aa r X i v : . [ m a t h . QA ] J un FORMAL OSCILLATORY DISTRIBUTIONS
ALEXANDER KARABEGOV
Abstract.
We introduce the notion of an oscillatory formal dis-tribution supported at a point. We prove that a formal distribu-tion is given by a formal oscillatory integral if and only if it isan oscillatory distribution that has a certain nondegeneracy prop-erty. We give an algorithm that recovers the jet of infinite orderof the integral kernel of a formal oscillatory integral at the criticalpoint from the corresponding formal distribution. We also provethat a star product ⋆ on a Poisson manifold M is natural in thesense of Gutt and Rawnsley if and only if the formal distribution f ⊗ g ( f ⋆ g )( x ) is oscillatory for every x ∈ M . Introduction
According to the stationary phase method, if φ is a real phase func-tion on R n which has a nondegenerate critical point x with zero criticalvalue, φ ( x ) = 0, and f is an amplitude supported near x , there existsan asymptotic expansion(1) (cid:18) i ~ (cid:19) n Z e i ~ φ ( x ) f ( x ) dx ∼ Λ ( f ) + ~ i Λ ( f ) + (cid:18) ~ i (cid:19) Λ ( f ) + . . . as ~ →
0, where Λ r are distributions supported at x (see [10]). Theformal distribution(2) Λ = Λ + ν Λ + ν Λ + . . . , where we use the formal parameter ν instead of ~ /i , is a formal os-cillatory integral (FOI) in the terminology of [8] and [7]. It can bedefined by simple algebraic axioms expressed in terms of the jet of in-finite order of the phase function φ at x . Moreover, the full jet of φ at x is uniquely determined by the formal distribution Λ. We build analgorithm that allows to recover this jet of infinite order from Λ. Mathematics Subject Classification.
Key words and phrases. formal oscillatory integral, oscillatory distribution, nat-ural deformation quantization.
The class of FOIs introduced in [8] is more general. It includes theasymptotic expansions of oscillatory integrals where the phase func-tion itself has an asymptotic expansion in ~ and can be complex, asexplained in Section 5.In this paper we answer the following question asked by Th. Voronov:given a formal distribution, how to determine whether it is a FOI? Tothis end we introduce the notion of an oscillatory distribution. It is aformal distribution Λ supported at a point x which in local coordinatesis given by the formula Λ( f ) = e ν − X f (cid:12)(cid:12) x = x , where X = ν X + ν X + . . . is a formal differential operator withconstant coefficients such that the order of the differential operator X r is at most r for all r ≥
2. It turns out that this property doesnot depend on the choice of local coordinates. We show that a formaldistribution is a FOI if and only if it is an oscillatory distribution thathas a certain nondegeneracy property.In [5] Gutt and Rawnsley singled out an important class of starproducts which they call natural . For each r ≥
1, the bidifferentialoperator C r for a natural star product is of order at most r in botharguments (see details in Section 4). All classical star products arenatural. We will prove that a star product ⋆ on a Poisson manifold M is natural if and only if the formal distributionΛ x ( f ⊗ g ) = ( f ⋆ g )( x )on M supported at ( x, x ) is oscillatory for every x .These results belong to the general framework of formal asymptoticLagrangian analysis. Various semiclassical and quantum aspects ofthis analysis are developed in the work on formal symplectic groupoidsby Cattaneo, Dherin, and Felder [2] and the author [6], symplecticmicrogeometry by Cattaneo, Dherin, and Weinstein [3], Lagrangiananalysis by Leray [10], the theory of oscillatory modules by Tsygan[11], and microformal analysis by Th. Voronov [12]. Acknowledgements
I am very grateful to A. Alekseev, H. Khu-daverdian, B. Tsygan, and Th. Voronov for important discussions andfor the opportunity to present a part of this work at two conferencesand during a visit to the University of Geneva in 2019.2.
Factorization
In this section we prove an elementary factorization result on pronilpo-tent Lie groups in filtered associative algebras which is the technicalbackbone of this paper.
ORMAL OSCILLATORY DISTRIBUTIONS 3
Let A be a filtered associative unital algebra over C with descendingfiltration A = A ⊃ A ⊃ . . . such that T i A i = { } . We denote by d ( a ) the filtration degree of a ∈ A so that d ( a ) = k for a ∈ A k \ A k +1 .We assume that this algebra is complete with respect to the norm | a | = 2 − d ( a ) . Then any series P i a i with a i ∈ A such that | a i | → g ⊂ A be a Lie algebra with respect to the commutator [ a, b ] = ab − ba . Then g is pronilpotent and exp g ⊂ A is the correspondingLie group. Each element g ∈ exp g is uniquely represented as(3) g = exp γ = ∞ X n =0 n ! γ n for some γ ∈ g . Then g − ∈ A and(4) γ = log(1 − (1 − g )) = − ∞ X n =1 n (1 − g ) n . We set g i := g ∩ A i for i ≥
1. The following statement is a conse-quence of formulas (3) and (4).
Lemma 2.1. If γ ∈ g , then (exp γ ) − ∈ A i if and only if γ ∈ g i . Suppose that g is a direct sum of subalgebras a and b such that g i = a i ⊕ b i , where a i := a ∩ A i and b i := b ∩ A i , for all i ≥ Proposition 2.1.
