Geometry of the high energy limit of differential operators on vector bundles
aa r X i v : . [ m a t h . SP ] S e p Geometry of the high energy limit of differentialoperators on vector bundles
Alexander StrohmaierNovember 21, 2018
Abstract
At high energies relativistic quantum systems describing scalar particles behave clas-sically. This observation plays an important role in the investigation of eigenfunctionsof the Laplace operator on manifolds for large energies and allows to establish relationsto the dynamics of the corresponding classical system. Relativistic quantum systemsdescribing particles with spin such as the Dirac equation do not behave classically athigh energies. Nonetheless, the dynamical properties of the classical frame flow deter-mine the behavior of eigensections of the corresponding operator for large energies. Wereview what a high energy limit is and how it can be described for geometric operators.
A quantum physical system is usually described by an algebra A of operators on a Hilbertspace H and the time evolution, which is a one-parameter group U ( t ) of unitary operators on H . An important example of such a system is the one describing the motion of a quantumparticle on a compact Riemannian manifold M . In this case the Hilbert space is L ( M, µ g ),where µ g is the Riemannian measure. The time evolution is described by the Schr¨odingerequation, which means that the unitary one-parameter group U ( t ) is given by U ( t ) = e − i tH , where H is the Schr¨odinger operator. For non-relativistic quantum systems H would typicallybe ∆ + V , where ∆ is the metric Laplace operator on M and V ∈ C ∞ ( M, R ) is a potential.1s we are interested in the high energy limit we will consider the Klein-Gordon time evolutionwhich describes relativistic particles. So we take H = (∆ + m + V ) / where m is a positive real number, the mass of the particle. We will assume here that m + V ( x ) ≥ H is a positive first order pseudodifferential operator with principalsymbol σ H ( ξ ) = k ξ k . The above group is defined by spectral calculus as∆ + m + V : H ( M ) → L ( M )is a self-adjoint operator with the Sobolev space H ( M ) as its domain. The algebra ofobservables would be a unital ∗ -subalgebra of B ( H ), the algebra of all bounded operators on H . A state on A is a linear functional ω : A → C such that(i) ω is complex linear,(ii) ω is positive, i.e. ω ( A ∗ A ) ≥ A ∈ A ,(iii) ω (1) = 1.The physical interpretation is that a state is a state of the system and assign to each observableits expectation value. An example is ω ( A ) = h ψ, Aψ i , where ψ is a vector of unit length in H . More generally, any trace class operator ρ with ρ ≥ ρ ) = 1 defines a state by ω ρ ( A ) = Tr( Aρ ) . These states are often referred to as normal. If A is a self-adjoint element in A then A has aspectral decomposition A = Z R λdE λ . If the system is in the normal state ω ρ then the probability that a measurement of theobservable A yields a value in the Borel measurable set O ⊂ R is given by ω ρ ( E O ) = tr( ρE O ) , where E O is the spectral projection onto O E O = Z O dE λ . U is a subset of M then the characteristic function χ U corresponds to the binary experiment that measureswhether the particle is in the region U . The probability of finding the particle in U is thereforegiven by ω ρ ( χ U ) or if the state is of the form ω ψ with ψ ∈ L ( M ) then of course we get forthe probability of finding the particle in U Z U | ψ ( x ) | dx. Whereas in the literature B ( H ) itself is often chosen as the algebra of observables, thischoice is sometimes not very convenient for practical purposes. On the physical side it isimpossible to build a detector that measures χ O . The reason is that such a measurementwould involve a detector that near the boundary of O had an arbitrary high resolution. If wewanted to be more realistic we would restrict ourself to algebras that contain functions thatare only smooth or continuous. On the mathematical side it is much easier to specify a stateon a smaller algebra rather than on the full algebra of bounded operators. Knowledge of thestate on the smaller algebra is often sufficient to extend it uniquely to a larger subalgebra.So which algebra to choose for the particle on the manifold?The state of a classical particle is completely determined by its momentum and its position.We expect the same to be true for quantum particles. In order to measure the position wetake the algebra C ∞ ( M ). Measurement of the momentum involves unbounded operatorsof the