Coherent states, quantum gravity and the Born-Oppenheimer approximation, II: Compact Lie Groups
CCoherent states, quantum gravity and the Born-Oppenheimerapproximation, II: Compact Lie Groups
Alexander Stottmeister a) and Thomas Thiemann b) Institut für Quantengravitation, Lehrstuhl für Theoretische Physik III,Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7/B2, D-91058 Erlangen,Germany
In this article, the second of three, we discuss and develop the basis of a Weyl quantisationfor compact Lie groups aiming at loop quantum gravity-type models. This Weyl quantisa-tion may serve as the main mathematical tool to implement the program of space adiabaticperturbation theory in such models. As we already argued in our first article, space adia-batic perturbation theory offers an ideal framework to overcome the obstacles that hinderthe direct implementation of the conventional Born-Oppenheimer approach in the canon-ical formulation of loop quantum gravity. Additionally, we conjecture the existence of anew form of the Segal-Bargmann-Hall “coherent state” transform for compact Lie groups G , which we prove for G = U (1) n and support by numerical evidence for G = SU (2).The reason for conjoining this conjecture with the main topic of this article originates inthe observation, that the coherent state transform can be used as a basic building blockof a coherent state quantisation (Berezin quantisation) for compact Lie groups G . But,as Weyl and Berezin quantisation for R d are intimately related by heat kernel evolution,it is natural to ask, whether a similar connection exists for compact Lie groups, as well.Moreover, since the formulation of space adiabatic perturbation theory requires a (deforma-tion) quantisation as minimal input, we analyse the question to what extent the coherentstate quantisation, defined by the Segal-Bargmann-Hall transform, can serve as basis of theformer. CONTENTS
I. Introduction II. A new look at the Segal-Bargmann-Hall “coherent state” transform
III. On Weyl and Kohn-Nirenberg calculi for compact Lie groups G = U (1) and an extension to R Bohr a) Electronic mail: [email protected] b) Electronic mail: [email protected] a r X i v : . [ m a t h - ph ] A p r IV. Conclusions and perspectives Acknowledgments V. Bibliography I. INTRODUCTION
We have argued in our first article that a realisation of the (time-dependent) Born-Oppenheimerapproximation for multi-scale quantum dynamical systems, which are modelled by techniques usedin loop quantum gravity, might be achieved along the lines of space adiabatic perturbation theory .This is, because space adiabatic perturbation theory avoids some technical limitations of the orig-inal Born-Oppenheimer approach, which in turn allows us to circumvent the so-called “problemof non-commutative fast-slow coupling” (originally pointed out in the context of loop quantumgravity ). The main technical tool, necessary for a successful implementation of the ideas behindspace adiabatic perturbation theory, is a Weyl quantisation associated with part of the multi-scalequantum system. More precisely, if the total quantum system is described as a coupled systemdecomposing into two sectors (for simplicity), one of which is called the slow sector, and the otherone the fast sector (thinking of different relevant time scales), we will require the existence of aWeyl quantisation (in the sense of a real deformation quantisation) for the description of the slowsubsystem. Then, if we further assume that the Weyl quantisation is scalable with a parameter ε quantifying the separation of scales between the subsystems, we can introduce a systematic per-turbation theory in the sense of Born and Oppenheimer by means of dequantising the slow sectorand exploiting the induced ε -dependent ? -product in the resulting function spaces (in analogy withstandard pseudo-differential operators and their symbolic calculus).This said, it is the primary purpose of the present article to investigate the possibility of formulatinga Weyl quantisation suitable for phase spaces of the type T ∗ G , G a compact Lie group. The reasonbehind this objective is that such phase space serve as the main building block of loop quantumgravity-type models.Before we come to the main part of the article, which is composed of three sections, let us brieflyoutline its structure and content:Section II introduces a new form of the Segal-Bargmann-Hall “coherent state” transform for com-pact Lie groups. This is motivated by the fact that unitary maps of this type provide the basisfor the construction of a coherent state quantisation (also known as Berezin or Wick/Anti-Wickquantisation) of the co-tangent bundle, T ∗ G , of a compact Lie group G . But, before we enterinto the discussion of the coherent state transform, we recall the definition of crossed product C ∗ -algebras, paying special attention to the transformation group C ∗ -algebra C ( G ) (cid:111) L G , as the latteris intimately connected with the Hilbert space L ( G ), of which the coherent states are specific ele-ments. Moreover, in foresight of section III, introducing the C ∗ -algebra C ( G ) (cid:111) L G is convenient,because it is fundamental to the construction of the Weyl quantisation for compact Lie groups, weare aiming at. A relation between coherent state quantisation and Weyl quantisation for compactLie groups will be established subsection III C (with regard to their application to space adiabaticperturbation theory). For completeness, we provide the essential ingredients necessary to definethe coherent states for compact Lie groups introduced by Hall , as well.In section III, we present possible ways to obtain Weyl (and Kohn-Nirenberg) quantisations for com-pact Lie groups. To this end, we follow the common philosophy of constructing quantisationsof functions (pseudo-differential operators) from left (or right) convolution kernels. Although, ourconstructions, which are based on results of Turunen and Rhuzhansky and Landsman , succeedto some extent, we are forced to deal with the dichotomy of choosing between local and globalstructures at various points. The latter can be traced back to the fact that the exponential mapof a compact Lie group G , while still being onto, is no longer a diffeomorphims like in the case of R d (or nilpotent Lie groups in general). It appears, that the global formulas are easier to handle,when it comes to the composition of pseudo-differential operators (at least in the Kohn-Nirenbergformalism). But, it is the local setting, which is well adapted to deal with a semi-classical approxi-mation of the commutation relations underlying the transformation group algebra C ( G ) (cid:111) L G , andallows for a more direct analogy with the original treatment of space adiabatic perturbation theoryin . The main difficulty with the global formulas can be reduced to a lack of scale transformationscompatible with the quantisation formulas and the algebraic structure of C ( G ) (cid:111) L G , i.e. we aremissing a simple relation between pseudo-differential operators with different values of the quan-tisation parameter ε (the adiabatic parameter in space adiabatic perturbation theory). Therefore,while asymptotic expansions of pseudo-differential operators are still conceivable in the global set-ting, a simple ordering in terms of powers of ε is bound to fail. In subsection III B and subsectionIII D, we further elaborate on this aspect: Namely, we relate the latter to the necessity of having a ε -scaleable Fourier transform on G . Then, we show that a partial solution can be achieved, if theso-called Stratonovich-Weyl transform for G is invoked, but, that a general solution seems to beobstructed by the rigidity of the representation theory of G (integrality of highest weights). Only,in the case of G = U (1) n , we are able to proceed further by, first, lifting everything to the universalcovering group R n , and, second, passing to the Bohr compactification R n Bohr . The resulting theoryof pseudo-differential operators connected with R Bohr is discussed and compared to the theory ofalmost-periodic pseudo-differential operators in subsection III D, as well.Finally, we present some concluding remarks and perspectives in section IV.
II. A NEW LOOK AT THE SEGAL-BARGMANN-HALL “COHERENT STATE” TRANSFORM
Although, this article is mainly concerned with the development of a Weyl quantisation forloop quantum gravity-type models, it was already noted in the preceding article (cf. also )that the (Stratonovich-)Weyl and Wick/Anti-Wick (de-)quantisations are closely related from aconceptual point of view (cf. for a lucid overview), especially regarding their potential applicabilityin the context of space adiabatic perturbation theory (cf. ). In respect of their importance forWick/Anti-Wick correspondences, we give a short review of the construction of coherent states oncompact Lie Groups by Hall and conjecture a new version of the Segal-Bargmann-Hall transform(or resolution of unity) that is free from the “dual” heat kernel measure ν on the complexification G C Lie group G . We prove the conjecture in the case G = U (1) and support it by some numericalevidence for G = SU (2).Since our constructions can be related to the so-called crossed product C ∗ -algebra C ( G ) (cid:111) L G ,which will also be important for the other sections, we briefly recall its construction (cf. ). A. Locally compact groups and crossed products
Given a locally compact group G , a C ∗ -algebra A and a (strongly) continuous representation α : G → Aut( A ), one makes the Definition II.1:
The triple ( A , G, α ) is called a ( C ∗ -) dynamical system . A covariant representation of ( A , G, α ) isa triple ( H , π, U ) consisting of a (non-degenerate) representation π : A → B ( H ) and (strongly)continuous unitary representation U : G → U ( H ) , s.t. ∀ g ∈ G, A ∈ A : π ( α g ( A )) = U g π ( A ) U ∗ g . (2.1)Let us illustrate this definition by giving a few Examples II.2:
1. Clearly, ( A , { e } , id) and ( C , G, id) are trivial examples of dynamical systems. The covariantrepresentations of these correspond to (non-degenerate) representations of A in the former,and to strongly continuous unitary representation of G in the latter case.2. The left action L of G on itself gives rise to a continuous representation α L : G → Aut( C ( G ))on the C ∗ -algebra of continuous functions on G vanishing at infinity by (cf. , Lemma 2.5.) ∀ f ∈ C ( G ) , g ∈ G : α L ( g )( f ) = L ∗ g − f. (2.2)An important covariant representation of the dynamical system ( C ( G ) , G, α L ) comes fromthe multiplier representation M : C ( M ) → B ( L ( G )) and the left regular representation λ : G → U ( L ( G )), where L ( G ) is defined with respect to a (left) Haar measure. Thecompatibility of the pair ( M, λ ) reflects the covariance condition (2.1): ∀ f ∈ C ( G ) , g ∈ G : M( α L ( g )( f )) = λ g M( f ) λ ∗ g . (2.3)3. In analogy with the preceding example, we consider the C ∗ -algebra C b ( R ) of bounded con-tinuous function on R , and the (left) action τ of R on itself by translations. The triple( C b ( R ) , R , ( τ − ) ∗ ) forms a dynamical system, and the triple ( L ( R ) , M , λ ) is a covariant rep-resentation. Now, let us introduce the notation U ( x ) := M( e ix ( . ) ) , V ( ξ ) := λ − ξ , x, ξ ∈ R .This family of unitary operators in L ( R ) satisfies the canonical commutation relations inWeyl form: V ( ξ ) U ( x ) V ( ξ ) ∗ = λ − ξ M( e ix ( . ) ) λ ξ (2.4)= M( α L ( − ξ )( e ix ( . ) ))= M( e ix (( . )+ ξ ) )= e ixξ M( e ix ( . ) )= e ixξ U ( x ) . Thus, we obtain the Weyl algebra of 1-particle quantum systems as a dynamical system.The importance of example 2 will become clear after the definition of the crossed product C ∗ -algebra A (cid:111) α G associated with a dynamical system, and the statement of theorem II.7, which willbe related to our conjecture of a new Segal-Bargmann-Hall transform for compact Lie groups. But,before we turn to the definition of the crossed product, we add an remark on dynamical systemswith commutative A (cf. , especially Proposition 2.7.). Remark II.3:
The Gelfand-Naimark theorem (cf. ) tells us that A ∼ = C ( X ), where X is the set of characters of A with the locally compact Hausdorff weak ∗ -topology. Thus, the dynamical system ( A , G, α ) is iso-morphic to the dynamical system ( C ( X ) , G, α ). But, a dynamical system of the form ( C ( X ) , G, α )comes from a left G -space ( X, G ), which is why these are called transformation group C ∗ -algebras ,and we may consider the fibration, p : X → G \ X , of X over the space of (left) orbits G \ X . Since ev-ery G -orbit G · x, x ∈ X can be identified with a quotient G/H x , where H x is the stabiliser subgroupof x ∈ X , it is possible to associate dynamical systems ( C ( G/H G · x ) , G, α G/H G · x ) , G · x ∈ G \ X with sufficiently regular fibrations.Interestingly, example 3, above, tells us that the associated R -space is not R acting on itself bytranslations, but rather ( β R , R ), i.e. an action of R on its Stone-Čech compactification β R , since C b ( R ) ∼ = C ( β R ).Coming to the discussion of the crossed product of a dynamical system ( A , G, α ), we indicate that itwill be a C ∗ -algebra, denoted by A (cid:111) α G , such that its (non-degenerate) representations correspondin a one-to-one fashion to covariant representations of the dynamical system. Furthermore, thecrossed product is built in close analogy with the group C ∗ -algebra C ∗ ( G ), which turns out to bethe special case C (cid:111) id G . Definition II.4 (cf. , Definition 2.7.2.): Given a dynamical system ( A , G, α ) , we denote by dg and ∆ , respectively, a (left) Haar measureand its modular function on G . The completion of the pre-Banach *-algebra C c ( G, A ) , equippedwith1. (multiplication, twisted convolution) ( x ∗ y )( g ) := Z G x ( h ) α h ( y ( h − g )) dh, (2.5)
2. (involution) x ∗ ( g ) := ∆( g ) − α g ( x ( g − )) ∗ , (2.6)
3. (norm) || x || := Z G || x ( h ) || A dh, x, y ∈ C c ( G, A ) , g ∈ G, (2.7) is L ( G, A ) , the convolution Banach *-algebra of ( A , G, α ) . Next, we need a C ∗ -norm on L ( G, A ) to define A (cid:111) α G . Lemma II.5 (cf. , p. 138, and , p. 52): || x || := sup {|| π ( x ) || | π : L ( G, A ) → B ( H π ) − a Hilbert space representation } (2.8) defines a C ∗ -seminorm on L ( G, A ) , called the universal norm . The universal norm || . || isdominated by || . || . The completion of L ( G, A ) w.r.t. || . || is a C ∗ -algebra. Definition II.6:
The C ∗ -algebra L ( G, A ) || . || is called the ( C ∗ -)crossed product , A (cid:111) α G , of ( A , G, α ) . The one-to-one correspondence between covariant representations ( H , π, U ) of ( A , G, α ) and (non-degenerate) representations ( H , ρ ) of A (cid:111) α G is achieved via the integrated form of the former (cf. ,Proposition 2.40.): ρ ( x ) := Z G π ( x ( h )) U h dh, x ∈ C c ( G, A ) . (2.9)As we already mentioned above, example 2 is closely related to our conjectured new Segal-Bargmann-Hall transform for L ( G ), where G is a compact Lie group, which is why we statethe following theorem concerning the properties of C ( G ) (cid:111) L G , and its representation coming fromthe covariant representation ( L ( G ) , M , λ ). Theorem II.7 (Stone-von Neumann, cf. , Theorem 4.24.): Given a locally compact group G , we have C ( G ) (cid:111) L G ∼ = K ( L ( G )) . (2.10) Moreover, the integrated form of the covariant representation ( L ( G ) , M , λ ) of ( C ( G ) , G, α L ) is afaithful irreducible representation of C ( G ) (cid:111) L G onto K ( L ( G )) . B. Covariant coherent states
In this subsection, we recall the definition of the coherent states given by Hall for compactLie groups. To this end, let G denote an arbitrary compact Lie group (dim G = n ) with Lie algebra g . On g , we fix an Ad -invariant inner product h , i g , e.g. the negative of the Killing form, andwe normalise the Haar measure dg on G to coincide with the Riemannian volume measure comingfrom former. The inner product on g gives rise to a Laplace-Beltrami operator ∆ on G , which is aCasimir operator for G , and the associated heat equation, ∂ t f t = 12 ∆ f t , (2.11)has a fundamental solution ρ t , t >
0, the heat kernel , at the identity in G . ρ t has a series expansionin terms of representation theoretical data of G : ρ t ( g ) = X π ∈ ˆ G d π e − t λ π χ π ( g ) , (2.12)where ˆ G is the set of isomorphism classes of irreducible unitary representations of G , d π is thedimension, λ π the value of the Casimir operator ∆ and χ π the character of π ∈ ˆ G . An importantproperty of ρ t is, that it is a strictly positive C ∞ -class-function on G for t > G has a unique complexification G C (cf. ) with Lie algebra g C , which is thecomplexification of g . The inner product h , i g extends to a real-valued inner product on g C via h X + iY , X + iY i g C = h X , X i g + h Y , Y i g , X , X , Y , Y ∈ g , (2.13)giving rise to left-invariant Riemannian metric on G C . As for G , the Haar measure dz on G C isnormalised w.r.t. to the Riemannian volume measure coming from h , i g C . There is also a uniqueantiholomorphic antiautomorphism ∀ z ∈ G C : z z ∗ extending the inversion ∀ g ∈ G : g g − on G . It is related to the inversion on G C via complex conjugation ∀ z ∈ G C : z ∗ = ¯ z − .The coherent states for G are constructed as result of the observation that ρ t admits a uniqueanalytic continuation from G to G C (proved in ). In terms of the series expansion, one makes the Definition II.8:
The functions Ψ tz ( g ) := ρ t ( g − z ) = X π ∈ ˆ G d π e − t λ π χ π ( g − z ) , g ∈ G, z ∈ G C . (2.14) are called (covariant) coherent states for G . Here, “covariance” refers to the behaviour of Ψ tz underthe (left) regular representation of G : ( λ h Ψ tz )( g ) = Ψ tz ( h − g ) (2.15)= ρ t (( h − g ) − z )= ρ t ( g − ( hz ))= Ψ thz ( g ) . Clearly, this G -action on the set coherent states extends to a (simply) transitive, although non-unitary w.r.t. L ( G ) , G C -action. Before we comment on further properties of these functions, we need to introduce some furthernotation (following closely ). Namely, we need an analogue of ρ t on the G C -homogeneous space G C /G . The latter admits a unique G C -invariant Riemannian structure, which agrees at the identitycoset with the restriction of h , i g C to i g ⊂ g C . The analogue of ρ t is the fundamental solution ν t at the identity coset of the heat equation, ∂ t f t = 14 ∆ f t , (2.16)on G C /G . ν t can be identified with a left and right G -invariant function, also denoted ν t , on G C ,which is normalised as Z G C ν t ( z ) dz = vol( G ) . (2.17)In addition to the structures introduced so far, we need some typical objects from the structuretheory of compact Lie groups. That is, we fix a maximal torus T ⊂ G with Lie algebra t , and realroots R ⊂ t ∗ ∼ = t , α ∈ R : ⇔ α = 0 , ∃ = X ∈ g C : ∀ H ∈ t : [ H, X ] = 2 πiα ( H ) X. (2.18)Furthermore, we pick a set of positive roots R + and denote by δ = P α ∈ R + α half the sum of thepositive roots. W is the Weyl group of T , C a fundamental Weyl chamber and Γ ⊂ t the kernel ofthe exponential map restricted to t .For G C , we invoke the (right) polar decomposition z = ge iX ∈ G C , g ∈ G, X ∈ g , which gives adiffeomorphism, Φ : T ∗ G ∼ = G × g −→ G C , Φ( g, X ) = ge iX , (2.19)that turns the “phase space”, T ∗ G , in a natural way into a Kähler manifold. The Haar measure dz on G C and the Liouville measure dg dX on T ∗ G ∼ = G × g (by right translation), the latter beingthe product of the Haar measure on G and the Lebesgue dX , normalised by means of h , i , on g ,fit together in the following way (cf. , Lemma 5): ∀ f ∈ C c ( G C ) : Z G C f ( z ) dz = Z G Z g f ( ge iX ) dg σ ( X ) dX. (2.20)Here, σ is the Ad - G -invariant function on g determined by ∀ H ∈ t : σ ( H ) = Y α ∈ R + (cid:18) sinh α ( H ) α ( H ) (cid:19) . (2.21)The explicit formula for the measure ν t ( z ) dz under (right) polar decomposition is (cf. , Lemma5): ν t ( z ) dz = 1( πt ) n e −| δ | g ∗ t e − t | X | g η ( X ) dg dX, (2.22)where η is an analytic square root of σ : ∀ H ∈ t : η ( H ) = Y α ∈ R + sinh α ( H ) α ( H ) . (2.23)Finally, it is important to observe that every element z ∈ G C has as decomposition of the form z = ge iH h, g, h ∈ G, H ∈ t , (2.24)because every element in G is conjugate to some element in the maximal torus T .Let us now come back to the coherent states (2.14) and their properties. Firstly, they belong tothe Hilbert space L ( G ), which follows from the (analytically continued) heat kernel identity (cf. ,Theorem 6): (Ψ tz , Ψ tz ) L = Z G ρ t ( g − z ) ρ t ( g − z ) dg (2.25)= ρ t ( z ¯ z ) < ∞ . Moreover, one has an explicit formula for the norm of the coherent states (cf. , Eq. 8) derivedfrom Urakawa’s Poisson summation formula for the restriction of ρ t to the maximal torus T :(Ψ tz , Ψ tz ) L = ρ t ( z − ¯ z ) = ρ t (( z ∗ z ) − ) (2.26) (2.24) = ρ t ( h − e − iH h )= ρ t ( e − iH ) , ρ t ( e − iH ) = 1(4 πt ) n e | δ | g ∗ e t | H | g η ( H ) − (2.27) × X γ ∈ Γ ∩ C e iδ ( γ ) e − t | γ | g P γ ∈ W · γ e − it h γ,H i g Q α ∈ R + α (cid:0) H + i γ (cid:1)Q α ∈ R + α ( H ) . Secondly, the coherent states provide a resolution of unity in L ( G ), thus providing a unitarytransformation, the (anti-)Segal-Bargmann-Hall transform, L ( G ) → H L ( G C , ν t ) , ∀ z ∈ G C : Φ (Ψ tz , Φ) L , (2.28)which maps square integrable functions on G isometrically onto antiholomorphic square integrable,w.r.t. ν t ( z ) dz , functions on G C : ∀ Φ , Φ ∈ L ( G ) : (Φ , Φ ) L = Z G C (Φ , Ψ tz ) L (Ψ tz , Φ ) L ν t ( z ) dz. (2.29)Alternatively, we will use the mnemonic (in the weak sense, using Dirac’s notation): = Z G C | Ψ tz ih Ψ tz | ν t ( z ) dz. (2.30)
1. A new Segal-Bargmann-Hall “coherent state” transform
Although, the coherent states (2.14) can be thought of as a generalisation of the standard coherentstates in L ( R n ) there is a certain asymmetry, already pointed out in , which results from thecoherent states not being normalised, as would be standard in quantum physical treatments due tothe need for a probabilistic interpretation of the “overlap functions” (Φ , Φ ) L , Φ , Φ ∈ L ( G ).Furthermore, the measure ν t ( z ) dz is not proportional to t -scaled Liouville measure (2 πt ) − n dg dX as one would expect in relation to the “correspondence principle”. With regard to the resolution ofunity (2.30), one has in the R n -case ( R n C = C n , z = x + ip ): = Z C n | Ψ tz ih Ψ tz | ν t ( z ) dz (2.31)= C t Z C n | Ψ tz ih Ψ tz | ( h Ψ tz | Ψ tz i ) − dz, where Ψ tz ( x ) = (2 πt ) − n e − t ( z − x ) , ν t ( z ) = (4 πt ) − n e − t = ( z ) , dz = d n x n d n p and C t = (4 πt ) − n .The equality between the first and second line in (2.31) follows immediately, because C t ((Ψ tz , Ψ tz ) L ) − = ν t ( z ) . (2.32)But, the interesting point about (2.31) is, that we already know that the first line carries over toarbitrary compact Lie groups, while the second line can be written down for arbitrary compact Liegroups as well, as it involves no additional structures. Moreover the constant C t may be computefrom the norm of the coherent states (Ψ tz , Ψ tz ) L as a function of z ∈ C n , explicitly it is obtained0as an integral over the imaginary directions in C : C − t = Z R n ((Ψ tz , Ψ tz ) L ) − n d n = z. (2.33)Unfortunately, equation (2.32) is not valid for arbitrary compact Lie groups, but, as we will arguebelow, the bounds on (Ψ tz , Ψ tz ) L given in suffice to make sense out of an analogue of the secondline of (2.31). Having made this observation, we come to the main conjecture of this section. Conjecture II.9:
Given an arbitrary Lie group G , there exists a resolution of unity = C t Z G Z g | Ψ t Φ( g,X ) ih Ψ t Φ( g,X ) | ( h Ψ t Φ( g,X ) | Ψ t Φ( g,X ) i ) − dg dX, (2.34) C − t = vol( G ) Z g ( h Ψ t Φ( g,X ) | Ψ t Φ( g,X ) i ) − dX ∝ t − n . for small enough t > . For commutative G or G = SU (2) (2.34) holds for all t > . We notice, that in contrast to (2.30) the resolution of unity (2.34) lives on the phase space T ∗ G ∼ = G × g , which is natural from a quantum physical perspective.A possible strategy for a proof could be provided by the fact, that L ( G ) is an irreducible rep-resentation of the transformation group C ∗ -algebra C ( G ) (cid:111) L G (see theorem II.7). Thus, if theoperator defined by the right hand side of the first line of (2.34) commuted with all representativesof C ( G ) (cid:111) L G , the conjecture would be (partly) proved by an appeal to Schur’s lemma.Before, we argue in the favour of the conjecture, and prove it for G = U (1), we obtain as a corollaryan extension to (connected) Lie groups of compact type, i.e. those that admit an Ad -invariant innerproduct on their Lie algebras. The structure of these Lie groups is clarified by Proposition II.10 (cf. , Proposition 2.2.): Given a connected Lie group K of compact type with Ad -invariant inner product h , i on its Liealgebra k , there exists a compact connected Lie group G and natural number n ∈ N , such that K ∼ = G × R n (2.35) as Lie groups and the associated Lie algebra isomorphism k ∼ = g × R n is orthogonal. This implies
Corollary II.11:
The resolution of unity (2.34) holds for arbitrary (connected) Lie groups K ∼ = G × R n of compacttype, if the coherent states for K are chosen as product states of the coherent states for G and thestandard coherent states for R n . The Liouville measure on K is then the product measure of theLiouville measures on G and R n , respectively. We conclude this section with an extended
Remark II.12 (on Conjecture II.9):
From (2.20), (2.22), (2.26) and (2.27), we have the following formula for the product of (Ψ tz , Ψ tz ) L ν t ( z ):(Ψ tz , Ψ tz ) L ν t ( z ) z = ge iH h = (2 πt ) − n σ ( H ) − (2.36) × X γ ∈ Γ ∩ C e iδ ( γ ) e − t | γ | g P γ ∈ W · γ e − it h γ,H i g Q α ∈ R + α (cid:0) H + i γ (cid:1)Q α ∈ R + α ( H ) . Now, Hall shows in , Proposition 3, that the absolute value of sum over γ ∈ W · γ is boundedfrom above by an expression P ( | γ | g / √ t ) Q α ∈ R + α ( H ), where P is a polynomial of degree equal totwice the number of positive roots R + . This, immediately, leads to the conclusion that, for smallenough t >
0, we have constants a t , b t > t → a t = 1 = lim t → b t exponentially fast ,such that (2 πt ) − n b t ≤ (Ψ t Φ( g,X ) , Ψ t Φ( g,X ) ) L ν t ( X ) σ ( X ) ≤ (2 πt ) − n a t . (2.37)This shows the equivalence of the measures ν t ( X ) σ ( X ) dg dX and ((Ψ t Φ( g,X ) , Ψ t Φ( g,X ) ) L ) − dg dX on G × g , and due to the finiteness and positivity of a t , b t , we know that the integrals in (2.34)makes sense. Namely, for all Φ ∈ L ( G ) we have:(2 πt ) n a t || (Ψ t ( . ) , Φ ) L || L ( G C ,ν t ) ≤ || (Ψ t Φ( . ) , Φ ) L || L ( G × g , ( || Ψ t || L ) − ) (2.38) ≤ (2 πt ) n b t || (Ψ t ( . ) , Φ ) L || L ( G C ,ν t ) ⇒ (Ψ t ( . ) , Φ ) L ∈ H L ( G C , ν t ) ⇔ (Ψ t Φ( . ) , Φ ) L ∈ H L ( G × g , ( || Ψ t Φ( . ) || L ) − )Next, we analyse the operator A t := C t Z G Z g | Ψ t Φ( g,X ) ih Ψ t Φ( g,X ) | ( h Ψ t Φ( g,X ) | Ψ t Φ( g,X ) i ) − dg dX (2.39)in some detail. To this end, we introduce the representative functions ∀ π ∈ ˆ G, m, n = 1 , ..., d π : h π, m, n | g i = π ( g ) mn , g ∈ G, (2.40)and find, because h π, m, n | Ψ t Φ( g,X ) i = e − t λ π π ( ge iX ) mn , h π, m, n | A t | π , m , n i (2.41)= C t Z G Z g h π, m, n | Ψ t Φ( g,X ) ih Ψ t Φ( g,X ) | π , m , n i ( h Ψ t Φ( g,X ) | Ψ t Φ( g,X ) i ) − dg dX = C t Z G Z g ( ρ t ( e − i X )) − e − t ( λ π + λ π ) π ( ge iX ) mn π ( ge iX ) m n dg dX C t Z g ( ρ t ( e − i X )) − e − t ( λ π + λ π ) d π ,d π X k,k =1 π ( e iX ) kn π ( e iX ) k n Z G π ( g ) mk π ( g ) m k dg dX = C t vol( G ) d − π δ π,π δ m,m Z g ( ρ t ( e − i X )) − e − tλ π π ( e i X ) n n dX. Defining an operator A tπ ∈ End( V π ) by the matrix elements( A tπ ) n n = Z g ( ρ t ( e − i X )) − e − tλ π π ( e i X ) n n dX, n, n = 1 , ..., d π , (2.42)we see that it commutes with π ( g ) , g ∈ G , because ρ t is an even class function and dX is Ad -invariant:( π ( g ) A tπ ) mn = d π X k =1 π ( g ) mk ( A tπ ) kn = d π X k =1 π ( g ) mk Z g ( ρ t ( e − i X )) − e − tλ π π ( e i X ) kn dX (2.43)= d π X k =1 Z g ( ρ t ( e − i X )) − e − tλ π π ( ge i X g − ) mk π ( g ) kn dX = d π X k =1 Z g ( ρ t ( e − i X )) − e − tλ π π ( e i Ad g ( X ) ) mk π ( g ) kn dX = d π X k =1 Z g ( ρ t ( e − i Ad g − ( X ) )) − e − tλ π π ( e i X ) mk π ( g ) kn dX = d π X k =1 Z g ( ρ t ( g − e − i X g )) − e − tλ π π ( e i X ) mk π ( g ) kn dX = d π X k =1 Z g ( ρ t ( e i X )) − e − tλ π π ( e i X ) mk π ( g ) kn dX = d π X k =1 ( A tπ ) mk π ( g ) kn = ( A tπ π ( g )) mn . Thus, by Schur’s lemma we have A tπ = d − π tr( A tπ ) , and our conjecture is equivalent to the formula: C − t d π = tr( A tπ ) = Z g ( ρ t ( e i X )) − e − tλ π χ π ( e i X ) dX = Z g e − tλ π χ π ( e i X ) P π ∈ ˆ G d π e − tλ π χ π ( e i X ) dX (2.44)= Z g χ π ( e t ∆ e i X ) P π ∈ ˆ G d π χ π ( e t ∆ e i X ) dX = Z g χ π ( e t ∆+ i X ) P π ∈ ˆ G d π χ π ( e ∆+ i X ) dX = Z g χ π ( e ( tτ + iX )2 t + X t ) P π ∈ ˆ G d π χ π ( e ( tτ + iX )2 t + X t ) dX = Z g χ π ( e ( tτ + iX )2 t ) P π ∈ ˆ G d π χ π ( e ( tτ + iX )2 t ) dX Z g tr V π ( e ( tτ + iX )2 t )tr L ( G ) ( e ( tτ + iX )2 t ) dX. Here, we introduced the object τ = P ni =1 τ i ⊗ τ i for some orthonormal basis { τ i } i =1 ,...,n ⊂ g , suchthat τ · τ = P ni,j =1 τ i τ j h τ i , τ j i g = P ni =1 τ i = ∆ and τ · X = P ni =1 τ i X j h τ i , τ j i g . From a physicist’spoint of view, the last line in (2.44) is especially attractive, because it resembles the average over g of the Boltzmann-like distribution p ( V π | L ( G )) ( X ) := tr V π ( e ( tτ + iX )2 t )tr L ( G ) ( e ( tτ + iX )2 t ) , X π ∈ ˆ G d π p ( V π | L ( G )) ( X ) = 1 X ∈ g . (2.45)Since characters, χ π , π ∈ ˆ G , and the (analytically continued) heat kernel ρ t are class functions, wemay further simplify the expression (2.44) by Weyl’s integration formula on g (cf. ): ∀ f ∈ C c ( g ) : Z g f ( X ) dX = Z C ∗ Y α ∈ R + α ( H ) Z G f ( Ad g ( H )) dg dH (2.46)= 1 | W | Z t Y α ∈ R + α ( H ) Z G f ( Ad g ( H )) dg dH. Here, C ∗ ⊂ t corresponds to the Weyl chamber C via h , i , | W | is the number of elements in W ,and dH is a suitably normalised Lebesgue measure on t . Thus, we find for (2.44): C − t d π = tr( A tπ ) = Z g ( ρ t ( e i X )) − e − tλ π χ π ( e i X ) dX (2.47)= vol( G ) | W | Z t Y α ∈ R + α ( H ) ! ( ρ t ( e i H )) − e − tλ π χ π ( e i H ) dH. To provide further evidence for our conjecture, we will prove it for G = U (1), which also impliesthe conjecture for G = U (1) n , n ∈ N . Subsequently, we give numerical results that suggest thevalidity of the conjecture for G = SU (2).If G = U (1), we have G C = C ∗ . The coherent state overlap function is found to be (cf. ): h Ψ tξ | Ψ tξ i = X j ∈ Z e − tj e jl = ϑ (cid:18) ilπ (cid:12)(cid:12)(cid:12)(cid:12) itπ (cid:19) = ϑ (cid:18) lt (cid:12)(cid:12)(cid:12)(cid:12) iπt (cid:19) r πt e t l , (2.48)where we chose coordinates ξ = e i ( ϕ + il ) , ( ϕ, l ) ∈ [0 , π ) × R and ϑ denotes the third Jacobitheta function. The characters of the irreducible representations of U (1) can be labeled by j ∈ Z , χ j ( ξ ) = ξ l , d j = 1, and the eigenvalues of the Casimir operator are λ j = j . Putting everythingtogether, (2.44) gives: C − t = Z R e − tj e − jl (cid:18) ϑ (cid:18) lt (cid:12)(cid:12)(cid:12)(cid:12) iπt (cid:19) r πt e t l (cid:19) − dl = Z R r tπ e − t ( l + tj ) (cid:18) ϑ (cid:18) lt (cid:12)(cid:12)(cid:12)(cid:12) iπt (cid:19)(cid:19) − dl (2.49)4= Z R r tπ e − t l (cid:18) ϑ (cid:18) lt (cid:12)(cid:12)(cid:12)(cid:12) iπt (cid:19)(cid:19) − dl = t Z R X n ∈ Z e ( l + i πn √ t ) ! − dl. The last line follows from the Z -invariance of ϑ in the first argument. Clearly, this establishes theconjecture for G = U (1). A Numerical evaluation of (2.49) indicates C t = t .For G = SU (2), the formulas become slightly more involved, although we end up with a1-dimensional integral as rank( SU (2)) = 1. The characters of irreducible representations of SU (2) are labeled by n ∈ N , χ n ( e iH ) = sinh(2 np )sinh(2 p ) , for some suitable coordinate p on t ∼ = R . Thedimension and the eigenvalue of the Casimir operator are d n = n and λ n = n − , respectively.The positive root is given by α ( H ) = p . This allows for the computation of all objects involved in(2.47): C − t d n = 2 π Z R p e − t ( n − sinh(2 np )sinh(2 p ) X m ∈ N me − t ( m − sinh(2 mp )sinh 2 p ! − dp (2.50)= π Z R p e − t ( p − t n ) X m ∈ Z me − t ( p − t m ) ! − dp = π Z R (cid:18) p + t n (cid:19) e − t p X m ∈ Z ( m + n ) e − t ( p − t m ) ! − dp = π Z R p e − t ( p − t n ) (cid:18) e − t p ( ∂ p ϑ ) (cid:16) p πi (cid:12)(cid:12)(cid:12) it π (cid:19)(cid:19) − dp = i π Z R p e − t ( p − t n ) (cid:18) e − t p ϑ (cid:16) p πi (cid:12)(cid:12)(cid:12) it π (cid:19)(cid:19) − dp. Again, we need to prove an integral formula involving the Jacobi theta function ϑ . Only this time,we have do deal with the derivative of ϑ , which is why a simple shift p p + t n does not sufficeto prove the formula. Nevertheless, a numerical evaluation of the integral (see table I and figures 1& 2) I ( t, n ) := 2 i Z R p e − t ( p − t n ) (cid:18) e − t p ϑ (cid:16) p πi (cid:12)(cid:12)(cid:12) it π (cid:19)(cid:19) − dp (2.51)hints at the correctness of conjecture II.9 for G = SU (2), since the integral relation between I ( t, n ) = nI ( t,
1) seems to be a rather strong requirement. Furthermore, table I indicates therelation I ( t, n ) = t I (1 , n ), and thus I ( t, n ) = t n ⇒ C t = πt ) . III. ON WEYL AND KOHN-NIRENBERG CALCULI FOR COMPACT LIE GROUPS
In this section, we discuss local and global Weyl and Kohn-Nirenberg calculi for compact Liegroups G (dim G = n ) to provide a (pseudo-differential) framework for the Born-Oppenheimerapproximation or space-adiabatic perturbation theory in loop quantum gravity (see also section5t n 1 2 3 4 51 0 .