Any element g ∈ exp g can be uniquely factorized as g = ab with a ∈ exp a and b ∈ exp b .Proof. Given g = exp γ ∈ exp g for some γ ∈ g = g , we can represent γ uniquely as γ = α + β for some α ∈ a and β ∈ b . It followsfrom Lemma 2.1 that e − α e γ e − β = e γ for some γ ∈ g . Then γ = α + β for α ∈ a and β ∈ b .Repeating this process, we obtain sequences { α i } , { β i } , and { γ i } with α i ∈ a i , β i ∈ b i , and γ i ∈ g i such that γ i = α i + β i and e − α i e γ i e − β i = e γ i +1 . We get that g = e γ = e α e γ e β = e α e α e γ e β e β = . . . It follows that g = ab , where a ∈ exp a and b ∈ exp b are given by theconvergent infinite products a = e α e α e α . . . and b = . . . e β e β e β . The representation g = ab is unique because exp a ∩ exp b = { } . (cid:3) ALEXANDER KARABEGOV
Throughout this paper, we will apply Proposition 2.1 several timesin different contexts. Each time we will reuse the same notations for afiltered associative algebra A and a pronilpotent Lie algebra g ⊂ A .3. Some classes of formal distributions and operators
Let M be a real manifold and x be a point in M . We denoteby D ( M ) the algebra of differential operators on M , by D x ( M ) thespace of all distributions on M supported at x , and by δ x the Diracdistribution at x ( δ x ( f ) = f ( x )). The mapping A δ x ◦ A, from D ( M ) to D x ( M ) is surjective.Let ν be a formal parameter. We say that a ν -formal differentialoperator A = A + νA + . . . ∈ D ( M )[[ ν ]]is natural if the order of A r is at most r for all r ≥
0. If U is a coordinatechart on M with coordinates { x i } , a natural operator A on U can beuniquely written as A = ∞ X r =0 f i ...i r r ( ν, x ) ( ν∂ i ) . . . ( ν∂ i r ) , where f i ...i r r ∈ C ∞ ( U )[[ ν ]] is symmetric in i , . . . , i r for each r ≥ ∂ i = ∂/∂x i .The natural operators on M form an associative algebra. If A and B are natural operators, then the operator ν − [ A, B ] is natu-ral. Therefore, the formal differential operators of the form ν − A ,where A is natural, form a Lie algebra with respect to the commu-tator [ A, B ] = AB − BA . Definition 3.1.
A formal differential operator A ∈ D ( M )[[ ν ]] is calledoscillatory if it is represented as A = exp( ν − X ) , where X = ν X + ν X + . . . is a natural operator. Definition 3.2.
A formal distribution Λ ∈ D x ( M )[[ ν ]] is called oscil-latory if there exists an oscillatory operator A such that Λ = δ x ◦ A . Assume that Λ = Λ + ν Λ + . . . is an oscillatory distribution on M supported at x and represented as Λ = δ x ◦ exp( ν − X ), where X = ν X + ν X + . . . is natural. Then Λ = δ x and Λ = δ x ◦ X . Since X is a differential operator of order at most 2, there exists a uniquesymmetric bilinear form β Λ on T ∗ x M such that β Λ ( df ( x ) , dg ( x )) = Λ ( f g ) ORMAL OSCILLATORY DISTRIBUTIONS 5 for any functions f and g on M such that f ( x ) = g ( x ) = 0. The form β Λ is a coordinate-free object. Let U ⊂ M be a coordinate neighbor-hood of x with coordinates { x i } . If X = a ij ∂ i ∂ j + b i ∂ i + c , then β Λ ( df ( x ) , dg ( x )) = 2 a ij ∂ i f ∂ j g (cid:12)(cid:12) x = x . The form β Λ is thus given by the tensor 2 a ij ( x ). Definition 3.3.
An oscillatory distribution Λ is called nondegenerateif the bilinear form β Λ is nondegenerate. If Λ is a distribution on a coordinate neighborhood U of x supportedat x , there exists a unique differential operator C with constant coef-ficients such that Λ = δ x ◦ C . We will need the following fact. Lemma 3.1.
Any differential operator A on U can be uniquely repre-sented as a sum A = B + C of differential operators such that δ x ◦ B = 0 and C has constant coefficients.Proof. Let C be the unique differential operator with constant coeffi-cients such that δ x ◦ C = δ x ◦ A. Set B := A − C . Then δ x ◦ B = 0 and A = B + C . (cid:3) Any differential operator A on U can be uniquely represented in thenormal form, A = N X r =0 A i ...i r ( x ) ∂ i . . . ∂ i r , where A i ...i r ( x ) ∈ C ∞ ( U ) is symmetric in i , . . . , i r . Then A = B + C ,where B = N X r =0 (cid:0) A i ...i r ( x ) − A i ...i r ( x ) (cid:1) ∂ i . . . ∂ i r is such that δ x ◦ B = 0 and C = N X r =0 A i ...i r ( x ) ∂ i . . . ∂ i r has constant coefficients.We fix a coordinate chart U and consider the algebra A := D ( U )[[ ν ]]of formal differential operators on U equipped with the ν -filtration (thefiltration degree of ν is 1). Let g ⊂ A be the Lie algebra of formaldifferential operators on U of the form ν − X , where X = ν X + ν X + . . . is a natural operator. This is a pronilpotent Lie algebrawith respect to the ν -filtration. A distribution Λ on U supported ata point x is oscillatory if there exists an element A ∈ g such that ALEXANDER KARABEGOV
Λ = δ x ◦ exp( A ). The following proposition provides a criterion thata given formal distribution supported at a point is oscillatory. Proposition 3.1.
