Coherent states, quantum gravity and the Born-Oppenheimer approximation, I: General considerations
CCoherent states, quantum gravity and the Born-Oppenheimerapproximation, I: General considerations
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
This article, as the first of three, aims at establishing the (time-dependent) Born-Oppenheimer approximation, in the sense of space adiabatic perturbation theory, for quan-tum systems constructed by techniques of the loop quantum gravity framework, especiallythe canonical formulation of the latter. The analysis presented here fits into a rather generalframework, and offers a solution to the problem of applying the usual Born-Oppenheimeransatz for molecular (or structurally analogous) systems to more general quantum systems(e.g. spin-orbit models) by means of space adiabatic perturbation theory. The proposedsolution is applied to a simple, finite dimensional model of interacting spin systems, whichserves as a non-trivial, minimal model of the aforesaid problem. Furthermore, it is ex-plained how the content of this article, and its companion, affect the possible extractionof quantum field theory on curved spacetime from loop quantum gravity (including matterfields).
CONTENTS
I. Introduction II. On the Born-Oppenheimer ansatz
III. Weyl quantisation and space adiabatic perturbation theory IV. A model with non-commutative slow variables: Spin-orbit coupling V. Conclusions & perspectives Acknowledgments VI. Bibliography I. INTRODUCTION
In this article, we begin our investigations into a framework that allows for the formulation ofthe (time-dependent) Born-Oppenheimer approximation for physical models of the type of loop a) Electronic mail: [email protected] b) Electronic mail: [email protected] a r X i v : . [ m a t h - ph ] A p r quantum gravity .To this end, we continue and generalise certain ideas and proposals for the extraction of quan-tum field theory on curved spacetimes from models of loop quantum gravity present in theliterature , and try to put them into a rigorous and (computationally) effective mathematicalframework. The latter is provided by space adiabatic perturbation theory as developed by Pa-nati, Teufel and Spohn in which is a mathematically precise formulation, by means of pseudo-differential calculus, of the intuitive content of the Born-Oppenheimer approximation in a time-dependent setting. Namely, the quantum system under consideration is split into slow and fastsubsystems, and (partially) dequantised in the slow sector (deformation quantisation). As a result,an algebra of functions on a manifold (representing the slow subsystem) taking values in operatorson the fast subsystem’s Hilbert space is obtained. Moreover, the algebra of functions admits anon-commutative ? -product, which captures the operator product of the slow subsystem and hasan expansion that exploits the separation of scales between the slow and fast sector. Then, assum-ing the spectral problem of the function, more precisely its principal part in an expansion w.r.tthe separation of scales, quantising to the Hamiltonian of the system is under sufficient control,the time evolution operator is analysed by means of the ? -product and its expansion (see for ageneral method). This latter analysis corresponds to the perturbative analysis of molecular spectraby means of instantaneous electron configuration for fixed distributions of nuclei in the originalBorn-Oppenheimer setup.For the time-independent Born-Oppenheimer approximation originally formulated in , as opposedto the time-dependent setting, the use of pseudo-differential operators to construct locally isospec-tral effective Hamiltonians has some tradition. Classical works in this respect are . A generalreduction scheme in the time-independent case, which is mathematically similar to space adiabaticperturbation theory, and therefore works for the more general type of models, we have in mind, aswell, is depicted in .Now, let us describe, in a little more detail, in which sense space adiabatic perturbation theoryprovides a suitable framework for the (time-dependent) Born-Oppenheimer approximation in mod-els à la loop quantum gravity, and how quantum field theory on curved spacetimes fits into thisperspective:The Born-Oppenheimer has a long tradition in applications to quantum gravity . But, its usein the context of loop quantum gravity is only quite recent, accompanying the construction of(effective) models. A first attempt to incorporate the Born-Oppenheimer approximation into thecanonical formulation of loop quantum gravity was made in (an application in covariant loop quan-tum gravity/spin foam models can be found in ).It is suggested in , that the connection between loop quantum gravity models with matter con-tent and quantum field theory on curved spacetimes, roughly, arises in the following way: Firstly,a deparametrised Hamiltonian formulation of the considered model in the presence of additional,so-called dust fields , which provide a (physical) system of spacetime coordinates , is chosen andquantised by the methods of loop quantum gravity. The deparametrisation is a generalisation of asimilar procedure used to obtain the Friedmann equations in cosmological models, where the (ap-proximately) homogeneous and isotropic distribution of (super-)galactic structures in the universeserves as a dust field. Secondly, the quantum system is separated into slow and fast subsystemsin accordance with the splitting into a gravitational and matter sector (in general, a mixing ofgravitational and matter degrees of freedom is conceivable). Such a separation is motivated, onthe one hand, by the fact that the natural mass scales, set by the coupling constants κ (Einstein’sconstant) and λ (scalar field), are typically well-separated, which is captured by the (small) di-mensionless parameter ε = κλ , and, on the other hand, by the observation that all experiments ongravity-matter systems, performed so far, are well described by treating the gravitational field asa classical entity (notably, the accordance of (classical) Λ-CDM model with recent observations ofthe cosmic microwave background by PLANCK). In specific models, a further investigation of theseparation of scales is necessary, as it has to be ensured that the physical states subject to analysisrespect this formal argument . Thirdly, the Born-Oppenheimer ansatz is invoked to obtain effec-tive Hamiltonians for the (fast) matter sector, which are parametrised by (classical) configurations(or states) of the slow system. Ideally, these effective Hamiltonians define quantum field theoriesfor given (external) classical gravitational fields, i.e. quantum field theories on curved spacetimes.Finally, information on the spectral problem of the effective Hamiltonians is used as input for thedescription of the total quantum system.As pointed out in , two main obstacles to a successful implementation of the outlined programpresent themselves in the following form:1. Non-commutative fast-slow coupling:
The third step, i.e. applying the Born-Oppenheimer ansatz in the construction of effectiveHamiltonians, requires a peculiar structure of the Hamiltonian of the coupled quantum system:The part of the Hamiltonian modelling the coupling between the slow and fast subsystemsneeds to implemented by a family of mutually commuting, self-adjoint operators w.r.t. theslow variables. This property implies that the coupling-part of the Hamiltonian admits adescription as a fibred operator over some parameter space connected with the slow variables,i.e. the common spectrum of the slow-sector operators mediating the coupling.Due to the structure of the quantum algebra, the holonomy-flux algebra (or its spin-offs),constructed in the quantisation of the gravitational field along the lines of loop quantumgravity, the Hamiltonians of the models, we are interested, do not have this feature: Theoperators representing the (spatial) metric, the flux operators (short: fluxes), generate a non-commutative (sub)algebra, and it is the (spatial) metric that couples to the matter fields ingeneric gravity-matter Hamiltonians.2.
Continuum limit:
Quantisations à la loop quantum gravity of classical field theories including gravity are mod-elled on a projective limit, Γ = lim ←− i ∈ I Γ i , of truncated configuration or phase spaces, Γ i , i ∈ I .Typically, Γ has an interpretation as a distributional completion of the classical smooth con-figuration or phase space, Γ. Thus, in the quantum theory only Γ is naturally accessible, andthe specification of elements of Γ, which contains the classical gravitational field configura-tions or states, has to be achieved via observables admitting a suitable continuum limit.It is, therefore, a minimal requirement that we find a generalisation of the Born-Oppenheimeransatz, which is compatible with the projective limit structure arising in the quantisationprocess.In the present article, we focus on a possible resolution of the first issue on rather general grounds bymeans of space adiabatic perturbation theory, which offers a more flexible framework than the orig-inal Born-Oppenheimer ansatz. The second problem will be (partly) addressed in our second andthird article , where we also establish the mathematical basis necessary to realise the programof space adiabatic perturbation theory in models, which are structurally similar to loop quantumgravity.The remainder of the article is structured as follows:In section II, we recall, in an informal way, the Born-Oppenheimer ansatz (e.g. ), as it usually pre-sented for molecular Hamiltonians (or systems analogous to those), and the derivation of effectiveHamiltonians, which govern the motion of the fast subsystem inside the adiabatically decoupledsubspaces and allow for the derivation of effective equations for the slow variables in semi-classicallimit. Following this, we argue that the treatment of more general quantum systems, which allowfor a splitting into slow and fast degrees of freedom, but do not have the peculiar form commonto molecular Hamiltonians, requires a generalisation of the Born-Oppenheimer ansatz. This gener-alisation manifests itself in extending the (de-)quantisations procedure in terms of orthogonal (inthe generalised sense) pure state families ( fibered or direct integral representations of operators ),which is at the heart of the original Born-Oppenheimer ansatz, to more general deformation (de-)quantisations, e.g. coherent pure state quantisations ( Wick/Anti-Wick or Berezin quantisations )or
Kohn-Nirenberg and
Weyl quantisations .In section III, we formulate the framework of space adiabatic perturbation in rather general terms,rather focusing on structural aspects than on technical details, due to the fact that its originalformulation is given in the context of Weyl quantisation on R d for operators on L ( R d ), whichis a setting to narrow for the applications that we have in mind.In section IV, we apply the general framework of the previous section to a simple, finite-dimensionalmodel of two coupled spin systems introduced by Faure and Zhilinskii in a discussion of topolog-ical aspects of the Born-Oppenheimer approximation. The (de-)quantisation we choose to analysethis model of coupled spin systems is the so-called Stratonovich-Weyl quantisation for the 2-sphere S , which is a direct analogue of the Weyl quantisation on R . The reason, why we dis-cuss this model, is that it constitutes a sort of minimal representative of a quantum system, thatis not amenable to the usual Born-Oppenheimer approximation because of the structure of the su -algebras describing its observables. Furthermore, the non-trivial topology of the manifold S affects the applicability of space adiabatic perturbation theory in an interesting way – a featurethat is expected for loop quantum gravity models, which are based on projective limits of co-tangentbundles T ∗ G of compact Lie groups G , as well. Due to compactness of S , reflecting the finitedimensionality of the model, it is not of vital importance to pay to much attention to the technicaldetails of the (de-)quantisation procedure (all operators are bounded, C ∞ ( S ) = C ∞ b ( S )).Finally, we conclude the article in section V, and comment on the implications of our findings,especially in respect of our companion articles . II. ON THE BORN-OPPENHEIMER ANSATZ
In this section, we discuss aspects of the time-dependent Born-Oppenheimer approximation inthe analysis of coupled quantum systems, H = H s ⊗ H f , (2.1)consisting of two sectors characterised by well-separated interaction/time scales (captured by a“small” parameter ε ), hereafter called the slow sector or slow degrees of freedom , H s , and the fastsector or fast degrees of freedom , H f . We explain on a rather formal level in which sense the conven-tional Born-Oppenheimer approximation fits into the picture of pure state (de-)quantisation w.r.t. to a (generalised) orthogonal family of pure states in H s (or a suitable extension H f ⊂ S f ),which is adapted to the operators of the slow sector that couple non-trivially to the fast sector. Asobserved in , it turns out, that it is of vital importance, that the coupling operators are assumedto be mutually commuting, for this approach to work.Following this, we argue that a treatment of coupled quantum systems, where this restrictive as-sumption is not satisfied (e.g. the Dirac equation with slowly varying external fields , spin-orbitcoupling ), requires another type of (de-)quantisation, presumably not even by pure states. A. The Born-Oppenheimer ansatz
The ansatz of Born and Oppenheimer is usually derived in the context of molecular Hamilto-nians with external magnetic field ,ˆ H mol = 12 m nuc (cid:16) ˆ P + A ( ˆ Q ) (cid:17) + H e ( ˆ Q ; ˆ q, ˆ p ) , (2.2)defined on a dense domain D ( H mol ) ⊂ L ( R d , H f ) ∼ = L ( R d ) ⊗ H f and self-adjoint there. The slownuclei are modelled on { ( Q, P ) , L ( R d ) } and the fast electrons constitute the fibre Hilbert space { ( q, p ) , H f } .For simplicity, we do not include any (possibly internal) degrees of freedom besides position andmomentum of the nuclei and electrons into the discussion.Due to the rather special form of (2.2), i.e. the coupling between slow and fast degrees of freedomhappens solely via the vector of mutually commuting operators ˆ Q , and H e ( ˆ Q ; ˆ q, ˆ p ) is fibered overthe spectrum σ ( Q ) = R d of Q , it is possible to analyse the spectral properties of ˆ H mol by meansof spectral decomposition of the (self-adjoint) electronic Hamiltonians H ( Q ; ˆ q, ˆ p ) , Q ∈ R n for fixedconfigurations of the nuclei. Namely, we introduce the a Q -dependent orthonormal basis of H f (orat least of a subspace of bound states H bf ): H ( Q ; ˆ q, ˆ p ) ψ n,d n ( Q ) = e n ( Q ) ψ n,d n ( Q ) , ψ n,d n ( Q ) ∈ H f , Q ∈ R d , n ∈ N, d n ∈ D N , (2.3)where a discrete, possibly degenerate, fibered spectrum, {{ e n ( Q ) } n ∈ N } Q ∈ R d , without eigenvaluecrossings is assumed to exists for the family { H ( Q ; ˆ q, ˆ p ) } Q ∈ R d . { e n ( Q ) } Q ∈ R d is called the n-thelectronic band . Next, we introduce the (generalised) complete product basis { δ ( d ) Q ⊗ ψ n,d n ( Q ) } Q ∈ R d ⊂ L ( R d , H f ) , (2.4)and project to the component equations of the eigenvalue equation,ˆ H mol Ψ E = E Ψ E , Ψ E ∈ L ( R d , H f ) , (2.5)w.r.t. this basis ( (cid:126) = 1): E Ψ En,d n ( Q ) = E (cid:16) δ ( d ) Q ⊗ ψ n,d n ( Q ) , Ψ E (cid:17) = (cid:16) δ ( d ) Q ⊗ ψ n,d n ( Q ) , ˆ H mol Ψ E (cid:17) (2.6)= 12 m nuc (cid:18) δ ( d ) Q ⊗ ψ n,d n ( Q ) , (cid:16) ˆ P + A ( ˆ Q ) (cid:17) Ψ E (cid:19) + (cid:16) δ ( d ) Q ⊗ ψ n,d n ( Q ) , ˆ H e ( ˆ Q ; ˆ q, ˆ p )Ψ E (cid:17) = X n ,d n − m nuc X n ,d n D n ,d n n,d n · D n ,d n n ,d n + e n ( Q ) δ n,n δ d n ,d n Ψ En ,d n ( Q ) , where we used the resolution of unity L ( R d , H f ) = Z R d dQ X n,d n ( δ ( d ) Q ⊗ ψ n,d n ( Q )) ⊗ (cid:16) δ ( d ) Q ⊗ ψ n,d n ( Q ) , . (cid:17) , (2.7)and defined D n ,d n n,d n := δ n,n δ d n ,d n ( ∇ Q + iA ( Q )) − i A ( Q ) n ,d n n,d n , (2.8) A ( Q ) n ,d n n,d n := i (cid:0) ψ n,d n ( Q ) , ( ∇ Q ψ n ,d n )( Q ) (cid:1) H f . (2.9)The adiabatic Born-Oppenheimer approximation of (2.6), improved by the Berry-Simon connec-tion or Mead potential , consists in ignoring the inter-band terms of the operator D , whichyields (in electronic units m e = 1 , m e m nuc = ε ) E Ψ En,d n ( Q ) = X d n − ε X d n ( D n ) d n d n · ( D n ) d n