Pseudodifferential Weyl Calculus on (Pseudo-)Riemannian Manifolds

One can argue that on flat space $\mathbb{R}^d$ the Weyl quantization is the most natural choice and that it has the best properties (e.g. symplectic covariance, real symbols correspond to Hermitian operators). On a generic manifold, there is no distinguished quantization, and a quantization is typically defined chart-wise. Here we introduce a quantization that, we believe, has the best properties for studying natural operators on pseudo-Riemannian manifolds. It is a generalization of the Weyl quantization - we call it the balanced geodesic Weyl quantization. Among other things, we prove that it maps square integrable symbols to Hilbert-Schmidt operators, and that even (resp. odd) polynomials are mapped to even (resp. odd) differential operators. We also present a formula for the corresponding star product and give its asymptotic expansion up to the 4th order in Planck's constant.


Introduction
Quantization means representing operators by their classical symbols, that is, functions on phase space. This concept first appeared in quantum physics in the early 20th century. Only later this idea was adopted in pure mathematics. Mathematicians, in particular, introduced the so-called pseudodifferential operators, defined as quantizations of certain symbol classes. Pseudodifferential calculus is nowadays a major tool in the study of partial differential equations.
Pseudodifferential calculus is useful both on R d and on manifolds. In the case of R d , one can use its special structure to define several distinguished quantizations. One of them is the Weyl or Weyl-Wigner quantization. One can argue that it is the most natural quantization and that it has the best properties. Let us list some of its advantages: 1. The Weyl quantization is invariant with respect to the group of linear symplectic transformations of the phase space T˚R d " R d ' R d .
2. Real symbols correspond to Hermitian operators.
3. Error terms in various formulas have a smaller order in if we use the Weyl quantization than if we use other kinds of quantizations.
4. The Weyl quantization of even, resp. odd polynomials is an even, resp. odd differential operator. 5. The Weyl quantization is proportional to a unitary operator if symbols are equipped with the natural scalar product and operators are equipped with the Hilbert-Schmidt scalar product.
In order to define a quantization on a generic manifold M , one typically covers it by local charts, and then uses the formalism from the flat case within each chart. This construction obviously depends on coordinates and thus there is no distinguished quantization on a generic manifold. In particular, considering the natural symplectic structure of the cotangent bundle T˚M , quantizations are symplectically invariant only on the level of the so-called principal symbol. This is not a problem for many applications of pseudodifferential calculus, which are quite rough and qualitative. In such applications, it is even not very important which kind of quantization one uses. In more quantitative applications, it is more important to choose a good quantization, which usually means a version of the Weyl quantization.
Suppose in addition that M is pseudo-Riemannian. One can try to use its structure to define a quantization that depends only on the geometry of M . In order to discuss various possibilities, let us first assume that M is geodesically simple, that is, each pair of points x, y P M can be joined by exactly one geodesic.
Here is one possible proposal of a geometric 1 generalization of the Weyl quantization: given a function T˚M Q pz, pq Þ Ñ bpz, pq, we define its naive geodesic Weyl quantization to be the operator with the integral kernel, which for x, y P M is defined as Op naive pbqpx, yq :" ż Tz M bpz, pqe´i u¨p dp p2πq d , (1.1) where z is the middle point of the geodesic joining x and y, u is the tangent vector at z such that x " exp z p´1 2 uq and y " exp z p 1 2 uq, and we integrate over the variable p in the cotangent space Tz M . The quantization Op naive is geometric (does not depend on coordinates) and it reduces to the Weyl quantization if M " R d . However, there exist better definitions, as we argue below.
In our paper we propose to multiply the right-hand side of (1.1) by |gpxq| 1 4 |gpyq| 1 4 |gpzq| 1 2 , (1.2) the product of the appropriate roots of the determinants of the metric at x, y and z. With this factor we obtain an integral kernel which is a half-density in both x and y. Then we multiply it by the biscalar ∆px, yq the square root of the so-called Van Vleck-Morette determinant. 2 We obtain Oppbqpx, yq :" ∆px, yq bpz, pqe´i u¨p dp p2πq d , (1.4) which we call the balanced geodesic Weyl quantization of the symbol b. This quantization belongs to a family of quantizations considered in [14], although for scalar functions instead of half-densities. Note that the Van Vleck-Morette determinant is a biscalar, and therefore Oppbqpx, yq is a half-density in both x and y, which is appropriate for the integral kernel of an operator acting on smooth, compactly supported half-densities on M . Moreover, the balanced geodesic Weyl quantization has a remarkable property: it satisfies Tr`Oppaq˚Oppbq˘" ż T˚M apz, pqbpz, pq dz dp p2πq d . (1.5) This is the analog of property (5) from the list of advantages of the Weyl quantization in the flat case. Actually, the balanced geodesic Weyl quantization has all the advantages from that list except for (1). It is easy to see that a quantization which reduces to the Weyl quantization in the flat case and satisfies (1.5) is essentially unique and is given by (1.4).
In general pseudo-Riemannian manifolds, there can be no or many geodesics joining pairs of points. However, there always exists a certain neighborhood Ω of the diagonal such that each pair px, yq P Ω is joined by a distinguished geodesic. Therefore, on a general pseudo-Riemannian manifold the definition (1.4) (and also (1.1)) makes sense only inside Ω. To make it global, one can insert a smooth cutoff supported in Ω, equal to 1 in a neighborhood of the diagonal. This is not a serious drawback, since in practice pseudodifferential calculus is mostly used to study properties of operators close to the diagonal. Note also that this cutoff does not affect Oppaq if a is a polynomial in the momenta, since the kernel of Oppaq is then supported inside the diagonal.
We are convinced that our quantization is a good tool for studying natural operators on M , such as the Laplace-Beltrami operator ∆ in the Riemannian case and the d'Alembert (wave) operator l in the Lorentzian case. In our future papers, we plan to describe applications of this quantization to computing singularities at the diagonal of the kernel of the heat semigroup e τ ∆ , Green's operator p∆`m 2 q´1, the proper time dynamics e iτ l , the Feynman propagator pl´i0q´1, etc.
Besides the geodesic Weyl quantization, we introduce also a whole family of quantizations parametrized by τ P r0, 1s. The Weyl quantization corresponds to τ " 1 2 . All of them satisfy obvious analogs of the identity (1.5). One can argue that the cases τ " 0 and τ " 1 are also of practical interest. However, the case τ " 1 2 typically leads to the most symmetric algebraic expressions. As mentioned above, the main application of our pseudodifferential calculus is to obtain asymptotic expansions of integral kernels Bpx, yq around the diagonal x " y. The geodesic Weyl quantization gives such expansions around the middle point of the geodesic joining x and y. If we use the τ " 0 quantization, then the expansion uses x as the central point, which is less symmetric and involves less cancellations.