form i X , where X is a vector field. In other words we would need an algebra ofoperators that contains enough bounded functions of X so that we can approximate thespectral projections of i X by elements in our algebra. If we choose a classical symbol p oforder 0 on R then p ( X ) is a classical pseudodifferential operator of order 0. So if we choosethe algebra of pseudodifferential operators ΨDO cl ( M ) on M this algebra contains enoughobservables to measure the location and the momentum of our particle up to some arbitrarysmall error. And, indeed, the restriction of a normal state ω ρ to the algebra A determinesthis state completely. To see this note that normal states are continuous in the weak- ∗ -topology on B ( H ) and ΨDO cl ( M ) is weak- ∗ -dense in B ( H ). Since states are automaticallynorm continuous any state can by continuity be uniquely extended to a state on the normclosure it is reasonable to use as the algebra of observables the norm closure of the algebra ofpseudodifferential operators, that is A = ΨDO cl ( M ) . V which generates the unitary one-parameter group U ( t ). So we would think of a normal state ω ρ as a state with energy larger than λ if ω ρ ( E [ −∞ ,λ ) ) = 0where E is the spectral projection of ∆ + V .Note that the set of states is weak- ∗ -compact and the algebra A is separable. Therefore,the set of states is sequentially compact in the weak- ∗ -topology. That is every sequenceof normal states has weak- ∗ limit points. Suppose ω ρ n is a sequence of normal states thatconverges to a not necessarily normal state ω ∞ in the weak- ∗ -topology. Then we think of ω ∞ as a high energy limit if ω ρ n ( E [ −∞ ,λ ) ) → λ >
0. Using spectral calculus one finds that this is equivalent to ω ∞ (( | ∆ + V | + 1) − ) = 0 . Now suppose that K is a pseudodifferential operator of order −
1. Then, K p | ∆ + V | + 1 isa pseudodifferential operator of order 0 and is therefore bounded. By the Cauchy-Schwarzinequality | ω ∞ ( K ) | ≤ ω ∞ (cid:18)(cid:12)(cid:12)(cid:12) K p | ∆ + V | + 1 (cid:12)(cid:12)(cid:12) (cid:19) · ω ∞ (cid:16) ( p | ∆ + V | + 1) − (cid:17) and therefore ω ∞ ( K ) = 0for any high energy limit ω ∞ . Therefore, ω ∞ vanishes on the algebra of pseudodifferentialoperators of order − K . Consequently, at high energies the states become states on the quotient algebra A / K . Now it is well known (e.g. [Se65], Th. 11.1) that the principal symbol map σ : ΨDO cl ( M ) → C ∞ ( S ∗ M )4xtends continuously to a map ˆ σ : A → C ( S ∗ M )and ker ˆ σ = K . This means that A / K is naturally isomorphic to the commutative algebra C ( S ∗ M ). States on C ( S ∗ M ) are bythe Riesz representation theorem in one to one correspondence to regular Borel probabilitymeasures on S ∗ M . So for every high energy limit ω ∞ there exists a unique probability measure µ on S ∗ M such that ω ∞ ( A ) = Z S ∗ M ˆ σ A ( ξ ) dµ ( ξ ) . The fact that the high energy limit states are actually states on an abelian algebra canbe interpreted as the passage from quantum to classical mechanics. The system behavesclassically for very large energies. That the quantum mechanical time evolution becomes theclassical motion along geodesics is now a consequence of Egorov’s theorem. Namely, if A ∈ A then also A ( t ) = U ( − t ) AU ( t ) ∈ A and ˆ σ A ( t ) = G ∗ t (ˆ σ A ) , where G t is the geodesic flow on S ∗ M and G ∗ t is its pull-back acting on functions. This meansthat the group of ∗ -automorphisms α t ( A ) = U ( − t ) AU ( t )which describes the quantum mechanical time evolution on the level of observables (the so-called Heisenberg picture) factors to A / K ∼ = C ( S ∗ M ) and becomes there the geodesic flow.This is a very concise way of saying that in the limit of high energy the quantum systembecomes classical and the time-evolution becomes the motion along geodesics with constantspeed.Note that in the high energy limit the potential does not play a role any more. From thephysical point of view this is expected as particles with high energy do not ”feel” a potentialand move at the speed of light along lightlike geodesics.Interestingly some high energy limit can be computed explicitly. Let φ j be a completeorthonormal sequence of eigenfunctions of ∆ such that∆ φ j = λ j φ j , λ ≤ λ ≤ λ ≤ . . . . ω N defined by ω N ( A ) = 1 N N X j =1 h φ j , Aφ j i has a high energy limit. It follows from the classical Tauberian theorem of Karamata thatthe limit of this sequence is given by ω ∞ ( A ) = lim t → + Tr( Ae − t (∆+ V ) )Tr( e − t (∆+ V ) ) . The (microlocal) heat kernel expansion