d n + e n ( Q ) δ d n ,d n ! Ψ En,d n ( Q ) , (2.10)where D n is the diagonal part of D in n-th electronic band. Additionally, a perturbative ex-pansion in the adiabatic parameter ε of the eigenvalues E and eigenvector coefficients Ψ En,d n ( Q ) isperformed .A slightly more educated guess, which is in spirit of Kato’s time-adiabatic theorem, proceeds byprojecting out the intra-band parts of the operator on the right hand side of (2.6). This amountsto using the band projections,ˆΠ n = Z R d dQ Π n ( Q ) δ ( d ) Q ⊗ (cid:16) δ ( d ) Q , . (cid:17) , Π n ( Q ) = X d n ψ n,d n ( Q ) ⊗ ( ψ n,d n ( Q ) , . ) H f , (2.11)which commute with H e ( ˆ Q ; ˆ q, ˆ p ), to define the intra-band effective Hamiltonians:ˆ H effmol ,n = ˆΠ n ˆ H mol ˆΠ n (2.12)= Z R d dQ X d n ,d n (cid:16) δ ( d ) Q ⊗ ψ n,d n (cid:17) (cid:16) ˆΠ n ˆ H mol ˆΠ n (cid:17) d n d n ( Q, − iε ∇ Q ) (cid:16) δ ( d ) Q ⊗ ψ n,d n , . (cid:17) , where (cid:16) ˆΠ n ˆ H mol ˆΠ n (cid:17) d n d n ( Q, − iε ∇ Q ) = (cid:0) − iε ∇ Q + εA ( Q ) − ε A ( Q ) n (cid:1) (2.13)+ ε X n = n (cid:16) A ( Q ) n n · A ( Q ) nn (cid:17) + e n ( Q ) ! d n d n . A ( Q ) n is the quantum-geometric magnetic potential and Φ( Q ) n = P n = n A ( Q ) n n · A ( Q ) nn is thequantum-geometric electric potential. The relevance of the band projections and the effectiveHamiltonians lies within the fact, that the band subspaces are invariant to first order in ε for statesof bounded kinetic energy, i.e. [ ˆ H mol , ˆΠ n ] = O ( ε ) . (2.14)Although, there is a subtlety associated with (2.14), because the relevant time scale for the slowsector is of order ε − ( Duhamel’s formula and gap conditions are important here ).First order time-adiabatic theorems concerning the approximation of the intra-band dynamics bymeans of the first order expansion of (2.12) in ε (or its generalisations to collections of electronicbands) can be found in e.g. . A systematic treatment of higher order corrections requires morerefined techniques to be discussed in section III. Especially, the naive hope that the second orderexpansion of (2.12) in ε is compatible with a second order adiabatic theorem is not justified .Fortunately, these techniques also lift the restriction of the above considerations to couplings ofslow and fast degrees of freedom via mutually commuting operators in the slow sector.Semi-classical approximations to the dynamics of the slow variables can be obtained by ε -dependentpseudo-differential techniques ( Egorov’s theorem ), yielding in zeroth order in ε classical dy-namics governed by a Hamiltonian with potential energy given by the electronic energy associatedwith the band ( Peierls substitution ), i.e. H mol , ( P, Q ) = 12 P + e n ( Q ) . (2.15)In the case of coupled observables on the slow and fast sector, we have to take into account firstoder corrections to (2.15) (matrix-valued in the presence of degeneracies): H mol , (1) ( P, Q ) = 12 P + 2 εP ( A ( Q ) − A ( Q ) n ) + e n ( Q ) . (2.16) B. The coherent-state Born-Oppenheimer ansatz
As pointed out in the previous section, the conventional Born-Oppenheimer approximation ex-ploits to some extent the special structure of (2.2) as being a perturbation, which acts on the slowdegrees of freedom, of an operator, H e ( ˆ Q ; ˆ q, ˆ p ), fibered over the common spectrum of the mutuallycommuting operators ˆ Q i , i = 1 , ..., d, of the slow sector, which provide the coupling to the fastdegrees of freedom.But, for the applications, that we have in mind, it would be advantageous to lift the restriction onthe mutual commutativity of the coupling operators Q i , i = 1 , ..., d . Clearly, if we do not requirethe vector Q to have mutually commuting components, we will have to find a viable substitute forthe (generalised) product basis (2.4). A property that makes (2.4) special, is that it provides adiagonalisation of the slow-fast coupling operator H e ( ˆ Q ; ˆ q, ˆ p ), which supports the idea that elec-tronic configuration ψ n,d n ( Q ) for fixed configuration of nuclei Q ∈ R d can be used to determine theproperties of a molecule.One possibility to generalise this aspect to encompass the non-commutative setting is to move froma configuration space approach to the slow variables to a phase space approach. More precisely,if we treat H e ( . ˆ q, ˆ p ) : R d → L ( H f ) as a function on configuration space R d subjected to the(operator-valued) pure state quantisation , F ( R d , L ( H f )) → L ( L ( R d , H f )) , f ( . ˆ q, ˆ p ) Z R dQ f ( Q, ˆ q, ˆ p ) δ ( d ) Q ⊗ (cid:16) δ ( d ) Q , . (cid:17) , (2.17)w.r.t. the system of (generalised orthonormal) pure states { δ ( d ) Q } Q ∈ R d , we replace the system ofpure states or even the type of quantisation to handle (operator-valued) functions F ( T ∗ R d , L ( H f ))on phase space T ∗ R d ∼ = R d . The reason for this lies within the fact that phase space quantisationtypically allows for a wider range of operators to be covered, as the orthogonality relation at theheart of (2.17) are lifted . Another important point to make in this respect is, that we are notso much interested in a quantisation scheme, but a de-quantisation scheme, i.e. a way to assignfunctions (also called symbols ), f ∈ F ( T ∗ R d , L ( H f )), on the phase space of the slow degrees offreedom with values in operators on the Hilbert space of the fast sector.The goal is then, assuming the spectral problem of the operators { f ( Q, P ; ˆ q, ˆ p ) } ( Q,P ) ∈ R d is un-der sufficient control, to systematically approximate the full spectral problem and the dynamics of f ( ˆ Q, ˆ P ; ˆ q, ˆ p ), in terms of configurations of the fast degrees of freedom for fixed phases or instanta-neous states of the (classical) slow sector: f ( Q, P ; ˆ q, ˆ p ) ψ n,d n ( Q, P ) = e n ( Q, P ) ψ n,d n ( Q, P ) (2.18)for ψ n,d n ( Q, P ) ∈ H f , ( Q, P ) ∈ R d , n ∈ N, d n ∈ D N , in analogy with (2.3).Before, we turn to the discussion of a systematic treatment of the outlined ideas in the framework of(space) adiabatic perturbation theory , we take a look at what will happen to (2.6), if we replace(2.17) by a coherent state quantisation for electronic energies depending on the momenta ofthe nuclei as well, H e ( ˆ Q, ˆ P ; ˆ q, ˆ p ).To this end, we assume that H e ( ˆ Q, ˆ P ; ˆ q, ˆ p ) admits an upper or contravariant symbol , H e,ε ( . ; ˆ q, ˆ p ),w.r.t. the ( ε -dependent) standard coherent states ζ ( ε ) Z ∈ L ( R d ) , Z ∈ C d , for the CCR algebra { ˆ A ε = √ ( ˆ Q + i ˆ P ) , ˆ A ∗ ε = √ ( ˆ Q − i ˆ P ) , [ ˆ A ε , ˆ A ∗ ε ] = ε } associated with the ε -scaled slow sector { ˆ Q ε , ˆ P ε , [ ˆ P , ˆ Q ] = − iε } : H e ( ˆ Q ε , ˆ P ε ; ˆ q, ˆ p ) = H e ( ˆ A ε , ˆ A ∗ ε ; ˆ q, ˆ p ) = Z C d d Z ( επ ) d H e,ε ( Z, Z ; ˆ q, ˆ p ) ζ ( ε ) Z ⊗ (cid:16) ζ ( ε ) Z , . (cid:17) L ( R d ) . (2.19)Next, we assume the existence of a family of bases { ψ ( ε ) n,d n ( Z, Z ) } Z ∈ C d of H f (or a subspace thereof)adapted to the symbol H e,ε ( . ; ˆ q, ˆ p ) in the sense of (2.18), from which construct the (overcomplete)family of product states { ζ ( ε ) Z ⊗ ψ ( ε ) n,d n ( Z, Z ) } Z ∈ C d ⊂ L ( R d , H f ). Stressing the analogy with theprevious subsection further, we introduce the (self-adjoint) complete family of operators:ˆΠ εn = Z C d d Z ( επ ) d Π ( ε ) n ( Z, Z ) ζ ( ε ) Z ⊗ (cid:16) ζ ( ε ) Z , . (cid:17) L ( R d ) , (2.20)Π ( ε ) n ( Z, Z ) = X d n ψ ( ε ) n,d n ( Z, Z ) ⊗ (cid:16) ψ ( ε ) n,d n ( Z, Z ) , . (cid:17) H f , X n ˆΠ n = L ( R , H f ) , which are expected to be almost projections in the O ( ε )-sense, because the coherent state family { ζ ( ε ) Z } Z ∈ C d becomes orthogonal in this limit:( ˆΠ εn ) = ˆΠ εn + O ( ε ) . (2.21)On the same grounds, the electronic energy and the almost projections commute in the O ( ε )-sense,[ H e ( ˆ Q ε , ˆ P ε ; ˆ q, ˆ p ) , ˆΠ εn ] = O ( ε ) , (2.22)and we may hope for (cp. (2.14)): [ ˆ H mol , ˆΠ εn ] = O ( ε ) (2.23)on a suitable domain of bounded energy states.Regarding the computation of effective Hamiltonians for the (almost) subspaces img ˆΠ εn in the senseof (2.12), there is another catch, due to the coherent states not being orthogonal: In the conventionalBorn-Oppenheimer ansatz the expression for the effective Hamiltonian can be obtained from therestriction