For many authors, the philosophy of quantization is quite different from ours. Some authors study quantization as an end in itself. Others are interested only in applications which are quite robust and to a large extent insensitive to the choice of the quantization. Such applications include propagation of singularities, elliptic regularity, computation of the index of various operators. Other applications, such as spectral asymptotics, are more demanding in the choice of quantization. Our aim is to have an efficient tool for computing the asymptotics of various operators, giving an expansion which is as simple as possible. We have already checked this when computing the Feynman propagator on a Lorentzian manifold. We started from the naive geodesic Weyl quantization, and we discovered empirically that inserting the prefactors (1.2) and (1.3) decreases substantially proliferation of various error terms.
There is another argument why the balanced Weyl quantization is superior to the naive one: If one computes asymptotics of heat kernels or Feynman propagators using the traditional methods, without the momentum variables, as, e.g., in [3,4], then the prefactors (1.2) and (1.3) appear. So these prefactors simplify the expressions one looks for.
Our original motivation for introducing the balanced geodesic Weyl quantization comes from quantum field theory on curved spacetimes. One would like to define renormalized Wick powers of fields and its derivatives using a scheme that depends only on the local geometry. On a (flat) Minkowski space, renormalization is usually done in the momentum representation. There are in fact several (essentially equivalent) schemes for renormalization that use momentum representation. It is usually stated that on curved spaces the momentum representation is not available, and one has to use the position representation, which is much more complicated. The main tool for renormalization is then the asymptotics around the diagonal of the Feynman propagator, or what is equivalent, of the so-called Hadamard state defining the two point function. The use of a quantization allows us to use the momentum representation for renormalization on curved spacetimes. In our opinion, the balanced geodesic Weyl quantization will lead to the simplest computations in this context. Quantization and pseudodifferential calculus is an old subject with an interesting history and large literature. The Weyl quantization was first proposed by Weyl in 1927 [37]. Wigner was the first who considered its inverse, called sometimes the Wigner function or the Wigner transform [39]. The star product (the product of two operators on the level of symbols) and the identity (1.5) were first described by Moyal [24].
Several quantizations on manifolds were considered before in the literature. Working either with the Levi-Civita connection on a Riemannian manifold, with an arbitrary connection, or with the so-called linearization introduced in [5,6], most works attempt to generalize the τ " 0 or τ " 1 quantization, see e.g. [5,6,15,38], the more recent works [7,28,29,[31][32][33] and the references therein. An extensive introduction to early results on this topic can be found in [15]. Besides addressing the intricacies of defining a quantization on a manifold (e.g., caustics of geodesics), some of these works discuss symbol classes, the star product, heat kernel and resolvent computations (although only to low order).
A generalization of the Weyl quantization to manifolds with a connection was advocated by Safarov [22,31] and is essentially equivalent to what we call the naive geodesic Weyl quantization. Similar definitions can be found also in other places in the literature, e.g. [25]. More recently, Levy [20] considered a similar generalization for manifolds with linearization. Like in our manuscript, these papers consider the whole class of τ -quantizations on manifolds. They, however, do not use the geometric factor involving the Van-Vleck-Morette determinant, which as we argued above, improves the properties of a quantization.
Ideas very close to those of our paper were discussed by Fulling [14], who analyzed the effect of various powers of the Van Vleck-Morette determinant on the quantization. Fulling, in particular, remarked that the square root of this determinant may be viewed as the distinguished choice. He also noticed the term 1 6 R, which appears when one tries to define the Laplace-Beltrami or d'Alembert operator using the Weyl-type quantization involving the Van Vleck-Morette determinant. There is one difference between our approach and Fulling's: he used scalars, whereas we use half-densities -of course, this is a minor difference, since it is easy to pass from one framework to the other. We go much further than Fulling in the analysis of the balanced geodesic Weyl quantization: we prove the identity (1.5), which we view as a key advantage of this quantization, and we analyze the corresponding star product.
Güntürk in his PhD thesis [16] attempted to develop a Weyl calculus on Riemannian manifolds, including the star product. His approach, however, involved some constructions depending on coordinates, hence it was not fully geometric.
As we stressed above, the balanced geodesic Weyl quantization is geometric -by this we mean that it essentially depends only on the geometry of a pseudo-Riemannian manifold M . However, one should mention that the name geometric quantization has already a well-established meaning, which involves a somewhat different setting. The usual starting point of geometric quantization is a symplectic or, more generally, a Poisson manifold. Then one tries to define a noncommutative associative algebra which is a deformation of the commutative algebra of functions on this manifold. This deformation is often performed only on the level of formal power series in a small parameter, usually called Planck's constant . The resulting construction goes under the name of (formal) deformation quantization. See e.g. [1,13,40] for an overview on geometric quantization and [23] for its relation to deformation quantization.
In our construction, the symplectic manifold is always T˚M , the cotangent bundle to a pseudo-Riemannian manifold M . We obtain a certain unique realization of deformation quantization -a formal power series in that gives an associative product on functions on T˚M , which depends only on the geometry of M .
The plan of the paper is as follows: In Sect. 2 we introduce notation for various objects of differential geometry of pseudo-Riemannian manifolds, which we will use when presenting our results. In Sect. 3 we define a family of quantizations depending on a τ P r0, 1s. The most important is the geodesic Weyl quantization, which corresponds to τ " 1 2 . We discuss its basic properties. Among these properties the most demanding technically is the formula for the star product. We give its expansion up to the terms of the 4th order. In Sect. 4 we explain the methods for the derivation of the expansion of the star product. This is the most technical part of our paper. The methods that we use are essentially known from the works of Synge [35], DeWitt [12] (see also [9]), Avramidi [2,3] and others.
There are several systems of notation in differential geometry. We will use more than one. For the presentation of our results, we use mostly a coordinatefree and index-free notation, which is rather concise and transparent. It works especially well around the diagonal and is convenient for presentation of the results of our work. However, to compute quantities it is preferable to use other notations, which typically involve coordinates and indices.