Let Λ be a formal distribution on U supported at apoint x . If C is the unique formal differential operator with constantcoefficients such that Λ = δ x ◦ exp( C ) , then Λ is oscillatory if and only if C ∈ g .Proof. If C ∈ g , then Λ is oscillatory. Now assume that Λ is oscillatory.Let b be the Lie algebra of formal differential operators A ∈ g suchthat δ x ◦ A = 0. Denote by c the Lie algebra of the formal differentialoperators with constant coefficients from g . Lemma 3.1 implies that g = b ⊕ c and g i = b i ⊕ c i for all i ≥ ν -filtrationspaces. Notice that the algebras g and b are coordinate-free objects,while the complementary algebra c depends on the choice of coordinateson U . Since Λ is oscillatory, Λ = δ x ◦ exp( A ) for some A ∈ g . It followsfrom Proposition 2.1 that there exist unique elements B ∈ b and C ∈ c such that e A = e B e C . Then δ x ◦ exp B = δ x andΛ = δ x ◦ exp( A ) = δ x ◦ (exp( B ) exp( C )) = δ x ◦ exp( C ) . (cid:3) Natural star products
Given a vector space V , we denote by V (( ν )) the space of formalvectors v = ν r v r + ν r +1 v r +1 + . . . , where r ∈ Z and v i ∈ V for all i ≥ r .Let M be a Poisson manifold with Poisson bracket {· , ·} . A starproduct ⋆ on M is an associative product on C ∞ ( M )(( ν )) given by theformula(5) f ⋆ g = f g + ∞ X r =1 ν r C r ( f, g ) , where C r are bidifferential operators on M for r ≥ C ( f, g ) − C ( g, f ) = { f, g } (see [1]). We assume that the unit constant 1 is theunity for the star product, f ⋆ f = 1 ⋆ f for all f . Given f, g ∈ C ∞ ( M )(( ν )), denote by L f the operator of left star multiplication by f and by R g the operator of right star multiplication by g so that L f g = f ⋆ g = R g f. The associativity of the star product ⋆ is equivalent to the conditionthat [ L f , R g ] = 0 for any f, g . The mapping f L f is an injective ORMAL OSCILLATORY DISTRIBUTIONS 7 homomorphism from the star algebra ( C ∞ ( M )(( ν )) , ⋆ ) to the algebra D ( M )(( ν )) of formal differential operators on M . It has a left inversemapping A A D ( M )(( ν ))), L f L f f ⋆ f. Gutt and Rawnsley introduced in [5] an important notion of a naturalstar product. A star product (5) is natural if the bidifferential operator C r is of order not greater than r in both arguments for every r ≥ ⋆ is natural if the operators L f and R f are natural for all f ∈ C ∞ ( M ). Then L f and R f are natural forall f ∈ C ∞ ( M )[[ ν ]]. All classical star products (Moyal-Weyl, Wick,Fedosov, and Kontsevich star products) are natural (see [5], [4], and[9]). We give an equivalent description of natural star products interms of oscillatory distributions in Theorem 4.1 below. To prove thistheorem, we need some preparations.Let t , . . . , t n be formal parameters, where n is any number, and A := ( D ( M )(( ν ))) [[ t , . . . , t n ]]be the associative algebra of formal differential operators on M of theform(6) A = ∞ X k =0 t j . . . t j k A j ...j k , where A j ...j k ∈ D (( ν )) are ν -formal differential operators on M sym-metric in j , . . . , j k . We equip A with the t -filtration {A i } for whichthe filtration degree of t i is 1 for every i (and the filtration degreeof ν is zero). We say that an operator (6) is natural if all oper-ators A j ...j k are natural. The algebra A acts on the space F :=( C ∞ ( M )(( ν ))) [[ t , . . . , t n ]] equipped with the t -filtration {F i } . Thespace F is a commutative algebra with respect to the “pointwise” mul-tiplication of formal series. Given f ∈ F , we denote by m f the multi-plication operator by f . Then m f ∈ A and m f f . Each operator A ∈ A is uniquely represented as the sum(7) A = m A + ( A − m A ) , where A − m A annihilates constants, ( A − m A )1 = 0.Let g ⊂ A be the Lie algebra of operators of positive t -filtrationdegree of the form ν − A , where A ∈ A is natural. The Lie algebra g ispronilpotent with respect to the t -filtration { g i } , where i ≥
1. Its Liegroup is exp g ⊂ A .Denote by a the commutative subalgebra of g of multiplication opera-tors and by b the subalgebra of g of operators that annihilate constants. ALEXANDER KARABEGOV
Then g = a ⊕ b and g i = a i ⊕ b i for all i ≥ G be the set of formal functions f = ν − f − + f + νf + . . . from F . Then a = { m f | f ∈ G} . Given f ∈ G , the exponential series e f = 1 + f + 12 f + . . . defines an element of F and exp a = { m e f | f ∈ G} . We setexp G := { e f | f ∈ G} ⊂ F . It is the Lie group of the commutative Lie algebra G . The mapping a a a onto exp G . Lemma 4.1.
For each g ∈ exp g , the operator g leaves invariant theset exp G . In particular, g ∈ exp G .Proof. Assume that g ∈ exp g and f ∈ G . Then m e f ∈ exp a and gm e f ∈ exp g . By Proposition 2.1, the element gm e f is uniquely repre-sented as a product gm e f = ab , where a ∈ exp a and b ∈ exp b . Then a ∈ exp G and b g to thefunction e f , we get g ( e f ) = ( gm e f )1 = ( ab )1 = a ∈ exp G . Thus, g (exp G ) ⊂ exp G and therefore g ∈ exp G . (cid:3) Let ⋆ be a natural star product on M . We extend it to F so that L t i = R t i = t i be the “pointwise” multiplication operator by t i forevery i . The space G ⊂ F is a Lie algebra with respect to the star-commutator [ f, g ] ⋆ = f ⋆ g − g ⋆ f . This Lie algebra is pronilpotentwith respect to the t -filtration {G i } , where i ≥
1. Given f ∈ G , theexponential series exp ⋆ f = 1 + f + 12 f ⋆ f + . . . defines an element of F . We setexp ⋆ G := { exp ⋆ f | f ∈ G} ⊂ F . This is the Lie group of the Lie algebra ( G , [ · , · ] ⋆ ). Lemma 4.2.