of (2.6) to the n-th electronic band. This is no longer the case in the coherent stateframework. While the explicit expression for the effective Hamiltonian, ˆ H eff ,ε mol ,n = Π εn ˆ H mol Π εn ,becomes more involved because of the absence of orthogonality relations, the analogue of (2.6) isstill rather simple ( A = 0), E Ψ E, ( ε ) n,d n ( Z ; Z, Z ) (2.24)= X n ,d n n ,d n (cid:18) ε (cid:0) ∂ A (cid:1) n ,d n n,d n − Zδ n,n δ d n ,d n (cid:19) · (cid:18) ε (cid:0) ∂ A (cid:1) n ,d n n ,d n − Zδ n ,n δ d n ,d n (cid:19) Ψ E, ( ε ) n ,d n ( Z ; Z, Z )+ Z C d d Z ( επ ) d X n ,d n e n,ε ( Z , Z ) K ( ε ) ( Z, Z ; ( Z, Z ) , ( Z , Z )) n ,d n n,d n Ψ E, ( ε ) n ,d n ( Z ; Z , Z ) , where we introduced the following objects:Ψ E, ( ε ) n,d n ( Z ; Z, Z ) = (cid:16) ζ ( ε ) Z ⊗ ψ ( ε ) n,d n ( Z, Z ) , Ψ E (cid:17) L ( R d , H f ) , (2.25) (cid:0) ∂ A (cid:1) n ,d n n,d n = δ n,n δ d n ,d n ∇ Z − i A ( Z, Z ) n ,d n n,d n , A ( Z, Z ) n ,d n n,d n = i (cid:16) ψ ( ε ) n,d n ( Z, Z ) , ∇ Z ψ ( ε ) n ,d n ( Z, Z ) (cid:17) H f , K ( ε ) ( Z, Z ; ( Z, Z ) , ( Z , Z )) n ,d n n,d n = ( ζ ( ε ) Z , ζ ( ε ) Z ) L ( R d ) ( ψ ( ε ) n,d n ( Z, Z ) , ψ ( ε ) n ,d n ( Z , Z )) H f . Here, K ( ε ) is a reproducing kernel in the Segal-Bargmann space
S B ε ( C d , H f ) defined by the co-herent states ζ ( ε ) Z , Z ∈ C d :Ψ ( ε ) n,d n ( Z ; Z, Z ) = Z C d d Z ( επ ) d K ( ε ) ( Z, Z ; ( Z, Z ) , ( Z , Z )) n ,d n n,d n X n ,d n Ψ ( ε ) n ,d n ( Z ; Z , Z ) , (2.26)and A is (a part of) a phase space generalisation (in complex ( Z, Z )-coordinates) of the Berry-Simon connection (cp. (2.9)). Since the coherent states ζ ( ε ) Z , Z ∈ C d , are holomorphic up toa factor e − ε Z · Z , (2.24) has to be supplemented by a “flatness constrain”, which characterises0elements of S B ε ( C d , H f ) (apart from L -integrability): X n ,d n ε (cid:16) δ n,n δ d n ,d n ∇ Z − i A ( Z, Z ) n ,d n n,d n (cid:17)| {z } :=( ∂ A ) n ,dn n,dn Ψ E, ( ε ) n ,d n ( Z ; Z, Z ) = − Z Ψ E, ( ε ) n,d n ( Z ; Z, Z ) . (2.27)Related to this aspect of the coherent state ansatz is the observation that the (upper) symbol of acoherent state de-quantized operator is not immediately obtained from the (diagonal) expectationvalues of the operator w.r.t. the coherent states, i.e. the lower symbol . Instead, we find: L εf ( Z, Z ) n,d n n ,d n (2.28)= (cid:16) ζ ( ε ) Z ⊗ ψ ( ε ) n,d n ( Z, Z ) , f ( ˆ A ε , ˆ A ∗ ε ; ˆ q, ˆ p ) ζ ( ε ) Z ⊗ ψ ( ε ) n ,d n ( Z, Z ) (cid:17) L ( R d , H f ) = Z C d d Z ( επ ) d X n ,d n n ,d n (cid:12)(cid:12)(cid:12) ( ζ ( ε ) Z , ζ ( ε ) Z ) L ( R d ) (cid:12)(cid:12)(cid:12) f ( Z, Z ; ˆ q, ˆ p ) n ,d n n ,d n × ( ψ ( ε ) n,d n ( Z, Z ) , ψ ( ε ) n ,d n ( Z , Z )) H f ( ψ ( ε ) n ,d n ( Z , Z ) , ψ ( ε ) n ,d n ( Z, Z )) H f . Therefore, the task of computing the upper symbol of the effective Hamiltonian, as a candidatefor a generator of the semi-classical dynamics of the slow degrees of freedom, is a rather cumber-some task, and might not even be possible. Moreover, while the lower symbol of an operator isunambigously defined by the coherent states (if it exists), and can even be argued to characterisethe operator uniquely for certain kinds of complete families of coherent states (not to be confusedwith completeness in the Hilbert basis sense ), the upper symbol will only be unique, if a suit-able class of symbols is identified. The dichotomy between upper and lower symbol can be furtherexemplified by their duality regarding the trace:tr L ( R d , H f ) (cid:16) f ( ˆ A ε , ˆ A ∗ ε ) ˆ F (cid:17) = Z C d d Z ( επ ) d tr H f (cid:0) f ( Z, Z ) L εF ( Z, Z ) (cid:1) , (2.29)for f ∈ L ( C d , S ( H f )) and ˆ F ∈ B ( L ( R d , H f )) .Let us now turn to the apparent questions and problems the sketched approach raises.Firstly, we may ask, why we should only subject the electronic part of ˆ H mol to the coherent state (de-)quantisation, as this is not forced upon us, in comparison with the conventional Born-Oppenheimeransatz, where the range of the pure state quantisation associated with the mutually commutingcoupling operators Q i , i = 1 , ..., d, is severely restricted by orthogonality relations. Secondly, weshould wonder, how we are supposed to obtain a systematic perturbation theory of the involved ε -dependent objects, as already the leading order approximation of the restricted dynamics (toimg ˆΠ εn ) presumably requires control of the first order ε -expansion of the effective Hamiltonian.Thirdly, we might want to achieve a more symmetric form of (de-)quantisation, especially regarding(2.29).In view of the next subsection and the fact that we should not expect a splitting, as in the exampleat hand, for Hamiltonians describing more general systems, the answer to the first question is thatwe should not do so (a very instructive example is provided by the Dirac equation with slowlyvarying potentials ).Addressing the second problem is slightly more subtle, but a practicable answer, also justified bythe successes of adiabatic perturbation theory, is that we should not only look for a phase space1quantisation, but a deformation quantisation with sufficiently broad range to cover interestingoperators, i.e. we would like to have a de-quantisation of a large enough (depending on the problem)algebra of operators on the total Hilbert space ( L ( R d , H f ) in our example) such that the operatorproduct (in the slow sector) gets mapped to a non-commutative ( ? ε -)product on a suitable class ofphase space functions (with values in operators on H f ). Moreover, to be able to control the errors,arising in the perturbative expansion in ε , a certain notion of continuity of the quantisation shouldbe available. The third point, can dealt with by replacing the coherent state quantisation with theWeyl quantisation, and turns out to be intimately connected with the second point. III. WEYL QUANTISATION AND SPACE ADIABATIC PERTURBATION THEORY
The issues raised at the end of the previous subsection, can be addressed in the setting of spaceadiabatic perturbation theory, which was developed by Panati, Spohn and Teufel in .We recall in this section the main steps and ideas of this program, and what are the necessaryingredients to implement them. The original work of Panati, Spohn and Teufel is phrased in termsof (equivariant) Weyl quantisation, when the phase space of the slow variables can be realised as T ∗ R d ∼ = R d (or a quotient thereof by a lattice Γ ⊂ R d in the equivariant case).To begin with, the quantum dynamical system, ( H , ( ˆ H, D ( ˆ H ))), to be considered, given in terms ofa (self-adjoint) Hamiltonian ˆ H acting on a (dense) domain D ( ˆ H ) in Hilbert space H , should admitthe following general description:(a) There is a splitting of the Hilbert space, H , into slow, H s , and fast, H f , degrees of freedom.The separation the two sectors is controlled by a (small) dimensionless parameter ε .(b) There is a (continuous) deformation quantisation (with deformation parameter ε ), b . ε : S ∞ ( ε ; Γ , B ( H f )) ⊂ C ∞ (Γ , B ( H f )) −→ L ( H ) , (3.1)of the (classical) phase space, Γ, of the slow variables with values in linear operators, L ( H ), on H . Here, S ∞ ( ε ; Γ , B ( H f )) is a class of ( ε -dependent) quantisable functions, the semi-classicalsymbols , on Γ with values in bounded operators, B ( H f ), on H f . The operator product in L ( H ) is reflected in a (continuous) ? ε -product on S ∞ ( ε ; Γ , B ( H f )). Elements of the latteradmit an asymptotic expansion in ε , increasing the regularity (boundedness) of (operator-)contributions with increasing order, and compatible with a similar expansion of ? ε ( Moyalproduct ). Quantisations of O ( ε ∞ )-elements in S ∞ ( ε ; Γ , B ( H f )) are “small” bounded operators( smoothing operators ). Moreover, the quantisation encompasses the Hamiltonian ˆ H , i.e. thereis a semi-classical symbol H ε ∈ S ∞ ( ε ; Γ , B ( H f )) (taking values in self-adjoint operators on H f ) with asymptotic expansion H ε ∼ ∞ X k =0 ε k H k , ∀ k ∈ N : H k ∈ C ∞ (Γ , B ( H f )) , (3.2)such that c H εε = ˆ H .(c) There is a relevant part, σ ∗ ( H ), of the (point-wise) spectrum, σ ( H ) = { σ ( H ( γ )) } γ ∈ Γ , ofthe principal symbol H , that is isolated from the (point-wise) remainder, σ c ∗ ( H ), by a finitegap (global over Γ) .2Let us briefly explain the meaning the three conditions just stated:Clearly, (a) describes the identification of the (two) sectors of the quantum system, which are tobe considered as being separated by different time scales w.r.t. the dynamics ( ε quantifies theseparation).(b) provides a kind of minimal (formal) framework, that is necessary to establish a systematic per-turbation theory that exploits the separation of scales defined by (a) (adiabatic perturbation theoryin orders of ε ). Thus, the existence of an appropriate deformation (de-)quantisation procedure (anda compatible symbolic calculus) to handle operators in the slow sectors is the main technical build-ing block, upon which the whole program of space adiabatic perturbation theory rests.(c) defines the starting point of the perturbation theory in the limit of infinitely separated timesscales ( ε →