Basic notation
Let M be a connected manifold. The tangent and cotangent space at x P M are denoted T x M and TxM , respectively. T p,q x M will denote the space of pcontravariant and q-covariant tensors at x.
Often we will use a coordinate dependent notation, which involves indices, denoted by Greek letters. Sometimes we will also use multiindices, which will be indicated by boldface letters. For instance, α " pα 1 , . . . , α n q and |α| " n. Thus T P T p,q x M after fixing a system of coordinates can be written as T β α with |α| " q, |β| " p.
From now on we assume that M is a (pseudo-)Riemannian manifold M with the metric tensor g. For any x P M let U x Ă T x M be the set of vectors u such that the inextendible geodesic γ u pτ q starting at x with initial velocity u is defined at least for τ P r0, 1s. The exponential map is then defined as For brevity we will often write x`u :" exp x puq, x´u :" exp x p´uq.
We say that M is geodesically complete if for any x P M we have U x " T x M .

Bitensors
Given two points x, y P M , a bitensor is an element T P T p,q x M b T t,s y M for some p, q, t, s. A bitensor field is a function MˆM Q px, yq Þ Ñ T px, yq such that T px, yq is a bitensor. In other words, a bitensor field T is a section of the exterior tensor product bundle T p,q M b T t,s M . Below will generally not distinguish between bitensors and bitensor fields in notation, and call both simply 'bitensors'. If we use coordinate notation, we distinguish indices belonging to the second point by primes. For example, T px, yq β ν 1 α µ 1 is a |β|-contravariant, |α|-covariant tensor at x and |ν|-contravariant, |µ|-covariant tensor at y. Note that in the context of bitensors the prime is not a part of the name of the corresponding indices, and only the indication to which point they belong.
As another example, consider a bitensor T px, yq µν 1 and two vector fields v µ and w µ . Then f px, yq " T px, yq µν 1 vpxq µ wpyq ν is a biscalar, i.e., a scalar in x and y.
If no ambiguity arises, we will often omit the dependence on x, y.
For the coincidence limit y Ñ x we use Synge's bracket notation whenever the limit exists and is independent of the path y Ñ x.