The subsets exp ⋆ G and exp G of F coincide.Proof. Given f ∈ G , the operator νL f = L νf is natural and therefore L f ∈ g . Thus, exp L f ∈ exp g . By Lemma 4.1, the operator exp L f with f ∈ G leaves invariant the set exp G . Given f, g ∈ G , we have(exp ⋆ f ) ⋆ e g = (cid:0) L exp ⋆ f (cid:1) e g = (exp L f ) e g ∈ exp G . ORMAL OSCILLATORY DISTRIBUTIONS 9
Taking g = 0, we get that exp ⋆ f ∈ exp G . Hence, exp ⋆ G ⊂ exp G .Given u ∈ G i , there exists v ∈ G such that e v = exp ⋆ ( − u ) ⋆ e u . Since e u = 1 + u (mod F i ) and exp ⋆ ( − u ) = 1 − u (mod F i ) , we see that e v ∈ F i and therefore v ∈ G i .Let f ∈ G = G . We will show that e f ∈ exp ⋆ G . We construct asequence { f k } , k ≥
0, in G such that f = f ∈ G and e f k +1 = exp ⋆ ( − f k ) ⋆ e f k for k ≥
0. We have f k ∈ G k for all k ≥
0. Observe that e f = e f = (exp ⋆ f ) ⋆ e f = (exp ⋆ f ) ⋆ (exp ⋆ f ) ⋆ e f = . . . Since e f k → k → ∞ in the topology induced by the t -filtration,we get that e f = (exp ⋆ f ) ⋆ (exp ⋆ f ) ⋆ . . . ∈ exp ⋆ G . It follows that exp ⋆ G = exp G . (cid:3) We give some basic facts on full symbols of formal differential oper-ators. Let U be a coordinate chart with coordinates { x i } , i = 1 , . . . , n ,and let { ξ i } be the dual fiber coordinates on T ∗ U which are treated asformal parameters. A formal differential operator A ∈ D ( U )(( ν )) canbe written in the normal form as A = ∞ X j = k ν j N j X r =0 A i ...i r j ( x ) ∂ i . . . ∂ i r , where k ∈ Z , A i ...i r j ( x ) ∈ C ∞ ( U ) is symmetric in i , . . . , i r for all j and r , and ∂ i = ∂/∂x i . The full symbol of the operator A is the formalseries S ( A ) = ∞ X j = k N j X r =0 ν j − r A i ...i r j ( x ) ξ i . . . ξ i r , which is an element of ( C ∞ ( U )(( ν ))) [[ ξ , . . . , ξ n ]], because for a fixed r the power of ν is bounded below by k − r . The operator A is naturalif and only if N j ≤ j for all j or, equivalently, S ( A ) does not containnegative powers of ν . It is well-known that(8) S ( A ) = e − ν x i ξ i A (cid:16) e ν x i ξ i (cid:17) = (cid:16) e − ν x i ξ i Ae ν x i ξ i (cid:17) . The C (( ν ))-linear mapping A S ( A ) restricted to the formal differen-tial operators with constant coefficients is an algebra homomorphism:if A and B have constant coefficients, then S ( AB ) = S ( A ) S ( B ). For every x ∈ M there exists a formal distribution Λ x on M sup-ported at ( x, x ) such thatΛ x ( f ⊗ g ) = ( f ⋆ g )( x )for all f, g ∈ C ∞ ( M ). Theorem 4.1.
A star product ⋆ on a manifold M is natural if andonly if the formal distribution Λ x is oscillatory for all x ∈ M .Example . Let ( π ij ) be an n × n matrix with constant coefficients.The star product f ⋆ g = ∞ X r =0 ν r r ! π i j . . . π i r j r ∂ r f∂x i . . . ∂x i r ∂ r g∂x j . . . ∂x j r on R n is natural. If the matrix ( π ij ) is skew-symmetric and nonde-generate, this is the Moyal-Weyl star product. Consider the naturaloperator A := ν π ij ∂ ∂y i ∂z j on R n . The formulaΛ x ( f ⊗ g ) = ( f ⋆ g )( x ) = e ν − A ( f ( y ) g ( z )) (cid:12)(cid:12) y = z = x , where f, g ∈ C ∞ ( R n ), shows that the formal distribution Λ x is oscil-latory for any x . It is nondegenerate if and only if the matrix ( π ij ) isnondegenerate.Now we proceed with a proof of Theorem 4.1. Proof.