0, frozen dynamics of the slow system). That is, the spectral problem of the fastvariables is assumed to be under sufficient control for fixed (classical) states of the slow variables,and is used as input for an analysis of the dynamics of the coupled system (this is analogous toelectronic structure calculations in the conventional Born-Oppenheimer approach). A gap, isolatingan interesting part, σ ∗ ( H ), of the spectrum of H , is typically necessary to control the error inperturbation theory coming from non-adiabatic transitions, as these are dealt with by bounds onthe (local) resolvents of c H ε w.r.t. σ ∗ ( H )).Assuming that the above conditions are satisfied, the program consists roughly of four steps:1. Denoting the (smooth ) spectral projection of H onto the relevant part σ ∗ by π , an almostinvariant projection ˆΠ ε , i.e. [ ˆ H, ˆΠ ε ] = O ( ε ∞ ) , (3.3)is constructed, such that ˆΠ ε = b π εε + O ( ε ∞ ) is close to the quantisation of a (bounded) semi-classical symbol π ε ∈ S ∞ ( ε ; Γ , B ( H f )). The notation O ( ε ∞ ) indicates that the operatornorm of the left hand side should be bounded by any power of ε (uniformly for ε ∈ (0 , ε ] forsome ε > .The semi-classical symbol π ε has an asymptotic expansion with principal symbol π , π ε ∼ ∞ X k =0 ε k π k , (3.4)that qualifies as an invariant projection relative to Moyal product (characterising it uniquely),i.e. π ε ? ε π ε = π ε , π ∗ ε = π ε , [ H ε , π ε ] ? ε = 0 . (3.5)The subspace ˆΠ ε H ⊂ H is called almost invariant subspace , as it remains approximatelyinvariant w.r.t. the dynamics ( Duhamel’s formula ):[ e − i ˆ Hs , ˆΠ ε ] = O ( | s | ε ∞ ) . (3.6)The limit ε → ε H .2. Next, a unitary operator ˆ U ε ∈ B ( H ) is constructed, that identifies the almost invariantsubspace ˆΠ ε H with an ε -independent reference (sub)space ˆΠ r H , which allows for a simple3description. Similar to ˆΠ ε , ˆ U ε is O ( ε ∞ )-close to the quantisation of a semi-classical symbol u ε ∈ S ∞ ( ε ; Γ , B ( H f )). The latter has an asymptotic expansion, u ε ∼ ∞ X k =0 ε k u k , (3.7)with a (smooth) unitary-valued principal symbol u , that defines a reference projection π r ∈ B ( H f ) by u ( γ ) π ( γ ) u ( γ ) ∗ = π r . (3.8) u is called the reference unitary . The quantisation, ˆΠ r = H s ⊗ π r , of the reference projection π r defines the reference space ˆΠ r H . It holds, that:ˆΠ r = ˆ U ε ˆΠ ε ( ˆ U ε ) ∗ . (3.9)The asymptotic expansion of u ε is characterised (although not uniquely) by the followingproperties w.r.t. the Moyal product: u ε ? ε u ∗ ε = 1 = u ∗ ε ? ε u ε , u ε ? ε π ε ? ε u ∗ ε = π r . (3.10)Clearly, u gives a (global) trivialisation of the adiabatic bundle π H → Γ.3. In a third step, the dynamics generated by ˆ H (almost) inside ˆΠ ε H is mapped to the referencespace ˆΠ r H , where it is generated by a (self-adjoint) effective Hamiltonian ˆ h . Due to the factthat ˆΠ ε and ˆ U ε are O ( ε ∞ )-close to quantisations of a Moyal projection π ε and a Moyalunitary u ε , respectively, it is possible to define ˆ h as the quantisation of a self-adjoint semi-classical symbol h ε ∈ S ∞ ( ε ; Γ , B ( H f )), the effective Hamiltonian symbol : h ε ∼ u ε ? ε H ε ? ε u ∗ ε . (3.11)or π r ? ε h ε ? ε π r ∼ π r ? ε u ε ? ε H ε ? ε u ∗ ε ? ε π r (3.12) ∼ u ε ? ε π ε ? ε H ε ? ε π ε ? ε u ∗ ε , which makes a computation of the O ( ε n )-truncations h ε, ( n ) and π r ? ε h ε, ( n ) ? ε π r computa-tionally feasible. The effective Hamiltonian ˆ h satisfies by construction:[ˆ h, ˆΠ r ] = 0 (3.13) e − i ˆ Hs − ( c u εε ) ∗ e − i ˆ hs c u εε = O ( | s | ε ∞ ) ,e − i ˆ Hs − ( ˆ U ε ) ∗ e − i ˆ hs ˆ U ε = O ((1 + | s | ) ε ∞ ) , which entails the space adiabatic theorem with time scale t > e − i ˆ Hs ˆΠ ε − ( (cid:92) u ε, ( n ) ε ) ∗ e − i (cid:92) h ε, ( n + k ) ε s ˆΠ r (cid:92) u ε, ( n ) ε = O ((1 + | t | ) ε ∞ ) , (3.14)4for large enough n, k ∈ N , | s | ≤ ε − k t . Here, (cid:92) u ε, ( n ) ε and (cid:92) h ε, ( n + k ) ε are the quantisations of the O ( ε n +1 )- and O ( ε n + k +1 )-truncation of (3.7) and (3.11), respectively.The meaning of (3.14) is that the error of the adiabatic approximation is controlled, not only,by the order of the expansion of the effective Hamiltonian, but, to the same extent, by theorder of the expansion of the reference projection ˆΠ ε and its associated unitary ˆ U ε . Solelyexpanding the effective Hamiltonian at a given level of the error, improves the time scale onwhich the adiabatic approximation remains valid.The requirement to choose n, k ∈ N large enough in the expansions of c u εε and c h εε is necessary,because the leading orders of c h εε are typically unbounded upon quantisation.4. Finally, if σ ∗ ( H ) = { E ∗ } consists of a single (possibly degenerate) eigenvalue, a semi-classicalapproximation to the dynamics inside the almost invariant subspace can be made. From Heisenberg’s equation on the reference space ˆΠ r H , c O εε ( t ) = e iε ˆ ht c O εε e − iε ˆ ht , (cid:16) ∂ t c O εε (cid:17) ( t ) = iε [ˆ h, c O εε ( t )] , (3.15) Egorov’s hierachy is derived for semi-classical observables O ε ∈ S ∞ ( ε ; Γ , B ( π r H f )) (expansionon the level of symbols): ⇒ ( ∂ t O ) ( t ) = { E ∗ , O ( t ) } + i [ h , O ( t )] (3.16)( ∂ t O ) ( t ) = { E ∗ , O ( t ) } + i [ h , O ( t )] + ( { h , O ( t ) } − { O ( t ) , h } ) (3.17)+ i [ h , O ( t )]...The latter can be solved for O n , n ∈ N iteratively, because the equation at the n -th levelonly depends on O m , m ≤ n . The solution at the lowest order, (3.16), i.e. the time evolutionof the principal symbol O , is determined by the classical flow generated by the Hamiltonianfunction h = E ∗ , and, in the case of a degenerate eigenvalue, the unitary evolution generatedby h transported along the flow of h : O ( γ, t ) = V ( γ, t ) ∗ O (Φ t ( γ )) V ( γ, t ) , O ( γ,
0) = O ( γ ) , γ ∈ Γ , (3.18)where ∂ t Φ t ( γ ) = X E ∗ ( γ ) , ∂ t V ( γ, t ) = − ih (Φ t ( γ )) V ( γ, t ) (3.19)Φ ( γ ) = γ, V ( γ,
0) = π r H f , with X E ∗ denoting the Hamiltonian vector field of E ∗ w.r.t. the symplectic structure on Γ.For scalar principal symbols O = o π r H f (3.18) reduces to o ( γ, t ) = o (Φ t ( γ )) , o ( γ,
0) = o ( γ ) , γ ∈ Γ , (3.20)hence the name semi-classical approximation.Under suitable assumptions (gap conditions) the quantisation of the O ( ε n +1 )-expansion of asemi-classical observable O ε , subject to semi-classical time-evolution, can be related to thequantum dynamics in the reference space. For example, regarding the lowest order semi-5classical flow (3.19), a first order Egorov’s theorem for the principal part O and its quanti-sation is conceivable : ∀ T ∈ R ≥ : ∃ C T > ∀ t ∈ [ − T, T ] : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e iε ˆ ht c O ε e − iε ˆ ht − (cid:92) O ( t ) ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B (ˆΠ r H ) ≤ ε C T . (3.21)In view of the time evolution described by (3.15), it has to be kept in mind, that this equa-tion encodes the O ( ε ∞ )-approximation of the original quantum dynamics generated by ˆ H ,after mapping it to the reference space ˆΠ r H . The upshot of this is, that semi-classical ob-servables inside the reference space, O ε ∈ S ∞ ( ε ; Γ , B ( π r H f )), correspond (up to O ( ε ∞ )) tosemi-classical observables inside the almost invariant subspace ˆΠ ε H . More precisely, semi-classical observables w.r.t. ˆΠ ε H are modelled by operators ˆ O ∈ L ( H ) that are almost diagonalw.r.t. ˆΠ ε : [ ˆ O, ˆΠ ε ] = O ( ε ∞ ) . (3.22)The dynamics of general observables ˆ O ∈ L ( H ) can be considered in the weak sense of re-stricting to expectation values w.r.t. states of the physical system in ˆΠ ε H . This amounts toprojecting ˆ O to the almost invariant subspace:ˆ O | ˆΠ ε H = ˆΠ ε ˆ O ˆΠ ε . (3.23)Remarkably, 3. and 4. show that the adiabatic and semi-classical limit are completely decoupled inspace adiabatic perturbation theory: While the third step fully takes place in the quantum domain,and the ε -quantisation is merely a technical tool to control the perturbation theory (adiabatic limit),the fourth step invokes the in build semi-classical properties of the (de-)quantisation procedure toestablish a connection between the classical and quantum domains (semi-classical limit). IV. A MODEL WITH NON-COMMUTATIVE SLOW VARIABLES: SPIN-ORBIT COUPLING