then shows thatlim t → + Tr( Ae − t (∆+ V ) )Tr( e − t (∆+ V ) ) = Z S ∗ ˆ σ A ( ξ ) dµ L ( ξ ) , where µ L is the normalized Liouville measure on S ∗ M .The state ω t defined by ω t ( A ) = Tr( Ae − t (∆+ V ) )Tr( e − t (∆+ V ) )is the KMS-state with temperature t − describing a quantum system at temperature t − inthermal equilibrium. In the limit as the temperature goes to infinity the state converges tothe Liouville measure on the unit-cotangent bundle.The sequence of eigenstates h φ j , · φ j i is a sequence of invariant states and any weak- ∗ -limitpoint is a therefore an invariant high energy limit. The above says that on average thesestates converge to the Liouville measure. If the Liouville measure is ergodic with respect tothe geodesic flow this means that the tracial state ω ∞ defined above is ergodic. This meansthere is no non-trivial decomposition of ω ∞ into a convex combination of invariant states.From this one can conclude that any subsequence of h φ j , · φ j i that does not have the state ω ∞ as a weak- ∗ -limit has to have counting density zero, as otherwise it would give rise to adecomposition of ω ∞ into invariant states. A more careful argument along these lines (see[Shn74, Shn93, CV85, Zel87]) shows that in fact there is a subsequence of counting densityone of eigenfunctions φ j ( k ) such thatlim k →∞ h φ j ( k ) , Aφ j ( k ) i = ω ∞ ( A ) , for all A ∈ A . This is usually referred to as Quantum ergodicity.6 The Dirac equation and Laplace type operators
Relativistic quantum systems that describe particles with spin like electrons, neutrinos are notdescribed by the Klein-Gordon equation. As the particles have an internal degree of freedom,the spin, the Hilbert space will consist of vector valued functions and the observable algebraneeds to include operators that detect these internal degrees of freedom. This is appropriatelydescribed by the following construction. Suppose that E → M is a complex hermitian vectorbundle over M we take as the Hilbert space the space of square integrable sections of E H = L ( M ; E )and as an algebra of observables we take the norm closure of the space of zero order classicalpseudodifferential operators acting on sections of this vector bundle A = ΨDO cl ( M ; E ) . Now a second order differential operator P is said to be of Laplace-type if in local coordinatesit has the form P = − X ij g ij ∂ ∂x i ∂x j + B, where B has order 1, or in other words if and only if σ P ( ξ ) = g ( ξ, ξ ) . Similarly, a first order differential operator D is said to be of Dirac type if and only if σ D ( ξ ) = g ( ξ, ξ ) . Of course a first order operator is of Dirac type if and only if its square is a Laplace typeoperator.As we saw in the previous section the Klein-Gordon operator ∆ + V + m is a Laplace typeoperator acting on the trivial vector bundle. So the time evolution in this case is describedby the square root of a Laplace type operator. To describe electrons one typically chooses aspin structure on M and then the complex vector bundle is the associated spinor bundle S .The algebra of observables in this case is the algebra of zero order classical pseudodifferentialoperators acting on sections of the spinor bundle. The time-evolution is described by theDirac operator D acting on sections of the spinor bundle. This operator will however notbe positive any more, which is a typical feature of relativistic quantum theory. The system7escribes electrons and positrons at the same time. Pure electron states are states that aresupported in the positive spectral subspace of D . Since on this subspace the operators D and | D | coincide the time evolution of such states may as well be described by the operator H = | D | . Whereas from the viewpoint of a one-particle theory it might seem strange to takethe operator | D | as the generator of the time evolution this is perfectly justified in a fullyquantized theory. The generator of the time-evolution of the fully quantized free electron-positron field restricted to the one-particle subspace is given by | D | rather than D . Theapparent violation of Einstein causality by the infinite propagation speed of the operatorexp( − i t | D | ) is resolved in the fully quantized theory and is not causing a problem there (seee.g. [Th92] for details).Note that if one chooses the group generated by D instead of | D | the time evolutiondoes not leave the space of operators invariant. Instead of passing to the | D | one can alsorestrict the algebra of observables. This approach in favored in [Co83] and also