Assume that a star product ⋆ on M is such that the distributionΛ x is oscillatory for all x ∈ M . Let U be a coordinate chart on M withcoordinates { x i } . Then for each x ∈ U there exists a unique naturaloperator with constant coefficients(9) A ( x ) = ∞ X r =2 ν r X k + l ≤ r F i ...i k j ...j l r,k,l ( x ) ∂ k ∂y i . . . ∂y i k ∂ l ∂z j . . . ∂z j l such that(10) ( f ⋆ g )( x ) = e ν − A ( x ) ( f ( y ) g ( z )) (cid:12)(cid:12) y = z = x . Since ( f ⋆ x ) = f ( x ), we get thatexp ∞ X r =2 ν r − X k ≤ r F i ...i k r,k, ( x ) ∂ k ∂y i . . . ∂y i k ! f ( y ) (cid:12)(cid:12)(cid:12)(cid:12) y = x = f ( x )for any f ( x ). Hence, F i ...i k r,k, ( x ) = 0 for all r and k . Similarly, F j ...j l r, ,l ( x ) =0 for all r and l . ORMAL OSCILLATORY DISTRIBUTIONS 11
Given f ∈ C ∞ ( U ), we will prove that the operator L f is natural. Tothis end, we will calculate its full symbol S ( L f ) using (8) and (10). Wewill show that it does not contain negative powers of ν . We have S ( L f ) = e − ν − x i ξ i L f (cid:16) e ν − x i ξ i (cid:17) = e − ν − x i ξ i (cid:16) f ⋆ e ν − x i ξ i (cid:17) = e − ν − x i ξ i e ν − A ( x ) (cid:16) f ( y ) e ν − z i ξ i (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) y = z = x = (cid:16) e − ν − z i ξ i e ν − A ( x ) e ν − z i ξ i (cid:17) f ( y ) (cid:12)(cid:12)(cid:12)(cid:12) y = z = x =exp (cid:16) e − ν − z i ξ i (cid:0) ν − A ( x ) (cid:1) e ν − z i ξ i (cid:17) f ( y ) (cid:12)(cid:12)(cid:12)(cid:12) y = z = x . It suffices to prove that the operator e − ν − z i ξ i ( ν − A ( x )) e ν − z i ξ i doesnot contain negative powers of ν . Using (9), we will write this operatoras follows, ∞ X r =2 ν r − X k + l ≤ r F i ...i k j ...j l r,k,l ∂ k ∂y i . . . ∂y i k (cid:18) ∂∂z j + 1 ν ξ i (cid:19) . . . (cid:18) ∂∂z j l + 1 ν ξ i l (cid:19) . Since F j ...j l r, ,l = 0 for all r and l , the condition k + l ≤ r in the secondsum implies that l ≤ r −
1, which proves the claim. One can showsimilarly that the operator R f is natural for f ∈ C ∞ ( U ). Since U isarbitrary, the star product ⋆ is natural on M .Now assume that ⋆ is a natural star product on M and U ⊂ M isan arbitrary coordinate chart. We will show that Λ x is oscillatory forevery x ∈ U . Let { ξ i } and { η i } be two sets of formal variables dualto { x i } . We extend the star product ⋆ to F := ( C ∞ ( U )(( ν )))[[ ξ, η ]]so that L ξ i = R ξ i = ξ i and L η i = R η i = η i for all i . Denote by G theLie algebra of functions from ν − C ∞ ( U )[[ ν, ξ, η ]] of positive filtrationdegree with respect to the variables ξ and η with the star commutator[ f, g ] ⋆ = f ⋆g − g⋆f as the Lie bracket. This is a pronilpotent Lie algebrawith the Lie group exp ⋆ G whose elements are the star exponentialsexp ⋆ f = 1 + f + 12 f ⋆ f + . . . of the elements of G . We can write the star product ⋆ as (10) with A ( x ) = ∞ X r =2 ν r X k + l ≤ N r F i ...i k j ...j l r,k,l ( x ) ∂ k ∂y i . . . ∂y i k ∂ l ∂z j . . . ∂z j l , where N r is some integer for each r ≥
2. We have to show that A ( x )is natural for every x ∈ U , i.e., that N r ≤ r for all r ≥
2. To this end, we consider two functions in exp G = { e f | f ∈ G} , f ( x ) := e ν − x i ξ i and g ( x ) := e ν − x i η i . By Lemma 4.2, f, g ∈ exp ⋆ G . Therefore, f ⋆ g ∈ exp ⋆ G = exp G . Using(8) and (10), we get that for x ∈ U ,( f ⋆ g )( x ) = e ν − A ( x ) (cid:16) e ν − ( y i ξ i + z i η i ) (cid:17) (cid:12)(cid:12) y = z = x = e ν − x i ( ξ i + η i ) (cid:16) e − ν − ( y i ξ i + z i η i ) e ν − A ( x ) e ν − ( y i ξ i + z i η i ) (cid:17) (cid:12)(cid:12) y = z = x = e ν − x i ( ξ i + η i ) S (cid:16) e ν − A ( x ) (cid:17) = e ν − ( x i ( ξ i + η i )+ S ( A ( x )) ) ∈ exp G , where S ( A ( x )) = ∞ X r =2 X k + l ≤ N r ν r − k − l F i ...i k j ...j l r,k,l ( x ) ξ i . . . ξ i k η j . . . η j l is the full symbol of A ( x ). Since ν − (cid:0) x i ( ξ i + η i ) + S ( A ( x )) (cid:1) ∈ G ,S ( A ( x )) does not contain negative powers of ν , which implies that A ( x ) is natural and therefore Λ x is oscillatory for any x ∈ U . Since U is arbitrary, Λ x is oscillatory for any x ∈ M . (cid:3) In [6] it was shown that the natural star products have a good semi-classical behavior. Theorem 4.1 relates these star products to oscilla-tory distributions which can be thought of as quantum objects.5.
Formal oscillatory integrals
Let M be a real n -dimensional manifold, x be a point in M , ϕ = ν − ϕ − + ϕ + νϕ + . . . be a formal complex-valued function and ρ = ρ + νρ + . . . be a formalcomplex-valued density on M such that x is a nondegenerate criticalpoint of ϕ − with zero critical value, ϕ − ( x ) = 0, and ρ ( x ) = 0.We call the pair ( ϕ, ρ ) a phase-density pair with the critical point x .A formal oscillatory integral (FOI) at x associated with the phase-density pair ( ϕ, ρ ) is a formal distributionΛ = Λ + ν Λ + . . . on M supported at x such that the value Λ( f ) for an amplitude f heuristically corresponds to the formal integral expression(11) ν − n Z e ϕ f ρ. ORMAL OSCILLATORY DISTRIBUTIONS 13
The distribution Λ is defined by certain algebraic axioms expressed interms of the pair ( ϕ, ρ ) which correspond to formal integral propertiesof (11). The full stationary phase expansion of an oscillatory integral(1) whose amplitude is supported near a nondegenerate critical pointof the phase function is given by a FOI. The notion of a FOI wasintroduced in [8] and developed further in [7].
Definition 5.1.