Parallel transport and covariant derivative
The metric defines the parallel transport along an arbitrary curve. We will most often use the parallel transport along geodesics. Given u P T x M and a tensor the tensor T parallel transported from x to the point x`u along the unique geodesic given by u.
In coordinates, the backward parallel transport on vectors is defined by the bitensor gpx, x`uq ν µ 1 at x and x`u, and on covectors by its inverse gpx, x`uq ν µ 1 . More precisely, for a tensor S ν and we use the Einstein summation convention. We have the identities The latter identity means that the metric is covariantly constant. Let M Q x Þ Ñ T pxq P T r,s x M be a tensor field, i.e., a tensor-valued function. The covariant derivative of T in direction u is defined as or, in coordinates, µν are the Christoffel symbols. We can also take covariant derivatives of bitensors. For that case, note that derivatives with respect to the two base points commute with each other. That is (suppressing all other indices), every bitensor field px, yq Þ Ñ T px, yq satisfies the identity T ;µν 1 " T ;ν 1 µ .
An important result concerning the covariant derivative of bitensors and their coincidence limit is Synge's rule. It states: We refer to Chap. I.4.2 of the excellent review article [30] for a proof.

Horizontal and vertical derivatives
Let T˚M Q px, pq Þ Ñ Spx, pq P T r,s x M be a tensor-valued function on the cotangent bundle. The horizontal derivative of S in direction u is defined as or, in coordinates, Note that the horizontal derivative can be viewed as a natural generalization of the covariant derivative. Therefore, it is natural to denote it by the same symbols -it will not lead to ambiguous expressions.
The vertical derivative of S at x in direction q P TxM is defined as or, in coordinates, Note that the vertical derivatives commutes with the horizontal derivative.

Geodesically convex neighbourhood of the diagonal
In general, a pair of points of M can have no joining geodesics, a single one or many. We will say that M is geodesically simple if every pair of points is joined by a unique geodesic γ x,y . Unfortunately, many interesting connected geodesically complete manifolds are not geodesically simple. However, in the general case we have a weaker property, which we describe below.
Let Diag :" tpx, xq | x P M u denote the diagonal. There exists a neighbourhood Ω Ă MˆM of Diag with the property: for all px, yq P Ω there is a unique geodesic r0, 1s Q τ Þ Ñ γ x,y pτ q P M joining x and y, and Such a neighbourhood will be called a geodesically convex neighbourhood of the diagonal.
For px, yq P Ω, we introduce the suggestive notation which is the tangent to the distinguished geodesic joining x and y. Note that py´xq is a bitensor. More precisely, it is a vector in x and a scalar in y, i.e., an Parallel transporting py´xq, we define for τ P r0, 1s as a short-hand. Clearly py´xq " py´xq 0 and py´xq τ " p1´τ q´1py´zq, z " x`τ py´xq for τ ‰ 1. Furthermore, note the coincidence limit which follows directly from (2.4).

Synge's world function
Synge's world function Ω Q px, yq Þ Ñ σpx, yq is defined as half the squared geodesic distance between x and y. That is, using the notation of (2.4), σpx, yq :" It is an example of a biscalar.

Bitensor of parallel transport
Another example of a bitensor is the bitensor of the parallel transport, g µ ν 1 , which transports T y M onto T x M along the geodesics γ y,x . In the notation of (2.1), for a vector field v, Note that in Subsect. 2.3 we considered a similar object g µ ν px, x`uq, except that there it was viewed as a function of x P M , u P T x M , and now it is viewed as a function of px, yq P Ω.
Equivalently, the bitensor g µ ν 1 is defined by the transport equations with the initial condition rg µ ν 1 s " δ µ ν .

Van Vleck-Morette determinant
If T P T 1,1 x M , we will write |T | for |det T |. Note that |T | is well defined independently of coordinates.
If S P T 0,2 x M or S P T 2,0 x M , we will use the same notation |S| for the absolute value of the determinant. |S| may now depend on the coordinates, but it can still be a useful object. For instance, the metric gpxq belongs to T 0,2 x , however |gpxq| will play a considerable role in our analysis.
Let T px, yq µ ν 1 be a bitensor, contravariant in x and covariant in y. Then it is easy to see that |T px, yq| |gpxq| Proof. We use the symmetry of the world-function and then we raise and lower the indices: which shows the symmetry of ∆px, yq.
We have py´xq τ " vpy´xqw τ py´xq . Therefore, where we only parallel transport the part of the bitensor at x, that is, Using (2.10), we obtain for the determinant . Now (2.12a) and (2.12b) follow.
The Jacobian (2.13) will play an important role in our construction. For brevity we therefore define the geometric factor (z τ " x`τ px´yq as before) Υ τ px, yq :" ∆px, yq |gpz τ q| 1 2 , which will appear in several of our proofs.