used in[BoK98, BoK99, Bol01, BoG04, BoG04.2] in order to investigate the semi-classical limit ofthe Dirac operator.Spin 1 particles like photons and mesons are described by Maxwell’s equation or the Procaequation. For example the quantum system describing photons is given as follows. The vectorbundle is the complexified co-tangent bundle Λ M = T ∗ M . The Hilbert space is the closureof the space of co-closed 1-forms in L ( M ; E ) and the space of observables is the algebra A = P ΨDO cl ( M ; Λ M ) P , where ΨDO cl ( M ; Λ M ) is the algebra of pseudodifferential operators and P is the orthogonalprojection onto the space of co-closed 1-forms. The relativistic time evolution is given by theone parameter group generated by the Laplace-Beltrami operator ∆ acting on one-forms.As we can see the time evolution is in these examples given by a Laplace-type operatoracting on the sections of a vector bundle. The algebra of observables is either the full algebraof pseudodifferential operators or an appropriate subalgebra that is invariant under the time-evolution. The symbol map σ is now a mapΨDO cl ( M ; E ) → C ∞ ( S ∗ M ; π ∗ End( E ))from the pseudodifferential operators of order 0 to the smooth functions on S ∗ M with valuesin the bundle End( E ). Here π ∗ E denotes the pull back of the bundle E on M under theprojection π : S ∗ M → M . As in the scalar case the symbol map has a continuous extensionˆ σ to a map A → C ( S ∗ M ; π ∗ End( E ))8here A = ΨDO cl ( M ; E ) is the norm closure of the space of pseudodifferential operatorsacting on L ( M ; E ).In contrast to the Egorov theorem for scalar pseudodifferential operators the Egorov the-orem for matrix valued pseudodifferential operators involves terms of the time-evolution thatare of lower order ([D82, EW96]). Let us give some invariant meaning to this. If ∆ E is aself-adjoint Laplace type operator acting on the sections of some hermitian vector bundle E ,then there exists a unique connection ∇ E on E and a unique potential V ∈ C ∞ ( M, End( E ))such that ∆ E = ∇ ∗ E ∇ E + V. The locally defined connection-1-form can be interpreted as the sub-principal symbol of ∆ E (see [JS06]). The connection ∇ E of course defines a connection ∇ End( E ) on End( E ). Thisconnection can be used to extend the geodesic flow on S ∗ M to a flow on π ∗ (End( E )) byparallel translation. We will denote the induced action on the sections of π ∗ (End( E )) by β t .It is easy to check that β t is a one-parameter group of ∗ -automorphisms on C ( S ∗ M ; π ∗ (End( E ))) . The analog of Egorov’s theorem is now as follows. Let A be the algebra ΨDO cl ( M ; E ). Then,if U ( t ) = e − i t √ ∆ E and A ∈ A we have A t = U ( − t ) AU ( t ) ∈ A , ˆ σ A t = β t ( σ A ) . In other words β t is the high energy limit of the quantum time evolution. A proof can befound in [BuO06] and in [JS06]. Most geometric operators like the Dirac operator and the Laplace-Beltrami operator are actingon sections of vector bundles that are constructed in a geometric way from the manifold. Weassume here that M is oriented and that F M is the bundle of oriented orthonormal frames.We will show that in many cases the bundle End( E ) as a hermitian vector bundle withconnection is isomorphic to an induced bundle of the frame bundle by some representation ρ : SO ( n ) → Aut(End( C m )) of SO ( n ) by ∗ -automorphisms of End( C m )End( E ) = F M × ρ End( C m )9ith connection induced by the Levi-Civita connection on F M . Example 3.1 (Dirac operators) . Suppose that D is the Dirac operator associated with a spin-structure or spin c -structure acting on the sections of the associated spinor bundle S . Then theaction of the complex Clifford algebra bundle Cl(
T M ) on S is irreducible and therefore End( S ) is a quotient of the bundle Cl(
T M ) . The connection on S is compatible with the Clifford actionand therefore, the induced connection on End( S ) is compatible with the Clifford connectionon Cl(
T M ) . But the Clifford algebra bundle is as a hermitian vector bundle with connectionobtained as an associated bundle Cl(
T M ) =
F M × ρ Cl( R n ) , where ρ is the canonical representation of SO ( n ) on Cl( R n ) . Note that the spinor bundle itselfis not an associated bundle of F M , but
End( S ) nevertheless is. By the Bochner-Lichnerowicz-Weitzenb¨ock-Schr¨odinger formula D = ∇ ∗ ∇ + V, where V is some potential (for example R in the case of a spin structure). Thus, D is aLaplace-type operator and the corresponding connection on S and on End( S ) is the Levi-Civitaconnection. Example 3.2.