125 0 .
25 0 .
375 0 . . e ∼ . ∼ . ∼ . ∼ . ∼ . π ∼ . ∼ . ∼ . ∼ . ∼ . Table I. Numerical evaluation of I ( t, n ). Figure 1. Plots of I ( t, n ) ( t fixed): I (1 , n ), I (2 , n ), I ( e, n ) and I ( π, n ) (bottom to top). II of the third article in this series ). The need for local as well as global calculi is due to thefact that the exponential map is, while still onto, no longer a diffeomorphism for compact groups.Both, local and global calculi, are advantageous in certain situations: On the one hand, it is quitenatural to handle the “semi-classical limit” of the (quantum) commutation relations,[ f, f ] = 0 , (3.1)[ P X , f ] = − iε R X f, [ P X , P Y ] = iε P [ X,Y ] , Figure 2. 3D Plot of I(n,t). where f, f ∈ C ∞ ( G ) , X, Y ∈ g and R X f = ddt | t =0 L ∗ e tX f , in local calculi via the Baker-Campbell-Hausdorff formula, on the other hand, the composition and computation of symbols of operators issimpler for the global calculi, and the class of admissible symbols is larger.The local calculi are based on a generalised Weyl quantisation introduced by Landsman (cf. )based on (strict) Rieffel deformations (cf. ), while the global calculi are closely related to theKohn-Nirenberg calculus of Ruzahnsky and Turunen (cf. ). In contrast to those existing accountson (pseudo-differential) quantisation on compact Lie groups, we arrive at the calculi from theperspective of dequantisation of the transformation group C ∗ -algebra C ( G ) (cid:111) L G ∼ = K ( L ( G )),which is a natural quantum algebra over a single edge of loop quantum gravity (see section II ofthe third article in this series ). Furthermore, it is important to note, from the point of view ofapplications, that the Weyl calculi are favored over the Kohn-Nirenberg calculi, because the formerare real, i.e. hermitean/self-adjoint operators tend to have hermitean/self-adjoint symbols (cf. ).We start, in subsection III A 1, with the definition of the global calculi. In subsection III A 2 weintroduce the local calculi and the “Paley-Wiener-Schwartz” symbol spaces S K,mρ,δ . For the latter,we prove a completeness result w.r.t. asymptotic expansions by adopting the method of kernelcut-off operators from the calculus of Volterra-Mellin operators (cf. ). For the global calculi weprovide a reformulation in terms of the Stratonovich-Weyl transform of Figueroa, Gracia-Bondíaand Várilly (cf. ) in subsection III B, which additionally gives rise to a scaled ε -scaled integraltransform on G . As a byproduct, we prove strictness of the Stratonovich-Weyl quantisation.Following this, we comment on the relation of the calculi to coherent state quantisation in subsectionIII C. We conclude the section with a closer look a the special case G = U (1) and a possible extension7of the global calculi to G = R Bohr , where we will argue that the global calculus is suitable for dealingwith symbols that are not analytic in the momenta.
A. Pseudo-Differential Operators on Compact Lie groups
Before we state the definitions for the types of pseudo-differential operators on compact Lie groupsthat we intend to discuss, we start with a (informal) motivation, i.e. we refrain from defining thefunction spaces on which the following formulae will be well-defined:For quantum mechanics on R n the commutation relations (3.1) correspond to the standard com-mutation relations for position and momentum,[ Q i , Q j ] = 0 = [ P i , P j ] (3.2)[ P i , Q j ] = − iε δ i,j , ∀ i, j = 1 , ..., n, which are often presented in their Weyl form obtained by considering the (formal) exponentials W ( x, ξ ) := e i ( x · Q + ξ · P ) , x, ξ ∈ R n , ( Weyl elements ). In the Schrödinger representation on L ( R n )the action of the Weyl elements is defined to be:( W ( x, ξ )Ψ)( q ) = e iε x · ξ e iξ · q Ψ( q + εξ ) , Ψ ∈ L ( R n ) . (3.3)Moreover, the Weyl elements provide a means of quantising functions σ on phase space T ∗ R n ∼ = R n × R n to operators A σ on L ( R n ) by Fourier transformation, i.e.: A σ := 1(2 π ) n Z R n dx dξ F [ σ ]( x, ξ ) W ( x, ξ ) = Z R n dq dp σ ( q, p ) ˆ W ( q, p ) = ( ˆ W , σ ) L ( R n ) , (3.4)where we introduce the Fourier-Weyl elements ˆ W ( q, p ) := π ) n R R n dx dξ e − i ( x · q + ξ · p ) W ( x, ξ ),ˆ W ( q, p ) ∗ = ˆ W ( q, p ), and F [ σ ]( x, ξ ) = π ) n R R n dq dp σ ( q, p ) e − i ( x · q + ξ · p ) is the Fourier transformof σ . Combining (3.3) and (3.4), we find( A σ Ψ)( q ) = 1(2 πε ) n Z R n dx dξ σ ( ( q + x ) , ξ ) e iε ξ · ( q − x ) Ψ( x ) . (3.5)Thus, A σ is pseudo-differential operator on L ( R n ) with amplitude oder symbol σ . In addition tothe quantisation formula (3.4), the Fourier-Weyl elements also give rise to a useful formula for thedequantisation of an operator A on L ( R n ), i.e. finding a symbol σ A s.t. A σ A = A , because of the(distributional) orthogonality relations (cid:16) ε π (cid:17) n ( W ( x, ξ ) , W ( x , ξ )) HS = (cid:16) ε π (cid:17) n tr( W ( x, ξ ) ∗ W ( x , ξ )) = δ ( n ) ( x − x ) δ ( n ) ( ξ − ξ ) (3.6)(2 πε ) n ( ˆ W ( q, p ) , ˆ W ( q , p )) HS = (2 πε ) n tr( ˆ W ( q, p ) ∗ ˆ W ( q , p )) = δ ( n ) ( q − q ) δ ( n ) ( p − p ) . Applying (3.6) to the product A ρ = A σ A τ of two operators A σ and A τ , we find the well-knownformula for the twisted or Moyal product ρ = σ ? ε τ of the symbols σ and τ , and its asymptotic8expansion (cf. ): ρ ( q, p ) = 1(2 π ) n Z R n dq dp e i ( q · q + p · p ) Z R n dq dp F [ σ ]( q , p ) F [ τ ]( q − q , p − p ) e iε ( q · p − p · q ) (3.7) ∼ exp (cid:18) − iε ∇ p · ∇ x − ∇ q · ∇ ξ ) (cid:19) | ( q,p )=( x,ξ ) σ ( q, p ) τ ( x, ξ ) . Clearly, to arrive at (3.7) we need to evaluate the tri-kernel tr( ˆ W ( q, p ) ∗ ˆ W ( q , p ) ˆ W ( q , p )), whichis possible, because linear combinations of (Fourier-)Weyl elements are closed under products.Thus, we conclude that the structures required for a theory of pseudo-differential operators adaptedto quantum systems defined by the commutation relations (3.1) or their exponential form (whichwe recognize as covariant representation of ( C ( G ) , G, α L ) , ),[ f, f ] = 0 , (3.8) U g f U ∗ g = L ∗ g − fU g U g = U gg , f, f ∈ C ( G ) , g, g ∈ G, are a set of (Fourier-)Weyl elements, closed under product if possible, and a notion of Fouriertransform. Moreover, to obtain a practical calculus for the application of the Born-Oppenheimerscheme and space-adiabatic perturbation theory we should require the existence of a formula similarto (3.7) for symbols of operator products, which admits a suitable asymptotic expansion.Let us also add a short comment on the choice of Weyl elements (3.3): Namely, we could make thealternative definitions W R ( x, ξ ) := e ix · Q e iξ · P , W L ( x, ξ ) := e iξ · P e ix · Q , (3.9)which lead to the Kohn-Nirenberg pseudo-differential operators A Lσ , A Rσ associated with a symbol σ (cf. ), which are standard in treatments of pseudo-differential operators on manifolds (cf. ):( A Rσ Ψ)( q ) = 1(2 πε ) n Z R n dx dξ σ ( q, ξ ) e iε ξ · ( q − x ) Ψ( x ) , (3.10)( A Lσ Ψ)( q ) = 1(2 πε ) n Z R n dx dξ σ ( x, ξ ) e iε ξ · ( q − x ) Ψ( x ) . (3.11)Although, the Kohn-Nirenberg pseudo-differential operators generalise in a much more straightfor-ward manner to manifolds, because (3.10) and (3.11) can be localised in the position variable q ,and thus to general (compact) Lie groups, they are disadvantageous w.r.t. to the Born-Oppenheimerscheme and space-adiabatic perturbation theory, because they do not provide a real quantisationin contrast to (3.5), i.e. (cid:0) A R/Lσ (cid:1) ∗ = A R/Lσ ∗ , due to the asymmetric treatment of Q and P .
1. A global calculus
Having identified the ingredients necessary for the definition of pseudo-differential operators, wewill now explain how these are realized for a compact Lie group G in a global fashion. With9regard to the notation, we stick to section II. As noted above, the commutation relations (3.1)correspond in exponential form (3.8) to a covariant representation of ( C ( G ) , G, α L ), and thus tothe transformation group C ∗ -algebra C ( G ) (cid:111) L G . In the (faithful) integrated representation ρ L on L ( G ) a function F ∈ C ( G × G ) ⊂ C ( G ) (cid:111) L G acts according to( ρ L ( F )Ψ)( g ) = Z G dh F ( h, g )( U h Ψ)( g ) = Z G dh F ( h, g )Ψ( h − g ) (3.12)= Z G dh F ( gh − , g )Ψ( h ) , Ψ ∈ L ( G ) . Clearly, (3.12) still to makes sense as a continuous operator ρ L ( F ) : C ∞ ( G ) → D ( G ) for F ∈ D ( G × G ) ∼ = D ( G ) ˆ ⊗ D ( G ), and if we restrict F to be in D ( G, C ∞ ( G )) ∼ = D ( G ) ˆ ⊗ C ∞ ( G ) itdefines a continuous operator on C ∞ ( G ) . Therefore, it is possible to make Definition III.1:
For F R ( π, m, n ; h ) := π ( . ) mn δ h , F L ( π, m, n ; h ) := π ( h − . ) mn δ h ∈ D ( G, C ∞ ( G )) , where π ∈ ˆ G , m, n = 1 , ..., d π is a unitary irreducible representation, we obtain the operators (cid:0) ρ L (cid:0) F R ( π, m, n ; h ) (cid:1) Ψ (cid:1) ( g ) = π ( g ) mn Ψ( h − g ) , (3.13) (cid:0) ρ L (cid:0) F L ( π, m, n ; h ) (cid:1) Ψ (cid:1) ( g ) = π ( h − g ) mn Ψ( h − g ) , Ψ ∈ C ∞ ( G ) , which extend to operators on L ( G ) by continuity. In view of (3.9), these operators generalise the Weyl elements used in the definition of the Kohn-Nirenberg pseudo-differential operators (3.10) and (3.11). In the following, we will restrict attentionto the (right) elements F R ( π, m, n ; h ), although all statements hold in slightly modified form for F L ( π, m, n ; h ), as well. The analogy with (3.10) and (3.11) becomes even clearer, when we considerthe Weyl elements in context of the (global) Fourier transform between G and its (unitary) dualˆ G , and its inverse, F [Ψ]( π ) = ˆΨ( π ) := Z G dg Ψ( g ) π ( g ) , Ψ ∈ L ( G ) , (3.14) F − [Φ]( g ) = ˇΦ( g ) := X π ∈ ˆ G d π tr( π ( g ) ∗ Φ( π )) , Φ ∈ L ( ˆ G ) :We define the Fourier-Weyl elementsˆ F R ( g ; π, m, n ) := X π ∈ ˆ G d π d π X m ,n =1 π ( g ) m n Z G dh π ( h ) mn F R ( π , m , n ; h ) (3.15)= δ g π ( . ) mn ∈ C ∞ ( G, D ( G )) . Then, for σ ∈ ˆ D ( ˆ G, C ∞ ( G )) = F ( D ( G, C ∞ ( G ))), we have F Rσ ( h, g ) = X π ∈ ˆ G d π d π X m,n =1 Z G dg σ ( π, g ) mn ˆ F R ( g ; π, m, n )( h, g ) (3.16)0= X π ∈ ˆ G d π tr( π ( h ) ∗ σ ( π, g )) = ˇ σ ( h, g ) , which is analogous to (3.4), and continuous to be well-defined in the distributional sense for σ ∈ F ( D ( G × G )). The representation of ˇ σ on C ∞ ( G ) via ρ L leads to the pseudo-differential operatorsof Ruzhansky and Turunen (up to the fact that those authors employ the right convolution algebra C ( G ) (cid:111) R G ): (cid:0) ρ L (cid:0) F Rσ (cid:1) Ψ (cid:1) ( g ) = Z G dh F Rσ ( gh − , g )Ψ( h ) (3.17)= Z G ˇ σ ( gh − , g )Ψ( h ) = X π ∈ ˆ G d π tr( π ( g ) ∗ σ ( π, g ) ˆΨ( π )) . As a consequence of the Peter-Weyl theorem and the fact that (left) convolution is well-defined on D ( G ) the (Fourier-)Weyl elements satisfy (distributional) orthogonality relations:tr (cid:0) ρ L (cid:0) F R ( π, m, n ; h ) ∗ (cid:1) ρ L (cid:0) F R ( π , m , n ; h ) (cid:1)(cid:1) = d − π δ π,π δ m,m δ n,n δ h ( h ) , (3.18)tr (cid:0) ρ L (cid:0) F R ( g ; π, m, n ) ∗ (cid:1) ρ L (cid:0) F R ( g ; π , m , n ) (cid:1)(cid:1) = δ g ( g ) d − π δ π,π δ m,m δ n,n , where the adjoint ( . ) ∗ has to be taken the sense of (2.6). Furthermore, the linear span of Weylelements is closed under products due to complete reducibility of unitary representations of G . infact, we have: (cid:0) F R ( π, m, n ; h ) ∗ L F R ( π , m , n ; h ) (cid:1) ( g, g ) (3.19)= π ( g ) mn π ( h − g ) δ hh ( g ) = d π X k =1 π ( h − ) m k π ( g ) mn π ( g ) k n δ hh ( g )= d π X k =1 π ( h − ) m k X π ∈ ˆ G N π π,π X s =0 d π X M,N =1 C ( π, m ; π , k | π , M ; s ) C ( π, n ; π , n | π , N ; s ) π ( g ) MN δ hh ( g )= d π X k =1 π ( h − ) m k X π ∈ ˆ G N π π,π X s =1 d π X M,N =1 C ( π, m ; π , k | π , M ; s ) C ( π, n ; π , n | π , N ; s ) × F R ( π , M, N ; hh )( g, g ) , where we denote by the C ’s the Clebsch-Gordan coefficients associated with the decomposition π ⊗ π ∼ = M π ∈ ˆ G N π π,π M s =1 π with multiplicities N π π,π ∈ N . (3.20)Dequantisation of a continuous operator A on C ∞ ( G ) takes the form: σ A ( π, g ) mn := tr (cid:16) ρ L (cid:16) ˆ F R ( g ; π, m, n ) ∗ (cid:17) A (cid:17) , (3.21)1which agrees with the formula of Ruzhansky and Turunen for A = ρ L ( F ) , F ∈ D ( G, C ∞ ( G )): σ F ( π, g ) mn = tr (cid:16) ρ L (cid:16) ˆ F R ( g ; π, m, n ) ∗ (cid:17) ρ L ( F ) (cid:17) = Z G dh F ( h, g ) π ( h ) mn = ˆ F ( π, g ) mn . (3.22)By the Schwartz kernel theorem (cf. ) this covers already the general case of continuous operatorson C ∞ ( G ).Before we try to deform the Kohn-Nirenberg quantisation, defined so far, to obtain a Weyl quanti-sation, i.e. a real quantisation, we collect some of the properties of the former in a Proposition III.2:
Given continuous operators
A, B : C ∞ ( G ) → D ( G ) with F A , F B ∈ D ( G × G ) , the symbols σ A , σ B ∈ F ( D ( G × G )) satisfy:1. The map A σ A is a linear homeomorphism between L ( C ∞ ( G ) , D ( G )) and ˆ D ( ˆ G × G ) with σ ( π, g ) = π ( e ) .2. σ A ∗ ( π, g ) = σ F ∗ A ( π, g ) = σ α − ( F A ) ( π, g ) ∗ , where ( α − ( F ))( h, g ) = F ( h, hg ) , F ∈ D ( G × G ) .3. σ U h AU ∗ h ( π, g ) = σ ( α h − × L h − ) ∗ F A ( π, g ) = π ( h ) σ A ( π, h − g ) π ( h ) ∗ , where α h ( g ) = hgh − .4. If U R : G → U ( L ( G )) denotes the right regular representation, i.e. ( U Rh Ψ)( g ) = Ψ( gh ) for Ψ ∈ L ( G ) , h, g ∈ G , we have: σ U Rh AU R ∗ h ( π, g ) = σ (id × R h ) ∗ F A ( π, g ) = σ A ( π, gh ) .5. ( A, B ) HS = tr( A ∗ B ) = ( F A , F B ) L ( G × G ) = ( σ A , σ B ) L ( ˆ G × G ) , if A, B are Hilbert-Schmidtoperators. Moreover, the maps HS ( L ( G )) A F A ∈ L ( G × G ) and HS ( L ( G )) A σ A ∈ L ( ˆ G × G ) are unitary.6. σ AB ( π, g ) = R G dh F A ( h, g ) π ( h ) σ B ( π, h − g ) , if A, B : C ∞ ( G ) → C ∞ ( G ) .7. The symbol σ A can be computed by σ A ( π, g ) = ( πAπ ∗ )( g ) = π ( g ) Z G dh F A ( gh − , g ) π ( h ) ∗ . (3.23) Proof:
1. The linearity of A σ A and σ ( π, g ) = π ( e ) are evident from the definition. The homeo-morphism property follows from the Schwartz kernel theorem and the fact that the (partial)Fourier transform sets up a homeomorphism between D ( G × G ) and ˆ D ( ˆ G × G ).2. For Ψ ∈ C ∞ ( G ), we have( σ A ∗ ( π ) , Ψ) = Z G dg Z G dh F A ∗ ( h, g ) π ( h )Ψ( g ) = Z G dg Z G dh F ∗ A ( h, g ) π ( h )Ψ( g ) (3.24)= Z G dg Z G dh F A ( h − , h − g ) π ( h )Ψ( g ) = Z G dg Z G dh F A ( h, hg ) π ( h ) ∗ Ψ( g )= Z G dg (cid:18)Z G dh F A ( h, hg ) π ( h ) (cid:19) ∗ Ψ( g )2= Z G dg (cid:18)Z G dh α − ( F A )( h, g ) π ( h ) (cid:19) ∗ Ψ( g ) .
3. Referring to the definition of A = ρ L ( F A ), we find: U h AU ∗ h = U h ρ L ( F A ) U ∗ h = ρ L (( α h − × L h − ) ∗ F A ) . (3.25)Therefore, we have: σ U h AU ∗ h ( π, g ) = σ ( α h − × L h − ) ∗ F A ( π, g ) = Z G dh ( α h − × L h − ) ∗ F A ( h , g ) π ( h ) (3.26)= Z G dh F A ( h , h − g ) π ( α h ( h )) = π ( h ) Z G dh F A ( h , h − g ) π ( h ) π ( h ) ∗ = π ( h ) σ A ( π, h − g ) π ( h ) ∗ .
4. This follows along the same lines as 3.5. It is well known (cf. ), that A, B are Hilbert-Schmidt operators if and only if F A , F B ∈ L ( G × G ), which gives the second equality and the unitarity of A F A . The third equalityand the unitarity of A σ A follow, because the (partial) Fourier transform is a unitary mapfrom L ( G × G ) to L ( ˆ G × G ).6. By assumption the product AB is well-defined and gives rise to a (left) convolution kernel F AB , which is found from AB = ρ L ( F A ) ρ L ( F B ) = ρ L ( F A ∗ L F B ) . (3.27)The formula for σ AB follows from direct computation: σ AB ( π, g ) = Z G dh ( F A ∗ L F B )( h, g ) π ( h ) (3.28)= Z G dh Z G dh F A ( h , g ) F B ( h h, h g ) π ( h )= Z G dh Z G dh F A ( h , g ) π ( h ) F B ( h h, h g ) π ( h h )= Z G dh F A ( h , g ) π ( h ) Z G dh F B ( h, h g ) π ( h )= Z G dh F A ( h, g ) π ( h ) σ B ( π, h − g ) .
7. Employing (3.22), we find:ˆ F A ( π, g ) = Z G dh F A ( h, g ) π ( h ) = π ( g ) Z G dh F A ( h, g ) π ( g − h ) (3.29)= π ( g ) Z G dh F A ( gh, g ) π ( h )3= π ( g ) Z G dh F A ( gh − , h ) π ( h ) ∗ = ( πAπ ∗ )( g ) . The last property is extremely useful in the actual computation of symbols σ A of operators A : C ∞ ( G ) → D ( G ), as it does not require the computation of the (left) convolution kernel F A .To see how this works, we compute the symbols of P X = − iεR X , X ∈ g , and f ∈ C ∞ ( G ): σ P X ( π, g ) = ( πP X π ∗ ) = π ( g )( P X π ∗ )( g ) = − iεπ ( g ) ddt | t =0 π ( e tX g ) ∗ (3.30)= − iεπ ( g ) ddt | t =0 π ( g ) ∗ π ( e − tX ) = iεdπ ( X ) ,σ f ( π, g ) = ( πf π ∗ )( g ) = π ( g ) f ( g ) π ( g ) ∗ = f ( g ) V π . Combining this with the sixth property, gives rise to the commutation relations (3.1): σ ff ( π, g ) = π ( g )( f f π ∗ )( g ) = f ( g ) f ( g ) V π = σ f ( π, g ) σ f ( π, g ) , (3.31) σ P X f = π ( g )( P X f π ∗ )( g ) = π ( g )( P X π ∗ )( g ) f ( g ) − iε ( R X f )( g ) V π = σ P X ( π, g ) σ f ( π, g ) − iεσ R X f ( π, g ) ,σ P X P Y ( π, g ) = π ( g )( P X P Y π ∗ )( g ) = iεπ ( g )( P X π ∗ )( g ) dπ ( Y ) = − ε dπ ( X ) dπ ( Y ) = σ P X σ P Y . The second property of the above proposition gives a measure to which extent the quantisationˆ D ( ˆ G × G ) σ ρ L ( F Rσ ) ∈ L ( C ∞ ( G ) , D ( G )) fails to be real. Interestingly, there is a simple wayto cure this, if we were allowed to take square roots in G , which is indeed possible for compact Liegroups by means of the exponential map exp : g → G , as the latter is onto (cf. ). Moreover, wecan define √ g, ∀ g ∈ G , s.t. p g − = √ g − , but there is also a price to pay: Namely, √ . : G → G is in general not a smooth homomorphism, but only a measurable map. Definition III.3:
For σ ∈ ˆ D ( ˆ G × G ) , s.t. α L (cid:0) F Rσ (cid:1) ( h, g ) = F Rσ ( h, √ h − g ) is in D ( G × G ) , e.g. if √ . is smooth on sing supp ( F Rσ ) , we define the Weyl quantisation F Wσ of σ by F Wσ ( h, g ) := α L (cid:0) F Rσ (cid:1) ( h, g ) . (3.32) The Weyl elements F W ( π, m, n ; h ) are the Weyl quantisations of the symbols ( σ ( π,m,n ; h ) ( π , h )) m n := d − π δ π,π δ m ,m δ n,n δ h ( h ) , (3.33) where π, π ∈ ˆ G, m, n = 1 , ..., d π , m , n = 1 , ..., d π , h, h ∈ G . Explicitly, we have F W ( π, m, n ; h )( h , g ) = π ( √ h − g ) mn δ h ( h ) (3.34)= d π X k =1 π ( √ h − ) mk π ( g ) kn δ h ( h )= d π X k =1 π ( √ h − ) mk F R ( π, k, n ; h )( h , g ) . Lemma III.4:
For σ ∈ ˆ D ( ˆ G × G ) as in definition III.3, we have ρ L (cid:0) F Wσ (cid:1) ∗ = ρ L (cid:0) F W ∗ σ (cid:1) = ρ L (cid:0) F Wσ ∗ (cid:1) . (3.35) Moreover, the Weyl elements satisfy tr (cid:0) ρ L (cid:0) F W ( π, m, n ; h ) ∗ (cid:1) ρ L (cid:0) F W ( π , m , n ; h ) (cid:1)(cid:1) = d − π δ π,π δ m,m δ n,n δ h ( h ) . (3.36) Proof:
The first statement is a consequence of the adjointness property (2.) in proposition III.2: F W ∗ σ ( h, g ) = F Wσ ( h − , h − g ) = F Rσ ( h − , √ hh − , g ) = F Rσ ( h − , √ h − g ) (3.37)= X π ∈ ˆ G d π tr (cid:16) π ( h ) σ ( π, √ h − g ) (cid:17) = X π ∈ ˆ G d π tr (cid:16) π ( h ) ∗ σ ( π, √ h − g ) ∗ (cid:17) = F Rσ ∗ ( h, √ h − g ) = F Wσ ∗ ( h, g ) . The second statement follows from the orthogonality relations (3.18) and (3.34):tr (cid:0) ρ L (cid:0) F W ( π, m, n ; h ) ∗ (cid:1) ρ L (cid:0) F W ( π , m , n ; h ) (cid:1)(cid:1) (3.38)= d π X k =1 d π X k =1 π ( √ h − ) mk π ( √ h ) m k tr (cid:0) ρ L (cid:0) F R ( π, k, n ; h ) ∗ (cid:1) ρ L (cid:0) F R ( π , k , n ; h ) (cid:1)(cid:1) = d − π δ π,π δ n,n δ h ( h ) d π X k =1 π ( √ h ) km π ( √ h − ) m k = d − π δ π,π δ m,m δ n,n δ h ( h ) , where the last line makes sense because √ . : G → G admits a unique pointwise evaluation. Remark III.5:
Unfortunately, the definition of Fourier-Weyl elements seems to be problematic, as can be seen froma (formal) calculation:ˆ F W ( g ; π, m, n )( h, g ) = X π ∈ ˆ G d π d π X m ,n =1 π ( g ) m n Z G dh π ( h ) mn F W ( π , m , n ; h )( h, g ) (3.39)= π ( h ) mn δ √ hg ( g )From the last line, we infer that the definition of ˆ F W ( g ; π, m, n ) would require the composition of δ with √ . , which is not necessarily well-defined, because √ . is in general not even continuous.5Still, we can define a dequantisation map, if we restrict ourselves to operators A : C ∞ ( G ) → D ( G ),s.t. α − L ( F A ) ( h, g ) = F A ( h, √ hg ) is in D ( G × G ), in analogy with definition III.3. Definition III.6:
The Weyl symbol of an operator A : C ∞ → D ( G ) , s.t. α − L ( F A ) ( h, g ) = F A ( h, √ hg ) is in D ( G × G ) , is σ WA ( π, g ) := F h α − L ( F A ) i ( π, g ) = Z G dh α − L ( F A ) ( h, g ) π ( h ) . (3.40)From the definitions it is obvious, that we have Corollary III.7:
The Weyl quantisation σ F Wσ and the Weyl symbol map A σ WA are inverse to each other. In accordance with proposition III.2, we collect some properties of the Weyl (de)quantisation,although, at this point, we refrain from specifying the sets of operators or symbols for which theprocedure is well-defined any further. We will cure this in the following subsection, where we definethe local calculi.
Proposition III.8:
Given continuous operators
A, B : C ∞ ( G ) → D ( G ) , s.t. the Weyl symbols σ WA , σ WB ∈ ˆ D ( ˆ G × G ) are well-defined, then we have1. σ WU h AU ∗ h ( π, g ) = π ( h ) σ W (id × L h − ) ∗ F A ( π, g ) π ( h ) ∗ .2. σ WU Rh AU R ∗ h ( π, g ) = σ WA ( π, gh ) .3. If the product AB and its Weyl symbol σ WAB are well-defined, we have σ WAB ( π, g ) = X π ,π ∈ ˆ G d π d π Z G dg Z G dg π ( g ) π ( g ) tr( π ( g ) ∗ σ WA ( π , p g p g g g )) (3.41) × tr( π ( g ) ∗ σ B ( π , p g g p g g g )) . Proof:
The properties 1.-3. are proved in a completely analogous way as the corresponding properties inproposition III.2.