Quantization on a flat space
In this subsection we collect well-known facts concerning the quantization on a flat space, which we would like to generalize to curved (pseudo-)Riemannian manifolds. Consider the vector space X " R d , with X # denoting the space dual to X . By the Schwartz Kernel Theorem, continuous operators B : SpX q Ñ S 1 pX q are defined by their kernels Bp¨,¨q P S 1 pXˆX q. An operator B on L 2 pX q is Hilbert-Schmidt if and only if its kernel satisfies Bp¨,¨q P L 2 pXˆX q. The Hilbert-Schmidt scalar product satisfies the identity TrpA˚Bq " ż XˆX Apx, yqBpx, yq dx dy.
Let us fix a positive number called Planck's constant. The parameter conveniently keeps track of the "order of semiclassical approximation".
Let b P S 1 pXˆX # q. For any τ P R, we associate with b the operator Op τ pbq : SpX q Ñ S 1 pX q, given by the kernel and called the τ -quantization of the symbol b.
The most natural quantization corresponds to τ " 1 2 and is called the Weyl quantization of the symbol b. Instead of Op 1 2 pbq we will simply write Oppbq. The quantizations corresponding to τ " 0 and τ " 1 are also useful. They are sometimes called the x, p and the p, x quantizations. In a part of the PDE literature, the x, p quantization is treated as the standard one. Quantizations corresponding to τ different from 0, 1 2 , 1 are of purely academic interest. If A, B are Hilbert-Schmidt operators on L 2 pM q such that A " Op τ pa τ q and B " Op τ pb τ q, then TrpA˚Bq " ż XˆX # a τ pz, pqb τ pz, pq dz dp p2π q d .
Suppose that the operators Oppaq and Oppbq can be composed. Then one defines the star product or the Moyal product a ‹ b by Oppa ‹ bq " Oppaq Oppbq.
Using the identity for an invertible nˆn matrix A with Re A ě 0, which holds for f P SpR n q but can also be understood in larger generality, one obtains two formulas for the star product: (3.2b)

Operators on a manifold
In this subsection, the (pseudo-)Riemannian structure of M is irrelevant.
If B is a continuous operator C 8 c pM q Ñ D 1 pM q, then its kernel is the distribution in D 1 pMˆM q, denoted Bp¨,¨q, such that xf |Bgy " For instance, the kernel of the identity is given by the delta distribution. We will treat elements of C 8 c pM q not as scalar functions, but as half-densities. With this convention, the kernel of an operator is a half-density on MˆM . Note that with our conventions we need not specify a density with respect to which we integrate.
If two operators A, B can be composed, then we have 3) is true, e.g., if both A and B are Hilbert-Schmidt, but of course it also holds in various other situations. Clearly, the space of square-integrable half-densities on M forms a Hilbert space which will be denoted L 2 pM q. It is well known that an operator B on L 2 pM q is Hilbert-Schmidt if and only if its kernel satisfies Bp¨,¨q P L 2 pMˆM q.
If two operators A, B are Hilbert-Schmidt, the Hilbert-Schmidt scalar product is given by

Balanced geodesic quantization
Consider a smooth function T˚M Q pz, pq Þ Ñ bpz, pq. (3.5) Note that T˚M possesses a natural density, independent of the (pseudo-)Riemannian structure. Hence b can be interpreted as we like -as a scalar, density or, which is the most relevant interpretation for us, as a half-density. Assume first that M is geodesically simple. Let τ P R. We associate with (3.5) an operator with the kernel Op τ pbqpx, yq :" ∆px, yq |gpz τ q| where z τ " x`τ py´xq P M and u τ " py´xq τ P T zτ M . We call Op τ pbq the geodesic τ -quantization of the symbol b.
If M is not geodesically simple, then the definition (3.6) needs to be modified. Recall that Ω is a geodesically convex neighbourhood of Diag. Choose Ω 1 , another geodesically convex neighbourhood of Diag such that the closure of Ω 1 is contained in Ω. Fix a function χ P C 8 pMˆM q such that χ " 1 on Ω 1 and supp χ Ă Ω. Then instead of (3.6) we set Op τ pbqpx, yq :" χpx, yq ∆px, yq |gpz τ q| 7) where z τ " x`τ py´xq P M and u τ " py´xq τ P T zτ M .
Most of the time we will use τ " 1 2 , which is the analog of the Weyl quantization, and then we will write simply Oppbq instead of Op 1 2 pbq. Oppbq will be called the balanced geodesic Weyl quantization of the symbol b.
Our quantization depends on a Planck constant ą 0. This is however a minor thing: if we set " 1, we can easily put it back in all formulas, by dividing all momenta except those appearing in the arguments of symbols by (i.e., replacing p by ´1 p). Therefore, for simplicity, in all proofs we will set " 1. However, in the statements of various properties, we will keep explicit. Conversely, suppose that we are given a kernel Bpx, yq supported in Ω 1 . Note that then we can drop χ from (3.7) and its τ -symbol is b τ pz, pq :" Bpx, y τ qe iuτ¨p dx for τ ‰ 0, where u τ "´τ´1pz´xq and y τ " z`p1´τ qu. Proof. On the one hand, if bpz, pq is a symbol with τ -quantization Op τ pbqpx, yq, where x " z´τ u and y " z`p1´τ qu. On the other hand, if Bpx, yq is a kernel and b τ pz, pq is its symbol given by (3.8), where u " py´xq τ , x 1 " z´τ v and y 1 " z`p1´τ qv.