The bundle Λ p M is an associated bundle of the frame bundle Λ p M = F M × σ Λ p C n where σ is the canonical representation of SO ( n ) on Λ p C n . This of course induces a connectionon Λ p M which is the Levi-Civita connection on forms. The Hodge-Laplace operator on p -forms ∆ p is then defined by ∆ p = dδ + δd, where d : C ∞ (Λ p M ) → C ∞ (Λ p +1 M ) is the exterior differential and δ : C ∞ (Λ p +1 M ) → C ∞ (Λ p M ) its formal adjoint. Again, by the Bochner–Weitzenb¨ock formula the ∆ p = ∇ ∗ ∇ + V, where V involves curvature terms and ∇ is the Levi-Civita connection. Note that End(Λ p M ) = F M × ρ End(Λ p C n ) , where ρ = σ ⊗ σ ∗ . E ) is an associatedbundle of F M . Consequently, the bundle π ∗ End( E ) → S ∗ M is an associated bundle of F M → S ∗ M , where the map π : F M → S ∗ M is defined by projecting onto the first vectorin the frame and identifying vectors and covectors using the metric. So, if we view F M asan SO ( n − S ∗ M then we can think of π ∗ End( E ) as the associatedbundle π ∗ End( E ) = F M × ˆ ρ End( C m ) , where ˆ ρ is the restriction of ρ : SO ( n ) → Aut(End( C m )) to the subgroup SO ( n −
1) whichwe think of as the subgroup that fixes the first vector in the standard representation on R n .Therefore, sections of π ∗ End( E ) can be identified with functions f on F M with values inEnd( C m ) satisfying the transformation property f ( x · g ) = ˆ ρ − ( g ) f ( x ) , (1)for all g ∈ SO ( n −
1) and x ∈ F M . The frame flow on
F M is the extension of the geodesicflow on S ∗ M by parallel translation to the space F M . More explicitly, if ( e , . . . , e n ) ∈ F M is an orthonormal frame, then the frame flow γ t ( e , . . . , e n ) = ( e ( t ) , . . . , e n ( t )) can be definedas follows. The vector e ( t ) is the tangent of the endpoint of the unique geodesic of length t with starting tangent vector e . In other words e ( t ) = G t ( e ). The rest of the frame( e , . . . , e n ) is parallel transported along this geodesic using the Levi-Civita connection togive the orthonormal basis ( e ( t ) , . . . , e n ( t )) in the orthogonal complement of e ( t ). It isimportant here that the Levi-Civita connection preserves angles so that the parallel transportof the frame yields a frame. The frame flow gives rise to a flow on the space of functions on F M by pull-back. By construction γ t commutes with the right action of SO ( n −
1) on
F M and therefore the space of functions satisfying the transformation property (1) is left invariant.Since all constructions are compatible it turns out that the flow β t originally constructed fromthe connection on E coincides with this flow. That is( β t f )( x ) = f ( γ − t x ) , where γ t is the frame flow. 11 The high energy limit for geometric operators
Let φ j be a complete orthonormal sequence of eigensections of a positive Laplace-type operator∆ E such that ∆ E φ j = λ j φ j , λ ≤ λ ≤ λ ≤ . . . . Then, the sequence of normal states ω N defined by ω N ( A ) = 1 N N X j =1 h φ j , Aφ j i has a high energy limit ω ∞ . As in the scalar case we have ω ∞ ( A ) = lim t → + Tr( Ae − t ∆ E )Tr( e − t ∆ E )and lim t → + Tr( Ae − t ∆ E )Tr( e − t ∆ E ) ) = 1rk E Z S ∗ Tr(ˆ σ A ( ξ )) dµ L ( ξ ) , where µ L is the normalized Liouville measure on S ∗ M and Tr is the trace. Indeed this stateis obviously invariant under the classical time evolution β t .In order to understand which quantum limit can be obtained from subsequences of non-zero counting density one needs to decompose the state ω ∞ into ergodic states with respectto the action β t .Indeed, one can prove the following theorem ([Zel96, JS06, JSZ08]) Theorem 4.1.