We apply the general scheme of space adiabatic perturbation theory to a simple finite dimensionalmodel, which describes the interaction of two spin systems ( spin-orbit coupling ), one of which isassumed to model the fast degrees of freedom, while the other represents the slow sector. Thechoice of coupled spin systems is, on the one hand, motivated by the fact that one part of thealgebra of loop quantum gravity takes values in su . On the other hand, the orbital angularmomentum operator constitutes an easily tractable model of a vector of coupling operators, suchthat its components are not mutually commuting. Additionally, the model allows us to study effectsof non-trivial topological structures of the adiabatic line bundles .Concretely, we choose a (slightly adapted) model used by Faure and Zhilinskii to discuss themanifestation of topological indices (e.g. the Chern number) of the adiabatic line bundle, to whichthe Berry-Simon connection is associated, in the spectrum of the coupled system:We consider two spin systems ( su -algebras), { J, [ J i , J j ] = i(cid:15) ijk J k } and { S, [ S i , S j ] = i(cid:15) ijk S k } ,(irreducibly) represented on finite-dimensional Hilbert spaces H j and H s ( d j := 2 j +1 , d s := 2 s +1 ∈ N , dim H j = d j and dim H s = d s , d j > d s ). The Hamiltonian governing the dynamics of the coupled6system H = H j ⊗ H s is: ˆ H λd j = (1 − λ ) H j ⊗ S + λ d j J · S, λ ∈ [0 , . (4.1)Faure and Zhilinskii use the pre-factor j , instead of d j , in front of the coupling term J · S . But,the factor d j turns out to be the expansion parameter of the ? -product to be introduced in thiscontext , and is therefore better suited for our purposes . Anyway, in the (adiabatic) limit j → ∞ , the difference of the two factors becomes negligible.For the purpose of de-quantisation of the slow sector, we employ the formalism of Stratonovich-Weylquantisation , which can be summarised as follows :1. There is a B ( H j )-valued function ∆ j on the Poisson manifold S = { n ∈ R | n = 1 } (twosphere), which can be used to quantise functions f ∈ C ∞ ( S ) via the formula:ˆ A d j f = d j π Z S d n f ( n )∆ j ( n ) ∈ B ( H j ) , (4.2)where d n denotes the 4 π -normalised surface measure on S .2. The de-quantisation A SW d j ∈ C ∞ ( S ) of an operator ˆ A ∈ B ( H j ), called the Stratonovich-Weylsymbol (for short: symbol), is achieved by A SW d j ( n ) = tr H j (cid:16) ∆ j ( n ) ˆ A (cid:17) . (4.3)3. ∆ j has the properties:(a) ∀ n ∈ S : ∆ j ( n ) ∗ = ∆ j ( n ),(b) d j π R S d n ∆ j ( n ) = H j ,(c) ∀ n ∈ S : d j π R S d m tr H j (cid:0) ∆ j ( m )∆ j ( n ) (cid:1) ∆ j ( m ) = ∆ j ( n ),(d) ∀ A, B ∈ B ( H j ) : tr H j (cid:16) ˆ A ˆ B (cid:17) = d j π R S d n A SW d j ( n ) B SW d j ( n ),(e) ∀ g ∈ SU (2) : π j ( g )∆ j ( n ) π j ( g ) ∗ = ∆ j ( Ad g ( n )),where π j : SU (2) → B ( H j ) is the irreducible representation of SU (2) defining H j , and Ad : SU (2) → SO (3) is the adjoint action of SU (2) under the identification su ∼ = ( R , × ).4. The ? -product of two symbols A SW d j and B SW d j is given by: (cid:16) A SW d j ? B SW d j (cid:17) ( n ) = d j π Z S d m d j π Z S d k tr H j (cid:0) ∆ j ( n )∆ j ( m )∆ j ( k ) (cid:1) A SW d j ( m ) B SW d j ( k ) . (4.4)It should be noted that the quantisation, f ˆ A d j f , is not injective, because the range of thede-quantisation, ˆ A A SW d j , is the d j -dimensional subalgebra C ∞ d j ( S ) of C ∞ ( S ) generated byspherical harmonics { Y lm } l ∈ N ,m = − l,..,l ⊂ C ∞ ( S ) with l ≤ j . This observation is in accordance7with the fact that the spherical-harmonic tensor operators { ˆ Y lm } l ∈ N ,m = − l,..,l constitute a basis for B ( H j ) .Nevertheless, the d j -expansion of the ? -product is computed w.r.t. C ∞ ( S ) = S d j ∈ N C ∞ d j ( S ), sincethe quantisation (4.2) projects out contribution from spherical harmonics with l > j anyway. The d j -expansion of the ? -product (4.4) to order O ( d − j ) follows from the techniques in , although thefirst order expansion displayed there is incorrect (see equation (53)): (cid:16) A SW d j ? B SW d j (cid:17) ( n ) (4.5) ∼ (cid:16) A SW d j ? B SW d j (cid:17) ( n ) + d − j (cid:16) A SW d j ? B SW d j (cid:17) ( n ) + d − j (cid:16) A SW d j ? B SW d j (cid:17) ( n ) + O ( d − j )= A SW0 ( n ) B SW0 ( n )+ d − j (cid:0) A SW0 ( n ) B SW1 ( n ) + A SW1 ( n ) B SW0 ( n ) − A SW0 ( n ) B SW0 ( n )+ (cid:0) ( n × ∇ n ) A SW0 (cid:1) ( n ) B SW0 ( n ) + A SW0 ( n ) (cid:0) ( n × ∇ n ) B SW0 (cid:1) ( n )+ in · (cid:0)(cid:0) ∇ n A SW0 (cid:1) × (cid:0) ∇ n B SW0 (cid:1)(cid:1) ( n ) (cid:1) + d − j (cid:0) A SW0 ( n ) B SW2 ( n ) + A SW1 ( n ) B SW1 ( n ) + A SW2 ( n ) B SW0 ( n ) − (cid:0) ( n × ∇ n ) A SW0 (cid:1) ( n ) (cid:0) ( n × ∇ n ) B SW0 (cid:1) ( n )+ ( n × ∇ n ) (cid:0)(cid:0) ∇ n A SW0 (cid:1) · (cid:0) ∇ n B SW0 (cid:1)(cid:1) ( n ) − (cid:0)(cid:0)(cid:0) ∇ n ( n × ∇ n ) A SW0 (cid:1) · (cid:0) ∇ n B SW0 (cid:1)(cid:1) ( n ) + (cid:0)(cid:0) ∇ n A SW0 (cid:1) · (cid:0) ∇ n ( n × ∇ n ) B SW0 (cid:1)(cid:1) ( n ) (cid:1) − (cid:0)(cid:0) ∇ n A SW0 (cid:1) · (cid:0) ∇ n B SW0 (cid:1)(cid:1) ( n )+ (cid:0) ( n × ∇ n ) A SW0 (cid:1) ( n ) B SW1 ( n ) + (cid:0) ( n × ∇ n ) A SW1 (cid:1) ( n ) B SW0 ( n )+ A SW0 ( n ) (cid:0) ( n × ∇ n ) B SW1 (cid:1) ( n ) + A SW1 ( n ) (cid:0) ( n × ∇ n ) B SW0 (cid:1) ( n )+ in · (cid:0)(cid:0)(cid:0) ∇ n A SW0 (cid:1) × (cid:0) ∇ n B SW1 (cid:1)(cid:1) ( n ) + (cid:0)(cid:0) ∇ n A SW1 (cid:1) × (cid:0) ∇ n B SW0 (cid:1)(cid:1) ( n ) − (cid:0)(cid:0) ∇ n A SW0 (cid:1) × (cid:0) ∇ n B SW0 (cid:1)(cid:1) ( n )+ (cid:0)(cid:0) ∇ n ( n × ∇ n ) A SW0 (cid:1) × (cid:0) ∇ n B SW0 (cid:1)(cid:1) ( n ) + (cid:0)(cid:0) ∇ n A SW0 (cid:1) × (cid:0) ∇ n ( n × ∇ n ) B SW0 (cid:1)(cid:1) ( n ) (cid:1)(cid:1) + O ( d − j ) , where A SW d j ∼ P ∞ k =0 d − kj A SW k and B SW d j ∼ P ∞ k =0 d − kj B SW k are semi-classical symbols . Clearly, (4.5)gives the expected behaviour in the leading order of the ? -commutator, i.e. the imaginary unittimes the Poisson bracket on S : h A SW d j , B SW d j i ? ( j cl ) ∼ ij cl · (cid:0)(cid:0) ∇ j cl A SW0 (cid:1) × (cid:0) ∇ j cl B SW0 (cid:1)(cid:1) ( j cl ) + O ( d − j ) , (4.6)where j cl = d j n is the “classical” spin vector. In contrast to the Moyal product for R d , the O ( d − j )-contribution to the ? -commutator does not vanish, which can be traced back to the non-trivial geometry of S .All of the above immediately generalises to the case of B ( H s )-valued symbols, due to finite dimen-sionality. But, we have to be cautious about the ordering of symbols, as they are operator valued,e.g. (4.6) only holds for scalar symbols. But, (4.5) was derived without assuming commutativity ofthe point-wise product of symbols.8 Remark IV.1:
In principle, we can also define a Berezin- ? -product for a spin coherent state quantisation of the scaleof Poisson algebras C ∞ ( S ) = S d j ∈ N C ∞ d j ( S ), because a closed de-quantisation formula, similar to(4.3), exists . A d j -expansion of this ? -product is arrived at via an easy, but extremely tedious,calculation along the lines of . We state only the result up to O ( d − j ), as we will not make furtheruse of it: (cid:16) A SW d j ? B SW d j (cid:17) ( n ) (4.7) ∼ A SW0 ( n ) B SW0 ( n )+ d − j (cid:0) A SW0 ( n ) B SW1 ( n )+ A SW1 ( n ) B SW0 ( n ) − A SW0 ( n ) B SW0 ( n ) − (cid:0)(cid:0) ∇ n A SW0 (cid:1) · (cid:0) ∇ n B SW0 (cid:1)(cid:1) ( n )+ in · (cid:0)(cid:0) ∇ n A SW0 (cid:1) × (cid:0) ∇ n B SW0 (cid:1)(cid:1) ( n ) (cid:1) + d − j (cid:0) A SW0 ( n ) B SW2 ( n ) + A SW1 ( n ) B SW1 ( n ) + A SW2 ( n ) B SW0 ( n ) − (cid:0)(cid:0) ∇ n A SW0 (cid:1) · (cid:0) ∇ n B SW1 (cid:1)(cid:1) ( n ) − (cid:0)(cid:0) ∇ n A SW1 (cid:1) · (cid:0) ∇ n B SW0 (cid:1)(cid:1) ( n ) − (cid:0)(cid:0) ∇ n A SW0 (cid:1) · (cid:0) ∇ n B SW0 (cid:1)(cid:1) ( n )+ (cid:0)(cid:0) ( n × ∇ n ) A SW0 (cid:1) ( n ) B SW0 ( n ) + A SW0 ( n ) (cid:0) ( n × ∇ n ) B SW0 (cid:1) ( n ) (cid:1) − (cid:0) ( n × ∇ n ) A SW0 (cid:1) ( n ) (cid:0) ( n × ∇ n ) B SW0 (cid:1) ( n )+ ( n × ∇ n ) (cid:0)(cid:0) ∇ n A SW0 (cid:1) · (cid:0) ∇ n B SW0 (cid:1)(cid:1) ( n ) − (cid:0)(cid:0)(cid:0) ∇ n ( n × ∇ n ) A SW0 (cid:1) · (cid:0) ∇ n B SW0 (cid:1)(cid:1) ( n )+ (cid:0)(cid:0) ∇ n A SW0 (cid:1) · (cid:0) ∇ n ( n × ∇ n ) B SW0 (cid:1)(cid:1) ( n ) (cid:1) + in · (cid:0)(cid:0)(cid:0) ∇ n A SW0 (cid:1) × (cid:0) ∇ n B SW1 (cid:1)(cid:1) ( n ) + (cid:0)(cid:0) ∇ n A SW1 (cid:1) × (cid:0) ∇ n B SW0 (cid:1)(cid:1) ( n ) − (cid:0)(cid:0) ∇ n A SW0 (cid:1) × (cid:0) ∇ n B SW0 (cid:1)(cid:1) ( n )+ (cid:0)(cid:0) ∇ n ( n × ∇ n ) A SW0 (cid:1) × (cid:0) ∇ n B SW0 (cid:1)(cid:1) ( n )+ (cid:0)(cid:0) ∇ n A SW0 (cid:1) × (cid:0) ∇ n ( n × ∇ n ) B SW0 (cid:1)(cid:1) ( n ) (cid:1) − i ( n × ∇ n ) (cid:0) n · (cid:0)(cid:0) ∇ n A SW0 (cid:1) × (cid:0) ∇ n B SW0 (cid:1)(cid:1) ( n ) (cid:1)(cid:1) + O ( d − j ) . Now, let us elaborate on the model (4.1):The symbol H λd j of ˆ H λd j is : H λd j ( n ) = (1 − λ ) S + λ q − d − j n · S (4.8)= (1 − λ ) S + λn · S | {z } = H λ ( n ) + λ ∞ X k =1 (cid:18) kk (cid:19) (1 − k ) − (4 d j ) − k n · S. Here, the semi-classical expansion is exact. Although, Faure and Zhilinskii use the lower symbol, L d j H λ ( n ) = ( ζ d j n , ˆ H λd j ζ d j n ) H j = (1 − λ ) S + λ (1 − d − j ) n · S, (4.9)w.r.t. a family of spin coherent states { ζ d j n } n ∈ S ⊂ H j , their analysis regards only the principal part H λ , which is the same as ours. Therefore, all their findings apply to our case as well. Differencesarise, when it comes of next-to-leading-order corrections, because the Stratonovich-Weyl symbol9has contributions in all even orders of d − j , while the lower symbol acquires only a first ordercontribution. In view of the previous subsection, this is especially interesting in the context ofdynamics and the time-dependent Born-Oppenheimer approximation, because already the firstorder adiabatic theorem requires us to take the d − j -order into account. A similar observation couldbe made, if we were to use the upper symbol.Denoting the eigenvectors of S by ψ m , m = − s, ..., s, we can state spectral properties of H λ inthe following way: H λ ( n ) ψ m ( n, λ ) = E m ( n, λ ) ψ m ( n λ ) , ψ m ( n λ ) = u λ ( n ) ∗ ψ m , (4.10) E m ( n, λ ) = N ( n, λ ) m, (1 − λ ) e + λn = N ( n, λ ) n λ ,N ( n, λ ) = p λ + (1 − λ ) + 2 λ (1 − λ ) cos( θ ) , cos( θ ) = e · n. Thus, the spectrum of H λ ( n ) is non-degenerate for all n ∈ S , λ ∈ [0 , n = e , λ = , where a collective degeneracy appears, H λ = ( − e ) = 0 . Here, u λ ( n ) is a SU (2)-element corresponding to the rotation of n λ to e via the adjoint action. It can be obtainedexplicitly, e.g. in ZYZ-notation (standard spherical coordinates relative to { e , e , e } ⊂ R ), as: u λ ( n ) = e − iϕ ( n λ ) S e iθ ( n λ ) S e iϕ ( n λ ) S , (4.11)where the rotation angles can be read from n λ to be:cos( θ ( n λ )) = N ( n, λ ) − ((1 − λ ) + λ cos( θ )) , sin( θ ( n λ )) = N ( n, λ ) − sin( θ ) , (4.12) ϕ ( n λ ) = ϕ ( n λ =0 ) = ϕ. At this point, we should mention, that the spectral projections, π λm, ( n ) = ψ m ( n λ ) ⊗ ( ψ m ( n λ ) , . ) H s ,are smooth (in n ) and globally defined for all λ ∈ [0 , ) ∪ ( , u λ ( n ), as can be deduced from the findings of Faure and Zhilinskii. Namely, every spectralprojection π λm, gives rise to a line bundle, π λm, H s −→ S , (4.13)called the adiabatic line bundle of spectral index m . The natural connection in each of the linebundles is given by the Berry-Simon connection, A λm ( n ) = i ( ψ m ( n λ ) , dψ m ( n λ )) H s , (4.14)and its curvature, F λm ( n ) = dA λm ( n ) , (4.15)gives the (first) Chern number of the line bundle, C λm = 12 π Z S F λm ∈ Z . (4.16)0Its value was found by Faure and Zhilinskii to be: C λm = (cid:26) λ ∈ [0 , ) − m λ ∈ ( , . (4.17)In this sense, the degeneracy at λ = is said to have a topological charge . Since we are dealingwith line bundles, the Chern number is a complete (topological) invariant , i.e. only for λ < canthe line bundles (4.13) be (smoothly) trivial for all m = − s, ..., s .Thus, only for λ < can the unitary map u λ : S → U ( H s ), be smooth and globally defined,and the program of space adiabatic perturbation theory can be made sense of ( u λ is the naturalcandidate for the reference unitary ). For λ > , the program, presumably, has to be modified,e.g. by adapting the (de-)quantisation to the non-trivial Berry-Simon connection A λm , as was donein for the case of magnetic, periodic Schrödinger operators with non-trivial Bloch bands, i.e. non-trivial line bundles over the toric component in the Bloch-Floquet splitting of R d .That the non-triviality of the line bundles (4.13) for λ > , and the entailed non-existence of aglobally smooth reference unitary u λ , is not just a minor technical drawback, can be understoodfrom the results of Faure and Zhilinskii, as well:They argue that the Chern number manifests itself in the exact spectrum of the Hamiltonian ˆ H λd j of the coupled system in the sense of a topological quantum number, which measures the dimensionof the range of the projection, ˆΠ λ,d j m , onto the almost invariant subspace constructed from π λm, (which still exists) in the limit d j → ∞ :dim img ˆΠ λ,d j m ∼ d j →∞ d j − C λm = (2 j + 1) + 2 m. (4.18)For λ = 1, this formula gives the exact degeneracy of spectrum for (pure) spin-orbit coupling, i.e.the dimension of the dynamically stable subspaces.But, from the perspective of space adiabatic perturbation theory, the dimension of img ˆΠ λ,d j m wouldbe forced to be: dim img ˆΠ λ,d j m = d j = 2 j + 1 , (4.19)because of unitarily equivalence to the reference projection ˆΠ r = ˆ U λ,d j m ˆΠ λ,d j m (cid:16) ˆ U λ,d j m (cid:17) ∗ , which, byconstruction, satisfies: dim img ˆΠ r = d j . (4.20)Therefore, if we were to apply space adiabatic perturbation theory to ˆ H λd j with λ > , we wouldnecessarily fail to predict the correct almost invariant subspaces for the dynamics. This is mostprominently visible for (pure) spin-orbit coupling ( λ = 1).We conclude the discussion of the model by providing the first order expansion of the effectiveHamiltonian symbol h λm, (1) restricted to the reference spaces in the case of λ < and s = (thefundamental representation of SU (2):1The Hamiltonian symbol is given in terms of the Pauli matrices: H λd j ( n ) = 12 (cid:18) (1 − λ ) σ + λ q − d − j n · σ (cid:19) (4.21)= 12 (1 − λ ) + λn q − d − j λ ( n − in ) q − d − j λ ( n + in ) q − d − j − (1 − λ ) − λn q − d − j = 12 (cid:18) (1 − λ ) + λn λ ( n − in ) λ ( n + in ) − (1 − λ ) − λn (cid:19)| {z } = H λ ( n ) + O ( d − j ) , and its eigenvalues are E λ ± ( n ) = ± N ( n, λ ) , (4.22)which are globally separated by a gap | E λ + ( n ) − E λ − ( n ) | ≥ N ( n, λ ) ≥ min θ ∈ [0 ,π ) N ( n, λ ) =: g λ > λ = (fig. 1). Figure 1. Plot showing the behaviour of the spectral distance N ( n, λ ) as a function of θ ∈ [0 , π ) for λ = (1 ± ) , (1 ± − ) , (1 ± − ) and (“=” (1 ± −∞ )) (top to bottom). The eigenvectors, corresponding to m = ± (abbreviated: ± ), are (in standard spherical coordi-2nates): ψ + ( n λ ) = cos( θ ( n λ )2 ) e iϕ sin( θ ( n λ )2 ) ! , ψ − ( n λ ) = − e − iϕ sin( θ ( n λ )2 )cos( θ ( n λ )2 ) ! , (4.23)which are well-defined for λ < .For λ > , these expression remain valid away from θ = π ( n = − e ), where ψ ± are not uniquelydefined. The projections, π λ + , ( n ) = cos ( θ ( n λ )2 ) e − iϕ sin( θ ( n λ )2 ) cos( θ ( n λ )2 ) e iϕ sin( θ ( n λ )2 ) cos( θ ( n λ )2 ) sin ( θ ( n λ )2 ) ! , (4.24) π λ − , ( n ) = sin ( θ ( n λ )2 ) − e − iϕ sin( θ ( n λ )2 ) cos( θ ( n λ )2 ) − e iϕ sin( θ ( n λ )2 ) cos( θ ( n λ )2 ) cos ( θ ( n λ )2 ) ! , are well-defined for all λ ∈ [0 ,