Remark 3.2.
The drawback of (3.7) is the fact that Op τ pbq depends on the cutoff χ. It is possible to modify this definition so that it is purely geometric and this cutoff is not needed. In fact, recall that given z P M and u P T z M there exist´8 ď t´ă t`ď`8 such that st´, t`r Q t Þ Ñ exp z ptuq P M is an inextendible geodesics. We set Op τ,global pbqpx, yq :" ÿ pz,uqPΓτ px,yqˇB px, yq Bpz, uqˇˇˇˇ´1 where Γ τ px, yq :" pz, uq P TMˇˇexp z p´τ uq " x, exp z`p 1´τ qu˘" y ( .
If not, they may be different, and besides Op τ,global pbq does not depend on the cutoff χ. However, since we are mostly interested in properties of operators around the diagonal, we keep the definition (3.7).

Independence of the quantization on the cutoff
We have already mentioned that the dependence of the quantization on the cutoff χ is mild. This is of course not true if the symbol has low regularity. It can be however expected for various typical classes of symbols used in the pseudodifferential calculus. In this subsection we will illustrate this independence on the example of the most popular symbol class, S m pT˚M q.
Let m P R. The class S m pT˚M q consists of b P C 8 pT˚M q such that for any compact K Ă M and for arbitrary multiindices α, β sup pz,pqPT˚M,zPK xpy |α|´m |B p α ∇ β bpz, pq| ă 8. Note that the class S m pT˚M q is purely geometric (it does not depend on the choice of coordinates).
The following two propositions are versions of well known properties of the standard pseudodifferential calculus adapted to the balanced geodesic Weyl quantization. Proof. The function px, yq Þ Ñ`z τ px, yq, u τ px, yq˘is smooth. So are the cutoff function and the geometric prefactor. Since we are only interested in smoothness outside of the diagonal, i.e., for u ‰ 0, it is therefore enough to study the smoothness of the function pz, uq Þ Ñ u α ż bpz, pqe iu¨p dp p2π q d for some (sufficiently large) |α|. Now, ∇ β bpz, pqp´iB p q α p γ e iu¨p dp p2π q d " |α|´|γ| ż e iu¨p p γ piB p q α ∇ β bpz, pq dp p2π q d . (3.10) Integrand (and prefactor) of (3.10) grow at most as |α|´|γ|´d xpy m´|α|`|γ| . For large enough |α| it is thus integrable. This shows the smoothness of Op τ pbqpx, yq outside of the diagonal and that it is Op 8 q.
Proposition 3.4. Suppose that χ 1 , χ 2 are two cutoffs of the type described above and Op 1 , Op 2 be the quantizations corresponding to χ 1 , χ 2 . Let b P S m pT˚M q. Then Op 1 pbqpx, yq´Op 2 pbqpx, yq is smooth, and together with all its derivatives it is Op 8 q.
Proof. We repeat the arguments of Prop. 3.3.

Hilbert-Schmidt scalar product
Proposition 3.5. Suppose that A, B are Hilbert-Schmidt operators on L 2 pM q, whose kernels are supported in Ω 1 . Let A " Op τ pa τ q and B " Op τ pb τ q. Then TrpA˚Bq " ż T˚M a τ pz, pqb τ pz, pq dp p2π q d dz. (3.11)
Thus all quantizations that we defined are unitary (up to a natural coefficient), which, as we believe, is a strong argument in favor of them.
One can expect that the formula (3.11) holds for a large class of operators such that A˚B is trace class, even if A and B are not Hilbert-Schmidt. For example, denote by f pxq the operator of multiplication by a complex function M Q z Þ Ñ f pzq. Note that Op τ pf q " f pxq. Therefore, we obtain the formula Tr`f pxq Op τ pbq˘" ż T˚M f pzqbpz, pq dp p2π q d dz. (3.12) Note that the integral kernel of the operator f pxq is supported exactly at the diagonal, therefore in the identity (3.12) there is no dependence on the cutoff χ.

Translating between different quantizations
Suppose we are given a symbol for the geodesic τ -quantization and wish to find a corresponding symbol for the τ 1 -quantization. An asymptotic formula is given by the following proposition (see also [31], where an analogous formula is derived): Proposition 3.6. Let τ, τ 1 P R and consider symbols b τ , b τ 1 such that Op τ pb τ q " Op τ 1 pb τ 1 q.
Proof. Let u P T z M , p P Tz M and q 1 P T z 1 M with z 1 :" z`pτ 1´τ qu.
Then, setting

Quantization of polynomial symbols
In this section, we consider the quantization of symbols, which are polynomial in the momenta. Note that for polynomial symbols the integral kernel of their quantization is supported on the diagonal, therefore there are no problems with multiple geodesics between two points.
As usual, boldface Greek letters denote multiindices. We sum over repeated multiindices and write p α " p α 1¨¨¨p αn .
Proof. We calculatè After an integration by parts, this implies (3.14).
For the balanced geodesic Weyl quantization, (3.14) simplifies and we find: Consider a polynomial symbol (3.13), as in Prop. 3
Proof. The second part of the proposition follows from p´i∇q α p´i∇ 1 q β ∆px, yq´1 2ˇx "y " p´i∇q β p´i∇ 1 q α ∆px, yq´1 2ˇx "y due to the symmetry of the Van Vleck-Morette determinant ∆px, yq " ∆py, xq. Indeed, for odd |γ|, either |α| or |β| is odd and thus the sum (3.15) consists of terms of the form Consider the example of a quadratic symbol apz, pq " apzq µν p µ p ν . Then where we used the fact that Suppose that a µν " g µν is the inverse metric. Since the metric is covariantly constant, we obtain independently of τ . In four dimensions this is the conformally invariant Laplace-Beltrami operator (or its pseudo-Riemannian generalization).