Suppose that p , . . . , p r are projections in A which commute with ∆ E . Supposefurthermore that P ri =1 p i = id and that the decomposition ω ∞ ( · ) = r X i =1 ω ∞ ( p i · ) is a decomposition into ergodic states ω i ( A ) = 1 ω ∞ ( p i ) ω ∞ ( p i A ) . Then Shnirelman’s theorem holds in the subspaces onto which p i project. More precisely, if φ j is an orthonormal sequence of eigensections of ∆ E such that ∆ E φ j = λ j φ j ,λ ≤ λ ≤ λ ≤ . . . ,p i φ j = φ j , nd such that φ j span the range of p i . Then there is a subsequence of eigensections φ k ( j ) ofcounting density one such that lim j →∞ h φ k ( j ) , Aφ k ( j ) i = ω i ( A ) . Let τ : SO ( n − → Aut(End( C k )) be a representation of SO ( n −
1) by ∗ -automorphisms.Then there is a unique projective unitary representation ρ : SO ( n − → P U ( k )such that τ ( g )( x ) = ρ ( g ) − xρ ( g ) . As before the frame flow induces a flow β t on the space of sections of F = F M × τ End( C k )by ∗ -automorphisms. The following theorem is proved in ([JS06, JSZ08]). Theorem 5.1.
Suppose that the frame flow on
F M is ergodic and that ρ is irreducible. Thenthe tracial state ω ( A ) = 1 k Z S ∗ M Tr(ˆ σ A ( ξ )) dµ L ( ξ ) , on C ( S ∗ M ; F ) is ergodic. This gives us a strategy to decompose the tracial state on C ( S ∗ M, End( E )) into ergodicstates assuming that End( E ) is an associated bundle of the frame bundle. Namely, supposethat ρ : SO ( n − → P U ( k ) is a projective unitary representation which gives rise to arepresentation ρ = ρ ⊗ ρ ∗ on End( C k ) such thatEnd( E ) = F M × τ End( C k ) . Then decompose C k into invariant subspaces for ρ C k = V ⊕ . . . ⊕ V r . The orthogonal projection p i onto V i is a matrix in End( C k ) and we have P ri =1 p i = id. Thismatrix is invariant under the action of ρ and therefore, the constant function p i ∈ C ∞ ( F M )13an be understood as a section in End( E ) as it satisfies the transformation rule. By the abovetheorem the state ω i ( · ) = 1 ω ( p i ) ω ( p i A )is ergodic and we have constructed an ergodic decomposition of the tracial state. If there arepseudodifferential operators P i in ΨDO cl ( M, E ) that are mutually commuting and commutewith ∆ E such that σ P i = p i we can then apply the theorem above and conclude that quantumergodicity holds in subspaces onto which P i projects. As before assume that S is the spinor bundle of some Spin structure or Spin c structure. Let D be the Dirac operator and sign( D ) defined by spectral calculus. Then sign( D ) is a zeroorder pseudodifferential operator and its principal symbol is given by σ sign( D )( ξ ) = γ ξ , where γ ξ denotes Clifford multiplication with ξ . Then the projections P ± = 12 (1 ± sign( D ))are pseudodifferential operators that commute with | D | . Their symbols are elements in C ∞ ( S ∗ X, π ∗ End( S )) that are invariant under the flow β t . If we identify sections of thisbundle with functions on F M with values in the End( C [ n/ ) then this function correspondsto Clifford multiplication with first vector in the Cl( R n )-module C [ n/ . This projects ontoan irreducible subspace of the projective representation SO ( n − → P U (2 [ n/ ) . If the frame flow is ergodic quantum ergodicity holds in the positive spectral subspace andnegative spectral subspace respectively. p -forms Since the bundles Λ p T ∗ M → M are associated bundles of the representation Λ p ρ , if ρ : SO ( n ) → C n the bundle π ∗ Λ p M → S ∗ M is associated with the restriction of this representa-tion to the subgroup SO ( n − p ρ are irreducible14or p = n the restriction to the group SO ( n −