Given a phase-density pair ( ϕ, ρ ) with a critical point x on a manifold M , a formal distribution Λ = Λ + ν Λ + . . . on M sup-ported at x and such that Λ is nonzero is called a formal oscillatoryintegral (FOI) associated with the pair ( ϕ, ρ ) if (12) Λ( vf + ( vϕ + div ρ v ) f ) = 0 for any function f and any vector field v on M . In (12) div ρ v denotes the divergence of the vector field v with respectto ρ given by the formula div ρ v = L v ρρ , where L v is the Lie derivative with respect to v . Axiom (12) corre-sponds to the formal integral property ν − n Z L v ( e ϕ f ρ ) = 0 . Observe that the condition (12) is coordinate-independent. As shownin [7], a FOI Λ associated with ( ϕ, ρ ) satisfies the following properties.(1) Λ exists and is unique up to a multiplicative formal constant c = c + νc + . . . with c = 0.(2) Λ = αδ x for some nonzero complex constant α .(3) Λ is determined by the jets of infinite order of ϕ and ρ at x .(4) If u = u + νu + . . . is any formal function on M , then Λ isassociated with ( ϕ + u, e − u ρ ).(5) If Λ is associated with two pairs ( ϕ, ρ ) and ( ˜ ϕ, ρ ) which sharethe density ρ , then the full jet of ˜ ϕ − ϕ at x is a formal constant. Definition 5.2.
A FOI associated with a pair ( ϕ, ρ ) is strongly asso-ciated with it if (13) ddν Λ( f ) − Λ (cid:18) dfdν + (cid:18) dϕdν + dρ/dνρ − n ν (cid:19) f (cid:19) = 0 for any function f . The condition (13) is coordinate-independent. It corresponds to theformal property of (11) that integration commutes with differentiationwith respect to the formal parameter ν . A FOI Λ strongly associatedwith ( ϕ, ρ ) satisfies the following properties.(1) Λ exists and is unique up to a multiplicative nonzero complexconstant.(2) Λ is determined by the jets of infinite order of ϕ and ρ at x .(3) If u = u + νu + . . . is any formal function on M , then Λ isstrongly associated with ( ϕ + u, e − u ρ ).(4) If Λ is strongly associated with two pairs ( ϕ, ρ ) and ( ˜ ϕ, ρ ) whichshare the density ρ , then the full jet of ˜ ϕ − ϕ at x is a complexconstant.It follows that for any phase-density pair ( ϕ, ρ ) with a critical point x there exists a unique FOI Λ strongly associated with it and suchthat Λ = δ x . It is coordinate-independent because it is determinedby the coordinate-independent conditions (12) and (13). After somepreparations, we will give a formula for Λ in local coordinates.6. Operators on a space of formal jets
Let M be a real manifold of dimension n . Denote by J the space ofjets of infinite order on M supported at x ∈ M , which is equipped withthe decreasing filtration {J i } by the order of zero at x . The space J is complete with respect to this filtration. Denote by D ( k ) the space ofdifferential operators on J of order at most k . An element A ∈ D ( k ) isa linear mapping A : J → J such that ad( f ) . . . ad( f k ) A = 0 for any f i ∈ J , where ad( f ) A = [ f, A ] = f ◦ A − A ◦ f . Then D = ∞ [ k =0 D ( k ) is the algebra of differential operators of finite order on J . The filtra-tion on J induces a filtration {D i } , where i ∈ Z , on D . The filtrationdegree of an operator A ∈ D is the largest integer k such that A J r ⊂ J r + k for all r ≥
0. The filtration degree of a differential operator of order k is at least − k , D ( k ) ⊂ D − k . Each space D ( k ) is complete with respect tothis filtration, but D is not. The completion ˆ D of D contains differentialoperators of infinite order on J . Denote the filtration degree of f ∈ J and of A ∈ D by d ( f ) and d ( A ), respectively.Let N be the algebra of natural operators on J [[ ν ]], N := { A + νA + . . . | A r ∈ D ( r ) for all r ≥ } . ORMAL OSCILLATORY DISTRIBUTIONS 15
Clearly, ν k N ⊂ N for all k ≥
0. We consider the algebra N (( ν )) whoseelements are of the form ν k A , where k ∈ Z and A ∈ N , N (( ν )) = ∞ [ r =0 ν − r N . Notice that ν − N is a Lie algebra with respect to the commutator ofoperators and ν − N acts on N by the adjoint action: given A ∈ ν − N and B ∈ N , we have ad( A ) B = [ A, B ] ∈ N .We equip the algebra N (( ν )) with the following filtration. We set d ( ν ) = 2. The filtration degree of A ∈ ν r N written as A = ν r A + ν r +1 A + . . . with A k ∈ D ( k ) is d ( A ) = inf { r + k ) + d ( A k ) | k ≥ } . Since d ( A k ) ≥ − k , we get that 2( r + k ) + d ( A k ) ≥ r + k . Hence, d ( A ) ≥ r . We call this filtration on N (( ν )) and a similar filtrationon J (( ν )) the standard filtration. The algebra N is complete withrespect to the standard filtration, {N i } , but N (( ν )) and J (( ν )) arenot. Denote by A the completion of the algebra N (( ν )) with respectto the standard filtration and by F the completion of J (( ν )). Thealgebra A acts on F . The elements of A and F can be written ascertain series X r ∈ Z ν r A r and X r ∈ Z ν r f r , respectively, where A r ∈ ˆ D and f r ∈ J . Set g := { A ∈ ν − N | d ( A ) ≥ } ⊂ A . It is a pronilpotent Lie algebra whose Lie group exp g lies in A .Suppose that ( ϕ, ρ ) is a phase-density pair on M with a critical point x and U is a coordinate neighborhood of x with coordinates { x i } suchthat x i ( x ) = 0 for all i , that is, x = 0. We set h ij := ∂ϕ − ∂x i ∂x j (cid:12)(cid:12)(cid:12)(cid:12) x =0 . Then ( h ij ) is a symmetric nondegenerate complex matrix with constantentries. Let ( h ij ) be its inverse matrix. We set(14) ψ := 12 h ij x i x j and ∆ := − h ij ∂ ∂x i ∂x j . In [7], Lemma 9.1, we proved that the formal distribution(15) ˜Λ( f ) := e ν ∆ f (cid:12)(cid:12) x =0 is a FOI associated with the pair ( ν − ψ, dx ), where dx = dx . . . dx n isthe Lebesgue density on U . Lemma 6.1.