Remark III.9:
The Weyl symbols of the elementary operators appearing in (3.1) equal their Kohn-Nirenbergsymbols, since we have F P X ( h, g ) = − iε ( R X δ e )( h ) , X ∈ g , and F f ( h, g ) = δ e ( h ) f ( g ) , f ∈ C ∞ ( G ): α − L ( F f )( h, g ) = δ e ( h ) f ( √ hg ) = δ e ( h ) f ( g ) (3.42) ⇒ σ Wf ( π, g ) = f ( g ) V π α − L ( F P X )( h, g ) = − iε ( R X δ e )( h ) ⇒ σ WP X ( π, g ) = iεdπ ( X ) , where there first line makes sense, because √ . : G → G is uniquely defined everywhere, althoughit is not continuous.6 Remark III.10:
If we choose G = R n , and therefore ˆ G = R n , and σ A , σ B ∈ S ( R n ), we can make still make senseout of the product formula (3.41), and a simple calculation shows that it is equivalent to the twistedproduct (3.7): σ WAB ( p, x ) = Z R n dp dx (2 π ) n Z R n dp dx (2 π ) n e ip · ( x + x ) e − ip · x e − ip · x σ WA ( p , x + x ) (3.43) × σ WB ( p , − x + x ) x
7→ − x − x ) x x − x ) = Z R n dp dx (4 π ) n Z R n dp dx (4 π ) n e − i ( p − p ) · x e i ( p − p ) · x e i ( p − p ) · x e − i ( p − p ) · x × σ WA ( p , x ) σ WB ( p , x )= Z R n dp dx (4 π ) n Z R n dp dx (4 π ) n e i (( p − p ) · ( x − x ) − ( p − p ) · ( x − x )) σ WA ( p , x ) σ WB ( p , x ) , where we recognize the last line as an alternative formula for the twisted product (3.7) (cf. ).So far, we have not dealt with the question of the existence of an asymptotic expansion for thesymbol σ ( W ) AB of an operator product AB : C ∞ ( G ) → C ∞ ( G ). This will be done in the nextsubsection, where we introduce an ε -dependent expansion of σ ( W ) AB in the local calculi. In contrast,Ruzhansky and Turunen define a global symbolic calculus, but the ε -dependence remains ratheropaque in their setting. We will come back to this point in subsections III D.
2. A local calculus of Paley-Wiener-Schwartz symbols
Following Rieffel and Landsman , we localise the quantisations discussed in the previoussubsection in the sense, that we pass from global symbols σ ∈ ˆ D ( ˆ G × G ), living on G and itsunitary dual ˆ G , to local symbols σ ∈ D ( T ∗ G ), living on T ∗ G ∼ = G × g ∗ (by right translation), via theexponential map exp : g → G . More precisely, this works as follows: exp defines a diffeomorphismbetween an open neighbourhood U ⊂ g of 0 ∈ g (possibly Ad -invariant) and an open neighbourhood V ⊂ G of e ∈ G . Additionally, we define Fourier transform and its inverse between functions on g and g ∗ : F [ F ]( θ ) = ˆ F ( θ ) := Z g dX e − iθ ( X ) F ( X ) , F ∈ L ( g ) , θ ∈ g ∗ (3.44) F − [ σ ]( X ) = ˇ σ ( X ) := Z g ∗ dθ (2 π ) n e iθ ( X ) σ ( θ ) , σ ∈ L ( g ∗ ) , X ∈ g , where we fix the normalisation of the Lebesque measures dX and dθ via the exponential map andthe Haar measure dg on G (cp. section II): Z V dg f ( g ) = Z U dX j ( X ) f (exp( X )) , f ∈ C c ( V ) . (3.45)Here, j is the analytic function j ( H ) = Q α ∈ R + sin( α ( H )) α ( H ) , H ∈ t ⊂ g , in the notation of section II(see especially (2.20) and (2.21)). Now, we are in a position to make the7 Definition III.11:
Given a function σ ∈ C ∞ PW ,U ( g ∗ ) ˆ ⊗ C ∞ ( G ) s.t. ˇ σ , the inverse Fourier transform of σ in the firstvariable, is in D ( g ) ˆ ⊗ C ∞ ( G ) with supp (ˇ σ ) ⊂ U , we define F σ ∈ C ∞ ( G ) ˆ ⊗ C ∞ ( G ) by: F σ ( h, g ) := ˇ σ ( X h , g ) = Z g ∗ dθ (2 π ) n e iθ ( X h ) σ ( θ, g ) for X h := exp − ( h ) , (3.46) which is well-defined due to the support properties of ˇ σ . We also define an ε -scaled version of (3.46) : ∀ (cid:15) ∈ (0 ,
1] : F εσ ( h, g ) := ε − n ˇ σ ( ε − X h , g ) = Z g ∗ dθ (2 πε ) n e iε θ ( X h ) σ ( θ, g ) . (3.47) We call F ( ε ) σ the ( ε -scaled) Kohn-Nirenberg quantisation of σ , as it defines a compact operator on L ( G ) via the left integrated representation ρ L : C ( G ) (cid:111) L G → K ( L ( G )) . Clearly, the ε -scaledquantisation extends to those σ , s.t. supp (ˇ σ ) ⊂ ε − U =: U ε .By standard distributional reasoning, the quantisation extends to distributions σ ∈ ˆ E U ( g ∗ ) ˆ ⊗ D ( G ) ,i.e. ˇ σ ∈ E U ( g ) ˆ ⊗ D ( G ) , where E U ( g ) is the space of compactly supported distributions in U ⊂ g .Furthermore, if we define the (smooth) square root √ . : V ⊂ G → V ⊂ G by √ g = exp( X g ) , wecan deform the Kohn-Nirenberg quantisation to a Weyl quantisation in analogy with the previoussubsection: F W,εσ ( h, g ) := α L ( F εσ )( h, g ) = F εσ ( h, √ h − g ) . (3.48) The Weyl quantisation will, in general, not be well-defined for σ ∈ ˆ E U ( g ∗ ) ˆ ⊗ D ( G ) , but surely for σ ∈ ˆ E U ( g ∗ ) ˆ ⊗ C ∞ ( G ) . Corollary III.12:
The Kohn-Nirenberg and Weyl quantisation have following adjointness (in the sense of (2.6) ) andcovariance (w.r.t. to G ) properties (cp. proposition III.2 & III.8):1. (cid:16) F ( ε ) σ (cid:17) ∗ = α L ( F ( ε ) σ ) and (cid:16) F W, ( ε ) σ (cid:17) ∗ = F W, ( ε ) σ ,2. U h F ( ε ) σ U ∗ h = α L ( h ) (cid:18) F ( ε ) (cid:0) Ad ∗ h − (cid:1) ∗ σ (cid:19) & U h F W, ( ε ) σ U ∗ h = α L (cid:16) h p h − ( . ) h p ( . ) − (cid:17)(cid:18) F W, ( ε ) (cid:0) Ad ∗ h − (cid:1) ∗ σ (cid:19) ,3. U Rh F ( ε ) σ U R ∗ h = α R ( h ) (cid:16) F ( ε ) σ (cid:17) = F ( ε ) (cid:16) R ∗ g − (cid:17) ∗ σ & U Rh F W, ( ε ) σ U R ∗ h = α R ( h ) (cid:16) F W, ( ε ) σ (cid:17) = F W, ( ε ) (cid:16) R ∗ g − (cid:17) ∗ σ ,where σ ∈ ˆ E U ( g ∗ ) ˆ ⊗ D ( G ) , h ∈ G , α R is the right regular representation of G , R ∗ is the pullback(right) action on T ∗ G and Ad ∗ : G → GL ( g ∗ ) denotes the coadjoint action. Proof:
We prove the statements for σ ∈ C ∞ PW ,U ( g ∗ ) ˆ ⊗ C ∞ ( G ), which implies them by standard distributionalreasoning for σ ∈ ˆ E U ( g ∗ ) ˆ ⊗ D ( G ) or ˆ E U ( g ∗ ) ˆ ⊗ C ∞ ( G ).1. ( F εσ ) ∗ ( h, g ) = F εσ ( h − , h − g ) = ε − n ˇ σ ( ε − X h − , h − g ) (3.49)8= ε − n ˇ σ ( − ε − X h , h − g ) = ε − n ˇ σ ( ε − X h , h − g ) = α L ( F εσ )( h, g ) . Here, the third equality follows from h − = exp( X h ) − = exp( − X h ), while the fourth equalityfollows from the interplay of the inverse Fourier transform and complex conjugation. By thesame reasoning, we find: (cid:0) F W,εσ (cid:1) ∗ ( h, g ) = F εσ ( h − , √ hh − g ) = ε − n ˇ σ ( ε − X h − , √ h − g ) (3.50)= ε − n ˇ σ ( − ε − X h , √ h − g ) = ε − n ˇ σ ( ε − X h , √ h − g )= F W,εσ ( h, g ) .
2. ( U h F εσ U ∗ h )( h , g ) = F εσ ( h − h h, h − g ) = ε − n ˇ σ ( ε − X h − h h , h − g ) (3.51)= ε − n ˇ σ ( ε − Ad h − ( X h ) , h − g )= ε − n F − [( Ad ∗ h − ) ∗ σ ]( ε − X h , h − g )= α L ( h ) (cid:18) F ε (cid:0) Ad ∗ h − (cid:1) ∗ σ (cid:19) ( h , g ) . The third equality follows from the definition of the adjoint action Ad : G → GL ( g ), and thefourth equality follows from the Ad -invariance of the Lebesgue measure dθ and the definitionof the coadjoint action. Analogously, we find:( U h F W,εσ U ∗ h )( h , g ) = F εσ ( h − h h, p ( h − h h ) − h − g ) (3.52)= ε − n ˇ σ ( ε − X h − h h , √ h √ h p ( h − h h ) − h − g )= ε − n ˇ σ ( ε − Ad h − ( X h ) , √ h √ h p ( h − h h ) − h − g )= ε − n F − [( Ad ∗ h − ) ∗ σ ]( ε − X h , √ h √ h p ( h − h h ) − h − g )= α L ( h √ h − h h √ h ) (cid:18) F W,ε (cid:0) Ad ∗ h − (cid:1) ∗ σ (cid:19) ( h , g ) .
3. By commutativity of left and right action, it follows:( U h F εσ U ∗ h )( h , g ) = F εσ ( h , gh ) = ε − n ˇ σ ( ε − X h , gh ) = F ε (cid:16) R ∗ g − (cid:17) ∗ σ , (3.53)( U h F W,εσ U ∗ h )( h , g ) = F εσ ( h , √ h gh ) = ε − n ˇ σ ( ε − X h , √ h gh ) = F W,ε (cid:16) R ∗ g − (cid:17) ∗ σ . (3.54)Our definition of the Weyl quantisation is indeed equivalent to the one given by Landsman in termsof a “geodesic midpoint construction” (cf. , Definition II.3.4.4.) Lemma III.13:
The operator defined by the Weyl quantisation F W,εσ of σ ∈ C ∞ PW ,U ( g ∗ ) ˆ ⊗ C ∞ ( G ) is equivalent to the operator defined by the Weyl kernel K W, ( ε ) σ ( h, g ) = ε − n ˇ σ ( ε − ν − δ ( h, g )) , (3.55) where ν − δ : V × V → T V ∼ = V × g maps ( h, g ) to the tangent vector at the midpoint of the geodesicfrom h to g (w.r.t. an invariant metric on G ). Proof:
By definition the operator corresponding to the Weyl quantisation F W,εσ is: ∀ Ψ ∈ L ( G ) : (cid:0) ρ L (cid:0) F W,εσ (cid:1) Ψ (cid:1) ( g ) = Z G dh F W,εσ ( h, g )Ψ( h − g ) = Z G dh F W,εσ ( gh − , g )Ψ( h ) . (3.56)Thus, the kernel of ρ L (cid:0) F W,εσ (cid:1) is K ρ L ( F W,εσ )( h, g ) = F W,εσ ( gh − , g ) (3.57)= ε − n ˇ σ ( ε − X gh − , p gh − − g )= ε − n ˇ σ ( ε − X gh − , exp( − X gh − ) g ) . But, (exp( − X gh − ) g, X gh − ) is exactly the point in V × g corresponding to the tangent vector atthe midpoint of the geodesic γ h → g : [0 , → V, γ h → g ( t ) = exp( tX gh − ) h, under right translation,because exp( X gh − ) h = exp((1 − ) X gh − ) h = exp( − X gh − ) exp( X gh − ) h = exp( − X gh − ) g .We conclude: K ρ L ( F W,εσ )( h, g ) = ε − n ˇ σ ( ε − ν − δ ( h, g )) = K W, ( ε ) σ ( h, g ) . (3.58)Therefore, we have the following theorem proven by Landsman in the context of Riemannian man-ifolds. Theorem III.14 (cf. , Theorem II.3.5.1. & Theorem III.2.8.1): The composition of ρ L and the Weyl quantisation Q Wε := ρ L ◦ F W,ε ( . ) : C ∞ PW ,U ( g ∗ ) ˆ ⊗ C ∞ ( G ) → K ( L ( G )) (3.59) is a nondegenerate strict quantization of C ∞ PW ,U ( g ∗ ) ˆ ⊗ C ∞ ( G ) ⊂ C ( G × g ∗ ) ∼ = C ( T ∗ G ) on ε ∈ (0 , ,i.e. we have for all σ, τ ∈ C ∞ PW ,U ( g ∗ , R ) ˆ ⊗ C ∞ ( G, R ) :1. (nondegeneracy): ∀ (cid:15) ∈ (0 ,
1] : Q Wε ( σ ) = 0 ⇔ σ = 0 .2. (Rieffel’s condition): ε
7→ || Q Wε ( σ ) || is continuous on (0 , , especially lim ε → || Q Wε ( σ ) || = || σ || ∞ .3. (von Neumann’s condition): lim ε → (cid:12)(cid:12)(cid:12)(cid:12) (cid:0) Q Wε ( σ ) Q Wε ( τ ) + Q Wε ( τ ) Q Wε ( σ ) (cid:1) − Q Wε ( στ ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 .4. (Dirac’s condition): lim ε → (cid:12)(cid:12)(cid:12)(cid:12) iε (cid:2) Q Wε ( σ ) , Q Wε ( τ ) (cid:3) − Q Wε ( { σ, τ } T ∗ G ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 . Here, { , } T ∗ G is the canonical Poisson structure on T ∗ G , which takes the following for on G × g ∗ by right translation (cp. , Proposition III.1.4.1): ∀ σ, τ ∈ C ∞ ( T ∗ G ) : { σ, τ } T ∗ G = h ∂ θ σ, Rτ i − h Rσ, ∂ θ τ i + { σ, τ } − , (3.60) where ( h X, Rf i )( g ) = ( R X f )( g ) = ddt | t =0 f (exp( tX ) g ) is the right differential on G , and { f, f } − ( θ ) = − θ ([ ∂ θ f, ∂ θ f ]) is the (minus) Lie-Poisson structure on g ∗ . Remark III.15:
For σ ∈ C ∞ PW ,U ( g ∗ ) ˆ ⊗ C ∞ ( G ), we have K ρ L ( F ( W ) ,εσ ) ∈ C ∞ ( G ) ˆ ⊗ C ∞ ( G ) ∼ = C ∞ ( G × G ) by definition.Therefore, Q ( W ) ε ( σ ) is not only a compact operator on L ( G ), but preserves C ∞ ( G ) and extendsto a smoothing operator from D ( G ) to C ∞ ( G ) by the properties of convolution (cf. , Theorem4.1.1.).Before we introduce the Paley-Wiener-Schwartz symbol spaces S K,m PW ,ρ,δ ⊂ ˆ E U ( g ∗ ) ˆ ⊗ C ∞ ( G ), wediscuss the quantization of symbols that are polynomial in the momentum variables θ ∈ g ∗ . Tomake this precise, we recall that the left and right pullback actions of G on T ∗ G are stronglyHamiltonian and compatible, i.e. we have bi-equivariant Poisson momentum maps (cf. , sectionIII.1.4): J L ∗ ( . ) − : T ∗ ( G ) → g ∗ , (3.61) J L ∗ ( . ) − ( θ, g ) = θ, J L ∗ ( . ) − ( L ∗ h − ( θ, g )) = Ad ∗ h (cid:16) J L ∗ ( . ) − ( θ, g ) (cid:17) , { J L ∗ ( . ) − X , J L ∗ ( . ) − Y } T ∗ G = − J L ∗ ( . ) − [ X,Y ] , J L ∗ ( . ) − ( R ∗ h − ( θ, g )) = J L ∗ ( . ) − ( θ, g ) ,J R ∗ ( . ) − : T ∗ ( G ) → g ∗ ,J R ∗ ( . ) − ( θ, g ) = − Ad ∗ g − ( θ ) , J R ∗ ( . ) − ( R ∗ h − ( θ, g )) = Ad ∗ h (cid:16) J R ∗ ( . ) − ( θ, g ) (cid:17) , { J R ∗ ( . ) − X , J R ∗ ( . ) − Y } T ∗ G = − J R ∗ ( . ) − [ X,Y ] , J R ∗ ( . ) − ( L ∗ h − ( θ, g )) = J R ∗ ( . ) − ( θ, g ) , { J L ∗ ( . ) − X , J R ∗ ( . ) − Y } T ∗ G = 0 , X, Y ∈ g (3.62)where we identified T ∗ G ∼ = G × g ∗ by right translation, as above, and defined J • X ( θ, g ) := J • ( θ, g )( X ) , X ∈ g . This allows us, to make the notion of polynomial symbols precise: Definition III.16:
Given a (smooth) function σ on T ∗ G ∼ = G × g ∗ with values in multilinear maps from R ⊕ L Nn =1 g ⊕ n to R , N ∈ N , of the form σ ( θ, g )( ⊕ X ) = f ( g ) + N X n =1 X i ,...,i n f i ...i n ( g ) J L ∗ ( . ) − X i ( θ, g ) ...J L ∗ ( . ) − X in ( θ, g ) (3.63)= f ( g ) + N X n =1 X i ,...,i n f i ...i n ( g ) θ ( X i ) θ ( X i n )1 for ⊕ X ∈ L Nn =1 g ⊕ n , { f , f i ...i n } i ,...,i n ⊂ D ( G ) , we call each σ ( ⊕ X ) ∈ ˆ E U ( g ∗ ) ˆ ⊗ D ( G ) a poly-nomial symbol of degree ≤ N . If σ ( ⊕ X ) ∈ ˆ E U ( g ∗ ) ˆ ⊗ C ∞ ( G ) , we say that σ is smooth. The Paley-Wiener-Schwartz theorem allows us to characterise the quantisation of polynomialsymbols: Corollary III.17:
For N ∈ N : σ is a polynomial symbol of degree ≤ N ⇔ ˇ σ is a distribution of order ≤ N in g with values in D ( G ) and supp (ˇ σ ) ⊂ { } ⇔ F ( ε ) σ is a distribution of order ≤ N in G with valuesin D ( G ) and supp ( F ( ε ) σ ) ⊂ { e } . Among the polynomial symbols, we find the special cases σ f ( θ, g ) := f ( g ) , f ∈ C ∞ ( G ) , and σ X ( θ, g ) := J L ∗ ( . ) − X ( θ, g ), X ∈ g , . After a moments reflection, we see that the quantisation ofthese symbols gives rise to the commutation relations (3.1): ∀ Ψ ∈ C ∞ ( G ) : (cid:16) Q ( W ) ε ( σ f )Ψ (cid:17) ( g ) = f ( g )Ψ( g ) , (cid:16) Q ( W ) ε ( σ X )Ψ (cid:17) ( g ) = − iε ( R X Ψ)( G ) , (3.64) Q ( W ) ε ( { σ f , σ f } T ∗ G ) = iε [ Q ( W ) ε ( σ f ) , Q ( W ) ε ( σ f )] = 0 , (3.65) Q ( W ) ε ( { σ X , σ f } T ∗ G ) = iε [ Q ( W ) ε ( σ X ) , Q ( W ) ε ( σ f )] = R X f,Q ( W ) ε ( { σ X , σ Y } T ∗ G ) = iε [ Q ( W ) ε ( σ X ) , Q ( W ) ε ( σ Y )] = iεR [ X,Y ] . Let us come to the definition of the Paley-Wiener-Schwartz symbol spaces S K,m PW ,ρ,δ , which areanalogous to the (classical) symbol spaces S mρ,δ in the theory of pseudo-differential operators andWeyl quantisation on R n (cf. ). The main obstacle to such a definition is the fact that theexponential is no longer a diffeomorphism, which is why we need to deal with compactly supportedinstead of tempered distributions in g , and thus by the Paley-Wiener-Schwartz theorem with entireanalytic functions on g ∗ by means of the Fourier transform. Concerning the development of anasymptotic calculus, the analyticity requirement forbids the use of 0-excision functions, which arestandard in pseudo-differential calculus. Luckily, this alleged shortcoming can be dealt with by themethod of kernel cut-off operator from the theory of Volterra-Mellin pseudo-differential operators(cf. ). Definition III.18:
Let K (cid:60) g be a convex compact subset and m ∈ R , ≤ δ ≤ ρ ≤ . A function σ ∈ C ∞ ( g ∗ C , C ∞ ( G )) belongs to the space of Paley-Wiener-Schwartz symbols S K,m PW ,ρ,δ if the following conditions aresatisfied:1. σ : g ∗ C → C ∞ ( G ) is (weakly) holomorphic ,2. ∀ α, β ∈ N n : ∃ C αβ > ∀ θ ∈ g ∗ C : sup g ∈ G | ( R α ∂ βθ σ )( θ, g ) | ≤ C αβ h θ i m −| β | ρ + | α | δ e H K ( = ( θ )) ,where h θ i := (1 + | θ | g ∗ C ) is the standard regularized distance , H K ( = ( θ )) := sup X ∈ K = ( θ )( X ) is the supporting function of K , and we use the standard multi index notation (w.r.t. a fixed ordered basis { τ i } ni =1 ⊂ g and its dual in g ∗ ). Clearly, the definition is independent of the ordering of the rightmulti-differentials R α = R α ...R α n n , because the commutator [ R i , R j ] = f kij R k reduces the order and δ ≥ . S K,m PW ,ρ ( g ∗ C ) denotes the analogue of S K,m PW ,ρ,δ with C ∞ ( G ) replaced by C . ≤ N belong to S K,N PW , , for all K (cid:60) g . The followingrelations among the symbol spaces are immediate consequences of the definition: S K,m PW ,ρ,δ ⊂ S K,m PW ,ρ ,δ , ρ ≥ ρ , S K,m PW ,ρ,δ ⊂ S K,m PW ,ρ,δ , δ ≤ δ (3.66) S K,m PW ,ρ,δ ⊂ S K ,m PW ,ρ,δ , K ⊂ K , S K,m PW ,ρ,δ ⊂ S K,m PW ,ρ,δ , m ≤ m . This suggests the definition of the following spaces: S K, ∞ PW ,ρ,δ := [ m ∈ R S K,m PW ,ρ,δ , S K, −∞ PW := \ m ∈ R S K,m PW ,ρ,δ . (3.67)The Kohn-Nirenberg and Weyl quantisation of the restriction σ | g ∗ of σ ∈ S K,m PW ,ρ,δ to g ∗ ⊂ g ∗ C define continuous operators on C ∞ ( G ). In the following, we will abuse notation and suppress therestriction index. Corollary III.19:
Given σ ∈ S K,m PW ,ρ,δ , there exists ε ∈ (0 , s.t. K ⊂ U ε . Then F W,εσ defines a continuous operatoron C ∞ ( G ) via ρ L . Moreover, if σ ∈ S K, −∞ PW ,ρ,δ , then ρ L (cid:0) F W,εσ (cid:1) is a smoothing operator from D ( G ) to C ∞ ( G ) . Proof:
By the Paley-Wiener-Schwartz theorem, ˇ σ is a distribution of order ≤ N , for N ∈ N s.t. m ≤ N ,in g with values in C ∞ ( G ) and supp (ˇ σ ) ⊂ K , which implies the first statement. The secondstatement follows from remark III.15.The optimal constants C αβ > S K,m PW ,ρ,δ into Fréchetspaces. Proposition III.20:
Fix a convex compact subset K (cid:60) g and m ∈ R , ≤ δ ≤ ρ ≤ . The countable family of (ordered)seminorms || σ || ( K,m,ρ,δ ) k := sup α,β ∈ N n | α | + | β |≤ k sup ( g,θ ) ∈ G × g ∗ C h θ i − m + | β | ρ −| α | δ e − H K ( = ( θ )) | ( R α ∂ βθ σ )( θ, g ) | , (3.68) k ∈ N , σ ∈ S K,m PW ,ρ,δ , defines a Fréchet space topology on S K,m PW ,ρ,δ . Proof:
We show that S K,m PW ,ρ,δ is Hausdorff and complete in the locally convex topology defined by theseminorms. Since metrisability follows, because the family of seminorms is countable, we mayconclude that the spaces are Fréchet.1. Assuming || σ || ( K,m,ρ,δ ) k = 0 for some k ∈ N , we have || σ || ( K,m,ρ,δ )0 = 0, implying u ≡ S K,m PW ,ρ,δ is separated, and thus Hausdorff.2. Let us assume that { σ i } ∞ i =1 ⊂ S K,m PW ,ρ,δ is a Cauchy sequence, i.e. { σ i } ∞ i =1 is Cauchy forall seminorms || . || ( K,m,ρ,δ ) k , k ∈ N . By the definition of the seminorms, we know that the3convergence of { R α σ i } ∞ i =1 is uniform on compact sets in G × g ∗ C for all α ∈ N n , which implies theexistence of limit σ ∈ C ∞ ( g ∗ C , C ∞ ( G )) in this sense. Similarly, we have compact convergenceof { Λ( R α σ i ) } ∞ i =1 for all Λ ∈ E β ( G ) , α ∈ N n , where E β ( G ) is the dual of C ∞ ( G ) with its strongtopology (cf. ), which implies the existence (holomorphic) limits σ Λ ,α ∈ O ( g ∗ C ) with lineardependence on Λ ∈ E β ( G ). Next, we show that the maps Λ σ Λ ,α ( θ ) , θ ∈ g ∗ C , are bounded,which allows us to conclude that there exists σ α ( θ ) ∈ C ∞ ( G ) s.t. σ Λ ,α ( θ ) = Λ( σ α ( θ )), because E β ( G ) is Fréchet-Montel, and thus reflexive: ∀ θ ∈ g ∗ C : ∀ (cid:15) > ∃ i ∈ N : ∀ i ≥ i : | σ Λ ,α ( θ ) | ≤ | σ Λ ,α ( θ ) − σ Λ ,αi ( θ ) | + | σ Λ ,αi ( θ ) | ≤ (cid:15) + | σ Λ ,αi ( θ ) | (3.69) ≤ (cid:15) + sup σ k ( θ ) ∈{ σ j ( θ ) } ∞ j =1 k ∈ N | Λ(( R α σ k )( θ )) | . As { ( R α σ j )( θ ) } ∞ j =1 ⊂ C ∞ ( G ) is (weakly) bounded, the boundedness of the maps Λ σ Λ ,α ( θ ) , θ ∈ g ∗ C , follows: ∀ Λ ∈ E β ( G ) : ∃ k Λ ∈ N , C Λ ,α > σ i ( θ ) ∈{ σ j ( θ ) } ∞ j =1 i ∈ N | Λ( σ αi ( θ )) | ≤ C Λ ,α sup i ∈ N sup g ∈ G | γ |≤ k Λ | ( R α + γ σ i )( θ, g ) |≤ C Λ ,α h θ i m +( k Λ + | α | ) δ e H K ( = ( θ )) sup i ∈ N || σ i || ( K,m,ρ,δ ) k Λ + | α | ≤ h θ i m + k Λ δ e H K ( = ( θ )) M k Λ + | α | . By construction, the map θ σ ( θ ) is weakly holomorphic, and σ α ( θ ) = ( R α σ )( θ ), becausethe above implies: ∀ α ∈ N n , θ ∈ g ∗ C : ∀ (cid:15) > ∃ i ∈ N : ∀ i, j ≥ i : sup g ∈ G | ( R α σ i )( θ, g ) − ( R α σ )( θ, g ) | < (cid:15) (3.71) ⇒ ∀ Λ ∈ E β ( G ) , α ∈ N n , θ ∈ g ∗ C : ∀ (cid:15) > ∃ i ∈ N : ∀ i ≥ i : (3.72) | Λ( σ α ( θ )) − Λ(( R α σ )( θ )) | ≤ | Λ( σ α ( θ )) − Λ( σ αi ( θ )) | + | Λ(( R α σ i )( θ )) − Λ(( R α σ )( θ )) |≤ | Λ( σ α ( θ )) − Λ( σ αi ( θ )) | + C Λ ,α sup g ∈ G | γ |≤ k Λ | ( R α σ i )( θ, g ) − ( R α σ )( θ, g ) | < (cid:15) Furthermore, we know from the assumptions that the sequences { τ K,m,ρ,δα,β,i = h . i − m + ρ | β |− δ | α | e − H K ( = ( . )) ( R α ∂ βθ σ i ) } ∞ i =1 ⊂ C b ( g ∗ C × G ) (3.73)4uniformly converge to limits τ K,m,ρ,δα,β ∈ C b ( g ∗ C × G ). It remains to be concluded that: τ K,m,ρ,δα,β ( θ, g ) = h θ i − m + ρ | β |− δ | α | e − H K ( = ( θ )) ( R α ∂ βθ σ )( θ, g ) , (3.74)which follows from the convergence properties established so far: ∀ α, β ∈ N n , ( g, θ ) ∈ G × g ∗ C : ∀ (cid:15) > ∃ i ∈ N : ∀ i ≥ i : (3.75) | τ K,m,ρ,δα,β ( θ, g ) − h θ i − m + ρ | β |− δ | α | e − H K ( = ( θ )) ( R α ∂ βθ σ )( θ, g ) |≤ | τ K,m,ρ,δα,β ( θ, g ) − h θ i − m + ρ | β |− δ | α | e − H K ( = ( θ )) ( R α ∂ βθ σ i )( θ, g ) | + h θ i − m + ρ | β |− δ | α | e − H K ( = ( θ )) | ( R α ∂ βθ σ i )( θ, g ) − ( R α ∂ βθ σ )( θ, g ) | < (cid:15) Remark III.21:
By the preceding proposition, we can give S K, ∞ PW ,ρ,δ a strict inductive limit topology (cf. ), whichmakes it an LF-space. Lemma III.22:
For σ ∈ S K,m PW ,ρ,δ , τ ∈ S K ,m PW ,ρ ,δ s.t. max( δ, δ ) ≤ min( ρ, ρ ) , we have continuous maps:1. ∀ α, β ∈ N n : σ R α ∂ βθ σ ∈ S K,m −| β | ρ + | α | δ PW ,ρ,δ .2. ( σ, τ ) στ ∈ S K + K ,m + m PW , min( ρ,ρ ) , max( δ,δ ) .This implies that the Poisson bracket (3.60) defines a bilinear operation { , } : S K,m PW ,ρ,δ × S K ,m PW ,ρ,δ → S K + K ,m + m − min( ρ − δ, ρ − ,ρ,δ . (3.76) Proof:
1. From the commutation relations [ R i , R j ] = f kij R k and δ ≥
0, we conclude:sup g ∈ G | ( R γ ∂ (cid:15)θ ( R α ∂ βθ σ ))( θ, g ) | ≤ sup g ∈ G | ( R α + γ ∂ β + (cid:15)θ σ )( θ, g ) | (3.77)+ X (cid:15) ∈ N n | ζ | < | α | + | γ | sup g ∈ G | ( R ζ ∂ β + (cid:15)θ σ )( θ, g ) |≤ h θ i m −| β + (cid:15) | ρ e H K ( = ( θ )) (cid:18) C ( α + γ )( β + (cid:15) ) h θ i | α + γ | δ + X (cid:15) ∈ N n | ζ | < | α | + | γ | C ζ ( β + (cid:15) ) h θ i | ζ | δ (cid:19) ≤ C ( α + γ )( β + (cid:15) ) h θ i m −| β + (cid:15) | ρ + | α + γ | δ e H K ( = ( θ )) .
2. Using the Leibniz formula and the fact that H K + H K = H K + K , we find:sup g ∈ G | ( R α ∂ βθ ( στ ))( θ, g ) | (3.78)5 ≤ X γ ∈ N n γ ≤ α (cid:18) αγ (cid:19) X (cid:15) ∈ N n (cid:15) ≤ β (cid:18) β(cid:15) (cid:19) sup g ∈ G | ( R γ ∂ (cid:15)θ σ )( θ, g ) | sup g ∈ G | ( R α − γ ∂ β − (cid:15)θ τ )( θ, g ) |≤ X γ ∈ N n γ ≤ α (cid:18) αγ (cid:19) X (cid:15) ∈ N n (cid:15) ≤ β (cid:18) β(cid:15) (cid:19) C γ(cid:15) h θ i m −| (cid:15) | ρ + | γ | δ e H K ( = ( θ )) C ( α − γ )( β − (cid:15) ) h θ i m −| β − (cid:15) | ρ + | α − γ | δ e H K ( = ( θ )) ≤ C αβ e H K ( = ( θ ))+ H K ( = ( θ )) h θ i m + m X γ ∈ N n γ ≤ α (cid:18) αγ (cid:19) h θ i | γ | δ + | α − γ | δ X (cid:15) ∈ N n (cid:15) ≤ β (cid:18) β(cid:15) (cid:19) h θ i | (cid:15) | ρ −| β − (cid:15) | ρ ≤ C αβ h θ i m + m −| β | min( ρ,ρ )+ | α | max( δ,δ ) e H K + K ( = ( θ )) . As a preparation for the main theorem of this subsection, we define the kernel cut-off operator.