Balanced geodesic star product
In this section, we consider only the balanced geodesic Weyl quantization. We define the balanced geodesic star product by the identity Oppa ‹ bq " Oppaq Oppbq.
Let us first describe an explicit formula for the star product, which generalizes (3.2) from the flat case. For simplicity, we assume that we can take χ " 1.

Theorem 3.9. The balanced geodesic star product is given by the formulas
Λpz, u 1 , u 2 qe 2ipw`u 1´u2 q¨p e 2ipu 1¨p2´u2¨p1 q du 1 du 2 dp 1 dp 2 pπ q 2d , Figure 1: The geodesic triangle in the proof of Thm. 3.9. z, z 1 , z 2 are the middle-points of the three geodesics between the points x, y,z spanning the triangle. w is the tangent to the geodesic from z to y;w is the tangent to the geodesic from z toz; v 1 and v 2 are the tangents to the geodesics from z to z 1 resp. z 2 . u 1 and u 2 are the parallel transports of the tangents to the geodesics from z 2 resp. z 1 toz along v 1 resp. v 2 . In the flat case, (3.17) and the vectors u 1 , The schematic arrangement of these vectors can be seen in Fig. 1.
Proof. Let Cpx, yq be the integral kernel of Oppa ‹ bq. Clearly, Let z be the middle point between x and y. Let w,w P T z M be defined by x " z´w, y " z`w,z " z`w.
Then we can rewrite (3.19) as The star product of the symbols can then be found by applying (3.8): Oppaqpz´w, z`wq Oppbqpz`w, z`wqe 2iw¨p dw dw. (3.20) Consider next the vectors u 1 , u 2 , v 1 , v 2 P T z M defined in (3.18). z`v 1 is the middle point between z´w and z`w, and z`v 2 is the middle point between z`w and z`w. Therefore, Changing the integration variables in (3.20) from w,w to u 1 , u 2 , and inserting the formulas for Oppaq, Oppbq, we obtain Λpz, u 1 , u 2 qe 2ipw`u 1´u2 q¨p e 2ipu 1¨p2´u2¨p1 q du 1 du 2 dp 1 dp 2 π 2d , with Λ as defined in (3.17).

Remark 3.10.
A similar formula can be given also for τ ‰ 1 2 . For this purpose, make in (3.17) the substitutions z´w Þ Ñ z´2τ w, z`w Þ Ñ z`2p1´τ qw, (3.21) and replace (3.18) by Unlike in the flat case, all derivatives in (3.16b) remain also for τ " 0 and τ " 1 because of the non-trivial geometric factor Λpz, u 1 , u 2 q expp2ipw`u 1`u2 q¨pq and the dependence of v 1 and v 2 on both u 1 and u 2 . Note, however, that v 1 " 0 for τ " 0 and v 2 " 0 for τ " 1.

Asymptotic expansion of the geodesic star product
The geodesic star product can be expanded as a sum according to the order of Planck's constant. Note that each n-th term contains exactly n position derivatives, with Riemann tensors counting as two derivatives. It also contains exactly n momentum derivatives, with multiplication by p counting as´1 derivatives. The asymptotic expansion does not depend on the cutoff χ.
Due to the length of the expressions involved, we shall adopt the following notation for the derivatives of the symbols: lower indices always denote horizontal derivatives a α 1¨¨¨αn " ∇ αn¨¨¨∇α 1 a, and upper indices always denote vertical derivatives Since we only consider scalar symbols, no ambiguity arises. Recall also that horizontal and vertical derivatives commute so that their relative position is irrelevant.
In this notation, the five lowest order summands in (3.22) are pa ‹ bq 0 " ab,

Analysis of the expansion of the star product
In this subsection we analyze terms that appear in the expansion of the star product. The term that appears at r , that is pa‹bq r pz, pq is a linear combination with numerical coefficients of terms of the forḿ where we use the same notation for the derivatives of the symbols a and b as in the previous subsection. We have |ρ i | " 3, π i are single indices, i " 1, . . . s. The following identities are always satisfied: Observe that the first identity can be read off immediately from (3.16b) and then the second follows by the fact that all indices must be contracted. These terms can be divided into two kinds: 1. Terms that have the same form as the terms appearing in the star product on a flat space. They satisfy |η| " s " s ÿ j"1 |ν j | " 0, |β 1 | " |α 2 |, |β 2 | " |α 1 |.
For the star product of several symbols we also can write pa 1 ‹¨¨¨‹ a n qpz, pq " 8 ÿ r"0 r pa 1 ‹¨¨¨‹ a n q r pz, pq`Op 8 q, where pa 1 ‹¨¨¨‹ a n q r pz, pq is a linear combination with numerical coefficients of terms of the form´s ź j"1 Rpzq π j ρ j ;ν j¯pη a 1 pz, pq β 1 α 1¨¨¨a n pz, pq β n αn .
The identities and bounds for n " 2 generalize to this case:

Calculation of the star product
In this section we explain the methods that we used to obtain the expansion of the star product. We also give various intermediate results.