The FOI (15) is strongly associated with the pair ( ν − ψ, dx ) .Proof. It follows from formula (12) with v = x i ∂ i and ρ = dx that(16) ˜Λ (cid:0) x i ∂ i f + (cid:0) ν − ψ + n (cid:1) f (cid:1) = 0 , where we have used that vψ = 2 ψ and L v ρ = nρ . Replacing f with − h ij ∂ j f and setting v = ∂ i in (12), we get(17) ˜Λ (cid:18) ∆ f − ν − x i ∂ i f (cid:19) = 0 , where the summation on i is assumed. Dividing (16) by 2 ν and addingthe result to (17), we get(18) ˜Λ (cid:18) ∆ f + (cid:18) ν − ψ + 12 ν − n (cid:19) f (cid:19) = 0 . Now we verify (13) with ϕ = ν − ψ and ρ = dx using (18): ddν ˜Λ( f ) − ˜Λ (cid:18) dfdν − (cid:18) ν − ψ + 12 ν − n (cid:19) f (cid:19) =˜Λ (cid:18) ∆ f + ∂f∂ν (cid:19) − ˜Λ (cid:18) dfdν − (cid:18) ν − ψ + 12 ν − n (cid:19) f (cid:19) = 0 . (cid:3) Assume that locally ρ = e u dx, where u = u + νu + . . . ∈ C ∞ ( U )[[ ν ]]. We call the function χ ( x ) := ϕ ( x ) − ν − ψ − ϕ (0) + u ( x ) − u (0)the phase remainder . Since we will need only the jet of infinite orderof χ at x = 0, we identify χ = ν − χ − + χ + . . . with its jet. Theorder of zero of χ − and of χ at x = 0 is at least 3 and 1, respectively.Hence, χ ∈ F and therefore the operator exp χ acts on F . Since d ( ν ∆) = 0, the operator exp( ν ∆) acts on J (( ν )) and respects thestandard filtration. Thus, it also acts on F respecting the filtration.We define a formal distribution Λ on U supported at x = 0 by theformula(19) Λ( f ) := (cid:0) e ν ∆ e χ f (cid:1) (cid:12)(cid:12) x =0 . If f ∈ C ∞ ( U )[[ ν ]], then its jet at x = 0 lies in F . Hence, e ν ∆ e χ f ∈ F ,which implies that Λ( f ) ∈ C [[ ν ]] and therefore Λ = Λ + ν Λ + . . . (thecoefficients at the negative powers of ν in e ν ∆ e χ f vanish at x = 0because its filtration degree is nonnegative). ORMAL OSCILLATORY DISTRIBUTIONS 17
Proposition 6.1.
The formal distribution (19) is the unique FOI
Λ =Λ + ν Λ + . . . strongly associated with the pair ( ϕ, ρ ) and such that Λ = δ .Proof. It follows from [7], Theorem 9.1, that Λ is associated with thepair ( ϕ, ρ ) and Λ = δ . It remains to prove that it is strongly associatedwith ( ϕ, ρ ) or, equivalently, with the pair ( ν − ψ + χ, dx ). We will useLemma 6.1 and the fact that Λ( f ) = ˜Λ( e χ f ). We have ddν Λ( f ) = ddν ˜Λ( e χ f ) = ˜Λ (cid:18) ddν ( e χ f ) + (cid:18) − ψν − n ν (cid:19) ( e χ f ) (cid:19) =˜Λ (cid:18) e χ (cid:18) dfdν + (cid:18) − ψν + dχdν − n ν (cid:19) f (cid:19)(cid:19) =Λ (cid:18) dfdν + (cid:18) ddν ( ν − ψ + χ ) − n ν (cid:19) f (cid:19) . (cid:3) Identification of formal oscillatory integrals
Below we will prove the following theorem.
Theorem 7.1.
A formal distribution
Λ = Λ + ν Λ + . . . on a manifold M supported at a point x ∈ M is a FOI strongly associated with somepair ( ϕ, ρ ) with the critical point x and such that Λ = δ x if and onlyif Λ is a nondegenerate oscillatory distribution. Let ( h ij ) be a symmetric nondegenerate complex n × n matrix withconstant entries and ( h ij ) be its inverse matrix. We use the samenotations ψ and ∆ as in (14). Observe that ν ∆ and ν − ψ lie in ν − N and d ( ν ∆) = d ( ν − N ) = 0. Lemma 7.1.
The adjoint action of the operators ν ∆ and ν − ψ byderivations of the algebra N integrates to automorphisms of this alge-bra which respect the standard filtration and therefore extend to auto-morphisms of the algebras A and g and the Lie group exp g .Remark. The operator exp ν ∆ acts on the space F , but the operatorexp( ν − ψ ) is undefined on that space. Proof.