Definition III.23 (cp. , Definition 3.6): For ϕ ∈ C ∞ ( g ) , we define the kernel cut-off operator by ( H ( ϕ ) σ )( θ, g ) := ˇ σ ( e − iθ ( . ) ϕ ) = Z g dX e − iθ ( X ) ϕ ( X ) Z g ∗ dθ (2 π ) n e iθ ( X ) σ ( θ, g ) (3.79) for σ ∈ S K,m PW ,ρ,δ , ( g, θ ) ∈ G × g ∗ C . Remark III.24: If θ ∈ g ∗ ⊂ g ∗ C , we have( H ( ϕ ) σ )( θ, g ) = ˇ σ ( e − iθ ( . ) ϕ ) = ˇ σ ( e − iθ ( . ) ϕχ ) (3.80)= Z g dX e − iθ ( X ) ϕ ( X ) χ ( X ) Z g ∗ dθ (2 π ) n e iθ ( X ) σ ( θ, g )= Z g dX ϕ ( X ) χ ( X ) Z g ∗ dθ (2 π ) n e − iθ ( X ) σ ( θ − θ , g ) , for some cut-off function χ ∈ C ∞ c ( g ) about supp (ˇ σ ), i.e. χ ≡ U of supp (ˇ σ ) and χ ≡ g \ U for some relatively compact neighbourhood U ⊃ U . Clearly, both sides define holomorphic functions of θ , that are equal on g ∗ ⊂ g ∗ C , and thusare equal on g ∗ C . The holomorphicity of the last expression can be concluded from Z g dX ϕ ( X ) χ ( X ) Z g ∗ dθ (2 π ) n e − iθ ( X ) σ ( θ − θ , g ) = Z g ∗ dθ (2 π ) n e − iθ ( X ) σ ( θ − θ , g ) F [ φχ ]( θ ) (3.81)and the observation that differentiation under the integral is permitted, which follows from σ ( θ − ( . ) , g ) F [ φχ ] ∈ C ∞ PW ( g ∗ ). But, the last line in (3.80) is independent of χ due to the sup-port properties of ˇ σ , which gives us( H ( ϕ ) σ )( θ, g ) = Z g dX ϕ ( X ) Z g ∗ dθ (2 π ) n e − iθ ( X ) σ ( θ − θ , g ) . (3.82)Let us establish some important properties of the kernel cut-off operator.6 Theorem III.25 (cp. , Theorem 3.7.): The kernel cut-off operator H : C ∞ b ( g ) × S K,m PW ,ρ,δ → S K,m PW ,ρ,δ continuous. If ρ > we have theasymptotic expansion H ( ϕ ) σ ∼ X α ∈ N n ( − | α | α ! (( − i∂ X ) α ϕ )(0) ∂ αθ σ (3.83) in S K,m PW ,ρ,δ . Proof:
Since S K,m PW ,ρ,δ is a Fréchet space it suffices to prove that the || . || ( K,m,ρ,δ ) k -seminorms of H ( ϕ ) σ are bounded by the || . k| ( K,m,ρ,δ ) k -seminorms of σ and the || . || ∞ ,k -seminorms of ϕ . By standardregularization techniques for oscillatory integrals, we have, for large enough M ∈ N and all α, β ∈ N n : | ( R α ∂ βθ H ( ϕ ) σ )( θ, g ) | (3.84)= | ( H ( ϕ )( R α ∂ βθ σ ))( θ, g ) | = Z g dX h X i n (cid:16) (1 − ∆ X ) M ϕ (cid:17) ( X ) Z g ∗ dθ (2 π ) n e − iθ ( X ) (cid:16) (1 − ∆ θ ) n h θ i − M ( R α ∂ βθ σ )( θ − θ , g ) (cid:17) . The contribution of ϕ to the integral can be estimated by: (cid:12)(cid:12)(cid:12)(cid:16) (1 − ∆ X ) M ϕ (cid:17) ( X ) (cid:12)(cid:12)(cid:12) ≤ M X m =0 (cid:18) Mm (cid:19) | (∆ mX ϕ ) ( X ) | (3.85) ≤ M || ϕ || ∞ , M . Applying the Leibniz rule and the estimates | ∂ γθ h θ i − M | ≤ C γ,M h θ i − M −| γ | , (3.86) | ( R α ∂ β + γθ σ )( θ − θ , g ) | ≤ C α,β + γ e H K ( = )( θ − θ ) h θ − θ i m −| β + γ | ρ + | α | δ ≤ θ ∈ g ∗ Peetre’s ineq. C α,β + γ e H K ( = )( θ ) h θ i m −| β + γ | ρ + | α | δ h θ i | m −| β + γ | ρ + | α | δ | , we get: (cid:12)(cid:12)(cid:12)(cid:16) (1 − ∆ θ ) n h θ i − M ( R α ∂ βθ σ )( θ − θ , g ) (cid:17)(cid:12)(cid:12)(cid:12) (3.87) ≤ C α,β,n,M e H K ( = )( θ ) h θ i m −| β | ρ + | α | δ h θ i − M + | m | +(2 n + | β | ) ρ + | α | δ || ϕ || ∞ , M || σ || ( K,m,ρ,δ ) | α | + | β | +2 n , and thus the seminorm estimate: || H ( ϕ ) σ || ( K,m,ρ,δ ) k (3.88)7 ≤ C k,n,M (cid:18)Z g dX h X i n (cid:19)| {z } =: C n < ∞ (cid:18)Z g ∗ dθ (2 π ) n h θ i − M + | m | +(2 n + k ) ρ + kδ (cid:19)| {z } < ∞ for large M || ϕ || ∞ , M || σ || ( K,m,ρ,δ ) k +2 n . To obtain the asymptotic expansion, we consider the Taylor expansion of ϕ at X = 0 of order N − ϕ ( X ) = X | α |≤ N − α ! ( ∂ αX ϕ )(0) X α + 1( N − X | α | = N (cid:18) Nα (cid:19) X α Z ds (1 − s ) N − ( ∂ αX ϕ )( sX ) | {z } =: ϕ α ( N ) ( X ) ∈ C ∞ b ( g ) . (3.89)Plugging this expression into the kernel cut-off operator and integrating by parts, we find:( H ( ϕ ) σ )( θ, g ) = X | α |≤ N − α ! ( ∂ αX ϕ )(0) Z g dX Z g ∗ dθ (2 π ) n e − iθ ( X ) σ ( θ − θ , g ) X α (3.90)+ 1( N − X | α | = N (cid:18) Nα (cid:19) Z g dX Z g ∗ dθ (2 π ) n e − iθ ( X ) σ ( θ − θ , g ) X α × Z ds (1 − s ) N − ( ∂ αX ϕ )( sX ) | {z } =: ϕ α ( N ) ( X ) ∈ C ∞ b ( g ) = X | α |≤ N − ( − | α | α ! ( − i∂ αX ϕ )(0) Z g dX Z g ∗ dθ (2 π ) n e − iθ ( X ) ( ∂ αθ σ )( θ − θ , g )+ ( − i ) N ( N − X | α | = N (cid:18) Nα (cid:19) Z g dX Z g ∗ dθ (2 π ) n e − iθ ( X ) ( ∂ αθ σ )( θ − θ , g ) × Z ds (1 − s ) N − ( ∂ αX ϕ )( sX )= X | α |≤ N − ( − | α | α ! ( − i∂ αX ϕ )(0)( ∂ αθ σ ) | {z } ∈ S K,m −| α | ρ PW ,ρ,δ ( θ, g )+ ( − i ) N ( N − X | α | = N (cid:18) Nα (cid:19) ( H ( ϕ α ( N ) )(( ∂ αθ σ ))) | {z } ∈ S K,m − Nρ PW ,ρ,δ ( θ, g ) , where the statement in the last line follows from lemma III.22 and the continuity property of H ,which was shown before. The result follows by the definition of asymptotic expansions, i.e.:Given { m k } ∞ k =1 ⊂ R , s.t. lim k →∞ m k = −∞ and m := max k ∈ N m k , and σ k ∈ S K,m k PW ,ρ,δ , σ ∈ S K,m PW ,ρ,δ ,we say that P ∞ k =1 a k is asymptotic to a , a ∼ P ∞ k =1 a k , if ∀ M ∈ R : ∃ k ∈ N : ∀ k ≥ k : a − k X k =1 a k ∈ S K,M PW ,ρ,δ . (3.91) a is unique up to S K, −∞ PW ,ρ,δ (smoothing symbols).The preceding theorem implies the important8 Corollary III.26 (cp. , Corollary 3.8.): Given a cut-off function ϕ ∈ C ∞ c ( g ) around X = 0 , we have continuous operator id − H ( ϕ ) : S K,m PW ,ρ,δ −→ S K, −∞ PW ,ρ,δ . (3.92) Proof:
The Taylor expansion of 1 − ϕ at X = 0 vanishes to infinite order, and id − H ( ϕ ) = H (1 − ϕ ).Now, we can state the main theorem of this section (cp. , Theorem 3.16.). Theorem III.27 (Asymptotic completeness of the Paley-Wiener-Schwartz symbols):
The symbol spaces S K,m PW ,ρ,δ are asymptotically complete in the following sense:Given { m k } ∞ k =1 ⊂ R , s.t. lim k →∞ m k = −∞ and m := max k ∈ N m k , and σ k ∈ C ∞ ( G, S
K,m k PW ,ρ,δ ) ,there exists σ ∈ C ∞ ( G, S
K,m PW ,ρ,δ ) s.t. a ∼ P ∞ k =1 a k . We call a a resummation of P ∞ k =1 a k . Before we prove the theorem, we need some lemmata, following the idea of proof for Volterrasymbols by Krainer . Lemma III.28:
Given β ∈ N n , ϕ ∈ C ∞ c ( g ) and σ ∈ S K, − (2( n +1)+ | β | )PW ,ρ ( g ∗ C ) , we have: sup θ ∈ g ∗ C e − r |= ( θ ) | | ( H (( − i∂ X ) β ϕ c )( e irτ n ( . ) σ ))( θ ) | ≤ k n ( ϕ, β ) 1 c n +1 || e irτ n ( . ) σ || (2 r, − (2( n +1)+ | β | ) ,ρ,δ )0 , (3.93) for some constant k n ( ϕ, β ) > , where c ∈ [0 , ∞ ) , ϕ c ( X ) = ϕ ( cX ) , r := inf { r ∈ [0 , ∞ ) | K ⊂ B r (0) = { X ∈ g | | X | ≤ r }} and e irτ n ( . ) ( θ ) := e irθ ( τ n ) with τ n the nth basis vector of g . Proof:
First, observe that multiplication of σ with e irτ n ( . ) shifts the support of ˇ σ , F − [ e irτ n ( . ) σ ] =ˇ σ ( . + rτ n ) =: ˇ σ r , s.t. 0 ∈ g is not an interior point of supp( F − [ e irτ n ( . ) σ ]) ⊂ K − rτ n ⊂ B r (0).Second, we have the equivalent estimates:sup θ ∈ g ∗ C | α | = | β | +2( n +1) e − r |= ( θ ) | | θ α ( e irτ n ( . ) σ )( θ ) | < ∞ ⇔ sup θ ∈ g ∗ C e − r |= ( θ ) | h θ i | β | +2( n +1) | ( e irτ n ( . ) σ )( θ ) | < ∞ . (3.94)Now, the assumptions imply that:1. F − [ e irτ n ( . ) σ ] | g + ≡
0, where g + := { X ∈ g | θ n ( X ) > } ( θ n is dual to τ n ).2. F − [ e irτ n ( . ) σ ] ∈ C | β | + n +1 , since( ∂ αX F − [ e irτ n ( . ) σ ])( X ) = i | α | Z g ∗ dθ (2 π ) n θ α e iθ ( X ) e irθ ( τ n ) σ ( θ ) (3.95)= i | α | Z g ∗ dθ (2 π ) n h θ i − ( n +1) (cid:16) h θ i n +1 θ α e iθ ( X ) e irθ ( τ n ) σ ( θ ) (cid:17) | α | ≤ β + n + 1 by (3.94).The Taylor expansion of F − [ e irτ n ( . ) σ ] at X = 0 up to order | β | + n reads: F − [ e irτ n ( . ) σ ]( X ) = X | α | = | β | + n +1 R α (0) X α , | R α (0) | ≤ α ! sup X ∈ K − rτ n | γ | = | α | | ∂ γX F − [ e irτ n ( . ) σ ]( X ) | (3.96)= 1 α ! sup X ∈ K | γ | = | α | | ( ∂ γX ˇ σ )( X ) | since ∂ αX F − [ e irτ n ( . ) σ ] vanishes at X = 0 for all | α | < | β | + n + 1. Now, we come to the proof of(3.93):sup θ ∈ g ∗ C e − r |= ( θ ) | | ( H (( − i∂ X ) β ϕ c )( e irτ n ( . ) σ ))( θ ) | (3.97)= sup θ ∈ g ∗ C e − r |= ( θ ) | | Z g dX e − iθ ( X ) (( − i∂ X ) β ϕ c )( X )ˇ σ r ( X ) |≤ sup θ ∈ g ∗ C e − r |= ( θ ) | Z g dX e |= ( θ ) || X | | (( − i∂ X ) β ϕ c )( X )ˇ σ r ( X ) |≤ Z g dX | (( − i∂ X ) β ϕ c )( X )ˇ σ r ( X ) | = Z g dX h X i n +12 h X i n +12 | (( − i∂ X ) β ϕ c )( X )ˇ σ r ( X ) |≤ (cid:18)Z g dX h X i n +1 (cid:19) | {z } =: C n +1 < ∞ (cid:18)Z g dX h X i n +1 | (( − i∂ X ) β ϕ c )( X )ˇ σ r ( X ) | (cid:19) ≤ C n +1 (cid:18) Z g dX h X i n +1 | (( − i∂ X ) β ϕ c )( X ) | (cid:18) X | α | = | β | + n +1 | X α || R α (0) | (cid:19) (cid:19) ≤ C n +1 sup X ∈ K | γ | = | β | + n +1 | ( ∂ γX ˇ σ )( X ) | (cid:18) Z g dX h X i n +1 | (( − i∂ X ) β ϕ c )( X ) X | α | = | β | + n +1 α ! X α | (cid:19) ≤ C n +1 C n +1 (2 π ) n (cid:18) sup θ ∈ g ∗ C e − r |= ( θ ) | h θ i | β | +2( n +1) | ( e irτ n ( . ) σ )( θ ) | (cid:19) × (cid:18) Z g dX h X i n +1 | (( − i∂ X ) β ϕ c )( X ) X | α | = | β | + n +1 α ! X α | (cid:19) = C n +1 C n +1 (2 π ) n || e irτ n ( . ) σ || (2 r, − (2( n +1)+ | β | ) ,ρ,δ )0 × (cid:18) c − (3 n +2) Z g dX h c − X i n +1 | {z } ≤h X i n +1 | (( − i∂ X ) β ϕ )( X ) X | α | = | β | + n +1 α ! X α | (cid:19) ≤ C n +1 C n +1 (2 π ) n || e irτ n ( . ) σ || (2 r, − (2( n +1)+ | β | ) ,ρ,δ )0 × (cid:18) c − n +1) Z g dX h X i n +1 | (( − i∂ X ) β ϕ )( X ) X | α | = | β | + n +1 α ! X α | (cid:19) = C n +1 C n +1 (2 π ) n (cid:18) Z g dX h X i n +1 | (( − i∂ X ) β ϕ )( X ) X | α | = | β | + n +1 α ! X α | (cid:19) | {z } =: k n ( ϕ,β ) × c − ( n +1) || e irτ n ( . ) σ || (2 r, − (2( n +1)+ | β | ) ,ρ,δ )0 Lemma III.29:
Let N ∈ N , { M αβ } | α | + | β |≤ N ⊂ N , ϕ ∈ C ∞ c ( g ) and σ ∈ O ( g ∗ C , C ∞ ( G )) , s.t. sup ( g,θ ) ∈ G × g ∗ C | α | + | β |≤ N e − r |= ( θ ) | h θ i l Mαβ m +2( n +1) | ( R α ∂ βθ σ )( θ, g ) | < ∞ , (3.98) then we have: sup ( g,θ ) ∈ G × g ∗ C | α | + | β |≤ N e − r |= ( θ ) | h θ i M αβ | ( R α ∂ βθ H ( ϕ c )( e irτ n ( . ) σ ))( θ, g ) | (3.99) ≤ ˜ k n ( ϕ, N, { M αβ } ) 1 c n +1 sup ( g,θ ) ∈ G × g ∗ C | α | + | β |≤ N e − r |= ( θ ) | h θ i l Mαβ m +2( n +1) | ( R α ∂ βθ ( e irτ n ( . ) σ ))( θ, g ) | for some constant ˜ k n ( ϕ, M, N ) > , where the notation of lemma III.28 is employed. Proof:
The assumptions imply:sup ( g,θ ) ∈ G × g ∗ C | α | + | β |≤ N e − r |= ( θ ) | h θ i l Mαβ m +2( n +1) | ( R α ∂ βθ ( e irτ n ( . ) σ ))( θ, g ) | < ∞ , (3.100)as multiplication of σ with e irτ n ( . ) shifts the support of ˇ σ , and modifies the decay properties of σ in the imaginary directions of g ∗ C according to the Paley-Wiener-Schwartz theorem. Next, weobserve, that the following properties hold, due to the definition of H :1. R α ∂ βθ H ( ϕ c )( e irτ n ( . ) σ ) = H ( ϕ c )( R α ∂ βθ ( e irτ n ( . ) σ )).2. θ α ( H ( ϕ c )( e irτ n ( . ) σ ))( θ, g ) = P β ≤ α (cid:0) αβ (cid:1) H (( − i∂ X ) β ϕ c )(( . ) α − β e irτ n ( . ) σ )( θ, g ).3. h θ i M αβ ≤ Pl Mαβ m k =0 (cid:0)l Mαβ m k (cid:1) P | γ | = k (cid:0) kγ (cid:1) θ γ .1Combining these properties with lemma III.28, we find:sup ( g,θ ) ∈ G × g ∗ C | α | + | β |≤ N e − r |= ( θ ) | h θ i M αβ | ( R α ∂ βθ H ( ϕ c )( e irτ n ( . ) σ ))( θ, g ) | (3.101) ≤ sup ( g,θ ) ∈ G × g ∗ C | α | + | β |≤ N e − r |= ( θ ) | l Mαβ mX k =0 (cid:18)l M αβ m k (cid:19) X | γ | = k (cid:18) kγ (cid:19) | θ γ ( R α ∂ βθ H ( ϕ c )( e irτ n ( . ) σ ))( θ, g ) |≤ sup ( g,θ ) ∈ G × g ∗ C | α | + | β |≤ N e − r |= ( θ ) | l Mαβ mX k =0 (cid:18)l M αβ m k (cid:19) X | γ | = k (cid:18) kγ (cid:19) X ζ ≤ γ (cid:18) γζ (cid:19) × | H (( − i∂ X ) ζ ϕ c )(( . ) γ − ζ R α ∂ βθ ( e irτ n ( . ) σ ))( θ, g ) |≤ sup | α | + | β |≤ N e − r |= ( θ ) | l Mαβ mX k =0 (cid:18)l M αβ m k (cid:19) X | γ | = k (cid:18) kγ (cid:19) X ζ ≤ γ (cid:18) γζ (cid:19) × sup ( g,θ ) ∈ G × g ∗ C | H (( − i∂ X ) ζ ϕ c )(( . ) γ − ζ R α ∂ βθ ( e irτ n ( . ) σ ))( θ, g ) |≤ (3.93) sup | α | + | β |≤ N l Mαβ mX k =0 (cid:18)l M αβ m k (cid:19) X | γ | = k (cid:18) kγ (cid:19) X ζ ≤ γ (cid:18) γζ (cid:19) k ( ϕ, ζ ) c n +1 × sup ( g,θ ) ∈ G × g ∗ C e − r |= ( θ ) | h θ i | ζ | +2( n +1) | ( θ γ − ζ R α ∂ βθ ( e irτ n ( . ) σ ))( θ, g ) |≤ sup | α | + | β |≤ N l Mαβ mX k =0 (cid:18)l M αβ m k (cid:19) X | γ | = k (cid:18) kγ (cid:19)X ζ ≤ γ (cid:18) γζ (cid:19) k ( ϕ, ζ ) c n +1 | {z } ≤ cn +1 ˜ k ( ϕ,N, { M αβ } ) × sup ( g,θ ) ∈ G × g ∗ C e − r |= ( θ ) | h θ i l Mαβ m +2( n +1) | ( R α ∂ βθ ( e irτ n ( . ) σ ))( θ, g ) | , which concludes the proof.With the help of the preceding lemmata, we can prove a crucial convergence result for the symbolspaces (cp. , Proposition 3.14.). Proposition III.30:
Given { m k } ∞ k =1 ⊂ R , s.t. m k ≥ m k +1 −→ k →∞ −∞ , and a countable system of bounded sets { S k j } j ∈ N ⊂ S K,m k PW ,ρ,δ for every k ∈ N , then there exists a sequence { c i } ∞ i =1 ⊂ [1 , ∞ ) , with c i < c i +1 −→ i →∞ ∞ , s.t. ∞ X i = k sup σ ∈ S ij p ( H ( ϕ d i )( e irτ n ( . ) σ )) < ∞ , (3.102) for all j, k ∈ N , all continuous seminorms p on S B r (0) ,m k PW ,ρ,δ and all sequences { d i } ∞ i =1 ⊂ [1 , ∞ ) with ∀ i ∈ N : d i ≥ c i . Here, we use again the notation of lemma III.28. Proof:
Without loss of generality, we may assume that { m k } ∞ k =1 ⊂ R − and S k j ⊂ S k j +1 for all j, k ∈ N .For all l ∈ N , we define (ordered) seminorms q r,ρ,δl ( σ ) := sup ( g,θ ) ∈ G × g ∗ C | α | + | β |≤ l e − r |= ( θ ) | h θ i − m l + | β | ρ −| α | δ | ( R α ∂ βθ σ )( θ, g ) | , q r,ρ,δl ≤ q r,ρ,δl +1 . (3.103)Using the preceding lemmata, we find for suitable σ and c ∈ [1 , ∞ ): q r,ρ,δl ( H ( ϕ c )( e irτ n ( . ) σ )) = sup ( g,θ ) ∈ G × g ∗ C | α | + | β |≤ l e − r |= ( θ ) | h θ i − m l + | β | ρ −| α | δ | ( R α ∂ βθ H ( ϕ c )( e irτ n ( . ) σ ))( θ, g ) | (3.104) ≤ c n +1 ˜ k ( ϕ, l, { M αβ } ) × sup ( g,θ ) ∈ G × g ∗ C | α | + | β |≤ l e − r |= ( θ ) | h θ i l Mαβ m +2( n +1) | ( R α ∂ βθ ( e irτ n ( . ) σ ))( θ, g ) | , where M αβ := d− m l + | β | ρ − | α | δ e . By assumption on the sequence { m k } ∞ k =1 , we can find i ∈ N , i ≥ l , s.t. 2 l M αβ m +2( n +1) ≤ − m i + | β | ρ −| α | δ , for | α | + | β | ≤ l , i.e. we need m i +2( n +2)+1 ≤ m l . Thus, we get the estimate: ∀ l ∈ N : ∃ i ∈ N : ∀ i ≥ i : q r,ρ,δl ( H ( ϕ c )( e irτ n ( . ) σ )) ≤ ˜ k ( ϕ, l, { M αβ } ) c n +1 q r,ρ,δi ( e irτ n ( . ) σ ) , (3.105)for σ ∈ S K,m i PW ,ρ,δ . The existence of the sequence { c i } ∞ i =1 ⊂ [1 , ∞ ) with the prescribed propertiesfollows by induction. Following Krainer , we construct sequences { c l i } ∞ i =1 ⊂ [1 , ∞ ) for l ∈ N , andtake { c := c i i } ∞ i =1 :1. Let l = 1: By (3.105), we can find a sequence { c i } ∞ i =1 ⊂ [1 , ∞ ), c i < c i +1 −→ i →∞ ∞ , s.t. for all i ∈ N with m i + 2( n + 2) + 1 ≤ m sup σ ∈ S i q r,ρ,δ ( H ( ϕ d i )( e irτ n ( . ) σ )) < − i (3.106)holds for all { d i } ∞ i =1 ⊂ [1 , ∞ ) with ∀ i ∈ N : d i ≥ c i .2. Let { c l i } ∞ i =1 ⊂ [1 , ∞ ) be constructed: By (3.105), we find a subsequence { c ( l +1) i } ∞ i =1 ⊂ { c l i } ∞ i =1 , s.t. for all i ∈ N with m i + 2( n + 2) + 1 ≤ m l +1 sup σ ∈ S il +1 q r,ρ,δl +1 ( H ( ϕ d i )( e irτ n ( . ) σ )) < − i (3.107)holds for all { d i } ∞ i =1 ⊂ [1 , ∞ ) with ∀ i ∈ N : d i ≥ c ( l +1) i .3. By construction, the diagonal sequence { c := c i i } ∞ i =1 ⊂ [1 , ∞ ), has the property c i ≥ c l i for i ≥ l and c i < c i +1 −→ i →∞ ∞ .4. Let j, k ∈ N and p be a continuous seminorm on S B r (0) ,m k PW ,ρ,δ , then there exist l s.t. therestriction of p to S B r (0) ,m i PW ,ρ,δ is dominated by q r,ρ,δl with a constant independent of i and S i j ⊂ S i l for almost all i ∈ N . These assertions follow from the inclusion properties of thesymbol spaces (see (3.66)). Employing the continuity of H (see theorem III.25), we concludethat the series (3.102) is indeed convergent for given data j, k ∈ N and p . Proof (of Theorem III.27):
Without loss of generality, we may assume m k ≥ m k +1 −→ k →∞ −∞ . For j, k ∈ N , we define S k j := { ( R α σ k )( g ) | g ∈ G, | α | ≤ j } ⊂ S K,m k PW ,ρ,δ . Here, R is the right differential in the first group variable.Since σ k ∈ C ∞ ( G, S
K,m k PW ,ρ,δ ), we know that the sets S k j are bounded in S K,m k PW ,ρ,δ . Now, we choosea cut-off function ϕ ∈ C ∞ c ( g ) around X = 0, and apply proposition III.30 to obtain a sequence { c i } ∞ i =1 ⊂ [1 , ∞ ), s.t. ∞ X i = k sup n p (cid:16) ( H ( ϕ c i )( e irτ n ( . ) R α σ ))( g ) (cid:17) | g ∈ G, | α | ≤ j o < ∞ (3.108)for all continuous seminorms p on S B r (0) ,m k PW ,ρ,δ . Therefore, the sum a ( r ) := ∞ X i =1 H ( ϕ c i )( e irτ n ( . ) σ ) (3.109)is unconditionally convergent in C ∞ ( G, S K − rτ n ,m PW ,ρ,δ ). Now, we define: a := e − irτ n ( . ) a ( r ) , (3.110)which is in C ∞ ( G, S
K,m PW ,ρ,δ ). It follows that the a ∼ P ∞ k =1 a k : a − k X i =1 a i = e − irτ n ( . ) a r − k X i =1 e irτ n ( . ) a i ! (3.111)= e − irτ n ( . ) ∞ X i = k +1 H ( ϕ c i )( e irτ n ( . ) σ ) + e − irτ n ( . ) k X i =1 (id − H ( ϕ c i ))( e irτ n ( . ) σ )4= e − irτ n ( . ) ∞ X i = k +1 H ( ϕ c i )( e irτ n ( . ) σ ) | {z } ∈ C ∞ (cid:16) G,S
K,mk +1PW ,ρ,δ (cid:17) + e − irτ n ( . ) k X i =1 H (1 − ϕ c i )( e irτ n ( . ) σ ) | {z } ∈ C ∞ ( G,S K, −∞ PW ,ρ,δ ) Cor. III.26 . We conclude the subsection by showing that the operator product of two Weyl quantisations F W,εσ , F
W,ετ of Paley-Wiener-Schwartz symbols σ ∈ S K,m PW ,ρ,δ , τ ∈ S K ,m PW ,ρ,δ , assuming that ε ∈ (0 ,
1] issmall enough, has a formal expansion in ε that qualifies as an asymptotic series for certain values of0 ≤ δ ≤ ρ ≤
1. Moreover the series has finitely many non-vanishing terms if σ and τ are polynomial.Unfortunately, we can (so far) not establish that this series is asymptotic to the dequantisation ofthe operator product, because the image of the Weyl quantisation is not obviously closed underproducts . To this end, we recall that the group product in V ⊂ G can be pulled back to U ⊂ g ,and the Dynkin-Baker-Campbell-Hausdorff formula tells us, that we may write X hg = exp − ( hg ) = exp − ( X h ) ∗ exp − ( X g ) = X h + X g + ∞ X k =1 P k ( X h , X g ) (3.112)= X h + X g + 12 [ X h , X g ] + 112 ([ X h , [ X h , X g ]] + [ X g , [ X g , X h ]]) + higher orders , for sufficiently small X h , X g ∈ U (cf. for convergence properties of (3.112)). Here, the P k , k ∈ N , are Lie-Polynomials. The ε -scaled version of the product, exp( ε ( X h ∗ ε X g )) = exp( εX h ) exp( εX g ),is: ε − (( εX h ) ∗ ( εX g )) = X h ∗ ε X ε = X h + X g + ∞ X k =1 ε k P k ( X h , X g ) (3.113)= X h + X g + ε X h , X g ] + ε
12 ([ X h , [ X h , X g ]] + [ X g , [ X g , X h ]]) + O ( ε ) . If we apply this formula to the twisted convolution (2.5) of the Weyl quantisations of F W,εσ , F
W,ετ ,we get for ε small enough: (cid:0) F W,εσ ∗ F W,ετ (cid:1) ( h, g ) (3.114)= Z G dh F W,εσ ( h , g ) F W,ετ ( h h, h g )= Z G dh F εσ ( h , √ h g ) F ετ ( h h, √ h − h h g )= ε − n Z G dh ˇ σ (cid:0) ε − X h , exp( − X h ) g (cid:1) ˇ τ (cid:0) ε − X h h , exp( − X h h ) exp( − X h ) g (cid:1) = ε − n Z G dh ˇ σ (cid:0) ε − X h , exp( − X h ) g (cid:1) × ˇ τ (cid:0) ε − (( − X h ) ∗ X h ) , exp( − (( − X h ) ∗ X h )) exp( − X h ) g (cid:1) = ε − n Z g dX h j ( X h ) ˇ σ (cid:0) ε − X h , exp( − X h ) g (cid:1) × ˇ τ (cid:0) ε − (( − X h ) ∗ X h ) , exp( − (( − X h ) ∗ X h )) exp( − X h ) g (cid:1) ε − n Z g dX h j ( εX h ) ˇ σ (cid:0) X h , exp( − ε X h ) g (cid:1) × ˇ τ (cid:0) ε − (( − εX h ) ∗ X h ) , exp( − (( − εX h ) ∗ X h )) exp( − εX h ) g (cid:1) = ε − n Z g dX h j ( εX h ) ˇ σ (cid:0) X h , exp( − ε X h ) g (cid:1) × ˇ τ (cid:0) (( − X h ) ∗ ε ( ε − X h )) , exp( − ε (( − X h ) ∗ ε ( ε − X h ))) exp( − εX h ) g (cid:1) = ε − n Z g dX h j ( εX h ) ˇ σ (cid:0) X h , exp( − ε X h ) exp( ε ( ε − X h )) exp( − ( X h )) g (cid:1) × ˇ τ (cid:0) (( − X h ) ∗ ε ( ε − X h )) , exp( − ε (( − X h ) ∗ ε ( ε − X h ))) exp( − εX h ) exp( ε ( ε − X h )) exp( − ( X h )) g (cid:1) = ε − n Z g dX h j ( εX h ) ˇ σ (cid:0) X h , exp( ε (( − X h ) ∗ ε ( ( ε − X h )))) exp( − ( X h )) g (cid:1) × ˇ τ (cid:0) (( − X h ) ∗ ε ( ε − X h )) , exp( ε (( − (( − X h ) ∗ ε ( ε − X h ))) ∗ ε ((( − X h ) ∗ ε ( ( ε − X h )))))) exp( − ( X h )) g (cid:1) , where we switched integration from G to g by means of the exponential map (see (3.45)), which ispermitted due to the support properties of ˇ σ and ˇ τ , changed integration variables X h ε − X h ,and successively replaced the group product in V by its ε -scaled pullback ∗ (cid:15) in U .Now, we would like to write (cid:0) F W,εσ ∗ F W,ετ (cid:1) ( h, g ) = F W,(cid:15)ρ ( h, g ) for some ρ ∈ S K + K ,m + m PW ,ρ,δ , whichwould define the twisted product of symbols ρ = σ ? ε τ . The shift in the support ( K, K ) K + K is to be expected, because of the relation of ? (cid:15) to the convolution of ˇ σ and ˇ τ . Unfortunately,as already mentioned above, the last line in (3.114), is not obviously of the form F W,(cid:15)ρ ( h, g ). Nev-ertheless, if we apply (3.113), we can deduce an asymptotic series in S K + K ,m + m PW ,ρ,δ by successiveintegration by parts. To this end, we explicitly write out the inverse Fourier transforms in (3.114): (cid:0) F W,εσ ∗ F W,ετ (cid:1) ( h, g ) (3.115)= ε − n Z g dX h j ( εX h ) Z g ∗ dθ (2 π ) n e iθ ( X h ) σ ( θ, exp( ε (( − X h ) ∗ ε ( ( ε − X h )))) exp( − ( X h )) g ) × Z g ∗ dθ (2 π ) n e iθ ((( − X h ) ∗ ε ( ε − X h ))) × τ ( θ , exp( ε (( − (( − X h ) ∗ ε ( ε − X h ))) ∗ ε ((( − X h ) ∗ ε ( ( ε − X h )))))) exp( − ( X h )) g ) . From the Dynkin-Baker-Campbell-Hausdorff formula (3.112), and the fact that the P k , k ∈ N , areLie-Polynomials, we infer that( − X h ) ∗ ε ( ε − X h ) = − ( X h − ε − X h ) + ∞ X k =1 ε k P k ( X h , X h − ε − X h ) , (3.116)for some (new) Lie-Polynomials P k , k ∈ N .Therefore, we can achieve the following rewriting of e iθ ((( − X h ) ∗ ε ( ε − X h ))) : e iθ ((( − X h ) ∗ ε ( ε − X h ))) = e iθ ((( − X h ) ∗ ε ( ε − X h ))+( X h − ε − X h )) e − iθ ( X h − ε − X h ) (3.117)= ∞ Y k =1 e iε k θ ( P k ( X h ,X h − ε − X h )) e − iθ ( X h − ε − X h ) ∞ Y k =1 ε k Q k ( ε, X h , θ , ∂ θ ) e − iθ ( X h − ε − X h ) , where Q k ( ε, X h , θ , ∂ θ ) , k ∈ N , are differential operators of infinite order. If we continue byperforming Taylor expansions of j ( εX h ) , σ ( θ, exp( ε (( − X h ) ∗ ε ( ( ε − X h )))) exp( − ( X h )) g ) and τ ( θ , exp( ε (( − (( − X h ) ∗ ε ( ε − X h ))) ∗ ε ((( − X h ) ∗ ε ( ( ε − X h )))))) exp( − ( X h )) g ) in ε , trade all X h - and ( X h − ε − X h )-dependence by differentiation of e iθ ( X h ) and e − iθ ( X h − ε − X h ) for θ - and θ -differentials, and finally perform repeated integrations by parts, culminating in an expressionwhere the dX h integration gives an oscillatory integral representation of δ ( n ) ( θ − θ ), we will getan infinite sum in orders of ε (cid:0) F W,εσ ∗ L F W,ετ (cid:1) ? ∼ ∞ X k =0 ε k C W,εk ( σ, τ ) (3.118)consisting of expressions of the form C W,εk ( σ, τ ), where each order ε k can be written as products ofdifferentials of σ and τ of total order ≤ k each. Due to the fact, that j = 1 for general compactLie groups G , the differential operations on σ and τ will not solely be determined by the Dynkin-Baker-Campbell-Hausdorff formula, and thus by the Poisson bracket (3.60), but also be affectedby differential operators determined by the structure of the (positive) roots of G . Nevertheless,it follows from the Taylor series of j that those additional differential operators will lead to lowerorder terms in S K + K ,m + m PW ,ρ,δ , than those coming from the Poisson bracket, if ρ > δ . Taking a closerlook at the formula for the Poisson bracket (3.60) and (3.76), we realise that the condition ρ > δ ,known from the R n -case, has to be supplemented by ρ > for the expression (3.118) to qualify asan asymptotic sum. We note that, by theorem III.27, there exists a resummation of the right handside of (3.118), which is unique up to smoothing symbols in S K + K , −∞ PW ,ρ,δ . Thus, from a practicalpoint of view it would suffice to show that the difference between the operators defined by the lefthand side and the resummation of the right hand side is sufficiently “small” in a sense to be madeprecise, if asymptoticity in S K + K ,m + m PW ,ρ,δ were to fail.Finally, let us be a bit more explicit, and give the expression for (3.118) up to order ε . Additionally,we provide the expression up to order ε before partial integration: j ( εX h ) e iθ ((( − X h ) ∗ ε ( ε − X h ))) σ ( θ, exp( ε (( − X h ) ∗ ε ( ( ε − X h )))) exp( − ( X h )) g ) (3.119) × τ ( θ , exp( ε (( − (( − X h ) ∗ ε ( ε − X h ))) ∗ ε ((( − X h ) ∗ ε ( ( ε − X h )))))) exp( − ( X h )) g ) ∼ O ( ε ) e − iθ ( X h − ε − X h ) σ ( θ, √ h − g ) τ ( θ , √ h − g )+ ε (cid:18) iθ ([ X h , X h − ε − X h ]) σ ( θ, √ h − g ) τ ( θ , √ h − g ) − ( R X h − ε − X h σ )( θ, √ h − g ) τ ( θ , √ h − g ) − σ ( θ, √ h − g )( R X h τ )( θ , √ h − g ) (cid:19) + ε (cid:18)(cid:16) − X α ∈ R + α ( X h ) −
