → R results to be well-defined and smooth because it is a composition of smooth functions: Definition 7: If (M, g) is a smooth Riemannian or Lorentzian manifold, a strongly convex Existence of a strongly convex covering C is guaranteed when explicitly assuming Hausdorff and paracompactness hypotheses on M 5 . In fact, paracompactness possesses an important technical feature discovered by A.H. Stone [St49] (see also [Mi59]).
(Notice that V ⊂ U V ∈ C in particular, so that a * -refinement is a refinement as well). This theorem implies the existence of the desired well-behaved covering C of normal convex open sets of (M, g) (see also Lemma 10 in Chapter 5 of [ON83]). Proof. Using Theorem 5, consider the covering C 0 made of all normal convex neighbourhoods that are subsets of the elements of A. Exploiting Theorem 8, consider a refinement C * 0 of C 0 satisfying, for every V ∈ C * 0 , The proof concludes by defining C as the family of normal convex neighbourhoods contained within elements of C * 0 so that (a) is in particular true by construction. To prove (b), we start by observing that, if C, now comes easily using convex normality of C V . First the intersection C ∩ C ′ is open. Next, if p, q ∈ C ∩ C ′ , then the unique geodesic segment γ : [0, 1] → C V joining p and q is also completely included in C ∩ C ′ since it must simultaneously stay in C and C ′ , they being normal convex as well. As a consequence, if p ∈ C ∩ C ′ , it necessarily holds C ∩ C ′ = exp p (V In summary, C ∩ C ′ fulfils Definition 3 and the proof is over. Collecting all results, we are in a position to state the main theorem of this section, concening the existence of strongly convex coverings in particular, which also includes a (local) uniqueness statement.
Theorem 10: Let (M, g) be a smooth (Hausdorff paracompact) Riemannian or Lorentzian manifold, and C a strongly convex covering of M . Then the following facts hold.
Proof. (a) If A is as in (3), Γ : [0, 1] × A → M defined as in (2) and σ : A → R defined as in (5) are well-defined and smooth as discussed in the paragraph before Eq.
(3) and after Eq. (2). (b) Define a new covering C 1 (a simultaneous refinement of C and C ′ ) made of the sets C ∩ C ′ , for all choices of C ∈ C and C ′ ∈ C ′ . According to Proposition 9, define C ′′ as a refinement of C 1 made of normal convex neighborhoods such that U, U ′ ∈ C ′′ implies that U ∩ U ′ is empty or normal convex and define x, y). The same fact holds for σ and σ ′ in view of their definition (5) in terms of Γ and Γ ′ .
Definition 11: A triple (A , Γ, σ) as in in (a) of Theorem 10 is said to be subordinated to C.
Remark 12: Strongly convex coverings are not an ad-hoc artifact for the proposal of this work, but a natural and commonly used technical tool in Semi-Riemannian Geometry. The existence of this sort of geodesically convex coverings is a straightforward fact in Riemannian Geometry (see the elementary version of the sketch of proof above when h = g). The extension to Lorentzian manifolds is however not straightforward. In addition to the topological approach of Proposition 9, a purely geometric (in this sense perhaps more natural) proof of existence of strongly convex coverings for a Lorentzian geometry can be obtained along the following construction 6 . (What follows is however valid, with the same proof, when referring to the geodesic flow of a smooth affine connection Γ on M which is not the Levi-Civita connection of some metric.) Let (M, g) be a smooth connected (Hausdorff 2nd countable) Lorentzian manifold.
(i) Let h be an auxiliary Riemannian metric on M (which exists as a consequence of elementary results in Riemannian geometry [KN96]). h can be chosen in order that the Riemannian manifold (M, h) is complete [KO61] so that the h-injectivity radius at a given point p ∈ M is a continuous function of p (see, e.g., Prop. 10.37 in [Lee18]). Consider the atlas of Riemannian normal coordinates (U p , ψ p ) centered on every p ∈ M and referred to the Riemannian metric h.
(ii) Following the classic proof of the Whitehead theorem on the existence of the topological basis of convex normal neighborhoods of g [KN96], one sees that every ψ p -coordinate ball B h r (p) with center p is g-normal convex if the radius r p is sufficiently small. This would happen if referring to any atlas on M , but in the considered case the balls B h r (p) are also geodesical balls with respect to h and they are normal and convex with respect to h for sufficiently small r p . As is known, these balls are also metric balls of the natural metric space on M induced by h (the distance d(p, q) is the inf of the h-length of the smooth curves joining p and q).
(iii) Then, one can choose a continuous function µ : M → (0, +∞) -with 2µ smaller than the h-radius of injectivity at each point p ∈ M -such that the ball B h r (p) with 0 < r ≤ 2µ(p) is g-normal convex for all p ∈ M .