1] except λ = , as explained above. The reference projections areprovided by π + ,r = π λ =0+ , = (cid:18) (cid:19) , π − ,r = π λ =0 − , = (cid:18) (cid:19) , (4.25)and we choose u λ ( n ) = cos( θ ( n λ )2 ) e − iϕ sin( θ ( n λ )2 ) − e iϕ sin( θ ( n λ )2 ) cos( θ ( n λ )2 ) ! (4.26)as reference unitary. The Berry-Simon connection and curvature are: A λ ± ( n ) = ∓
12 (1 − cos( θ ( n λ ))) dϕ, F λ ± ( n ) = ∓
12 sin( θ ( n λ )) dθ ( n λ ) ∧ dϕ. (4.27)In accordance with the general statements above, we find that C λ< ± = 0 and C λ> ± = ∓
1. Becausethe projections (4.24) correspond to single, non-degenerate eigenvalues, E λ ± ( n ) = ± N ( n, λ ), of H λ , the effective Hamiltonian symbols are scalar, when restricted to the reference space, and thefirst order contributions, h λ ± , , can be found from the formula : h λ ± , (1) ( n ) ∼ h λ ± , ( n ) + d − j h λ ± , ( n ) + O ( d − j ) , (4.28) h λ ± , ( n ) = E λ ± ( n ) π ± ,r , (4.29) h λ ± , ( n ) = π ± ,r (( u λ ? H λ ) ( n ) − ( E λ ± ? u λ ) ( n )) u λ ( n ) ∗ π ± ,r (4.30)= E λ ± ( n ) π ± ,r (cid:0)(cid:0) ( n × ∇ n ) u λ (cid:1) ( n ) u λ ( n ) ∗ + u λ ( n ) (cid:0) ( n × ∇ n ) ( u λ ) ∗ (cid:1) ( n ) (cid:1) π ± ,r + iπ ± ,r (cid:0) n · (cid:0) ( ∇ n u λ ) × ( ∇ n H λ ) (cid:1) ( n ) − n · (cid:0) ( ∇ n E λ ± ) × ( ∇ n u λ ) (cid:1) ( n ) (cid:1) u λ ( n ) ∗ π ± ,r = E λ ± ( n ) π ± ,r (cid:0) ( ∇ n u λ )( n ) · ( ∇ n ( u λ ) ∗ )( n ) (cid:1) π ± ,r π ± ,r (2 in ) · (cid:0) ( ∇ n E λ ± ) × ( u λ ∇ n ( u λ ) ∗ ) (cid:1) ( n ) π ± ,r − π ± ,r ( in ) · (cid:0) ( ∇ n u λ )( n ) × (( H λ − E λ ± ) ∇ n ( u λ ) ∗ )( n ) (cid:1) π ± ,r = − E λ ± ( n ) π ± ,r (cid:0) ( u λ ∇ n ( u λ ) ∗ )( n ) · ( u λ ∇ n ( u λ ) ∗ )( n ) (cid:1) π ± ,r + π ± ,r (2 in ) · (cid:0) ( ∇ n E λ ± ) × ( u λ ∇ n ( u λ ) ∗ ) (cid:1) ( n ) π ± ,r + ( E λ ∓ ( n ) − E λ ± ( n )) π ± ,r ( in ) · (cid:0) ( u λ ∇ n ( u λ ) ∗ )( n ) × ((1 − π ± ,r ) u λ ∇ n ( u λ ) ∗ )( n ) (cid:1) π ± ,r = (cid:18) θ ) (cid:0) ∂ θ E λ ± )( n ) A λ ± ( n ) ϕ − ( E λ ∓ ( n ) − E λ ± ( n )) F λ ± ( n ) θϕ (cid:1) + ( θ ) E λ ± ( n ) (cid:18) (cid:16) E λ ± ( n ) λ F λ ± ( n ) θϕ (cid:17) ∓ A λ ± ( n ) ϕ (cid:19)(cid:19) π ± ,r = (cid:18) θ ) (cid:0) ∂ θ E λ ± )( n ) A λ ± ( n ) ϕ + 2 E λ ± ( n ) F λ ± ( n ) θϕ (cid:1) + ( θ ) E λ ± ( n ) (cid:18) (cid:16) E λ ± ( n ) λ F λ ± ( n ) θϕ (cid:17) ∓ A λ ± ( n ) ϕ (cid:19)(cid:19) π ± ,r . Here, we used in the first line of (4.30) that H λ = 0. Apart from this, the third line of (4.30) isstill valid in general, and we observe that in addition to a familiar term containing the Berry-Simonconnection (second term, cp. (2.16)) two further terms appear, which can be attributed to thenon-trivial geometry of S . The fourth to the sixth line are special to the model at hand, but wesee that the Berry-Simon curvature already affects the first order contribution. V. CONCLUSIONS & PERSPECTIVES
In sections II & III, we have seen how the original Born-Oppenheimer ansatz, and its restrictedapplicability to slow-fast couplings via orthogonal pure state quantisations (fibred operators) in theanalysis of multi-scale quantum systems, can be superseded by the more flexible space adiabaticperturbation theory, which is formulated by means of a suitable deformation quantisation, e.g.Weyl quantisation (for slow variables with a phase space isomorphic to R d ) or Stratonovich-Weylquantisation (for spin systems, see section IV). Thus, the non-commutativity obstacle raised in (see section I) is lifted in way structurally enriching and conceptually refining the perturbativetreatment of scale-separated, coupled quantum systems. It is worth to point out, that in space adi-abatic perturbation theory the (classical) parameter space of the slow variables has the structure ofa phase space, in contrast to the slow subsystem’s configuration space appearing in the conventionalBorn-Oppenheimer approach to molecular quantum systems. In view of the possible extraction ofquantum field theory on curved spacetimes from loop quantum gravity, the appearance of a phasespace structure in the treatment of the slow/gravitational subsystem is advantageous, because apoint in phase space corresponds to a spacetime metric via the effective classical time evolutionarising in the semi-classical approximation of the coupled system. But, it is precisely a spacetimemetric, which is necessary for the construction of a quantum field theory on a curved spacetime.Furthermore, this indicates that quantum field theory on curved spacetimes is expected to be ofrelevance to the fourth step, i.e. the semi-classical limit, of space adiabatic perturbation theory,when the latter is applied to loop quantum gravity, and not so much to the preceding three steps,which are dominated by kinematical considerations regarding the slow subsystem’s phase space.To elaborate on the last statement, we notice that without invoking dynamics the correspondence4between phase space point (initial data: spatial metric and extrinsic curvature) and time evolu-tion trajectories (spacetime metric) is lost. Another interesting aspect of the phase space pictureturning up in the semi-classical approximation, is, that the effective evolution equations are tiedto a certain adiabatically decoupled subspace, which is constructed from a spectral subspace (inthe fast subsystem) of the (partially) dequantised Hamiltonian. The upshot of it being, that, inapplications to quantised gravity-matter systems, the above mentioned emergent spacetime metricdepends on the choice of spectral subspace in fast sector. Such dependence of the spacetime metricon the spectral properties of the matter field is commonly referred to as a rainbow metric , andarises naturally in the context of space adiabatic perturbation theory.The spin-orbit model discussed in section IV provides an idealised, though explicitly realised,testing ground for the solution of the non-commutativity problem, while simultaneously showingthe interplay between non-trivial topological properties of the slow sector’s phase space and thestructure of the total Hamiltonian.Establishing parts of the main mathematical toolbox necessary to implement a deformation(de-)quantisation and an associated symbolic calculus for loop quantum gravity and other mod-els, that are based on projective limit phase spaces Γ = lim ←− i ∈ I Γ i built from co-tangent bundles,Γ i = T ∗ G i , i ∈ I , of compact Lie groups, G i , i ∈ I , is the main topic of our companion articles .In subsection II B, we discussed the possibility to employ a coherent state quantisation for the slowvariables to generalise the original Born-Oppenheimer ansatz. Noteworthy, the use of lower symbols(partial traces w.r.t. coherent states projections) to obtain effective Hamiltonians in loop quantumgravity models was already put forward in , although a systematic way to connect informationon the spectral problem of the effective Hamiltonians with the spectral analysis of the total Hamil-tonian was not explored therein. As we have seen in section III, the existence of a ? -product is ofvital importance to establish such a link, i.e. the (de-)quantisation of the slow sector has to be a(strict) deformation quantisation. Regarding the latter, we argue in our companion articles ,that coherent state quantisations are generically, i.e. in the case of a non-compact phase space forthe slow variables, to singular to serve as a basis for deformation (de-)quantisation . This explainswhy we focus on the development of a less singular Weyl quantisation as pointed out above.Nevertheless, we devote a section of our second article to the discussion of a new form of theSegal-Bargmann-Hall transform , because this unitary map, represented as a resolution of unity,is at the heart of a coherent state quantisation of the co-tangent bundle, T ∗ G , of a compact Liegroup G , and thus fits into the general discussion of phase space quantisations and their relevanceto generalised Born-Oppenheimer schemes (see subsection II B). 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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