12 ( θ ([ X h , X h − ε − X h ])) − iθ (2[ X h , X h − ε − X h ] (2) + [ X h − ε − X h , X h ] (2) ) (cid:17) σ ( θ, √ h − g ) τ ( θ , √ h − g )+ 12 (cid:16) σ ( θ, √ h − g ) (cid:0) ( R X h τ )( θ , √ h − g ) + ( R [ X h ,X h − ε − X h ] τ )( θ , √ h − g ) (cid:1) + (cid:0) ( R X h − ε − X h σ )( θ, √ h − g ) + ( R [ X h ,X h − ε − X h ] σ )( θ, √ h − g ) (cid:1) τ ( θ , √ h − g ) (cid:17) + ( R X h − ε − X h σ )( θ, √ h − g )( R X h τ )( θ , √ h − g ) − iθ ([ X h , X h − ε − X h ]) × (cid:16) ( R X h − ε − X h σ )( θ, √ h − g ) τ ( θ , √ h − g )+ σ ( θ, √ h − g )( R X h τ )( θ , √ h − g ) (cid:17)(cid:19)! . Integrating this expression at order ε , we have: (cid:0) F W,εσ ∗ F W,ετ (cid:1) ∼ O ( ε ) F W,εστ − iε F W,ε { σ,τ } T ∗ G , (3.120) σ ? ε τ ∼ O ( ε ) στ − iε { σ, τ } T ∗ G as expected from theorem III.14. B. Scaled Fourier transforms, the Stratonovich-Weyl transform and coadjoint orbits
The global and local definitions of pseudo-differential operators we introduced in the previoussubsection appear to be intimately tied to the natural representation L ( G ) of the transformationgroup C ∗ -algebra C ( G ) (cid:111) α L G , and its regularity properties w.r.t. the group translations U g , g ∈ G ,up to this point. Especially, the expansion (3.118) associates the ε -dependence with the “momentumvariables” P X , X ∈ g , in (3.1). But, in view of applications in Born-Oppenheimer reduction schemesand adiabatic perturbation theory it seems to be useful to be able to shift the ε -dependence to thevariables dual to the “momenta”, thereby changing from the microscopic scale to the macroscopicscale ( X g = εX g , g = exp( εX g )). Furthermore, it can be advantageous to switch the roles of“momenta” and “positions” by means of a suitable integral transform on L ( G ), as seen form (Section 5).To exemplify this point, we recall how the Fourier transform, and its ε -scaled version, affect thecase of pseudo-differential operators on R n . Namely, if we consider a symbol σ ∈ S mρ,δ or S ( R n ),and its action on S ( R n ) via Weyl quantisation,( A σ Ψ)( q ) = 1(2 πε ) n Z R n dx dξ σ ( ( q + x ) , ξ ) e iε ξ · ( q − x ) Ψ( x ) , (3.121)which is adapted to the representation of the commutation relations (3.2) by Q = q · and P = − iε ∇ q ,we may interchanges the roles of Q and P w.r.t. to the Weyl quantisation by applying the ε -scaledFourier transform, F ε [Ψ]( p ) = ˆΨ ( ε ) ( p ) := Z R n dq e − iε p · q Ψ( q ) , (3.122)8 F − ε [Φ]( q ) = ˇΦ ( ε ) ( q ) := Z R n dp (2 πε ) n e iε p · q Φ( p ) , to obtain ( ˆ A σ ˆΨ ( ε ) )( p ) = 1(2 πε ) n Z R n dx dξ σ ( x, ( p + ξ )) e − iε ( p − ξ ) · x ˆΨ ( ε ) ( ξ ) , (3.123)which is adapted to the representation of (3.2) by Q = − iε ∇ p and P = p · . Consider, for example,a standard Born-Oppenheimer type Hamiltonian H = − ε ∇ q + iA ( q )) ⊗ H f + V ( q ) (3.124)acting on the coupled quantum system H = L ( R n ) ⊗ H f . (3.124) is obtained as Weyl quantisation(3.121) of the ε -dependent symbol σ H ( q, p ) = 12 ( p − εA ( q )) ⊗ H f + V ( q ) . (3.125)On the Fourier transformed side, the Hamiltonian takes the form (3.123):ˆ H = ε p − A ( i ∇ p )) ⊗ H f + V ( i ∇ p ) (unscaled) , (3.126)ˆ H ε = 12 ( p − εA ( iε ∇ p )) ⊗ H f + V ( iε ∇ p ) ( ε -scaled) . While these observation are almost trivial at this level, equation (3.124) and (3.126) illustrate thefact, that it is beneficial to have suitable integral transforms, possibly with ε -scaling, at hand todecide whether a given operator can be written as Weyl quantisation of a, possibly operator valued,symbol to apply Born-Oppenheimer reduction or space adiabatic perturbation theory. A less trivialexample is given by a Hamiltonian with periodic potential V Γ (Γ ⊂ R n is the periodicity latticeof V Γ ) and slowly varying external electromagnetic fields A, φ considered by Panati, Teufel andSpohn : H = 12 ( − i ∇ q − A ( εq )) + V Γ ( q ) + φ ( εq ) , (3.127)which can be rewritten as Weyl quantisation of an operator valued symbol by means of the Bloch-Floquet transform U : L ( R n ) → H Γ ∗ (cf. , Chapter 1.10): U [Ψ]( p, q ) := X γ ∈ Γ e − ip · ( q + γ ) Ψ( q + γ ) , ( q, p ) ∈ R n , Ψ ∈ S ( R n ) , (3.128)where H Γ ∗ = { Φ ∈ L ( R n , L ( T n )) | Φ( p + γ ∗ , q ) = e − iγ ∗ · q Φ( p, q ) } and Γ ∗ is the lattice dual toΓ. Applying U to (3.127) gives: U HU ∗ = 12 ( − i ∇ per q + p + A ( iε ∇ p )) + V Γ ( q ) + φ ( iε ∇ p ) , (3.129)9which can be understood as the Weyl quantisation of a Γ ∗ -equivariant symbol with values inbounded operators from H ( T n ) to L ( T n ), and thus makes (3.127) accessible to space adiabaticperturbation theory.This said, we return to the case of compact and simply connected Lie groups , where we applythe Stratonovich-Weyl-Fourier transform introduced by Figueroa, Gracia-Bondía and Várilly ,based on ideas of Stratonovich , to the pseudo-differential operators defined in the previous sec-tion, which gives rise to an alternative to the common Fourier transform from L ( G ) to L ( ˆ G ), andadditionally makes the effect of (radial) scaling particularly transparent. Since this transform issomewhat non-standard, we recall the main steps of its construction in some detail:According to , we construct the Stratonovich-Weyl operator ∆ π : O π → End( V π ), which allows usto map functions on O π to operator in V π , via the coherent state formalism for compact semisimpleLie groups :1. For a unitary irreducible representation, π ∈ ˆ G , we choose the corresponding (real) highestweight λ π ∈ t ∗ ⊂ g ∗ . Let O π = { θ ∈ g ∗ | ∃ g ∈ G : θ = Ad ∗ g ( λ π ) } ⊂ g ∗ be the coadjoint orbitof G through λ π . O π ∼ = G/G λ π is homogeneous space, where G λ π is the stabiliser of λ π . Itis the content of the Borel-Weil theorem that the correspondence π ↔ λ π is one-to-one up tounitary equivalence .2. Next, we choose a normalised weight vector v π ∈ V π of λ π . Moreover, we define the equivariantmomentum map J π ( v )( X ) := 12 πi ( v, dπ ( X ) v ) V π , v ∈ V π , X ∈ g , (3.130)which satisfies J π ( π ( g ) v ) = Ad ∗ g ( J π ( v )) , g ∈ G, , J π ( v π ) = λ π and J − π ( { λ π } ) = { zv π | z ∈ C , | z | = 1 } because the weight space of λ π is one-dimensional. Then, we have J − π ( O π ) = { π ( g ) v π | g ∈ G } .3. For θ ∈ O π we choose g θ ∈ G s.t. Ad ∗ g θ ( λ π ) = θ and g λ π = e . θ g θ is a measurable sectionw.r.t. to the Liouville measure dµ π on O π ( µ π ( O π ) = dim V π = d π ) induced by the naturalinvariant symplectic form ω π .4. We define the coherent state v θ := π ( g θ ) v π ∈ V π for θ ∈ O π , which is uniquely determined by θ up to a phase, since J π ( v θ ) = θ .5. For an operator A ∈ End( V π ), we have the covariant or lower symbol L πA ( θ ) := ( v θ , Av θ ) V π , (3.131)which uniquely determines A , since the coherent states { v θ } θ ∈ O π are complete by means ofthe natural Kähler structure on O π (cf. ). The lower symbol is covariant w.r.t. G : L ππ ( g ) Aπ ( g ) ∗ ( θ ) = L πA ( Ad ∗ g − ( θ )) , g ∈ G. (3.132)6. By duality and the Riesz-Fréchet theorem ( d π < ∞ ), we obtain the contravariant or uppersymbol A U πA : ( A, B ) HS = tr( A ∗ B ) = Z O π dµ π ( θ ) U πA ( θ ) L πA ( θ ) . (3.133)0The normalisation of dµ λ ensures that U π Vπ = 1. The upper symbol is covariant, as well: U ππ ( g ) Aπ ( g ) ∗ ( θ ) = U πA ( Ad ∗ g − ( θ )) , g ∈ G. (3.134)8. The map U πA L πA defines a positive G -invariant invertible operator K π on the finitedimensional space of functions S π := span C { U πA | A ∈ End( V π ) } ⊂ L ( O π ). G -invariance is tobe understood w.r.t. the quasiregular representation ( ρ ( g ) f )( θ ) = f ( Ad ∗ g − ( θ )) , f ∈ L ( O π ).Form the definition of lower and upper symbols, we infer that the kernel of K π is determinedby the overlap function of the coherent states: L πA ( θ ) = ( v θ , Av θ ) V π = Z O π dµ π ( θ ) | ( v θ , v θ ) V π | U πA ( θ ) . (3.135)By means of K π , the Stratonovich-Weyl symbol of A is defined to be: W πA := K π U πA = K − π L πA , (3.136)where K π is the positive square root of K π . It has the following properties:(a) End( V π ) A W πA ∈ S π is linear and bijective,(b) W πA ∗ = W πA ,(c) W π Vπ = 1,(d) W ππ ( g ) Aπ ( g ) ∗ = ρ ( g )( W πA ),(e) ( A, B ) HS = tr( A ∗ B ) (3.137)= Z O π dµ π ( θ ) W πA ( θ ) W πB ( θ ) .
9. Finally, the
Stratonovich-Weyl operator ∆ π : O π → End( V π ) is constructed in spirit of the(Fourier-)Weyl elements, already familiar from the previous subsection (see (3.4)). Namely,we look for an operator valued function ∆ π = (∆ π ) ∗ s.t.: W πA ( θ ) = tr(∆ π ( θ ) A ) = (∆ π ( θ ) , A ) HS , (3.138) A = Z O π dµ π ( θ ) W πA ( θ )∆ π ( θ ) = (∆ π , W πA ) L ( O π ) . To construct ∆ π , we decompose S π ∼ = End( V π ) ∼ = V π ⊗ V π ∼ = L η V ⊕ N ηπ η ∈ ˆ G ∼ = L η ∈ ˆ G L N ηπ s =1 V η ,where N ηπ ∈ N is the multiplicity of η in π ⊗ π . Now, we can introduce the generalisedspin-weighted spherical harmonics as an orthonormal basis of S π : (cid:0) Y πηsk ( g θ ) l (cid:1) l =1 ,...,d η := ( η s ( g θ ) kl ) l =1 ,...,d η (3.139)for s = 1 , ..., N ηπ , k = 1 , ..., d η , θ ∈ O π , where the matrix elements are computed w.r.t. toa basis, { v η,si } d η i =1 , v η,sd η = v η,s , adapted to the weight decomposition V η,s = V λ η ⊕ L λ V λ V η in V π ⊗ V π . We use the same notation for any adapted basis for aunitary irreducible representation of V of G . Here, λ η denotes the highest weight of η , and v η is a normalised weight vector of λ η . The generalised spherical harmonics Y π ( η,l η ) sk ( θ ) := Y π ( η,l η ) sk ( g θ ) are obtained from the decomposition π ( g ) id π π ( g ) jd π = X η,l η ,s,k C ( π, i ; π, j | η, k ; s ) C ( π, d π ; π, d π | η, l η ; s ) Y π ( η,l η ) sk ( g ) , (3.140)since v π ⊗ v π has (real) weight 0. The C ( π, m ; ζ, n | η, k ; s )’s denote the Clebsch-Gordan co-efficients of the decomposition π ⊗ ζ ∼ = L η ∈ ˆ G L N ηπ,ζ s =1 V η . These function on O π (weight 0!)diagonalise the kernel K π (cf. ), K π ( θ, θ ) = | ( v θ , v θ ) | (3.141)= | ( π ( g θ ) v π , π ( g θ ) v π ) | = (cid:12)(cid:12)(cid:12) d π X i =1 π ( g θ ) id π π ( g θ ) jd π (cid:12)(cid:12)(cid:12) = X η,l η ,s,kη ,l η ,s ,k d π X i,j =1 C ( π, i ; π, j | η, k ; s ) C ( π, i ; π, j | η , k ; s ) × C ( π, d π ; π, d π | η, l η ; s ) C ( π, d π ; π, d π | η , l η ; s ) Y π ( η,l η ) sk ( g θ ) Y π ( η ,l η ) s k ( g θ )= X η,l η ,s,k C ( π, d π ; π, d π | η, l η ; s ) Y π ( η,l η ) sk ( θ ) Y π ( η,l η ) sk ( θ ) , θ, θ ∈ O π , and thus determine the Stratonovich-Weyl operator ∆ π ,∆ π ( θ ) = X η,l η ,s,k C ( π, d π ; π, d π | η, l η ; s ) − Y π ( η,l η ) sk ( θ ) Z O π dµ π ( θ ) Y π ( η,l η ) sk ( θ ) P θ (3.142)= K − π P θ . Here, P θ = v θ ⊗ v ∗ θ denotes the projection onto the coherent state v θ . The phase conventionfor the Clebsch-Gordan coefficients is chosen s.t. C ( π, d π ; π, d π | η, l η ; s ) > Remark III.31:
From (3.142), we see that the operator norm of ∆ π ( θ ) is uniformly bounded in θ ∈ O π . Therefore,the quantisation formula, A f := Z O π dµ π ( θ ) f ( θ )∆ π ( θ ) , (3.143)defines an element of End( V π ) for any f ∈ L ( O π ). Since the generalised spherical harmonicsare smooth, (3.143) even makes sense for f ∈ D ( O π ). By restricting to f ∈ S π ⊂ C ∞ ( O π ) thequantisation, f A f , becomes nondegenerate, but in contrast to C ∞ ( O π ), which can be interpretedas the analog of the space S ∞ ρ,δ , S π is not closed under multiplication.2Let us state the properties of the Stratonovich-Weyl quantisation (3.143) as a Theorem III.32:
The Stratonovich-Weyl quantisation Q SW ε ( f ) := Z O π dµ ε − π ( θ ) f ( θ ) ∆ ε − π ( θ ) , Q SW0 ( f ) := f, ε − ∈ N , (3.144) is a degenerate strict deformation quantisation of C ∞ ( O π ) into End( V ε − π ) in the sense oftheorem III.14 (cf. , Definition II.1.1.1.). Here, ε − π ∈ ˆ G is determined by the highest weight ε − λ π ∈ C ∩ I r , the intersection of the closed fundamental Weyl chamber C and the lattice of (real)integral weights I ∗ r . The Poisson structure on O π is induced from the (minus) Lie-Poisson structureon g ∗ . Proof:
Degeneracy of the quantisation follows, because dim(End( V ε − π )) < ∞ . For f ∈ C ∞ ( O π ), weobserve that the Stratonovich-Weyl quantisation Q SW ε ( f ) is related to Berezin quantisation Q B ε ( f ) = R O π dµ π ( θ ) f ( θ ) P θ (cf. , Section III.1.11) by the operator K ε − π : Q SW ε ( f ) = Q B ε ( K ε − π f ) . (3.145)But, Landsman proves in , section II.1.11, that Q B ε is a strict quantisation of C ∞ ( O π ). Although,we need to slightly correct the ε -expansion of Q B ε , which is erroneous in . Thus, if we controlledthe ε -expansion of K ε − π to order ε , we would be able to decide the strictness of Q SW ε from thestrictness of Q B ε . To find the required ε -expansion of K ε − π , we compute the ε -expansion of K ε − π , K ε − π = K (0) π + εK (1) π + O ( ε ) , (3.146)and apply functional calculus, i.e. K ε − π = (cid:0) id C ( O π ) + (cid:0) K ε − π − id C ( O π ) (cid:1)(cid:1) (3.147)= id C ( O π ) + 12 ( K ε − π −
1) + O ( ε ) = id C ( O π ) + 12 εK (1) π + O ( ε ) , since lim ε → K ε − π = K (0) π = id C ( O π ) (cf. , Theorem III.1.11.1.), and ∀ f ∈ C ( O π ) : || K ε − π f || ∞ ≤|| f || ∞ . Now, let us show that K ε − π actually has an expansion of the form (3.146). To this end,we analyse K ε − π in the form of (3.135):( K ε − π f ) ( θ ) = Z O π dµ ε − π ( θ ) | ( v θ , v θ ) V ε − π | f ( θ ) (3.148)= Z O π dµ ε − π ( θ ) | ( v ε − π , ( ε − π )( g − θ g θ ) v ε − π ) V ε − π | f ( θ )= d ε − π Z G λπ dh Z G/G λπ dg θ | ( v ε − π , ( ε − π )( g − θ g θ ) v ε − π ) V ε − π | F ( g θ h )= d ε − π Z G dg | ( v ε − π , π ( g − θ g ) v ε − π ) V ε − π | F ( g )3= d ε − π Z G dg | ( v ε − π , π ( g ) v ε − π ) V ε − π | F ( g θ g ) , where we have used the (right) G λ π -invariance of g
7→ | ( v ε − π , π ( g − θ g ) v ε − π ) V ε − π | , as v ε − π is ahighest weight vector. Here, F = f ◦ p ∈ C ∞ ( G ) is the (right) G λ π -invariant functions correspondingto f ∈ C ∞ ( O π ) via p : G → G/G λ π ∼ = O π . Again, exploiting the fact that v ε − π is a highest weightvector, we find: ∀ g ∈ G : ( v ε − π , π ( g ) v ε − π ) V ε − π = ( v π , π ( g ) v π ) ε − V π , (3.149)since the Cartan composite V λ + λ of two highest weights λ , λ ∈ C ∩ I ∗ r has multiplicity 1 in V λ ⊗ V λ (cf. , VI.2.8). This allows us to write (3.148) in the form( K ε − π f ) ( θ ) = d ε − π Z G dg | ( v π , π ( g ) v π ) V π | ε − F ( g θ g ) (3.150)= d ε − π Z G dg e − ε − S ( g ) f ( p ( g θ g ))= d ε − π Z G/G λπ dg θ e − ε − S ( g θ ) f ( p ( g θ g θ ))= d ε − π Z O π dµ π ( θ ) e − ε − S π ( θ ) f ( g θ · θ )= d ε − π Z U λπ dµ π ( θ ) e − ε − S π ( θ ) f ( g θ · θ ) + O ( ε ∞ ) , where S ( g ) := − log | ( v π , π ( g ) v π ) V π | , which descends to S π on O π by (right) G λ π -invariance. Therestriction to an arbitrary open neighbourhood U λ π of λ π is justified by the fact that the positivefunction S π assumes its sole absolute minimum at λ π , S π ( λ π ) = 0, and the simple estimate: (cid:12)(cid:12)(cid:12)(cid:12) d ε − π Z O π \ U λπ dµ π ( θ ) e − ε − S π ( θ ) f ( g θ · θ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ d ε − π Z O π \ U λπ dµ π ( θ ) e − ε − S π ( θ ) | f ( g θ · θ ) | (3.151) ≤ d ε − π || f ( g θ · ( . )) || ∞ e − ε inf θ O π \ Uλπ S π ( θ ) ∈ O ( ε ∞ ) , as d ε − π ∈ O ( ε − dim O π ) by Weyl’s formula (cf. , VI.1.7). Next, we choose U λ π small enoughto identify it with a neighbourhood W λ π ⊂ T λ π O π ∼ = g / g λ π of 0 via the exponential map. Dueto the decomposition of g C = t C ⊕ L α ∈ R + g α ⊕ g − α in to root spaces g ± α , α ∈ R + , we have g λ π = t ⊕ L α ∈ R + h λ π ,α i g ∗ =0 ( g ∩ ( g α ⊕ g − α )), leading to complex coordinates ( z, z ) = { ( z α , z α ) } α ∈ R + λπ via g θ ( z,z ) = exp X α ∈ R + λπ ( z α E α − z α E − α ) ! , (3.152)where we defined R + λ π := { α ∈ R + | h λ π , α i g ∗ = 0 } , and introduced a Cartan-Weyl basis { H i } ri =1 ⊂ t , { E α , E − α } α ∈ R + , E ± α ∈ g α , for g C (cf. ):[ H i , H j ] = 0 , i, j = 1 , ..., r , (3.153)4[ H i , E ± α ] = ± πiα ( H i ) E ± α , i = 1 , ..., r , α ∈ R + , [ E α , E − α ] = H α , α ∈ R + , [ E α , E β ] = N α,β E α + β , α, β ∈ R + ∪ ( − R + ) , α + β = 0 , N α,β = 0 iff α + β ∈ R + ∪ ( − R + ) . Here, H α = T α h T α ,T α i g is the co-root associated with α = πi h T α , . i g ∈ R +61 .Thus, we arrive at( K ε − π f ) ( θ ) = d ε − π Z W λπ ⊂ g / g λ (cid:16) Y α ∈ R + λπ dz α dz α π (cid:17)| {z } := √ π − dim( O π ) dzdz J ( z, z ) e − ε − S π ( z,z ) f ( g θ · ( z, z )) (3.154)+ O ( ε ∞ ) , where J is the Jacobian associated with the exponential map. With (3.154) at hand, we are in aposition to determine the ε -expansion of K ε − π to order ε by an appeal to Laplace’s method (cf. ),i.e. we insert the Taylor expansion of S π ( z, z ) = − log | ( v π , π ( g θ ( z,z ) ) v π ) V π | around ( z λ π , z λ π ) = 0up to fourth(!) order into (3.154), and invoke the unique extension of the unitary representation π to a holomorphic representation of G C :( v π , π ( g θ ( z,z ) ) v π ) V π (3.155)= ∞ X n =0 n ! (cid:0) v π , (cid:16) X α ∈ R + λπ ( z α dπ ( E α ) − z α dπ ( E − α )) (cid:17) n v π (cid:1) V π = 1 + X α ∈ R + λπ ( z α ( v π , dπ ( E α ) v π ) V π | {z } =0 − z α ( v π , dπ ( E − α ) v π ) V π | {z } =0 )+ 12 X α,β ∈ R + λπ (cid:16) z α z β ( v π , dπ ( E α ) dπ ( E β ) v π ) V π | {z } =0 + z α z β ( v π , dπ ( E − α ) dπ ( E − β ) v π ) V π | {z } =0 − z α z β ( v π , dπ ( E α ) dπ ( E − β ) v π ) V π | {z } = δ α,β ( v π ,dπ ( H α ) v π ) Vπ = δ α,β h λ π ,α i g ∗ ( h α,α i g ∗ ) − − z α z β ( v π , dπ ( E − α ) dπ ( E β ) v π ) V π | {z } =0 (cid:17) + 16 X α,β,γ ∈ R + λπ (cid:16) z α z β z γ ( v π , dπ ( E α ) dπ ( E β ) dπ ( E γ ) v π ) V π | {z } =0 − z α z β z γ ( v π , dπ ( E α ) dπ ( E β ) dπ ( E − γ ) v π ) V π | {z } = δ α,γ − β N β, − γ h λ π ,α i g ∗ ( h α,α i g ∗ ) − − z α z β z γ ( v π , dπ ( E α ) dπ ( E − β ) dπ ( E γ ) v π ) V π | {z } =0 + z α z β z γ ( v π , dπ ( E α ) dπ ( E − β ) dπ ( E − γ ) v π ) V π | {z } = δ α − β,γ N α, − β h λ π ,γ i g ∗ ( h γ,γ i g ∗ ) − − z α z β z γ ( v π , dπ ( E − α ) dπ ( E β ) dπ ( E γ ) v π ) V π | {z } =0 + z α z β z γ ( v π , dπ ( E − α ) dπ ( E β ) dπ ( E − γ ) v π ) V π | {z } =0 + z α z β z γ ( v π , dπ ( E − α ) dπ ( E − β ) dπ ( E γ ) v π ) V π | {z } =0 − z α z β z γ ( v π , dπ ( E − α ) dπ ( E − β ) dπ ( E − γ ) v π ) V π | {z } =0 (cid:17) + 124 X α,β,γ,ζ ∈ R + λπ (cid:16) z α z β z γ z ζ ( v π , dπ ( E α ) dπ ( E β ) dπ ( E γ ) dπ ( E ζ ) v π ) V π | {z } =0 − z α z β z γ z ζ ( v π , dπ ( E α ) dπ ( E − β ) dπ ( E γ ) dπ ( E ζ ) v π ) V π | {z } =0 z α z β z γ z ζ ( v π , dπ ( E α ) dπ ( E − β ) dπ ( E − γ ) dπ ( E ζ ) v π ) V π | {z } =0 − z α z β z γ z ζ ( v π , dπ ( E α ) dπ ( E − β ) dπ ( E − γ ) dπ ( E − ζ ) v π ) V π | {z } =0 + z α z β z γ z ζ ( v π , dπ ( E α ) dπ ( E − β ) dπ ( E γ ) dπ ( E − ζ ) v π ) V π | {z } =0 + z α z β z γ z ζ ( v π , dπ ( E α ) dπ ( E β ) dπ ( E − γ ) dπ ( E − ζ ) v π ) V π | {z } =0 − z α z β z γ z ζ ( v π , dπ ( E α ) dπ ( E β ) dπ ( E − γ ) dπ ( E ζ ) v π ) V π | {z } =0 − z α z β z γ z ζ ( v π , dπ ( E α ) dπ ( E β ) dπ ( E γ ) dπ ( E − ζ ) v π ) V π | {z } =0 + z α z β z γ z ζ ( v π , dπ ( E − α ) dπ ( E β ) dπ ( E γ ) dπ ( E ζ ) v π ) V π | {z } =0 − z − α z β z γ z ζ ( v π , dπ ( E α ) dπ ( E − β ) dπ ( E γ ) dπ ( E ζ ) v π ) V π | {z } =0 + z α z β z γ z ζ ( v π , dπ ( E − α ) dπ ( E − β ) dπ ( E − γ ) dπ ( E ζ ) v π ) V π | {z } =0 − z α z β z γ z ζ ( v π , dπ ( E − α ) dπ ( E − β ) dπ ( E − γ ) dπ ( E − ζ ) v π ) V π | {z } =0 + z α z β z γ z ζ ( v π , dπ ( E − α ) dπ ( E − β ) dπ ( E γ ) dπ ( E − ζ ) v π ) V π | {z } =0 + z α z β z γ z ζ ( v π , dπ ( E − α ) dπ ( E β ) dπ ( E − γ ) dπ ( E − ζ ) v π ) V π | {z } =0 − z α z β z γ z ζ ( v π , dπ ( E − α ) dπ ( E β ) dπ ( E − γ ) dπ ( E ζ ) v π ) V π | {z } =0 − z α z β z γ z ζ ( v π , dπ ( E − α ) dπ ( E β ) dπ ( E γ ) dπ ( E − ζ ) v π ) V π | {z } =0 (cid:17) + O (( z, z ) )where we repeatedly exploited the fact that v π is a highest weight vector, the commutation rela-tions (3.153) and the (unitary) representation π implements the adjointness relations dπ ( E α ) ∗ = dπ ( E − α ) , α ∈ R +63 . At this point it is important to note that non-zero contributions at order( z, z ) only arise for terms with α = β, α = γ, β = γ due to the geometry of root systems, i.e. theonly multiples of a root α occurring in the decomposition of g C are ± α . (3.155) and (15) imply: | ( v π , π ( g θ ( z,z ) ) v π ) V π | = 1 − X α ∈ R + λπ h λ π , α i g ∗ h α, α i g ∗ z α z α (3.156)6+ 16 X α,β,γ ∈ R + λπ α = β,α = γ,β = γ (cid:0) − z α z β z γ ( v π , dπ ( E α ) dπ ( E β ) dπ ( E − γ ) v π ) V π + z α z β z γ ( v π , dπ ( E α ) dπ ( E − β ) dπ ( E − γ ) v π ) V π (cid:1) + X α,β,γ ∈ R + λπ α = β,α = γ,β = γ (cid:0) − z α z β z γ ( v π , dπ ( E α ) dπ ( E β ) dπ ( E − γ ) v π ) V π !| {z } =0 , due to dπ ( E α ) ∗ = dπ ( E − α ) , α ∈ R + + z α z β z γ ( v π , dπ ( E α ) dπ ( E − β ) dπ ( E − γ ) v π ) V π (cid:1) + 112 X α,β ∈ R + λπ h λ π , α i g ∗ h α, α i g ∗ h λ π , β i g ∗ h β, β i g ∗ z α z α z β z β + X α,β,γ,ζ ∈ R + λπ ( − z α z β z γ z ζ ( v π , dπ ( E α ) dπ ( E − β ) dπ ( E − γ ) dπ ( E − ζ ) v π ) V π )+ z α z β z γ z ζ ( v π , dπ ( E α ) dπ ( E − β ) dπ ( E γ ) dπ ( E − ζ ) v π ) V π + z α z β z γ z ζ ( v π , dπ ( E α ) dπ ( E β ) dπ ( E − γ ) dπ ( E − ζ ) v π ) V π − z α z β z γ z ζ ( v π , dπ ( E α ) dπ ( E β ) dπ ( E γ ) dπ ( E − ζ ) v π ) V π ! + O (( z, z ) ) . (3.157)Thus, we find by expanding the logarithm in S π : S π ( z, z ) = 12 X α ∈ R + λπ h λ π , α i g ∗ h α, α i g ∗ z α z α (3.158)+ 124 X α,β ∈ R + λπ h λ π , α i g ∗ h α, α i g ∗ h λ π , β i g ∗ h β, β i g ∗ z α z α z β z β − X α,β,γ,ζ ∈ R + λπ (cid:0) − z α z β z γ z ζ ( v π , dπ ( E α ) dπ ( E − β ) dπ ( E − γ ) dπ ( E − ζ ) v π ) V π ! + z α z β z γ z ζ ( v π , dπ ( E α ) dπ ( E − β ) dπ ( E γ ) dπ ( E − ζ ) v π ) V π + z α z β z γ z ζ ( v π , dπ ( E α ) dπ ( E β ) dπ ( E − γ ) dπ ( E − ζ ) v π ) V π − z α z β z γ z ζ ( v π , dπ ( E α ) dπ ( E β ) dπ ( E γ ) dπ ( E − ζ ) v π ) V π (cid:1) + O (( z, z ) )= 12 X α ∈ R + λπ h λ π , α i g ∗ h α, α i g ∗ z α z α + P ( z, z ) + O (( z, z ) ) . The integration in (3.154) can be extended from W λ π to g / g λ π at the expense of a term of order ε ∞ by an estimate similar to (3.151), if the integrands are suitably continued beyond W λ π , whichleads to ( h λ π , α i g ∗ > α ∈ R + λ π ):( K ε − π f ) ( θ ) (3.159)= √ π − dim( O π ) d ε − π Z g / g λ dz dz J ( z, z ) e − ε − S π ( z,z ) f ( g θ · ( z, z )) + O ( ε ∞ )7= √ π − dim( O π ) d ε − π Z g / g λ dz dz J ( z, z ) e − ε − ( P α ∈ R + λπ h λπ,α i g ∗h α,α i g ∗ z α z α +2 P ( z,z )+ O (( z,z ) )) f ( g θ · ( z, z ))+ O ( ε ∞ )= r ε π dim( O π ) d ε − π Z g / g λ dz dz J ( √ εz, √ εz ) e − ( P α ∈ R + λπ h λπ,α i g ∗h α,α i g ∗ z α z α +2 εP ( z,z )) f ( g θ · ( √ εz, √ εz ))+ O ( ε )= r ε dim( O π ) d ε − π (cid:18) Y α ∈ R + λπ h α, α i g ∗ h λ π , α i g ∗ (cid:19) J (0 , f ( g θ · (0 , ε (cid:18) X α ∈ R + λπ h α, α i g ∗ h λ π , α i g ∗ ∂ z α ∂ z α ( Jf ( g θ · ( . )))(0 ,