A strong g-convex covering is made of the family of all the balls B h µ(p) (p), p ∈ M . Indeed, from elementary properties of balls in metric spaces, the intersection (assumed to be non-empty) of a pair of such balls with centers p 1 and p 2 is included in the ball of the center p 1 and (bigger) radius 2µ(p). This ball is g-convex by construction.

An issue with the global Hadamard condition and Hadamard parametrix
Before addressing another issue still related to the well-definedness of σ and associated structures, we summarize the relevant notions introduced by the milestone paper [KW91] where, for the first time, a rigorous definition of a Hadamard state was proposed and used by B.S. Kay and R.M. Wald. The definition was used in [KW91] (relying on previous work as [FSW78] and [Ka85]) to establish some important uniqueness results of QFT on a spacetime equipped with a bifurcate Killing horizon related to the KMS states of a real Klein-Gordon scalar field with the Hawking temperature. However, the definition of Hadamard state discussed therein applies to every (four-dimensional) globally hyperbolic spacetime. The notion of Hadamard states in Kay-Wald's approach relies upon the notion of Hadamard parametrix. The Hadamard condition on states can be nowadays formulated without a Hadamard parametrix using microlocal techniques as we shall briefly discuss later. It is however worth stressing that the notion of Hadamard parametrix remains a crucial technical tool for the construction of other important mathematical objects in QFT as the Wick powers in the locally covariant formulation (see in particular [HW01,KM15]).

Hadamard states according to [KW91]
If (M, g) is a time-oriented smooth spacetime and x, y ∈ M , where J ± (S) are defined as in [ON83]. We say that x, y are causally related in (M, g) if J(x, y) = ∅. We henceforth assume that (M, g) is four dimensional and globally hyperbolic.
Remark 13: If x, y ∈ M are causally related in a globally-hyperbolic spacetime (M, g), then there is a causal geodesic segment joining them in view of Proposition 19 in Chapter 14 of [ON83]. This fact has a crucial consequence. If x, y are causally related and both U ⊃ J(x, y), U ′ ⊃ J(x, y) for convex normal neighbourhoods U, U ′ , then σ U (x, y) = σ U ′ (x, y). Indeed, the unique geodesic segments parametrized on [0, 1] connecting x and y respectively in U and U ′ must belong to J(x, y) ⊂ U ∩ U ′ and thus they must coincide. This fact is throughout exploited in [KW91] and provides a well-defined notion of signed squared geodesic distance σ( The definition of Hadamard state according [KW91] passes through the following four steps. (H1) The so-called (global) Hadamard parametrix is defined in [KW91], for every natural n and ǫ > 0, as is an open set supposed to exist where σ and G T,n ǫ are well defined, t(x, y) := T (x) − T (y), where the smooth function T : M → R is a Cauchy temporal function 7 increasing towards the future, the branch cut of the logarithm is taken along the negative real axis, and the function ∆(x, y) and v n (x, y) are known and defined in terms of σ(x, y) and known recursion integrals along the geodesic segment γ xy connecting x and y (see, e.g., Appendix A of [Mo03] and [HM12]).

Remark 14:
If σ(x, y) and the geodesic segment γ xy connecting x and y are well defined in some neighborhood, then ∆(x, y) and v n (x, y) are completely determined in that neighbourhood. This happens in particular for x, y ∈ U with U normal convex neighbourhood. Definition 15: An algebraic state ω on the (Weyl C * or * ) algebra of a real scalar Klein-Gordon field on (M, g) is said to be globally Hadamard according to [KW91] if the associated two-point function, i.e., a certain bilinear map [KW91] Λ ω : C ∞ 0 (M )×C ∞ 0 (M ) → C, satisfies the following requirement where µ g is the natural measure induced by g on M and Λ T,n ǫ (x, y) = χ(x, y)G T,n ǫ (x, y) + H n (x, y) , for every natural n and some associated functions H n ∈ C n (N × N ). 7 I.e., dT = 0 is everywhere past-directed and T −1 (r) is a smooth spacelike Cauchy surface for every r ∈ R.
Remark 16: In [KW91], it is proved that Definition 15 is independent of O, N, χ, Σ. Yet, that independence proof assumes at various steps that G T,n ǫ (x, y) is well defined, not only on Z M , but also on O (and O ′ ). In particular, σ(x, y) is expected to have the standard behaviour in O: σ(x, y) > 0 if x = y are not causally related. More precisely, σ(x, y) is supposed to take the standard form σ(x, x ′ ) = −(y 0 (x ′ )) 2 + 3 α=1 (y α (x ′ )) 2 in Riemannian normal coordinates y 0 , y 1 , y 2 , y 3 centered at one of the entries (here x) also for non-causally related arguments.