Coincidence limits of Synge's world function
In our calculations, we will need a certain family of coincidence limits of covariant derivatives of the Synge's function. More precisely, we will need rσ µ pα 1 qpβ 1 q s " rσ µ n q s, where parentheses around indices indicate symmetrization.
For notational simplicity, below we shall use the same index repeatedly to indicate symmetrization, e.g.
This notation is very convenient for the compact representation of tensorial expressions with multiple overlapping symmetries.
The result for n ą 2 is then obtained from (4.3) by induction.
Next note that to compute (4.1) it is enough to do it first for all primed or all unprimed indices: rσ α 1¨¨¨αn s " rσ α 1 1¨¨¨α 1 n s. Note that σ α 1¨¨¨αn is not symmetric with respect to permutation of indices except for the first two since covariant derivatives do not in general commute. For instance, we have For more complicated cases, we have the following lemma.
Proof. We have where αpi, µq is the multiindex coinciding with α except that on the ith place there is µ instead of α i . This proves the lemma for m " 0.
Then we apply the covariant derivatives ∇ νm¨¨¨∇ν 1 to both sides of (4.6). We obtain Clearly, after applying the Leibniz rule, the second term on the right of (4.7) will not contain σ's with more than n`m indices.
Let us now describe a recursive procedure to compute (4.4) relying on one of the identities (2.8), viz., Applying n´1 covariant derivatives, this yields σ α " pσ βα 1 σ β q ;α 2¨¨¨αn with α " pα 1 , . . . , α n q. Therefore, by the Leibniz rule and Lemma (4.2), where, as above, αpi, βq is the multiindex coinciding with α except that on the ith place there is β instead of α i , and¨¨¨indicates terms where no factor of σ has more than n´2 indices. Then we take the coincidence limit and use the basic coincidence limits (4.2) to obtain rσ α s " 0`nrσ α s`r¨¨¨s, where r¨¨¨s indicates the coincidence limit of the¨¨¨terms in (4.8). Since r¨¨¨s contains no factor of σ with more than n´2 indices, this gives the desired recursion.

Covariant Taylor expansion
Let M Q x Þ Ñ T pxq P T p,q x M be a tensor field. Note also the following important fact: d n dτ n vT px`τ uqw τ uˇτ "0 " pu¨∇q n T pxq.
Therefore, a (covariant) Taylor expansion for T is given by where it is necessary to first parallel transport T px`uq to x. Let us rewrite (4.9) in coordinates: where u β " u β 1¨¨¨u βn . We remark that formulas for remainder term are analogous to the usual Taylor expansion.
We will be especially interested in expansions of bitensors around the diagonal. Let px, yq Þ Ñ T px, yq be a bitensor. Then (4.9) can be rewritten as vT px, x`uqw u " rexppu¨∇ 1 qT spxq " ÿ n 1 n! rpu¨∇ 1 q n T spxq, where ∇ 1 denotes covariant differentiation with respect to the second argument. In coordinates, this can also be written as A particularly efficient approach to calculating coefficients of covariant expansions of many important bitensors is Avramidi's method [2,3,10], especially the semi-recursive variant presented in [27]. Avramidi's method relies on deriving recursion relations for the coefficients from certain transport equations (such as (2.8)) for the bitensor. We refer to [27] for a full explanation and several examples.
The Taylor expansion for tensor fields (4.9) generalizes to tensor-valued functions on the cotangent bundle via the horizontal derivative. (4.10)

Geodesic triangle
Consider three points x, y, z in a geodesically convex neighbourhood of the diagonal. By connecting these three points by the distinguished geodesics between them, we obtain a geodesic triangle. We define the following vectors: v :" py´xq P T x M, w :" pz´xq P T x M, The arrangement of these vectors is schematically depicted in Fig. 2.
While w " u`v on flat spaces, due to the effects of curvature on parallel transport this is no longer true on generic curved spaces, i.e., the triangle formed by these vectors does not close.
We will consider two cases: First, suppose that we are given v, w, and u is unknown. In other words, or, in terms of the world function with y " x`v, z " x`w, u µ "´gpx, yq µ α 1 σpy, zq α .
We perform two covariant expansions (with base point x) to find u µ "´gpx, yq µ α 1 ÿ n 1 n! pw¨∇ 1 q n σpy, xq α z x y v u 1 w u Figure 2: A geodesic triangle spanned by three points x, y, z. v is the tangent at x to the geodesic from x to y; w is the tangent at x to the geodesic from x to z; u 1 is the tangent at y to the geodesic from y to z; u is the parallel transport of u 1 from y to x along the geodesic given by v.
where we used (2.9). Secondly, suppose that we are given u, v, and w is unknown. In other words, or in terms of the world function with z " px`vq`vuw v , w µ "´σpx, zq µ .