0) + J (0 , f ( g θ · (0 , C ( λ π ) (cid:19)! + O ( ε )= (cid:18) Y α ∈ R + λπ h α, α i g ∗ h δ, α i g ∗ (cid:19) J (0 , f ( g θ · (0 , ε (cid:18) X α ∈ R + λπ h α, α i g ∗ h λ π , α i g ∗ ∂ z α ∂ z α ( Jf ( g θ · ( . )))(0 , J (0 , f ( g θ · (0 , (cid:18) C ( λ π ) + X α ∈ R + λπ h δ, α i g ∗ h λ π , α i g ∗ (cid:19)(cid:19)! + O ( ε ) . Here, C ( λ π ) results from contribution of e − εP ( z,z ) to the order ε . The half-integer orders of ε coming from the Taylor expansions of J and f ( g θ · ( . )) vanish, because Z g / g λπ dz dz e − P α ∈ R + λπ z α z α z β z γ = √ π dim( O π ) β ! δ β,γ , β, γ ∈ N dim( O π )0 . (3.160)Furthermore, we have J (0 ,
0) = (cid:18) Y α ∈ R + λπ h α, α i g ∗ h δ, α i g ∗ (cid:19) − , (3.161)as we know that lim ε → K ε − π = id C ( O π ) . This allows us to (3.159) into a slightly simpler form:( K ε − π f ) ( θ ) = f ( θ ) + ε (cid:18) Y α ∈ R + λπ h α, α i g ∗ h δ, α i g ∗ (cid:19) X α ∈ R + λπ h α, α i g ∗ h λ π , α i g ∗ ∂ z α ∂ z α ( Jf ( g θ · ( . )))(0 ,
0) (3.162)+ f ( θ ) (cid:18) C ( λ π ) + X α ∈ R + λπ h δ, α i g ∗ h λ π , α i g ∗ (cid:19)! + O ( ε )= f ( θ ) + ε (cid:16) K (1) π f (cid:17) ( θ ) + O ( ε ) . Now, following Landsman (cf. , Theorem III.1.11.4.), given any Φ ε ∈ V ε − π , || Φ ε || = 1 , we have:(Φ ε , ( Q SW ε ( f ) Q SW ε ( f ) − Q SW ε ( f f ))Φ ε ) V ε − π (3.163)8= (Φ ε , ( Q B ε ( K ε − π f ) Q B ε ( K ε − π f ) − Q B ε ( K ε − π ( f f )))Φ ε ) V ε − π = (Φ ε , ( Q B ε ( f ) Q B ε ( f ) − Q B ε ( f f ))Φ ε ) V ε − π + ε ε , ( Q B ε ( K (1) π f ) Q B ε ( f ) + Q B ε ( f ) Q B ε ( K (1) π f ) − Q B ε ( K (1) π ( f f )))Φ ε ) V ε − π + O ( ε )for all f, f ∈ C ∞ ( O π , R ).At this point, it is important to recall the equality || A || = sup || Φ ε || =1 | (Φ ε , A Φ ε ) V ε − π | for A ∈ End( V ε − π ) , A ∗ = A . The terms in O ( ε ) satisfy bounds of the form Cε k || f || ∞ ,m || f || ∞ ,n || Φ ε || for k ∈ N , m, n ∈ N , C >
0, which will imply Dirac’s condition (see theorem III.14) in the proofof strictness of Q SW ε , if Q B ε is strict. Similarly, Rieffel’s and von Neumann’s conditions will followfrom the strictness of Q B ε due to:(Φ ε , Q SW ε ( f )Φ ε ) V ε − π (3.164)= (Φ ε , Q B ε ( f )Φ ε ) V ε − π + ε (Φ ε , Q B ε ( K (1) π f )Φ ε ) V ε − π + O ( ε ) , (Φ ε , ( iε [ Q SW ε ( f ) , Q SW ε ( f )] − Q SW ε ( { f, f } − ))Φ ε ) V ε − π = (Φ ε , ( iε [ Q B ε ( f ) , Q B ε ( f )] − Q B ε ( { f, f } − ))Φ ε ) V ε − π + O ( ε ) . Finally, the strictness of Q B ε can be concluded from Landsman’s argument subject to some minormodifications. More precisely, Landsman considers the first term in the last line of (3.163) in theform: (Φ ε , ( Q B ε ( f ) Q B ε ( f ) − Q B ε ( f f ))Φ ε ) V ε − π (3.165)= d ε − π Z G dg F ( g )(Φ ε , ( ε − π )( g ) v ε − π ) V ε − π I ε ( g ) ,I ε ( g ) := d ε − π Z G dh ( v π , π ( h ) v π ) ε − V π F ε − π ( g, h ) ,F ε − π ( g, h ) := (( ε − π )( gh ) v ε − π , Φ ε ) V ε − π ( F ( gh ) − F ( g )) , and then subjects I ε ( g ) to an asymptotic expansion by Laplace’s method analogous to that of( K ε − π f ) ( θ ). In contrast to the previous calculation, neither I ε nor F ε − π are (right) G λ π -invariant,but are only (right) G λ π -equivariant: F ε − π ( g, hg λ π ) = e − iε φ ( g λπ ) F ε − π ( g, h ) , F ε − π ( gg λ π , h ) = e − iε φ ( g λπ ) F ε − π ( g, g λ π hg − λ π ) , (3.166) I ε ( gg λ π ) = e − iε φ ( g λπ ) I ε ( g ) , g λ π ∈ G λ π , φ : G λ π → R which ensures the invariance of (3.165). Nonetheless, the functions h ( v π , π ( h ) v π ) ε − V π F ε − π ( g, h )are (right) G λ π -invariant for every g ∈ G . By the same arguments used in (3.150), (3.151) and(3.154), as | F ε − π ( g, h ) | ≤ || f || ∞ , we have I ε ( g ) = √ π − dim( O π ) d ε − π Z g / g λπ dz dz J ( z, z ) e − ε ( − log( v π ,π ( g θ ( z,z ) ) v π ) Vπ ) F ε − π ( g, g θ ( z,z ) ) (3.167)+ O ( ε ∞ ) . − log( v π , π ( g θ ( z,z ) ) v π ) V π (3.168)= − ( (cid:0) v π , π ( g θ ( z,z ) ) v π ) V π − (cid:1) − (cid:0) ( v π , π ( g θ ( z,z ) ) v π ) V π − (cid:1) + O (( z, z ) )= 12 X α ∈ R + λπ h λ π , α i g ∗ h α, α i g ∗ z α z α − X α,β,γ ∈ R + λπ α = β,α = γ,β = γ (cid:0) − z α z β z γ δ α,γ − β N β, − γ h λ π , α i g ∗ ( h α, α i g ∗ ) − + z α z β z γ δ α − β,γ N α, − β h λ π , γ i g ∗ ( h γ, γ i g ∗ ) − (cid:1) + 124 X α,β ∈ R + λπ h λ π , α i g ∗ h α, α i g ∗ h λ π , β i g ∗ h β, β i g ∗ z α z α z β z β − X α,β,γ,ζ ∈ R + λπ (cid:0) − z α z β z γ z ζ ( v π , dπ ( E α ) dπ ( E − β ) dπ ( E − γ ) dπ ( E − ζ ) v π ) V π ! + z α z β z γ z ζ ( v π , dπ ( E α ) dπ ( E − β ) dπ ( E γ ) dπ ( E − ζ ) v π ) V π + z α z β z γ z ζ ( v π , dπ ( E α ) dπ ( E β ) dπ ( E − γ ) dπ ( E − ζ ) v π ) V π − z α z β z γ z ζ ( v π , dπ ( E α ) dπ ( E β ) dπ ( E γ ) dπ ( E − ζ ) v π ) V π (cid:1) + O (( z, z ) ) , which yields the corrected expansion of I ε ( g ) to order ε : I ε ( g ) = (cid:18) Y α ∈ R + λπ h α, α i g ∗ h δ, α i g ∗ (cid:19) J (0 , F ε − π ( g, e ) | {z } =0 (3.169)+ ε (cid:18) X α ∈ R + λπ h α, α i g ∗ h λ π , α i g ∗ ∂ z α ∂ z α ( JF ε − π ( g, g θ ( . , . ) ))(0 , J (0 , F ε − π ( g, e ) | {z } =0 (cid:18) C ( λ π ) + 2 C ( λ π ) + X α ∈ R + λπ h δ, α i g ∗ h λ π , α i g ∗ (cid:19)(cid:19)! + O ( ε )= ε √ dim( O π ) J (0 , − X α ∈ R + λπ h α, α i g ∗ h λ π , α i g ∗ ∂ z α ∂ z α ( JF ε − π ( g, g θ ( . , . ) ))(0 , ! + O ( ε )= ( ∂ zα J )(0 , ∂ zα J )(0 , ε √ dim( O π ) X α ∈ R + λπ h α, α i g ∗ h λ π , α i g ∗ ∂ z α ∂ z α ( F ε − π ( g, g θ ( . , . ) ))(0 , ! + O ( ε )The terms of order ( z, z ) do not yield contributions containing first derivatives of F ε − π , because ofthe constraints α = β, α = γ, β = γ and (3.160). Clearly, the expansion is compatible with (right) G λ π -equivariance, as can be seen from (3.166) and the G λ π -invariance of the differential operator:∆ λ π := (cid:18) X α ∈ R + λπ h α, α i g ∗ h λ π , α i g ∗ ∂ z α ∂ z α (cid:19) | z =0 ,z =0 . (3.170)0Now, strictness of Q B ε follows from Landsman’s argument. Remark III.33:
The Stratonovich-Weyl quantisation on coadjoint orbits can be interpreted as the analog of Weylquantisation on R n . In view of (3.137) it is distinguished from Berezin quantisation by the “tracialproperty” leading to the Stratonovich-Weyl-Fourier transform as pointed out by Figueroa, Gracia-Bondía and Várilly .Now, we are in a position to define the Stratonovich-Weyl-Fourier transform: Definition III.34 (cf. ): The
Stratonovich-Weyl-Fourier transform is the composition of the Fourier transform F : L ( G ) ⊂ L ( G ) → L ( ˆ G ) with the Stratonovich-Weyl symbol map W : ˆ G → L ( S π ∈ ˆ G O π ) =: L ( O G ) , i.e.: F SW [Ψ]( π, θ ) = ˆΨ SW ( π, θ ) (3.171)= W π F [Ψ] ( θ )= Z G dg Ψ( g ) tr(∆ π ( θ ) π ( g ))= Z G dg Ψ( g ) E ( g ; π, θ ) , Ψ ∈ L ( G ) , π ∈ ˆ G, θ ∈ O π , where we introduced the integral kernel E ( g ; π, θ ) = tr(∆ π ( θ ) π ( g )) = W ππ ( g ) ( θ ) . The space of integralcoadjoint orbits , O G , is endowed with the integral: Z O G dµ O G ( π, θ )Φ( π, θ ) := X π ∈ ˆ G d π Z O π dµ π ( θ ) Φ( π, θ ) , Φ ∈ L ( O G ) . (3.172)It follows from the next theorem that the Stratonovich-Weyl-Fourier transform is well-defined andisometric from L ( G ) to L ( O G ) (the range is L π ∈ ˆ G S π ). Moreover, it intertwines the convolutionproduct and the twisted product coming from: W πAB ( θ ) = ( W πA ? W πB )( θ ) = Z O π dµ π ( θ ) Z O π dµ π ( θ ) tr(∆ π ( θ )∆ π ( θ )∆ π ( θ )) W πA ( θ ) W πB ( θ ) . (3.173) Theorem III.35 (cf , Theorem 5): The Stratonovich-Weyl-Fourier transform satisfies the inversion formula Ψ( g ) = Z O G dµ O G ( π, θ ) E ( g ; π, θ ) F SW [Ψ]( π, θ ) , (3.174) and the Parseval-Plancherel identity Z G dg | Ψ( g ) | = Z O G dµ O G ( π, θ ) | F SW [Ψ]( π, θ ) | . (3.175)1 Furthermore, the convolution product ∗ on L ( G ) is intertwined with the twisted product ? : F SW [Ψ ∗ Φ] = F SW [Ψ] ? F SW [Φ] . (3.176)The integral kernel E : G × O G → C has important properties that entail its independence of therepresentative in π ∈ ˆ G (property 2). Proposition III.36 (cf. , Theorem 4): The integral kernel E satisfies:1. E ( g ; π, θ ) = E ( g − ; π, θ ) ,2. E ( α h ( g ); π, θ ) = E ( g ; π, Ad ∗ h − ( θ )) ,3. R O π dµ π ( θ ) E ( g ; π, θ ) = tr( π ( g )) = χ π ( g ) ,4. R G dg E ( g ; π, θ ) E ( g ; π, θ ) = d − π I π ( θ, θ ) ,5. ( E ( g ; . ) ? F SW [Ψ])( π, θ ) = F SW [ U g Ψ]( π, θ ) , Ψ ∈ L ( G ) ,6. E ( g ; . ) ? E ( h ; . ) = E ( gh ; . ) . Remark III.37:
Property 5 & 6 of proposition III.36 tell us that the action of the exponentials U g , g ∈ G, of the“momenta” P X , X ∈ g , in (3.1) is turned in to (non-commutative) multiplication with the functions E ( g ; . ) , g ∈ G by the Stratonovich-Weyl-Fourier transform. Thus, the range of the latter qualifiesas (non-commutative) flux representation for loop quantum gravity in the terminology of .As mentioned in the beginning of this subsection, the Stratonovich-Weyl-Fourier transform allowsus to study the effect of scaling in the global Fourier correspondence for compact Lie groups. Tothis end, we distinguish to types of ε -scaled Stratonovich-Weyl-Fourier transforms ( ε = k , k ∈ N ):1. (“position” scaling): F ε SW [Φ]( π, θ ) := R G dg E ( g ε ; π, θ )Ψ( g ),2. (“momentum” scaling): F SW ,ε [Φ]( π, θ ) := R G dg E ( g ; ε − π, θ )Ψ( g ),where ε − π is the unitary irreducible representation of G with highest weight ε − λ π .As above, the restriction to discrete scalings, ε = k , k ∈ N , is necessary as otherwise the powers g ε would not be uniquely defined, and ε − λ π would not belong to C ∩ I ∗ r . Clearly, these transformationscoincide for commutative Lie groups, where the Stratonovich-Weyl-Fourier transform coincides withusual Fourier transform. Structurally, the “momentum” scaled transform seems to be favoured, asit can be expressed in terms of the unscaled transform: F SW ,ε [Φ]( π, θ ) = Z G dg E ( g ; ε − π, θ )Ψ( g ) = F SW [Ψ]( ε − π, θ ) . (3.177)In contrast, the “position” scaled transform is not related to unscaled transform, because the ε -thpower is not a diffeomorphism of G (unless ε = 1). Therefore, we stick to the “momentum” scaledtransform in the following. Unfortunately, we immediately recognize, that the scaled transform nolonger defines an invertible map, as it restricts the Stratonovich-Weyl-Fourier transform to thoseintegral coadjoint orbits associated with the sub-lattice ε − I ∗ r ⊂ I ∗ r . Again, this is a feature forced2upon us by the rigidity of compact Lie groups. In subsection III D, we will discuss a possible wayto remove this restriction for G = U (1).Nonetheless, the Stratonovich-Weyl-Fourier transform provides us with a ε -scaled transform forsystems modelled on integral coadjoint orbits O π , which was already exploited in for G = SU (2).To see how this works in the general case, we note that λ π and ε − λ π have the same, possiblydegenerate, orbit type O π , since G λ π = G ε − λ π . Thus, we may study the “semiclassical” limit, ε →
0, of sequences of operators { A ε } ε − ∈ N , A ε ∈ End( V ε − π ) , in terms of their Stratonovich-Weylsymbols W ε − πA ε ∈ S ε − π ⊂ C ∞ ( O π ) ⊂ L ( O π ) and the twisted products ? ε , e.g. A ε = iε n X i =1 B i d ( ε − π )( τ i ) ⇔ W ε − πA ε ( θ ) = iε n X i =1 B i ddt | t =0 E ( e tτ i ; ε − π, θ ) , (3.178)which is a generalisation of the magnetic part of the Pauli Hamiltonian to G . Here, { τ i } ni =1 isa basis of g and B = ( B i ) i =1 ,..,n ∈ R n . Another physical application, apart from spin-orbitcoupling discussed in , where these structures feature prominently, is the description of classicaland quantum particles with internal symmetry in external gauge fields, which are governed by theclassical respectively quantum Wong equations (cf. ), and its relation to quantum chaos (cf. ).If we wanted to avoid working with degenerate coadjoint orbits, i.e. those corresponding to singularintegral weights λ, h α, λ i g ∗ = 0 for some α ∈ R + , we could incorporate the usual shift by half thesum of the positive roots, δ = P α ∈ R + α , in to the correspondence between dominant integralweights and integral coadjoint orbits, λ π → O δπ := { θ ∈ g ∗ | ∃ g ∈ G : θ = Ad ∗ g ( λ π + δ ) } , (3.179)which allows us to work solely with coadjoint orbits of strongly dominant integral weights, since λ ∈ C ∩ I ∗ r ⇔ λ + δ ∈ C ∩ I ∗ r (3.180)(cf. ). The coadjoint orbits of strongly dominant integral weights are isomorphic to the simplyconnected generalised flag variety G/T ∼ = G C /B + , where B + is the standard Borel subgroup asso-ciated with the positive roots R + (cf. ).Another advantage of the Stratonovich-Weyl-Fourier transform is that it enables us to pass frommatrix valued symbols on G × ˆ G in the definition of the global pseudo-differential calculus of subsec-tion (III A 1) to genuine functions on G × O G . Namely, the Stratonovich-Weyl symbol of a symbol σ ∈ ˆ D ( ˆ G × G ) is just the Stratonovich-Weyl-Fourier transform of its left convolution kernel F σ : σ SW ( π, θ ; g ) = W πσ ( π,g ) ( θ ) = tr(∆ π ( θ ) σ ( π, g )) (3.181)= Z G dh E ( h ; π, θ ) F σ ( h, g ) = ˆ F SW ( π, θ ; g ) . In this way, the action of the symbol σ on C ∞ ( G ) is related to the twisted product ? :( ρ L ( F σ )Ψ) ( g ) = Z O G dµ O G (cid:0) E ( g ; . ) ? σ SW ( . ; g ) (cid:1) ( π, θ ) ˆΨ SW ( π, θ ) . (3.182)3Also, the composition of two symbols σ, τ becomes expressible in terms of the twisted product (cp.proposition III.2, 6.): F SW [ F σ ∗ L F τ ]( π, θ ; g ) = Z G dh F σ ( h, g ) (cid:0) E ( h ; . ) ? τ SW ( . ; h − g ) (cid:1) ( π, θ ) . (3.183) C. Coherent states & quantisation
In the previous subsection, we have encountered the concept of Berezin quantisation w.r.t. a(complete) system of coherent states (cf. ), and its close relation to Stratonovich-Weyl quanti-sation on coadjoint orbits. As explained in section II B, we have a (complete) system of coherentstates for any compact Lie group G at our disposal, which naturally leads to the question howBerezin quantisation on T ∗ G in terms of these coherent states relates to the Kohn-Nirenberg andWeyl quantisation discussed in subsection III A. Moreover, it is important to understand, whetherthis Berezin quantisation of T ∗ G gives a suitable framework for the Born-Oppenheimer approxi-mation and adiabatic perturbation theory. As we will argue, the equivalent roles played by Berezinand (Stratonovich-)Weyl quantisation on coadjoint orbits with regard to such a framework is quitespecial to the case of compact phase spaces, and therefore finite dimensional Hilbert spaces, whilethe existence of an analog of the (smoothing) operator K π (3.136) will be the reason, why Weylquantisation is favoured, if the construction of a twisted product for phase space functions is in-tended.Let us start by defining Berezin quantisation on T ∗ G in terms of the coherent states from sectionII B. To this end, we note that the coherent states Ψ tz ∈ L ( G ) , t ∈ R , z ∈ G C , (2.14) provides uswith a coherent pure state quantisation of T ∗ G (in the sense of ). Therefore, the following definitionmakes sense for small enough t > ): Definition III.38:
Given f ∈ L p ( T ∗ G, dg dX ) , ≤ p ≤ ∞ , its Berezin quantisation is Q B t ( f ) := Z G C dz ν t ( z ) f (Φ − ( z )) | Ψ tz ih Ψ tz | , (3.184) which defines an operator in S p ( L ( G )) , the p th Schatten class in B ( L ( G )) , since tr (cid:0) | Q B t | p (cid:1) ≤|| f || pp ) for ≤ p < ∞ and || Q B t ( f ) || ≤ || f || ∞ . f ◦ Φ − is called the upper or contravariant symbol of the operator Q B t ( f ) . The lower or covariant symbol of an operator A ∈ B ( L ( G )) is L tA ( z, z ) := h Ψ tz | A | Ψ tz ih Ψ tz | Ψ tz i , (3.185) which satisfies || L tA || ∞ ≤ || A || , and || L tA | p || pp ≤ tr ( | A | p ) for A ∈ S p ( L ( G )) . Remark III.39:
Berezin quantisation would take a more natural form if the conjecture II.9 turned out to be true,as in that case the normalised projection P t ( z, z ) onto the coherent state vectors Ψ tz would provide4a resolution of unity w.r.t. to the Liouville measure on T ∗ G , and (3.184) would take the form: Q B t ( f ) = C t Z G Z g dg dX f ( g, X ) P t ( ge iX , ge − iX ) . (3.186) Corollary III.40 (cf. , Appendix 1): Berezin quantisation is real, Q B t ( f ) ∗ = Q B t ( f ) , and positive, Q B t ( f ) ≥ if f ≥ a.e.. The upperand lower symbols associated with Berezin quantisation are dual to one another , i.e. tr( AQ B t ( f )) = Z G C dz ν t ( z ) h Ψ tz | Ψ tz i L tA ( z, z ) f (Φ − ( z )) , A ∈ S q ( L ( G )) , f ∈ L p ( T ∗ G, dg dX ) , (3.187) for q + p = 1 . Moreover, the normalised coherent state projections P t ( z, z ) are complete, i.e. img( Q B t ) is sequentially strongly dense in B ( L ( G )) , due to the analyticity of the coherent states Ψ tz , as the latter implies ∀ A ∈ S ( L ( G )) : L tA = 0 if and only if A = 0 . While the corollary tells us, that the upper symbol exists for a sequentially strongly dense subspaceof operators in B ( L ( G )), it does not ensure that all operators in B ( L ( G )) can be obtained fromBerezin quantisation, as L ( G ) is infinite dimensional in contrast to the representation spaces V π considered in the previous subsection. Therefore, the existence of an upper symbol for a product ofBerezin quantisations Q B t ( f ) Q B t ( f ) cannot be concluded. Nevertheless, we might wonder whetherthere exists a set of functions S t B on T ∗ G , such that on the one hand Q B t : S t B → B ( L ( G )) isnondegenerate, and ∀ f, f ∈ S t B : ∃ f ∈ S t B : Q B t ( f ) Q B t ( f ) = Q B t ( f ) =: Q B t ( f ? t f ), while on theother hand Berezin quantisation of S t B encompasses sufficiently many “interesting” operators, and ? t can be (asymptotically) expanded w.r.t. the Poisson bracket on T ∗ G . To analyse this question insome detail, we first consider the relation between upper and lower symbol, if both exist, of a givenBerezin quantisation. Similar to (3.135), the lower symbol of Q B t ( f ) can be expressed in terms ofthe upper symbol and the overlap function of the coherent states: L tQ B t ( f ) ( z, z ) = ( h Ψ tz | Ψ tz i ) − Z G C dz ν t ( z ) |h Ψ tz | Ψ tz i| f (Φ − ( z )) . (3.188)This expression is also valid in the case of standard coherent states for R n , where it takes thewell-known explicit form (see the discussion at the beginning of subsection II B 1, cp. ): L tQ B t ( f ) ( z, z ) = Z C n d < ( z ) d = ( z )(2 πt ) n e − t ( z − z )( z − z ) f (Φ − ( z )) = (cid:0) e t∂ z ∂ z f ◦ Φ − (cid:1) ( z ) , (3.189)from which it can be inferred that L tQ B t is the restriction of an entire function on C n even if f ∈ S ( R n ) due to the smoothing nature of e t∂ z ∂ z = e t ∆ ( q,p ) , z = q + ip = Φ( q, p ). Moreover,Weyl quantisation (see (3.4)), Q B t ( f ) = A σ , fits into the picture in the same way as for the coadjointorbits (3.136) (cp. ): L tQ B t ( f ) ( q, p ) = (cid:16) e t ∆ ( q,p ) σ (cid:17) ( q, p ) , σ ( q, p ) = (cid:16) e t ∆ ( q,p ) f (cid:17) ( q, p ) , (3.190)which tells us that Weyl symbols are typically better behaved than upper symbols. Especially,since we would expect the (asymptotic) expansion of a twisted product ? t to be determined bylocal products of derivatives of the factors, which would be problematic in case of distributional5symbols.A similar situation is to be expected for a compact Lie group G , and the Berezin quantisation(3.184) of T ∗ G , as the latter is non-compact. We will show this explicitly for G = U (1) , or moreprecisely for U (1)-equivariant Berezin quantisation. The coherent states from section II B take thefollowing explicit form for G = U (1) (cp. ):Considering the Hilbert spaces H j := { Ψ ∈ L ( R ) | ∀ j ∈ Z : Ψ( ϕ + 2 πj ) = e πij j Ψ( ϕ ) } , j ∈ [0 , , (3.191)equipped with the scalar product(Ψ , Ψ ) j := Z [0 , π ) dϕ π Ψ ( ϕ )Ψ ( ϕ ) , (3.192)we have the following orthonormal bases of eigenfunction of J := − i∂ ϕ , D ( J ) := H j ∩ H ( R )(subject to the boundary conditions associated with (3.191)):Ψ j j ∈ H j : Ψ j j ( ϕ ) := e i ( j + j ) ϕ = h ϕ | j + j i , j ∈ Z . (3.193)Moreover, the Hilbert spaces H j are representation spaces of the U (1)-Weyl algebra (cp. (3.8))( V t ( β )Ψ)( ϕ ) := Ψ( ϕ + tβ ) , ( U t ( m )Ψ)( ϕ ) := e imϕ Ψ( ϕ ) , Ψ ∈ H j , (3.194) V t ( β ) U t ( m ) = e itmβ U t ( m ) V t ( β ) ,U t ( m ) U t ( n ) = U t ( m + n ) , V t ( β ) V t ( γ ) = V t ( β + γ ) ,U t (0) = = V t (0) , U t ( m ) ∗ = U t ( − m ) , V t ( β ) ∗ = V t ( − β ) , m, n ∈ Z , β, γ ∈ R . In terms of the bases the (equivariant) coherent states are:Ψ t,j ξ ∈ H j : Ψ t,j ξ ( ϕ ) = X j ∈ Z ( ξe tj ) − j e − t j Ψ j j ( ϕ ) (3.195)= h ϕ | ξ, j i t for ξ ∈ C ∗ = U (1) C ∼ = T ∗ U (1). These vectors are eigenfunctions of the annihilation operators X t := e − t U t (1) e − tJ , X t | ξ, j i t = ξ | ξ, j i t , (3.196)where the former can be obtained from the complexifier method, X t := e − t J U e t J (cf. ).Now, given a Berezin quantisation Q B t ( f ) of f ∈ C ∞ b ( T ∗ U (1)), the formula connecting the uppersymbol f ◦ Φ − , Φ( e iϕ , l ) = e iϕ e − l = e − l + iϕ , and the lower symbol L tQ B t ( f ) is: L tQ B t ( f ) ( ξ, ξ ) = Z C ∗ dξ ∧ dξ πi √ πt f (Φ − ( ξ )) | ξ | − e − (log | ξ j t )2 t | t h ξ, j | ξ , j i t | t h ξ, j | ξ, j i t (3.197)= Z [0 , π ) Z R dϕ dl π √ πt f ( ϕ , l ) e − ( l j t )2 t | t h ξ, j | Φ( e iϕ , l ) , j i t | t h ξ, j | ξ, j i t , ξ = e − l + iϕ and using t h Φ( e iϕ , l ) , j | Φ( e iϕ , l ) , j i t = X j ∈ Z e − tj e j (( l + l )+ i ( ϕ − ϕ ) − j t ) (3.198)= ϑ ( i π ( − ( l + l ) − i ( ϕ − ϕ ) + 2 j t ) | itπ )= r πt e t ( − ( l + l ) − i ( ϕ − ϕ )+2 j t ) ϑ ( t ( − ( l + l ) − i ( ϕ − ϕ )) + j | iπt )= r πt X j ∈ Z e t ((( l − j t )+( l − j t ))+ i ( ϕ − ϕ ) − πij ) , we have: L tQ B t ( f ) (Φ( e iϕ , l )) = e − ( l − j t )2 t ϑ ( lt − j | iπt ) Z [0 , π ) Z R dϕ dl πt f ( ϕ , l ) e − ( l j t )2 t (3.199) × X j,k ∈ Z e t ((( l − j t )+( l − j t ))+ i ( ϕ − ϕ ) − πij ) e t ((( l − j t )+( l − j t )) − i ( ϕ − ϕ )+2 πik ) = Z [0 , π ) Z R dϕ dl πt ϑ ( lt − j | iπt ) f ( ϕ , l ) e − t (( l − j t ) − ( l − j t )) × X j,k ∈ Z e − t ( ϕ − ϕ − πj ) e − t (2 π ( j − k )) e − t (2 π ( j − k ))( ϕ − ϕ − πj + i (( l − j t )+( l − j t ))) = n = j − k ( SL ( Z ) transform) Z [0 , π ) Z R dϕ dl πt ϑ ( lt − j | iπt ) f ( ϕ , l ) e − t (( l − j t ) − ( l − j t )) × X j,n ∈ Z e − t ( ϕ − ϕ − πj ) e − π t n e − πt n ( ϕ − ϕ − πj + i (( l − j t )+( l − j t ))) = X n ∈ Z e − π t n e − πin ( lt − j ) Z R dl πt ϑ ( lt − j | iπt ) Z [0 , π ) dϕ f ( ϕ , l ) × X j ∈ Z e − t (( l − l − iπn ) +( ϕ − ϕ + πn − πj ) ) = X n ∈ Z e − π t n e − πin ( lt − j ) | {z } = ϑ ( lt − j | iπt ) Z R dl πt ϑ ( lt − j | iπt ) Z [0 , π ) dϕ f ( ϕ , l ) × X j ∈ Z e − t (( l − l ) +( ϕ − ϕ +2 πj ) ) = 12 πt Z R dl X j ∈ Z Z [2 πj, π ( j +1) dϕ f ( ϕ − πj, l ) | {z } = f ( ϕ ,l ) e − t (( l − l ) +( ϕ − ϕ ) ) = 12 πt Z R dl Z R dϕ f ( ϕ , l ) e − t (( l − l ) +( ϕ − ϕ ) ) (cid:16) e t ∆ ( ϕ,l ) f (cid:17) ( ϕ, l ) , where we used the invariance of the measure dl ∧ dϕ = (2 πi | ξ | ) − dξ ∧ dξ under the substitution ϕ ϕ + πn, l l − iπn , because Φ( e iϕ , l ) = e − l + ϕ = e − ( l − iπn )+ i ( ϕ + πn ) = Φ( e ( − n iϕ , l − iπn ),and identified f with its periodic extension to T ∗ R . Remark III.41:
The relation L tQ B t ( f ) (Φ( e iϕ , l )) = ( e t ∆ ( ϕ,l ) f )( ϕ, l ) is in accordance with the covering R → R / π Z ∼ = U (1) and the observation that ϑ arises from a 2 π Z -periodisation of the Euclidean heat kernel, π ϑ ( l π | it π ) = P j ∈ Z √ πt e − ( l − πj )22 t . Furthermore, an analogous calculation as in (3.197) showsthat the lower symbol of a product of two Berezin quantisations Q B t ( f ) , Q B t ( f ) satisfies L tQ B t ( f ) Q B t ( f ) (Φ( e iϕ , l )) (3.200)= Z R dϕ dl πt e − t ( ϕ + il )( ϕ − il ) f ( ϕ + √ tϕ , l + √ tl ) × Z R dϕ dl πt e − t ( ϕ + il )( ϕ − il ) e − t ( ϕ − il )( ϕ + il ) f ( ϕ + √ tϕ , l + √ tl ) , which is familiar from the R n -case, as well.If we were to base the Berezin quantisation on the resolution of unity (2.34), L tQ B t ( f ) ( ξ, ξ ) = C t Z C ∗ dξ ∧ dξ πi | ξ | − f (Φ − ( ξ )) | t h ξ, j | ξ , j i t | t h ξ, j | ξ, j i t t h ξ , j | ξ , j i t | {z } =tr ( P j t ( ξ,ξ ) P j t ( ξ ,ξ ) ) , (3.201)which was already proven for G = U (1), we would obtain a similar result: L tQ B t ( f ) (Φ( e iϕ , l )) (3.202)= X n ∈ Z e − π t n e − πin ( lt − j ) ϑ ( lt − j | iπt ) Z R C t dl π ϑ ( l t − j | iπt ) Z [0 , π ) dϕ f ( ϕ , l ) X j ∈ Z e − t (( l − l − iπn ) +( ϕ − ϕ + πn − πj ) ) = C t π Z R dl ϑ (( lt − j ) + ( l t − j ) | i πt ) ϑ ( lt − j | iπt ) ϑ ( l t − j | iπt ) Z R dϕ f ( ϕ , l ) e − t (( l − l ) +( ϕ − ϕ ) ) . There is yet another interesting way to obtain the relation between the upper and lower symbols,namely via applying the commutation relation between creation and annihilation operators, X t X ∗ t = e t X ∗ t X t , (3.203)to any operator A ∈ B ( L ( U (1))) in (anti-)Wick-ordered form: A = X m,n ∈ Z (cid:0) A Wt (cid:1) mn ( X ∗ t ) m X nt (3.204)8= X m,n ∈ Z (cid:0) A Wt (cid:1) mn e − tmn | {z } =: ( A aWt ) mn X nt ( X ∗ t ) m . From the resolution of unity (2.30) and (3.196), we infer that the upper and lower symbol of A aregiven by the Laurent series: f A (Φ − ( ξ )) = A aWt ( ξ, ξ ) L tA ( ξ, ξ ) = A Wt ( ξ, ξ ) (3.205)= X m,n ∈ Z (cid:0) A aWt (cid:1) mn ξ m ξ n , = X m,n ∈ Z (cid:0) A Wt (cid:1) mn ξ m ξ n , = X m,n ∈ Z e − tmn (cid:0) A Wt (cid:1) mn ξ m ξ n which evidently satisfy L tA = e t ∆ f A . (3.206)Unfortunately, such simple reasoning is not available for general compact Lie groups, because thecommutation relations of the creation and annihilation operators do not close among themselves.To conclude this section, we also compute the analogs of the relations (3.190) for U (1)-equivariantKohn-Nirenberg symbols (cp. (3.15) and (3.17)). In this case, the Weyl elements are obtainedfrom the Weyl algebra (3.194): W j t ( ϕ, k ) := t π X m ∈ Z Z [0 , π/t ) dβ e − i ( mϕ + β ( k + j )) U t ( m ) V t ( β ) , (3.207)which satisfy the relationtr H j ( W j t ( ϕ, k ) ∗ W j t ( ϕ , k )) = 2 π X m ∈ Z δ ( ϕ − ϕ − πm ) δ k,k . (3.208)Now, Kohn-Nirenberg quantisation and dequantisation takes the form: A σ,t := 12 π X k ∈ Z Z [0 , π ) dϕ σ t ( ϕ, k ) W j t ( ϕ, k ) , σ A,t ( ϕ, k ) := tr H j ( W j t ( ϕ, k ) ∗ A t ) , (3.209)( A σ,t Ψ) ( ϕ ) = 12 π X k ∈ Z Z [0 , π ) dϕ e ik ( ϕ − ϕ ) σ t ( ϕ, k )Ψ( ϕ ) = X k ∈ Z e ikϕ σ t ( ϕ, k ) ˆΨ( k ) , Ψ ∈ C ∞ b ( R ) ∩ H j . There are natural symbol spaces associated with (3.209) (cf. ), which are in close analogy withthose familiar from pseudo-differential operators on R n ( m ∈ R , ≤ δ ≤ ρ ≤ σ ∈ S mρ,δ ( U (1) × Z ) : ⇔ ∀ k ∈ Z : σ ( . , k ) ∈ C ∞ ( U (1)) (3.210)& ∀ α, β ∈ N : ∀ ( ϕ, k ) ∈ U (1) × Z : ∃ C αβ > | ( ∂ αϕ ∆ βk σ )( ϕ, k ) | ≤ C αβ h k i m − ρβ + δα , S −∞ ( U (1) × Z ) := \ m ∈ R S mρ,δ ( U (1) × Z ) , S ∞ ρ,δ ( U (1) × Z ) := [ m ∈ R S mρ,δ ( U (1) × Z ) . Here, (∆ k f )( k ) := f ( k + 1) − f ( k ) for f : Z → C is the forward difference. If we allow for U (1)-equivariant symbols, i.e. σ ( ϕ + 2 πj, k ) = e πijj σ ( ϕ, k ) e − πijj , we can encompass operators A σ : H j → H j , as well.Clearly, we could have invoked U (1)-equivariant symbols, S mρ,δ, ( j ,j ) ( R ) = { σ ∈ S mρ,δ ( R ) | ∀ j ∈ Z : σ ( q + 2 πj, p ) = σ ( ϕ + 2 πj, k ) = e πijj σ ( q, p ) e − πijj }⊂ S mρ,δ ( R ) , (3.211)from the usual symbol classes instead of (3.210):( A σ,t Ψ)( q ) = 12 πt Z R dp Z [0 , π ) dq e it p ( q − q ) σ ( q, p )Ψ( q ) = 12 πt Z R dp e it pq σ ( q, p ) F t [Ψ]( p ) (3.212)for Ψ ∈ C ∞ b ( R ) ∩ H j . We will comment on a similar dichotomy for almost-periodic pseudo-differential operators in the following subsection III D. In the present case the distinction is onlyapparent, because symbols in S mρ,δ ( U (1) × Z ) can be interpolated by those in S mρ,δ ( R ) (cf. , Corollary4.6.13.).To conclude the present subsection, we provide the relations between upper, lower and Kohn-Nirenberg symbols, which display a smoothing from upper to Kohn-Nirenberg to lower symbolssimilar to (3.190): L tA σ,t (Φ( e iϕ , l )) = √ πϑ ( lt − j | iπt ) X k ∈ Z Z [0 , π ) dϕ σ t ( ϕ , k ) e − t ( l − t ( k + j )) (3.213) × X j ∈ Z e − t (( ϕ − ϕ − πj ) − i ( l − t ( k + j ))) = √ πϑ ( lt − j | iπt ) X k ∈ Z Z R dϕ σ t ( ϕ , k ) e − t ( l − t ( k + j )) e − t (( ϕ − ϕ ) − i ( l − t ( k + j ))) = √ πϑ ( lt − j | iπt ) X k ∈ Z Z R dϕ σ t ( ϕ , k ) e − t (( l − t ( k + j )) +( ϕ − ϕ ) ) e it ( ϕ − ϕ )( l − t ( k + j )) σ t ( ϕ, k ) = √ πt Z R dl Z R dϕ f A σ,t ( ϕ , l ) e − t (( t ( k + j ) − l ) +( ϕ − ϕ ) ) e − it ( ϕ − ϕ )( t ( k + j ) − l ) . These formulas are obtained by a calculation completely analogous to (3.199).
D. Scaled Fourier transforms for G = U (1) and an extension to R Bohr
In subsection III B, we have discussed the issue of defining a ε -scaled integral transform for acompact (simply connected) Lie group G by means of the Stratonovich-Weyl-Fourier transform.While the resulting transform (3.177) on L ( G ) is has its merits, when applied to systems modelled0on coadjoint orbits of G , its use is limited in the analysis of pseudo-differential operators on C ∞ ( G )as it is not invertible due to the fundamental discreteness inherent to representation theory of G ,i.e. any irreducible representation of G is uniquely (up to isomorphism) determined by an integraldominant (real) weight λ π ∈ C ∩ I ∗ r .In the following, first restricting to semisimple G , we will pursue the question, whether it is possibleto lift this discreteness by associating a representation π λ to any dominant weight λ ∈ C ( C ⊂ t ∗ admits a natural R + -action as it is a convex cone). To be a bit more precise, we will consider(complex linear) representations dπ λ of g C resp. U ( g C ) instead of G to lift the integrality condition.A natural way to achieve this is to exploit the construction of irreducible representations of G bymeans of Verma modules (cf. ). Since the representation theory of g C is typically formulatedwith respect to (infinitesimal) integral weights λ ∈ πiI ∗ r = I ∗ , and their complex linear extensionsto t ∗ C , instead of integral real weights, we will switch to using the former for the reminder of thesection. The same applies to the roots of g , and the subset of positive roots R + . Definition III.42 (cf. , I.1.3.): Given λ ∈ t ∗ , the U ( g C ) -module M ( λ ) := U ( g C ) ⊗ U ( b + ) C λ (3.214) is called the Verma module of highest weight λ . Here, b + := t C ⊕ M α ∈ R + g α | {z } =: n + (3.215) is the standard Borel subalgebra of g C associated with R + , and C λ is the 1-dimensional U ( b + ) -module defined by dπ λ ( H + N )1 := λ ( H )1 , H ∈ t C , N ∈ n + . (3.216) We denote by L ( λ ) := M ( λ ) /N ( λ ) the unique irreducible quotient module w.r.t. to the uniquemaximal submodule N ( λ ) ⊂ M ( λ ) (cf. , Theorem I.1.2.). Remark III.43:
The Verma module construction (3.214) also works for G = U (1) n . Since R + = ∅ and g C = t C ∼ = C n in this case, we have M ( λ ) = C λ , which is irreducible.For semisimple G , the irreducible representation L ( λ ) of g C is finite dimensional if and only if λ isdominant integral. M ( λ ) is freely generated by U ( n − )(1 ⊗ , n − := L α ∈ R + g − α .Unfortunately, the reason, why L ( λ ) does not integrate to a (unitary) representation of G , when λ is not dominant integral, can be traced back to the fact, that there is no Hilbert space structure on L ( λ ) compatible with the adjointness relations dπ λ ( E α ) ∗ = dπ λ ( E − α ) , α ∈ R + , and dπ λ ( H i ) ∗ = dπ λ ( H i ) , i = 1 , ..., r, for a choice of Cartan-Weyl basis (3.153) (cf. ) but only a Krein spacestructure. This is easily seen, in the case of g C = sl ( C ∩ I ∗ ∼ = N , R + = { } ):( dπ λ ( E − ) k v λ , dπ λ ( E − ) k v λ ) λ = ( v λ , dπ λ ( E ) k dπ λ ( E − ) k v λ ) λ (3.217)= ( v λ , dπ λ ( E ) k − ( dπ λ ( E − ) dπ λ ( E ) + dπ λ ( H )) dπ λ ( E − ) k − v λ ) λ = k ( λ ( H ) + 1 − k )( dπ λ ( E − ) k − v λ , dπ λ ( E − ) k − v λ ) λ k Y l =1 l ( λ ( H ) + 1 − l ) ! ( v λ , v λ ) λ , k ∈ N where we denoted the highest weight vector 1 ⊗ ∈ M ( λ ) by v λ , and invoked the commutation re-lations (3.153). The last line in (3.217) shows that the inner product of v kλ = dπ λ ( E − ) k v λ , k ∈ N , becomes negative for certain k > λ ( H )+1, if λ ( H ) / ∈ N . Moreover, the modules M ( λ ) , λ ( H ) / ∈ N , are irreducible implying that there is no compatible Hilbert space structure. For λ ( H ) ∈ N , weobtain the submodule N ( λ ) = span C { v kλ | k ≥ λ ( H ) + 1 } of null vectors, which can be factored out.A similar argument applies to general semisimple g C by an appeal to the sl -submodules generatedby { E α , E − α , H α } , α ∈ R + .In contrast, if G = U (1) n the set of roots is empty, and the standard inner product on C λ ∼ = C is compatible with the adjointness relations. Furthermore, the representation C λ integrates to a(unitary) representation of R n : π ( x · H ) λ e i P ni =1 x i λ ( H i ) , x · H = n X i =1 x i H i , x ∈ R n . (3.218)Thus, for G = U (1) n we may form the direct sums of Hilbert spaces H n := M { λ ( H i ) } ni =1 ∈ R n C λ , n ∈ N , H n ∼ = H ⊗ n . (3.219)The Hilbert space H = l ( R ) can be identified with L ( R Bohr ), the L -space on the Bohr compact-ification of R , which can be defined as the L -closure of span C { e λ : R → T | λ ∈ R } , e λ ( x ) := e iλx , w.r.t. the ergodic mean (the Haar measure on R Bohr )( f, f ) Bohr = Z R Bohr dµ Bohr ( x ) f ( x ) f ( x ) := lim T →∞ T Z [ − T,T ] dx f ( x ) f ( x ) . (3.220) L ( R Bohr ) is also naturally isomorphic with the space of Besicovitch almost periodic functions B ( R ) on R (cf. ). In analogy with the Bloch-Floquet transform of L ( R ), we have a direct sum(instead of direct integral) decomposition over the elementary cell [0 ,
1) (Brillouin zone) w.r.t. theHilbert spaces H j (see (3.191)): L ( R Bohr ) ∼ = M j ∈ [0 , H j . (3.221)Interestingly, a similar function space realization can be obtained for the simple quotients of theVerma modules of sl . Namely, we realise L ( λ ) , λ = λ ( H ) , as a subspace of L ( R ) by v λ ( x, y ) := e λ ( x ) , dπ λ ( H ) := − i ( ∂ x − ∂ y ) (3.222) dπ λ ( E ) := − ie ( x − y ) ∂ y , dπ λ ( E − ) := − ie − ( x − y ) ∂ x . L ( λ ) is spanned by the weight vectors: v kλ ( x, y ) := (cid:0) dπ λ ( E − ) k v λ (cid:1) ( x, y ) = k Y l =1 ( λ + 1 − l ) ! e λ − k ( x ) e k ( y ) , k ∈ N , (3.223) v λ ( x, y ) := v λ ( x, y ) , which satisfy v kλ ( x + 2 πm, y + 2 πn ) = e πij λ m v kλ ( x, y ) , j λ = λ mod 1 ∈ [0 , , m, n ∈ Z . Therefore, L ( λ ) constitutes a (diagonal) subspace of H j λ ⊗ H ∼ = { Ψ ∈ L ( R ) | ∀ m, n ∈ Z : Ψ( x + 2 πm, y +2 πn ) = e πij λ m Ψ( x, y ) } =: L j λ , ( R ). If L ( λ ) is endowed with the inner product ( . , . ) ( j λ , coming from H j λ ⊗ H , we will have:( v kλ , v k λ ) ( j λ , = (cid:18) k ! (cid:18) λk (cid:19)(cid:19) δ k,k , k, k ∈ N , (3.224)implying the (modified) adjointness relations dπ λ ( H ) ∗ = dπ λ ( H ) , (3.225) dπ λ ( E ) ∗ ( λ + dπ λ ( H )) v kλ = ( λ − dπ λ ( H )) dπ λ ( E − ) v kλ , k ∈ N . Here, we consistently set dπ λ ( E ) ∗ v λλ = 0 for λ ∈ N . Thus, as in (3.217), we find that the tensionbetween Hilbert space structure and adjointness relations is due to the root vectors E , E − , whilethe adjointness relation for the generator of the Cartan subalgebra H is the expected one. Thecompletion ˆ L ( λ ) of L ( λ ) w.r.t. the Hilbert space structure induced by ( . , . ) ( j λ , is given by(possibly finite) l -series w.r.t. the vectors v kλ , k ∈ N :Ψ ∈ ˆ L ( λ ) : ⇔ Ψ = X k ∈ N Ψ k (cid:18) k ! (cid:18) λk (cid:19)(cid:19) − v kλ , X k ∈ N | Ψ k | < ∞ . (3.226)The realisation (3.222) of L ( λ ) is closely related to the Schwinger representation of sl on C [ X, Y ]: a ∗ + = X := e ( x ) , a + = ∂ X := − ie − ( x ) ∂ x , (3.227) a ∗− = Y := e ( y ) , a − = ∂ Y := − ie − ( y ) ∂ y ,E ± . = a ∗± a ∓ , H . = a ∗ + a + − a ∗− a − , which can be readily generalised to abtritrary g C . Namely, given a (complex) linear, finite dimen-sional representation dπ : g C → End( V π ) and d π copies of the CCR-algebra, e.g. creation andannihilation operators on the (bosonic) Fock space F s ( C d π ) = L ∞ n =0 Sym n C d π , we define S g ,g π ( X ) := d π X m,n =1 dπ ( X ) mn ( a ∗ m a n + g a ∗ m + g a n + g g ) , X ∈ g C , g , g ∈ C , (3.228)which satisfies [ S g ,g π ( X ) , S g ,g ( Y )] = S g ,g π ([ X, Y ]) , X, Y ∈ g C , and coincides with the Schwingerrepresentation of sl for g = 0 = g and π = π , the fundamental representation .For the remainder of the subsection, we return to G = U (1) n , and further restrict to n = 1, in3view of (3.219) and ( R n ) Bohr ∼ = ( R Bohr ) n , since in this case the extension from integral weightsto arbitrary weights works to full extent. Moreover, the “decompactification” from the spaces H j , as representations of the U (1)-Weyl algebra (3.194), to the space H = l ( R ) ∼ = B ( R ) ∼ = L ( R Bohr ), which is a representation of the standard 1-particle Weyl algebra (obtained from thealgebraic state ω ( W ( α, β )) = δ α, ), not only lifts the problem of ε -scaling, but additionally allows tohandle the square root necessary for replacing the Kohn-Nirenberg quantisation by a genuine Weylquantisation. There are essentially two of the latter, the first is natural in view of the isomorphism l ( R ) ∼ = B ( R ) (cf. ), while the second is tied to the identification l ( R ) ∼ = L ( R Bohr ), and extendsthe one proposed by Fewster and Sahlmann (cf. ):The Weyl elements are as usual (cp. (3.194)): W ε ( α, β ) := e iε αβ U ε ( α ) V ε ( β ) , (3.229) W ε ( α, β ) ∗ = W ε ( − α, − β ) , W ε (0 ,
0) = ,W ε ( α, β ) W ε ( γ, δ ) = e − iε ( αδ − γβ ) W ε ( α + γ, β + δ ) , α, β, γ, δ ∈ R . Let us also introduce some frequently used test function and Sobolev spaces on R Bohr and its(topological) dual group R disc ( R with the discrete topology, cf. ): d ( R ) := { ˆΨ : R → C | supp( ˆΨ) finite } , ˇ d ( R ) := span C { e λ | λ ∈ R } (3.230)= Trig( R ) ,d ( R ) := C R , ˇ d ( R ) := { T = X λ ∈ R ˆ T ( λ ) e λ | ˆ T ∈ C R } = Trig ( R ) ,H sp ( R Bohr ) := Trig( R ) || . || ( s,p ) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X λ ∈ R ˆΨ( λ ) e λ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( s,p ) := (cid:18) X λ ∈ R ( h λ i s | ˆΨ( λ ) | ) p (cid:19) p ,H s ∞ ( R Bohr ) := { T ∈ Trig ( R ) | || T || s, ∞ < ∞} , || T || ( s, ∞ ) := sup λ ∈ R |h λ i s ˆ T ( λ ) | H ∞ p ( R Bohr ) := \ s ∈ R H sp ( R Bohr ) , H −∞ p ( R Bohr ) := [ s ∈ R H sp ( R Bohr ) . for s ∈ R , p ∈ [0 , ∞ ). The space d ( R ) bears some similarities with the space of test functions D ( R ) = C ∞ ( R ), especially we may endowed it with the strict inductive limit topology coming from d ( R ) = [ F ⊂ R | F | < ∞ d F ( R ) , d F ( R ) := { ˆΨ : R → C | supp( ˆΨ) ⊂ F } ∼ = C | F | , (3.231) || ˆΨ || F,k := max λ ∈ F |h λ i k ˆΨ( λ ) | , k ∈ N , although the limit is uncountable in this case. There is a natural isomorphism, Trig( R ) || . || ∞ = CAP ( R ) ∼ = C ( R Bohr ), between the uniformly almost periodic functions on R and the continuousfunction on R Bohr , which allows to define the space of smooth functions on R Bohr : C ∞ ( R Bohr ) =
CAP ∞ ( R ) := CAP ( R ) ∩ C ∞ b ( R ) , (3.232)4carrying the natural Fréchet space topology induced by C ∞ b ( R ). Furthermore, we have H ( R Bohr ) = L ( R Bohr ) ∼ = B ( R Bohr ), and H ( R Bohr ) ⊂ CAP ( R ) ⊂ H ( R Bohr ) , H ∞ ( R Bohr ) ⊂ CAP ∞ ( R ) ⊂ H ∞ ( R Bohr ) , H sp ( R Bohr ) ⊂ H s p ( R Bohr ) , p ≤ p , s ≤ s . But H ∞ p ( R Bohr ) is not embedded in
CAP ( R ) for p > R or T ), which canbe related to R disc not being σ -finite w.r.t. the counting measure, e.g. the function,ˆΨ( λ ) := (cid:26) n : λ = n , n ∈ N , (3.233)belongs to T p> H ∞ p ( R Bohr ), since ∀ n ∈ N : 2 −| ps | ≤ h n i ps ≤ | ps | , while it is not in H ∞ ( R Bohr ),because the harmonic series is divergent. The inner products of the various realizations of H inducedualities between H sp ( R Bohr ) , d ( R ) , Trig( R ) and H − sq ( R Bohr ) , d ( R ) , Trig ( R ) for p + q = 1, and b (or ˇ) in (3.230) indicates that the corresponding vectors or spaces are related via the (inverse)Fourier transform on R Bohr (or R disc ):ˆΨ( λ ) := ( e − λ , Ψ) Bohr , Ψ ∈ H ( R Bohr ) , ˇΦ( x ) := ( e x , Φ) l ( R ) , Φ ∈ l ( R ) . (3.234)The images of the spaces H sp ( R Bohr ) under the Fourier transform are denoted by h sp ( R ). Before weintroduce the aforementioned Weyl quantisations, we add a short Remark III.44:
We might ask, whether there is a analogue s ( R ) ⊂ l ( R ) of the Schwartz space S ( R ) ⊂ L ( R ).Since S ( R ) is typically defined in terms of the uniform boundedness of expression of the form( h x i m ∂ nx f )( x ) , m, n ∈ N , we need a viable substitute for the partial derivatives as elements of l ( R ) are never differentiable. Additionally, we expect ˇ s ( R ) ⊂ H ∞ p ( R Bohr ) ∀ p ∈ [1 , ∞ ] to hold,which requires a summability condition on the elements of s ( R ), because decay conditions at infinitydo not suffice in view of (3.233). A natural discretised replacement of the partial derivatives,already familiar from the context of s ( Z ) ⊂ l ( Z ), is the (scaled) forward difference (∆ x,x f )( x ) := f ( x + x ) − f ( x ), and because of the relative nesting of the Sobolev spaces, we could propose thefollowing definition: s ( R ) := { ˆΨ : R → C | X λ ∈ R h λ i m | (∆ nλ,λ ˆΨ)( λ ) | < ∞} , λ ∈ R , m, n ∈ N . (3.235)But, as (∆ nλ,λ ˆΨ)( λ ) = P nk =0 ( − n − k (cid:0) nk (cid:1) ˆΨ( λ + kλ ), it is easy to see that ˇ s ( R ) = H ∞ ( R Bohr )in this case. Similar observations hold if we replace the forward difference by the backward orcentral difference. Interestingly, there is a useful duality between S ( R ) ⊂ L ( R ) and s ( R ) ⊂ l ( R )respectively H ∞ ( R Bohr ) ⊂ L ( R Bohr ), which is compatible with inner products, Fourier transformsand multiplier actions (cp. ). Namely, for f ∈ S ( R ) and ˆΨ ∈ s ( R ) (Ψ ∈ H ∞ ( R Bohr )), we have( F [ f ] , ˆΨ) l ( R ) = X λ ∈ R F [ f ]( λ ) ˆΨ( λ ) , | ( F [ f ] , ˆΨ) l ( R ) | ≤ || Ψ || ( s, sup λ ∈ R h λ i − s | F [ f ]( λ ) | < ∞ , ( F [ f ] , ˆΨ) l ( R ) = ( ˇ F [ f ] , Ψ) Bohr (3.236)= ( f, Ψ) L ( R ) = ( F [ f ] , F [Ψ]) L ( R ) , · f ∈ S ( R ) , F [ f ] · ˆΨ ∈ s ( R ).Furthermore, the duality is compatible with the representations π F and π of the Weyl algebra on L ( R ) and l ( R ), respectively:( π F ( W ε ( α, β ) ∗ ) f, Ψ) L ( R ) = ( π F ( W ε ( β, − α ) ∗ ) F ε [ f ] , ˆΨ ( ε ) ) l ( R ) (3.237)= ( F ε [ f ] , π ( W ε ( β, − α )) ˆΨ ( ε ) ) l ( R ) = ( f, π ( W ε ( α, β ))Ψ) L ( R ) . Here, we used the ε -scaled version of (3.234): ˆΨ ( ε ) ( λ ) = ( e − λε , Ψ) Bohr (cp. also (3.122), althoughwe switched the sign of the Fourier exponential, as to fit with the use of left convolution kernels inthe case of compact Lie groups).The two choices of Weyl quantisation associated with B ( R ) and l ( R ), respectively, L ( R Bohr )resemble the dichotomy already mentioned in the previous subsection III C (see the comment below(3.210)), and arise from (so far formal) expressions:( A σ Ψ)( x ) = 12 πε Z R dλ Z R dx σ ( ( x + x ) , λ ) e − λ ( x − x ε )Ψ( x ) , Ψ ∈ Trig( R ) or CAP ∞ ( R ) , (3.238)and( A σ Φ)( λ ) = X λ ∈ R Z R Bohr dµ Bohr ( x ) σ ( x, ( λ + λ )) e λ − λ ε ( x )Φ( λ ) , Φ ∈ d ( R ) or ˇΦ ∈ H ∞ ( R Bohr ) , (3.239)( A σ Ψ)( x ) = X λ ∈ R Z R Bohr dµ Bohr ( x ) σ ( ( x + x ) , λ ) e − λ ( x − x ε )Ψ( x ) , Ψ ∈ Trig( R ) or H ∞ ( R Bohr ) . Fundamentals on the theory of operators defined by (3.238) in Kohn-Nirenberg form can be foundin the works of Shubin (cf. also ), which we recall to some extent, as to allow for immediatecomparison with operators of the form (3.239). Following this, we will make precise the definitionof the latter.Pseudo-differential operators like (3.238) with σ ∈ S mρ,δ ( R ) (Hörmander’s symbol classes) canbe defined on C ∞ b ( R ), which contains Trig( R ) and CAP ∞ ( R ), by the usual means of oscillatoryintegrals (cf. ). In the context of almost periodic functions, the admissible symbols AP S mρ,δ ( R ) ⊂ S mρ,δ ( R ) are adapted to preserve the property of almost periodicity, i.e. σ ∈ AP S mρ,δ ( R ) ⊂ C ∞ ( R ) : ⇔ R λ σ ( . , λ ) ∈ CAP ( R ) is continuous (3.240)& ∀ α, β ∈ N : ∀ ( x, λ ) ∈ R : ∃ C αβ > | ( ∂ αx ∂ βλ σ )( x, λ ) | ≤ C αβ h λ i m − ρβ + δα ,AP S −∞ ( R ) := \ m ∈ R AP S mρ,δ ( R ) , AP S ∞ ρ,δ ( R ) := [ m ∈ R AP S mρ,δ ( R )for m ∈ R , ≤ δ ≤ ρ ≤ Definition III.45:
An operator A σ on CAP ∞ ( R ) defined by (3.238) with σ ∈ AP S mρ,δ ( R ) is called an almost-periodicpseudo-differential operator. Statements familiar form the theory of pseudo-differential operators on R n about composition,adjoints, asymptotic expansions and continuity w.r.t. to CAP ∞ ( R ) and the scales of Sobolevspaces H sp ( R Bohr ) remain valid, as expected, because all necessary operations on the symbols pre-serve almost-periodicity. An especially interesting and useful property of almost-periodic pseudo-differential operators is
Proposition III.46 (cf. ): Given A σ with σ ∈ AP S mρ,δ ( R ) , δ < ρ, there exist the formal adjoint A ∗ σ w.r.t. to ( , ) L ( R ) and ( , ) Bohr , i.e. ( A ∗ σ Ψ , Φ) L ( R ) = (Ψ , A σ Φ) L ( R ) , Ψ , Φ ∈ S ( R ) , (3.241)( A ∗ σ Ψ , Φ) Bohr = (Ψ , A σ Φ) Bohr , Ψ ∈ H ∞ p ( R Bohr ) , Φ ∈ H ∞ q ( R Bohr ) ( p + q = 1) . (3.242) The symbol of A ∗ σ is σ . Furthermore, Shubin proves the equality || A σ || B ( L ( R )) = || A σ || B ( B ( R )) for bounded A σ , whichentails the equality of spectra spec B ( L ( R )) ( A σ ) = spec B ( B ( R )) ( A σ ) by the preceding proposition(this continues to hold for (hypo)elliptic A σ ). Nevertheless, the quality of the spectra w.r.t. L ( R )and B ( R ) can differ significantly, e.g. the spectrum of the Laplace operator ∆, which is essentiallyself-adjoint on S ( R ) and Trig( R ), respectively, is absolutely continuous in the first and pure pointin the second case.Now, we come to the definition of pseudo-differential operators in terms of (3.239), which we willcall Bohrian pseudo-differential operators . Let us first remark that the formulas (3.239) are closerin structure to those applied in the definition of pseudo-differential operators on compact Lie groups(see subsection III A), while the almost-periodic pseudo-differential operators heavily exploit thespecial relation between R and R Bohr . The same reasoning applies to U (1)-equivariant pseudo-differential operators (see subsection III C), where the special relation between R and U (1) viathe covering morphism R → R / π Z ∼ = U (1) is of avail. But, in contrast with U (1)-equivariantoperators, where the analogues, (3.209) and (3.212), of (3.238) and (3.239) are essentially equivalentdue to the sparseness of Z ⊂ R and the possibility of smooth interpolation, the situation will bedifferent in the present case. Definition III.47:
A function σ : R disc × R Bohr → C is called a symbol in S mρ,δ ( R Bohr × R disc ) = s mρ,δ , where m ∈ R , ≤ δ < ρ ≤ , if ∀ λ ∈ R : σ ( . , λ ) ∈ H ∞ ( R Bohr ) and ∀ α, β ∈ N : ∃ C α,β > s.t.: ∀ ( x, λ ) ∈ R Bohr × R disc : | ( ∂ αx ∆ βλ σ )( x, λ ) | ≤ C αβ h λ i m − ρβ + δα . (3.243) We set: s −∞ = \ m ∈ R s mρ,δ , s ∞ ρ,δ = [ m ∈ R s mρ,δ . (3.244)A simple application of the (discrete) Leibniz formula and Peetre’s inequality, ∀ r, λ, λ ∈ R : h λ + λ i r ≤ | r | h λ i | r | h λ i r , gives:7 Corollary III.48:
Let σ ∈ s mρ,δ and τ ∈ s m ρ ,δ . Then ∀ α, β ∈ N : ∂ αx ∆ βλ σ ∈ s m − ρβ + δα and στ ∈ s m + m min( ρ,ρ ) , max( δ,δ ) . In view of remark III.44, the definition of symbol classes s mρ,δ for (3.239) requires summabilityconditions or restrictions on the Fourier spectrum supplementing the usual decay conditions. Definition III.49:
A symbol σ ∈ s mρ,δ is said to have polynomially bounded spectral growth of order γ, γ ∈ R , if | supp(ˆ σ ( . , λ )) ∩ K λ | =: N λ ( λ ) ≤ C γ,γ h λ i γ h λ i γ for K λ = [ − λ , λ ] ⊂ R , λ ∈ R , where ˆ σ ( λ , λ ) := Z R Bohr dµ Bohr ( x ) e λ ( x ) σ ( x, λ ) . (3.245) Clearly, symbols of polynomially bounded spectral growth of order γ form a subspace s m, ( γ,γ ) ρ,δ ⊂ s mρ,δ . Furthermore, the property to be of polynomially bounded spectral growth is, on the one hand,preserved under multiplication in the Fourier domain and, on the other hand, not preserved underconvolution in the Fourier domain. An important subclass of symbols is given by
Definition III.50:
A symbol σ in s m, ( γ,γ =1) ρ,δ is called U (1) λ - equivariant , λ ∈ R , if ∀ λ ∈ R : supp(ˆ σ ( . , λ )) ⊂ Z j λ := λ ( Z + j ) for j ∈ [0 , . Similarly, T ∈ Trig ( R ) is U (1) λ -equivariant, if supp( ˆ T ) ⊂ Z j λ .The various Fourier images of the U (1) λ -equivariant versions of the function spaces on R Bohr (see (3.230) and below) are denoted by d ( Z j λ ) , d ( Z j λ ) , h sp ( Z j λ ) , etc. Next, we give meaning to (3.239) for σ ∈ s m, ( γ,γ ) ρ,δ , Φ ∈ d ( R ) in a similar fashion as one does for(3.238) with σ ∈ S mρ,δ , Ψ ∈ C ∞ b ( R ). To this end, we will work primarily with the first formula in(3.239), and interpret the second formula as a mnemonic for the dual operator defined by the ε -scaled Fourier transform. On the contrary, Fewster and Sahlmann construct operators by meansof the second formula. It is apparent that the first formula is natural, when working with the“volume representation” in loop quantum cosmological models. Proposition III.51:
Let σ ∈ s m, ( γ,γ ) ρ,δ , then A σ : d ( R ) → h ∞ ( R ) . If σ is U (1) λ -equivariant and λ ∈ R satisfies λ λ = pq ∈ Q , we have A σ : h ∞ ( Z j λ ) → h ∞ ( Z j λ ) for some λ ∈ R and j ∈ [0 , ( ε = 1 ).In general, if σ ∈ Trig ( R ) ⊗ d ( R ) , A σ defines a quadratic form Q σ on d ( R ) : Q σ (Φ , Φ ) := (Φ , A σ Φ ) l ( R ) , Φ , Φ ∈ d ( R ) . (3.246) The formal adjoint A ∗ σ , defined by ( A ∗ σ Φ , Φ ) l ( R ) = (Φ , A σ Φ ) l ( R ) , Φ , Φ ∈ d ( R ) , (3.247) has symbol σ . Proof:
For simplicity, we restrict to ε = 1, since the general case only leads to some trivial rescalingin the formula to follow. First, let σ ∈ s m,γρ,δ and Φ ∈ d ( R ), then we use e λ ( x ) = h λ i − M (1 − x ) M e λ ( x ) , M ∈ N , and the translation invariance of µ Bohr to regularize (3.239):( A σ Φ)( λ ) = X λ ∈ supp(Φ) Φ( λ ) Z R Bohr dµ Bohr ( x ) h λ − λ i − M ((1 − ∆ x ) M σ )( x , ( λ + λ ))) e λ − λ ( x ) . (3.248)By assumption, i.e. | supp(Φ) | < ∞ and σ ∈ s m, ( γ,γ ) ρ,δ , | supp( A σ Φ) ∩ K λ | ≤ C γ h λ i γ for some γ ∈ R . Therefore, for arbitrary s ∈ R and large enough M ∈ N || A σ Φ || ( s, (3.249) ≤ X λ ∈ supp( A σ Φ) h λ i s X λ ∈ supp(Φ) h λ − λ i − M M X α =0 (cid:18) Mα (cid:19) sup x ∈ R | ( ∂ αx σ )( x , ( λ + λ )) || Φ( λ ) |≤ M X α =0 (cid:18) Mα (cid:19) C α X λ ∈ supp(Φ) | Φ( λ ) | X λ ∈ supp( A σ Φ) h λ i s h λ − λ i − M h ( λ + λ ) i m +2 δα ≤ | s | +2 M M X α =0 (cid:18) Mα (cid:19) C α | m +2 δα | X λ ∈ supp(Φ) h λ i M + | m +2 δα | | Φ( λ ) | X λ ∈ supp( A σ Φ) h λ i − M − δα )+ s + m < ∞ . Here, we used Peetre’s inequality to arrive at the next to last line, and concluded finiteness of thelast sum from, using the polynomial growth bound on | supp( A σ Φ) ∩ K λ | , X λ ∈ supp( A σ Φ) h λ i − M − δα )+ s + m ≤ X λ ∈ supp( A σ Φ) h λ i − M (1 − δ )+ s + m (3.250)= lim n →∞ n X m =0 X λ ∈ supp( A σ Φ) ∩ K m \ K m − h λ i − M (1 − δ )+ s + m , K − = ∅≤ lim n →∞ n X m =0 h m − i − M (1 − δ )+ s + m ( N m − N m − ) ≤ C γ lim n →∞ X m =0 h m − i − M (1 − δ )+ s + m + γ , which is finite for large enough M , because δ <
1. To prove the second assertion, we observe thatfor Φ ∈ h ∞ ( Z j λ ) and U (1) λ -equivariant σ :( A σ Φ)( λ ) = 0 ⇔ λ = λ − λ | {z } ∈ Z j λ + λ |{z} ∈ Z j λ ∈ Z j λ + Z j λ . (3.251)But, Z j λ + Z j λ = { ( λ m + λ n ) + ( λ j + λ j ) | m, n ∈ Z } ⊂ λ q Z + λ q ( qj + pj ) . Thus, setting λ := λ q = λ p and j := ( qj + pj ) mod 1, we have A σ Φ ∈ d ( Z j λ ). Now, we regularise the9expression for A σ Φ in the same way as above to show that we even have A σ Φ ∈ h ∞ ( Z j λ ) ( s ∈ R ): || A σ Φ || ( s, (3.252) ≤ X λ ∈ Z j λ h λ i s X λ ∈ Z j λ h λ − λ i − M M X α =0 (cid:18) Mα (cid:19) sup x ∈ R | ( ∂ αx σ )( x , ( λ + λ )) || Φ( λ ) |≤ | s | +2 M M X α =0 (cid:18) Mα (cid:19) C α | m +2 δα | X λ ∈ Z j λ h λ i M + | m +2 δα | | Φ( λ ) | X λ ∈ Z j λ h λ i − M − δα )+ s + m < ∞ , where, again, we employed Peetre’s inequality, Φ ∈ h ∞ ( Z j λ ) and the finiteness of the last sum forlarge enough M ( δ < A σ and the formal adjoint A ∗ σ areobvious from the finiteness of all sums and the behaviour of the Fourier transform w.r.t. complexconjugation. Remark III.52:
Taking a look at (3.251), it becomes evident that the condition λ λ ∈ Q in proposition III.51 cannotbe relaxed easily, because two relatively irrational lattices may generated arbitrary dense supportfor A σ Φ in the bounded intervals K λ . Definition III.53:
Two operators A σ , A τ with U (1) λ σ -, respectively, U (1) λ τ -equivariant symbols σ, τ , defined on the(rational scales of) spaces h ∞ ( λ σ ) := [ λ ∈ R λ λσ ∈ Q [ j ∈ [0 , h ∞ ( Z j λ ) , h ∞ ( λ τ ) := [ λ ∈ R λ λτ ∈ Q [ j ∈ [0 , h ∞ ( Z j λ ) (3.253) are called relatively rational , if λ σ λ τ ∈ Q ( ε = 1 ). Clearly, in this case h ∞ ( λ σ ) = h ∞ ( λ τ ) (otherfunction space are defined similarly ). Corollary III.54:
Relatively rational operators generate an algebra with common domain h ∞ ( λ ) for some λ ∈ R . Corollary III.55:
Let σ be a U (1) λ -equivariant symbol, then A ( ε ) σ and A ( ε ) σ are relatively rational if and only if ε ε ∈ Q . Here, we made the ε -dependence of (3.239) explicit. If we want to use symbolic calculus in the analysis of Bohrian pseudo-differential operators, we willneed a statement concerning the asymptotic summation of symbols in s mρ,δ .0 Proposition III.56:
Let { m j } ∞ j =1 ⊂ R be such that lim j →∞ m j = −∞ , m := max j ∈ N m j , and σ j ∈ s m j ρ,δ for all j ∈ N .Then, there exists a symbol σ ∈ s mρ,δ , unique up to s −∞ , such that a ∼ P ∞ j =1 σ j , i.e.: ∀ n ∈ N : ∃ k n ∈ N : ∀ k ≥ k n : σ − k X j =1 σ j ∈ s m n ρ,δ . (3.254) If the symbols σ j , j ∈ N , are of polynomially bounded spectral growth of order ( γ, γ ) or U (1) λ -equivariant, then so is σ . Proof:
Using standard excision function techniques and literally repeating the argument in , Theorem4.4.1, accomplishes the proof. The statement concerning polynomially bounded spectral growthand equivariance follows, because the excision functions only touch the second argument of thesymbols.We close this section with a discussion of the differences between almost-periodic pseudo-differentialoperators and Bohrian pseudo-differential operators:Our first observation is, as already mentioned above, that almost-periodic pseudo-differential oper-ators are not equivalent to Bohrian pseudo-differential operators, at least not without extending thesymbol classes for almost-periodic pseudo-differential operators to include non-smooth elements, incontrast with the analogous situation in the U (1)-equivariant case. This is most easily inferred froman explicit example (important in loop quantum cosmology): A : h sp ( R ) → h s − p ( R ) , s ∈ R , ( A Φ)( λ ) := | λ | Φ( λ ) , Φ ∈ h s +1 p ( R ) , (3.255)which has the symbol σ A ( x, λ ) = | λ | , which makes sense for (3.238) as well as (3.239). Clearly, σ A ∈ s , (0 , , , but σ / ∈ AP S mρ,δ for any m, ρ, δ , because this require smoothness in the second ar-gument. Since an element Φ ∈ h sp ( R ) can have support at any point λ ∈ R , there is no smoothinterpolation of σ A (the same observation holds for the spaces h ∞ ( λ )).Second, since symbols of almost-periodic pseudo-differential operators form a subclass of Hörman-der’s symbols, it is possible to transfer much of the usual symbolic calculus to their setting. Thisremains partially true for Bohrian pseudo-differential operator, if we replace the symbolic calculuswith a discrete version familiar from the U (1)-equivariant case (cf. ). From a conceptional point ofview it is useful to introduce Fourier-Weyl elements, and the associated (de-)quantisation formulas(cp. (3.4), and below): ˆ W ε ( λ, x ) = X β ∈ R Z R Bohr dµ Bohr ( α ) e − λ ( α ) e − β ( x ) W ε ( α, β ) , (3.256)tr l ( R ) ( ˆ W ε ( λ, x ) ˆ W ε ( λ , x )) = δ λ,λ δ Bohr ( x − x ) ,A σ = X λ ∈ R Z R Bohr dµ Bohr ( x ) σ ( x, λ ) ˆ W ε ( λ, x ) ,σ A ( x, λ ) = tr l ( R ) ( ˆ W ε ( λ, x ) A ) , which are to interpreted in a distributional sense with δ Bohr = P λ ∈ R e − λ ∈ H ∞ ( R Bohr ). Theseformulas can be used to derive the product formula for symbols, ρ = σ ? ε τ , corresponding to the1operator product A ρ = A σ A τ (if defined): ρ ( x, λ ) = X λ ∈ R Z R Bohr dµ Bohr ( x ) e − λ ( x ) e λ ( x ) (3.257) × X λ ∈ R Z R Bohr dµ Bohr ( x )ˆ σ ( λ , x )ˆ τ ( λ − λ , x − x ) e − ε λ ( x − x ) e ε ( λ − λ ) ( x )= X λ ∈ R X λ ∈ R e − λ ( x ) e − λ ( x ) ˆ σ ( λ , λ + ε λ )ˆ τ ( λ , λ − ε λ ) , ˆ ρ ( λ , λ ) = X λ ∈ R ˆ σ ( λ , λ + ε ( λ − λ ))ˆ τ ( λ − λ , λ − ε λ ) , which is completely analogous with the formula for the standard Moyal product (3.7). For σ, τ ∈ Trig( R ) ⊗ d ( R ) the expression (3.257) is convergent, but in general, e.g. for A σ , A τ relativelyrational, it has to be interpreted in an oscillatory sense, i.e. it should be regularised in the way weused to define A σ . Remark III.57:
The formula (3.257) shows that the composition A ρ = A σ A τ of relatively rational operators A σ , A τ is also relatively rational to A σ , A τ .Finally, we look into possible asymptotic expansions of (3.257). Let us first assume that σ and τ are smooth in the second argument, and belong to Hörmander’s symbol classes. Then, we apply aTaylor expansion of the product of ˆ σ and ˆ τ in (3.257): ρ ( x, λ ) = X λ ∈ R e − λ ( x ) X λ ∈ R e − λ ( x ) (3.258) × N X n =0 n ! (cid:18) − iε (cid:19) n n X k =0 (cid:18) nk (cid:19) ( − n − k (cid:18) (cid:92) ( ∂ kx ∂ n − kλ σ ) ( λ , λ ) (cid:92) ( ∂ n − kx ∂ kλ τ ) ( λ , λ ) (cid:19) + (cid:18) − iε (cid:19) N +1 R ( N ) σ,τ ( x, λ ) , = N X n =0 n ! (cid:18) − iε (cid:19) n n X k =0 (cid:18) nk (cid:19) ( − n − k (cid:0) ( ∂ kx ∂ n − kλ σ )( λ , λ )( ∂ n − kx ∂ kλ τ )( λ , λ ) (cid:1) + (cid:18) − iε (cid:19) N +1 R ( N ) σ,τ ( x, λ ) ,R ( N ) σ,τ ( x, λ ) = 1 N ! X λ ∈ R e − λ ( x ) X λ ∈ R e − λ ( x ) N +1 X k =0 (cid:18) N + 1 k (cid:19) ( − N +1 − k × Z dt (1 − t ) N (cid:92) ( ∂ kx ∂ n − kλ σ ) ( λ , λ + tε λ ) (cid:92) ( ∂ n − kx ∂ kλ τ ) ( λ , λ − tε λ ) , which resembles the well-know formulas from the R n -case, besides the fact that we have to deal withthe Fourier series on R disc instead of the Fourier transform, which leads to different convergenceproperties (see above). If we do not want to impose smoothness of the symbols and work with the2classes s mρ,δ (see definition III.47), we will have to replace the Taylor expansion by some non-smoothanalogue. A least, in the case of relatively rational operators, this is achieved by the discrete Taylorexpansion or Newton series (cf. , Theorem 3.3.21), because this essentially reduces the situation tothe U (1)-equivariant case. For Φ : Z n → C , we have (in multi-index notation):Φ( λ + λ ) = X α ∈ N | α |≤ N α ! λ ( α ) (∆ αλ Φ)( λ ) + r N Φ ( λ, λ ) , λ ( α ) = n Y i =1 λ ( α i ) i , λ ( α i ) i := λ · ... · ( λ − α i + 1) , (3.259)= X α ∈ N | α |≤ N (cid:18) λ α (cid:19) (∆ αλ Φ)( λ ) + r N Φ ( λ, λ ) , = α ! (cid:18) λ α (cid:19) where the remainder r ( N )Φ ( λ, λ ) satisfies: | (∆ βλ r ( N )Φ )( λ, λ ) | ≤ max | α | = N +1 λ ∈ Q ( λ ) | λ ( α ) (∆ α + βλ Φ)( λ + λ ) | , (3.260) Q ( λ ) := { λ ∈ Z n | | λ i | ≤ | λ i | , i = 1 , ..., n } . Therefore, if A σ and A τ are relatively rational operators with σ ∈ s m σ ρ,δ and σ ∈ s m τ ρ,δ , their compo-sition A ρ = A σ A τ is defined, and we find: ρ ( x, λ ) = X λ ∈ Z jτ λτ X λ ∈ Z jσ λσ e − λ ( x ) e − λ ( x ) ˆ σ ( λ , λ + ε λ )ˆ τ ( λ , λ − ε λ ) (3.261)= e − ( λ σ j σ + λ τ j τ ) ( x ) X m τ ∈ Z X m σ ∈ Z e − ( λ σ m σ + λ τ m τ ) ( x ) ˆ σ ( λ σ ( m σ + j σ ) , λ + ε λ τ ( m τ + j τ )) × ˆ τ ( λ τ ( m τ + j τ ) , λ − ε λ σ ( m σ + j σ ))= e − ( λ σ j σ + λ τ j τ ) ( x ) X m τ ∈ Z X m σ ∈ Z e ( λ σ m σ − λ τ m τ ) ( x ) × N X n =0 n ! n X k =0 (cid:18) nk (cid:19) m ( k ) τ (cid:16) ∆ kλ, ε λ τ ˆ σ (cid:17) ( λ σ ( − m σ + j σ ) , λ + ε λ τ j τ ) × m ( n − k ) σ (cid:16) ∆ n − kλ, ε λ σ ˆ τ (cid:17) ( λ τ ( m τ + j τ ) , λ − ε λ σ j σ )+ r ( N ) σ,τ ( x, λ ; ε )= N X n =0 n ! n X k =0 (cid:18) nk (cid:19)(cid:18)(cid:16) D j σ x,λ σ (cid:17) ( n − k ) ∆ kλ, ε λ τ σ (cid:19) ( x, λ + ε λ τ j τ ) × (cid:18)(cid:16) D j τ x,λ τ (cid:17) ( k ) ∆ n − kλ, ε λ σ τ (cid:19) ( x, λ − ε λ σ j σ )+ r ( N ) σ,τ ( x, λ ; ε ) , where D j x,λ := − iλ ∂ x − j , and we employed the scaled forward difference (∆ λ,λ f )( λ ) = f ( λ + λ ) − f ( λ ). To arrive at the last line, we applied the inverse Fourier transform (the series are in3 h ∞ ( R )) together with the identity: λ ( k ) ˆΨ( λ ) = Z R Bohr dµ Bohr ( x ) e λ ( x )( i∂ x ) ( k ) Ψ( x ) , Ψ ∈ H ∞ ( R Bohr ) . (3.262)The formula (3.261) constitutes a discrete analogue of the usual asymptotic Weyl product formula.Although, the expansion is not manifestly given in orders of ε , it still has the crucial property thatthe contribution at order n belong to the symbol classes s m σ + m τ − n ( ρ − δ ) ρ,δ leading to contributions bystrictly smaller operators (assuming a suitable version of the Calderón-Vaillancourt theorem holds,cf. ) with every increase in the order of the expansion for δ < ρ (use corollary III.48). Finally,we would like to conclude that (3.261) qualifies as an asymptotic expansion, which would hold true,if we were to show that r ( N ) σ,τ ∈ s m σ + m τ − ( N +1)( ρ − δ ) ρ,δ . In view of the positive results of Ruzhanskyand Turunen for the U (1)-Kohn-Nirenberg calculus (cf. , Theorem 4.7.10), we fully expect this tobe the case.A simple boundedness theorem of Sobolev type for U (1) λ -equivariant operators A σ with σ ∈ s m , (this encompasses the important case σ ∈ s mρ, ⊂ s m , ) can be proved by means of the (discrete)Young’s inequality (cf. , where a similar reasoning is applied to U (1)-Kohn-Nirenberg operators,Proposition 4.2.3): Lemma III.58:
Given a function h : Z j λ × Z j λ → C , s.t. C := sup λ ∈ Z j λ X λ ∈ Z j λ | h ( λ, λ ) | < ∞ , C := sup λ ∈ Z j λ X λ ∈ Z j λ | h ( λ, λ ) | < ∞ , (3.263) we may define ( K h Φ) ( λ ) := P λ ∈ Z j λ h ( λ, λ )Φ( λ ) for all Φ ∈ h p ( Z j λ ) , and have: || K h Φ || (0 ,p ) ≤ C p C q || Φ || (0 ,p ) , ≤ p, q ≤ ∞ , p + q = 1 . (3.264) Proof:
The results is a simple repetition of the argument presented in for scales affine Z -lattices.Another useful result presented in , which generalises to our case, is Lemma III.59 (cp. , Lemma 4.2.1): Given σ ∈ s mρ,δ , its Fourier transform w.r.t. the first variable satisfies: ∀ r ∈ R ≥ , β ∈ N : (cid:12)(cid:12)(cid:12)(cid:16) ∆ βλ ˆ σ (cid:17) ( λ , λ ) (cid:12)(cid:12)(cid:12) ≤ C r,β h λ i − r h λ i m − ρβ + δr . (3.265)4 Proof:
In analogy with the usual regularisation arguments, we have for M ∈ N (cid:12)(cid:12)(cid:12)(cid:16) ∆ βλ ˆ σ (cid:17) ( λ , λ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z R Bohr dµ Bohr ( x ) e λ ( x ) σ ( x, λ ) (cid:12)(cid:12)(cid:12)(cid:12) (3.266)= (cid:12)(cid:12)(cid:12)(cid:12)Z R Bohr dµ Bohr ( x ) h λ i − M (cid:0) (1 − ∂ x ) M e λ ( x ) (cid:1) σ ( x, λ ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z R Bohr dµ Bohr ( x ) h λ i − M e λ ( x )(1 − ∂ x ) M σ ( x, λ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ h λ i − M M X α =0 (cid:18) Mα (cid:19) C αβ h λ i m − ρβ + δ α ≤ M X α =0 (cid:18) Mα (cid:19) C αβ !| {z } =: C M,β h λ i − M h λ i m − ρβ + δ M . The boundary term arising from the partial integrations vanishes, because ∀ λ ∈ R : σ ( . , λ ) ∈ C ∞ b ( R ). The result follows for M = pq ∈ Q ≥ from (cid:12)(cid:12)(cid:12)(cid:16) ∆ βλ ˆ σ (cid:17) ( λ , λ ) (cid:12)(cid:12)(cid:12) = (cid:18)(cid:12)(cid:12)(cid:12)(cid:16) ∆ βλ ˆ σ (cid:17) ( λ , λ ) (cid:12)(cid:12)(cid:12) q (cid:19) q (3.267) ≤ (cid:0) C p,β h λ i − p h λ i m − ρβ + δ p (cid:1) q (cid:0) C ,β h λ i m − ρβ (cid:1) q − q = C pq ,β h λ i − pq h λ i m − ρβ + δ pq , and for M = r ∈ R ≥ by continuity.Now, a Calderón-Vaillancourt type result can be proved: Theorem III.60: A U (1) λ -equivariant operator A σ : h ∞ ( Z j λ ) → h ∞ ( Z j λ ) as in proposition III.51 with σ ∈ s mρ,δ extends uniquely to bounded operator from h sp ( Z j λ ) to h s − tp ( Z j λ ) for p ∈ [1 , ∞ ) and any s, t ∈ R ,s.t. ∃ r ∈ R ≥ : δr ≤ t − m & (1 − δ ) r > ( | m | − | t | + | s − t | ) . (3.268) In particular, this justifies to call operators with σ ∈ s −∞ (infinitely) smoothing . Moreover, if δ = 0 , we can choose t = m . Proof:
For any t ∈ R , we estimate by Peetre’s inequality: || A σ Φ || p ( s − t,p ) = X λ ∈ Z j λ h λ i p ( s − t ) (cid:12)(cid:12)(cid:12)(cid:12) X λ ∈ Z j λ ˆ σ ( λ − λ , ( λ + λ ))Φ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) p (3.269)5 ≤ p | s − t | X λ ∈ Z j λ (cid:12)(cid:12)(cid:12)(cid:12) X λ ∈ Z j λ h λ − λ i | s − t | h λ i − t ˆ σ ( λ − λ , ( λ + λ )) h λ i s Φ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) p . Now, setting h σ ( λ , λ ) := h λ − λ i | s − t | h λ i − t ˆ σ ( λ − λ , ( λ + λ )), we would like to employYoung’s inequality (see lemma III.58), which will be possible since for any r ∈ R ≥ : C σ := sup λ ∈ Z j λ X λ ∈ Z j λ | h σ ( λ , λ ) | ≤ C r, sup λ ∈ Z j λ X λ ∈ Z j λ h λ − λ i | s − t | h λ i − t h λ − λ i − r h ( λ + λ ) i m + δr ≤ | m + δr | + | t | C r, sup λ ∈ Z j λ X λ ∈ Z j λ h λ − λ i − r (1 − δ )+ | m | + | t | + | s − t | h λ i − t + m + δr (3.270) ≤ | m + δr | + | t | C r, (cid:16) sup λ ∈ Z j λ h λ i − t + m + δr (cid:17) X λ ∈ Z j λ h λ i − r (1 − δ )+ | m | + | t | + | s − t | ,C σ := sup λ ∈ Z j λ X λ ∈ Z j λ | h σ ( λ , λ ) | ≤ C r, sup λ ∈ Z j λ X λ ∈ Z j λ h λ − λ i | s − t | h λ i − t h λ − λ i − r h ( λ + λ ) i m + δr ≤ | m + δr | C r, sup λ ∈ Z j λ X λ ∈ Z j λ h λ − λ i − r (1 − δ )+ | m | + | s − t | h λ i − t + m + δr ≤ | m + δr | C r, (cid:16) sup λ ∈ Z j λ h λ i − t + m + δr (cid:17) X λ ∈ Z j λ h λ i − r (1 − δ )+ | m | + | s − t | , where we used the estimate on ˆ σ from lemma III.59 and, repeatedly, Peetre’s inequality. In eachcase, the last line follows from the rationality of λ λ . Thus, if (3.268) holds, we will have C σ < ∞ and C σ < ∞ (0 ≤ δ < || A σ Φ || p ( s − m,p ) ≤ p | s − m | || K h σ ( h . i s Φ) || p ,p ≤ p | s − m | C σ ( C σ ) pq || Φ || ps,p . (3.271)Finally, because h ∞ ( R ) ⊂ h sp ( R ) is dense for s ∈ R , p ∈ [1 , ∞ ), the existence of a unique boundedextension of A σ follows. The remaining statements are obvious. IV. CONCLUSIONS AND PERSPECTIVES
In the main part of this article, we have introduced and analysed a Weyl quantisation for loopquantum gravity-type models aiming at the implementation of the program space adiabatic per-turbation theory in the latter. As we have argued, a complete implementation requires the Weylquantisation to be scalable with the quantisation parameter ε (also called adiabatic parameter) fora perturbation theory in orders of ε to be possible. Unfortunately, due to the fact, that models àla loop quantum gravity are constructed via projective limits of function algebras living on the co-6tangent bundles of compact Lie groups (in the phase space approach), there results a fundamentalasymmetry in the treatment of configuration and momentum space degrees of freedom in the quan-tum theory. This asymmetry entails the scalability (with ε ) of the local Weyl quantisation w.r.t. toAshtekar-Isham-Lewandowski representation, if and only if the adiabatic parameter ε is associatedwith the momentum variables: An effect that its easily understood from the observation that themomentum space degrees of freedom are modelled by the co-tangent space directions, which pos-sess a (local) vector space structure. In contrast, the configuration space degrees of freedom, whichare modelled by the compact Lie group underlying the co-tangent bundle, do not admit suitable,i.e. compatible with the (local) commutation relations defining the transformation group algebra, ε -scale transformations, because the existence of the latter would require exponential map to be ahomomorphism of Lie groups (existence of arbitrary (real) powers of group element in a homomor-phic way). In the global Weyl quantisation, where the co-tangent bundle spaces are replaced by therepresentation theoretical dual of the Lie group, the problem of scalability also shows up for themomentum space degrees of freedom (unitary equivalence classes of irreducible representations),and manifest itself in the rigid structure of the lattice of integral highest weights. Since the repre-sentation theoretical dual and the group itself are in a one-to-one correspondence (Tannaka-Kreinduality , Doplicher-Roberts theorem ), the question of scalability in the local and global settingcan be unified under the theme of ε -scalable Fourier transforms, which we discussed in subsectionIII B & III D).One reason, why we are not satisfied with the local Weyl quantisation, and its scalability w.r.t. tothe momentum variables, is exemplified by toy models of loop quantum cosmology-type. In those,we find that it is quite natural for the adiabatic parameter ε to be associated with the configurationspace degrees of freedom (holonomies of U (1)): A feature that may well persist in full loop quantumgravity-type models.Another reason is implicit in the construction of the (local) calculus Paley-Wiener-Schwartz sym-bols, which requires an analytic momentum dependence of the quantisable functions, because, aftera fibre-wise Fourier transform from the co-tangent bundle to the tangent bundle, the dual distri-bution is required to be of compact support. Thus, again the non-global nature of the exponentialmap of a compact Lie group is of some disadvantage, as it severely restricts (analyticity!) the classof quantisable function, and therefore the class of dequantisable operators.Such analyticity condition are not necessary in the global calculus, but the problems due to com-pactness show up in the way mentioned before.An almost satisfactory solution to the difficulties introduced by the compactness of the (truncated)configuration spaces, can, up to this point, only be achieved in U (1) n -models, where its possible toproceed from U (1) to the still compact, but representation theoretically less rigid, Bohr compact-ification of the reals R Bohr . Regrettably, this appears not to be a viable strategy in non-Abelianmodels .Since we also discussed the use of coherent state/Berezin quantisations for compact Lie groups(section II and subsection III C) in our research on the implementability of the ideas of Born andOppenheimer in loop quantum gravity-type models, it should be once more pointed out, that, un-less a suitable ? -product is constructed from these quantisations, the existence of which is highlyquestionable for non-compact (truncated) phase spaces, it seems to be very difficult to construct asystematic perturbation theory. Thus, while the use of coherent states is conceptually tempting, itseems to be technically and computationally disfavoured.7 ACKNOWLEDGMENTS
AS gratefully acknowledges financial support by the Ev. Studienwerk e.V.. This work wassupported in parts by funds from the Friedrich-Alexander-University, in the context of its EmergingField Initiative, to the Emerging Field Project “Quantum Geometry”.
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