Definition 15 was later proved to be equivalent to a certain microlocal version by a famous paper by M. Radzikowski [Rad96a], when assuming the requirement Λ ω ∈ D ′ (M × M ) (see (2) in Theorem 21 below). This second analytic version (extended to n-dimensional spacetimes with n ≥ 2) is the one usually nowadays adopted in perturbative aQFT, also including cosmological applications, starting form semiclassical versions of the Einstein equations (see [MPS21] for a recent application). The Hadamard parametrix plays a special role in the definition of locallycovariant Wick powers [HW01,KM15] and in the study of quantum energy inequalities [FS08]. Kay-Wald's version of the Hadamard condition has been used by R. Verch to prove physically important properties of Hadamard states at algebraic level, like local quasi equivalence and local definiteness [Ve94]. Using Kay-Wald's definition, Sahlmann and Verch [SV01] extended the formalism to vector-valued quantum fields in a globally hyperbolic spacetime of dimension n ≥ 2. There, also the equivalent microlocal formulation has been discussed and an extension of the theorem of propagation of Hadamard singularity has been established in the fashion of the original formulation [FSW78] of that property of Hadamard states. We recommend [BDFY15] for a recent account on the wide spectrum of applications of Hadamard states (a pedagogical introduction to quasifree Hadamard states and their relevance in aQFT takes place in [KM15] therein).
The specific use of the Hadamard condition in the study of Hawking radiation can be traced back to [FH90], already before that the precise form of the Hadamard parametrix was stated in [KW91]. Though the microlocal version has been recently employed in applications to aQFT in black-hole background [DMP11,Sa15], the originary [KW91] version of the Hadamard condition has continued to play a crucial role to discuss the Hawking radiation [Wa94], also in terms of a tunneling process [MP12,CMP14] (actually, those works only concern a local version of the Hadamard condition). See in particular the recent interesting work [KPV21] on the Hawking radiation (and partially on the black-hole entropy) for a collapsing black-hole spherically-symmetric spacetime, where the global Hadamard condition has been used.
An interesting global definition of Hadamard state has been recently discussed in [CDD20] in terms of pseudo differential operators and a different, but related, notion of global parametrix for globally hyperbolic spacetimes with compact Cauchy surfaces.

A gap in the definition of G T,n ǫ and a proposal of solution
The parametrix G T,n ǫ is evidently well defined on Z M , but there is no guarantee that it is also well defined on some open neighbourhood O ⊃ Z M . Indeed, the open set O must also contain pairs (p, q) which are not causally related and each such pair may be connected by many geodesic segments because Remark 13 does not apply. At this juncture, there is no explicit prescription to smoothly choose a unique geodesic segment for every such pair (p, q) in order to have a well defined σ(p, q), which, e.g., satisfies σ(x, y) > 0 when x = y are not causally related. The problem also arises in the definition of ∆(p, q) and v n (p, q) as they are computed using a geodesic segment joining p and q as said above.
Instead of attacking the problem directly by trying to establish the existence of a neighbourhood O ⊃ Z M where σ and G T,n ǫ are well defined, we adopt a different strategy to circumvent the gap by employing the achievements of Sect 2.2. The strategy relies on minimal modifications of original Kay-Wald's machinery. For this reason, in author's view, all important results established over the years that rely on Definition 15 (some of them quoted above) are correct.
Given a four-dimensional globally hyperbolic spacetime (M, g) with a time orientation, choose a strong convex covering C of M , define the triple (A , Γ, σ) subordinated to C as in Theorem 10 and the set Notice that A is an open neighborhood of Z C M by construction.
(H1)' Define a (global) Hadamard parametrix subordinated to C, for every natural n and ǫ > 0, as Above, t(x, y) := T (x) − T (y), where T : M → R is global smooth time function increasing towards the future, the branch cut of the logarithm is taken along the negative real axis, and the functions, σ, ∆ and v n are the ones constructed out of (A , Γ, σ) starting from C.
(H2)' Given a smooth spacelike Cauchy surface Σ of (M, g) (with dimension ≥ 2), a normal neighbourhood N C of Σ subordinated to C is an open set including Σ and such that (a) (N C , g| N C ) is a globally hyperbolic spacetime and Σ is a Cauchy surface of it; (b) (x, y) ∈ N C × N C are causally related in (M, g) iff (x, y) ∈ Z C M . Lemma 17: Given a strong convex covering of M , every smooth spacelike Cauchy surface of (M, g) admits a normal neighbourhood subordinated to C.
Proof. Use the same proof as the one of Lemma 2.2 of [KW91] with the only difference that all the used normal convex neighbourhoods must be taken in C.

(H3)' Consider an open set
, the set of causally related pairs (x, y) ∈ N C × N C ) and such that its closure in N C × N C satisfies Finally, taking advantage of the smooth Urysohn lemma, choose a smooth function χ : N C × N C → [0, 1] such that χ(x, y) = 1 for (x, y) ∈ A ′ N C ×N C and χ(x, y) = 0 for (x, y) ∈ A ∩ (N C × N C ).