Given A = A + νA + . . . ∈ N , we have d ( A r ) ≥ − r , hence d ( ν r A r ) ≥ r , and therefore ν r A r ∈ N r for all r ≥
0. The action ofexp(ad( ν ∆)) maps ν r A r to e ad( ν ∆) ( ν r A r ) = ∞ X s =0 ν r + s s ! (ad(∆)) s ( A r ) ∈ N r . The action of exp(ad( ν − ψ )) maps ν r A r to e ad( ν − ψ ) ( ν r A r ) = r X s =0 s ! (ad( ν − ψ )) s ( ν r A r ) ∈ N r . It follows that e ad( ν ∆) ( A ) and e ad( ν − ψ ) ( A ) are elements of N , because N is complete with respect to the standard filtration. (cid:3) Now we will give a proof of Theorem 7.1.
Proof.
Fix local coordinates { x i } around x such that x i ( x ) = 0 forall i . Denote by b the Lie algebra of operators A ∈ g such that δ ◦ A = 0and by c the Lie algebra of operators from g with constant coefficients.Then g = b ⊕ c . Let ( ϕ, ρ ) be a phase-density pair on M with thecritical point x = 0 and χ be the corresponding phase remainder.Then (19) is the unique FOI strongly associated with ( ϕ, ρ ) and suchthat Λ = δ . Lemma 7.1 implies that e ad( ν ∆) ( e χ ) ∈ exp g . By Proposition 2.1, there exist unique elements B ∈ b and C ∈ c suchthat(20) e ad( ν ∆) ( e χ ) = e B e C . It follows thatΛ( f ) = (cid:0) e ν ∆ e χ f (cid:1) (cid:12)(cid:12) x =0 = (cid:0) e ν ∆ e χ e − ν ∆ e ν ∆ f (cid:1) (cid:12)(cid:12) x =0 = (cid:0) e ad( ν ∆) ( e χ ) e ν ∆ f (cid:1) (cid:12)(cid:12) x =0 = (cid:0) e B e C e ν ∆ f (cid:1) (cid:12)(cid:12) x =0 = (cid:0) e ν ∆+ C f (cid:1) | x =0 , where we have used that the operators with constant coefficients ν ∆and C commute. The operator C can be written as C = ν − ( X + νX + . . . ) , where X r has constant coefficients, is of order at most r , and whosefiltration degree is at least 3 − r for all r . It follows that X = X = 0and X is of order at most 1. We see that ν ∆ + C = ν − (cid:0) ν (∆ + X ) + ν X + ν X + . . . (cid:1) ∈ ν − N and the operator ∆ + X can be written in coordinates as(21) − h ij ∂ i ∂ j + b i ∂ i + c. Since the matrix ( h ij ) is nondegenerate, the FOI Λ is a nondegenerateoscillatory distribution.Now suppose that Λ is a nondegenerate oscillatory distribution on amanifold M supported at x ∈ M . Fix local coordinates { x i } around x such that x i ( x ) = 0 for all i . According to Proposition 3.1, there ORMAL OSCILLATORY DISTRIBUTIONS 19 exists a unique natural operator with constant coefficients X = ν X + ν X + . . . such that Λ = δ ◦ exp( ν − X ) . If we write X as (21), where ( h ij ) is a symmetric matrix with constantentries, then this matrix is nondegenerate because Λ is a nondegenerateoscillatory distribution. We will have that C := ν − X + ν h ij ∂ i ∂ j ∈ c . Let ( h ij ) be the matrix inverse to ( h ij ). We will use the settings (14)and will show that there exists a ν -formal jet χ = ν − χ − + χ + . . . at x = 0 of positive filtration degree such that (20) holds for some B ∈ b . It will mean that Λ is a FOI at x = 0 strongly associated withthe phase-density pair ( ν − ψ + χ, dx ) .Denote by e the Lie algebra of operators from g that can be writtenas A = ∂ i ◦ A i for some formal differential operators A i . If we use the standard trans-position A A t of differential operators such that ( ∂ i ) t = − ∂ t and( x i ) t = x i , then A ∈ e if A ∈ g and A t annihilates constants, A t f the Lie algebra of multiplication operators from g . Then g = e ⊕ f . A simple calculation shows that e − ad( ν ∆) ( x k ) = x k + νh kl ∂∂x l and e ad( ν − ψ ) e − ad( ν ∆) ( x k ) = νh kl ∂∂x l . Therefore, the conjugation A e ad( ν − ψ ) e − ad( ν ∆) ( A )provides isomorphisms of the Lie algebra b onto e and of the Lie groupexp b onto exp e . By Proposition 2.1, there exist unique elements E ∈ e and χ ∈ f such that e ad( ν − ψ ) (cid:0) e C (cid:1) = e E e χ . Acting on both sides by exp(ad( ν ∆)) exp( − ad( ν − ψ )), we get e C = (cid:16) e ad( ν ∆) e ad( − ν − ψ ) (cid:0) e E (cid:1)(cid:17) (cid:0) e ad( ν ∆) ( e χ ) (cid:1) , which implies (20) if we set B := − e ad( ν ∆) e ad( − ν − ψ ) ( E ) ∈ b . It completes the proof of the theorem. (cid:3) By Borel’s lemma it suffices to give only the jet of infinite order of the phase at x = 0. It is interesting to notice that Theorem 7.1 and Proposition 3.1 in [7]imply that if Λ is a nondegenerate oscillatory distribution supportedat x , then the pairing f, g Λ( f g )on the space of formal jets J [[ ν ]] is nondegenerate.One of the consequences of Theorems 4.1 and 7.1 is that Fedosov’sstar product is given by some formal oscillatory integral (the distribu-tion Λ x for a Fedosov’s star product is nondegenerate for any x because C ( f, g ) = π ij ∂ i f ∂ j g , where π ij is a nondegenerate Poisson tensor).However, Fedosov’s construction does not use any oscillatory integralformulas. Only in the simplest case of the Moyal-Weyl star productit is given by the asymptotic expansion of a known oscillatory integral(and hence by a formal oscillatory integral). References [1] Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., and Sternheimer, D.:Deformation theory and quantization. I. Deformations of symplectic structures.
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