(H4)' With C, N C ,T , χ as above, we can give the definition of Hadamard state.
Definition 19: An algebraic state ω on the (Weyl C * or * ) algebra of a real scalar Klein-Gordon field on (M, g) is said to be globally Hadamard if the associated two-point function Λ ω : where µ g is the natural measure induced by g on M and for every natural n and some associated functions H n ∈ C n (N C × N C ).
Remark 20: An identity is of utmost physical interest: two parametrices subordinated to different strong convex coverings are however identical and also coincide with G T,n ǫ (x, y) in (8) when evaluated on causally related points (x, y) ∈ N C ∩ N C ′ . In fact, in the said hypothesis, it simultaneously holds (x, y) ∈ J(x, y) ⊂ C ∈ C and (x, y) ∈ J(x, y) ⊂ C ′ ∈ C ′ and thus, according to Remark 13, the geodesic segments joining x and y in C and C ′ , respectively, coincide. Finally the parametrices coincide as well in view of Remark 14. What happens to G T,n,C ǫ (x, y) for noncausally related points is physically irrelevant and it permits an arbitrary choice of the function χ appearing in χ(x, y)G T,n,C ǫ (x, y). A change of the function χ can be reabsorbed in a change of the functions H n . That is a consequence of the fact that, for x = y non-causally-related, σ(x, y) > 0 and no singularity (for ǫ → 0 + ) shows up in the parametrix G T,n,C ǫ (x, y). In other words, the parametrix viewed as a distribution is actually a smooth function for non-causallyrelated arguments. All that was discussed and clearly emphasized in [KW91] referring to the parametrix G T,n ǫ . Unfortunately these properties of G T,n ǫ rely also on a good behavior of σ in the whole open neighborhood O (and O ′ ) which is not proved to exist.
3.3 Independence of the choices of C, N C , T, χ and nice interplay with the microlocal formulation What remains to be demonstrated is that the given definition of Hadamard state does not depend on the choice of C, N C , T, χ and it corresponds to the microlocal formulation [Rad96a]. [Rad96a] aimed to establish that a state of a real Klein-Gordon field in a globally hyperbolic spacetime (M, g) is Hadamard in the sense of [KW91] if and only if it satisfies the microlocal spectral condition (14) below. (Actually it was done when also assuming the fair hypothesis that the two-point function Λ is a distribution of D ′ (M × M ).) As a matter of fact, this result gave rise to an alternative definition of Hadamard state.
The presentation of the Hadamard condition in the original sense of [KW91] in [Rad96a,Rad96b] is affected by the issue pointed out above (in the proof of Lemma 3.1 in [Rad96a] in particular) since [Rad96a] includes a faithful summary of relevant ideas and notions appearing in [KW91].
We argue that the statement of Theorem 5.1 in [Rad96a] which establishes the equivalence of the two formulations is however valid when assuming our definition of Hadamard state according to (H1)'-(H4)'. Let us re-state here part of Radzikowski's equivalence theorem (excerpt from Theorem 5.1 [Rad96a] with notations adapted to our paper). (2) Λ satisfies what follows.
((b) and (c) are in particular valid if Λ = Λ ω for an algebraic state ω on the (Weyl C * or * ) algebra of a real Klein Gordon quantum field.) Equivalence of (1) and (2) is still valid with the following changes. (b) and (c) in (1) are true mod C ∞ , (b) in (2) is true mod C ∞ , and (c) is omitted from (2).
Sketch of Proof. The proof of of Theorem 5.1 [Rad96a] uses both microlocal analysis arguments and some results from [KW91]. Concerning definitions and facts established in [KW91], it is assumed that (i) the parametrix G T,n ǫ (x, y) has the known structure in terms of σ, ∆, v n in a covering of normal convex neighbourhoods as stated in [KW91], (ii) the Hadamard expansion is well-behaved on the open neighborhood O ′ of the causally related points in N × N , where σ(x, y) > 0 for x = y which are not causally related (more precisely it takes the standard form σ(x, x ′ ) = −(y 0 (x ′ )) 2 + 3 α=1 (y α (x ′ )) 2 in Riemannian normal coordinates y 0 , y 1 , y 2 , y 3 centered at x), and (iii) the definition of Hadamard state according to [KW91] is independent from the choice of the Cauchy temporal function T . This proof of independence appears in Appendix B of [KW91] and it can be recast without changes for our definition of Hadamard state based on the parametrix G T,n,C ǫ (x, y) and a normal neighborhood N C . In summary, replacing O ′ for A ′ , using the fact that G T,n,C ǫ (x, y) has the same local structure as G T,n ǫ (x, y) in terms of σ and