1) is not irreducible unless p = 0 or p = n .The reason is that since C n = C ⊕ C n − is a decomposition into invariant subspaces of the SO ( n − p C n = Λ p C n − ⊕ Λ p − C n − is a decomposition into invariant subspaces. This decomposition is into irreducible subspacesunless p = n − in which case the first summand is not irreducible or p = n +12 in which casethe second summand is not irreducible.As in the case the Dirac operator one can find pseudodifferential operators of order zerothat have symbols that project onto these irreducible subspaces. Namely, define ∆ − p as theinverse of ∆ p on (ker ∆ p ) ⊥ and to be zero otherwise. Then P = ∆ − p δd,Q = ∆ − p dδ. are projections that commute with ∆ p . Their principal symbols are invariant elements in C ∞ ( S ∗ M, π ∗ End(Λ p )) that give rise to a decomposition of the tracial state as σ P + σ Q = 1 . Suppose now that the frame flow on
F M is ergodic. If p = n − then the state σ P ( · ) = 1 ω ( P ) ω ( P · )is ergodic as σ P corresponds to the projection onto the first summand in. If p = n − then thestate σ Q ( · ) = 1 ω ( Q ) ω ( Q · )is ergodic for the analogous reason. Thus, quantum ergodicity holds in the subspaces ontowhich P and Q project onto, which are the subspaces of co-exact and exact p -forms.In the case p = n − the state ω P is not ergodic. There is a further pseudodifferentialoperator commuting with ∆ p and with P , namely the operator R = ∆ − p ∗ d, where ∗ is the Hodge star operator. Note that R = 1 on rg( P ) and the decompositionof the state ω P into +1 and − R correspond to polarized forms. For example the case of n = 1, p =1 and co-closed 1-forms corresponds to electrodynamics in dimension 3. It is well knownthat electromagnetic waves can be decomposed into circular polarized waves and that thisdecomposition is invariant under the time-evolution determined by the Maxwell equation.Sequences of differently polarized eigensections give rise to different quantum limits as theobservable R that measures the polarization gives rise to an observable in the high energylimit, namely σ R , that distinguishes them. The two examples of the previous section may also be discussed in the category of K¨ahlermanifolds or other special geometries. A K¨ahler manifold can be thought of as a Riemannianmanifold of dimension 2 m such that the frame bundle can be reduced to a U ( m )-principalbundle in such a way that the parallel transport preserves the U ( m )-structure. This is equiv-alent to the existence of a covariantly constant complex structure. On K¨ahler manifolds theframe flow is not ergodic as the complex structure is preserved and gives rise to conservationlaws. It is much more natural however to consider the U ( m )-bundle U M of unitary frames in-stead and the restriction of the frame flow to it. Again, a lot of geometric constructions in thecategory of K¨ahler manifolds can be understood as associated bundle constructions startingfrom the unitary frame bundle. The bundles of ( p, q )-forms are associated bundles of
U M andnatural geometric operators to consider are the Dolbeault Dirac operator and the DolbeaultLaplace operator. Under the assumption that the unitary frame flow is ergodic the ergodicdecomposition of the tracial state can be found explicitly. It is closely related to the actionof a certain Lie-superalgebra on the space of exterior differential forms of a K¨ahler manifold.This action can be seen as the quantum counterpart of the classical symmetry that preventsthe frame flow on
F M to be ergodic. A detailed discussion of this and its implications canbe found in [JSZ08].
It was already found by [BoK98, BoK99, Bol01, BoG04, BoG04.2] for the Dirac operator in R n that the semi-classical limit for this operator can be described by a suitable extension ofthe classical flow and that ergodicity of that flow implies quantum ergodicity. The geometricframework discussed in the present article was introduced in [JS06] and further developed16n [JSZ08]. It deals with high energy limits rather than with the semi-classical limit so thatonly the non-trivial geometry of space contributes to the classical dynamics. A representationtheoretic framework generalizing the representation theoretic lift on locally symmetric spacesto induced bundles over locally symmetric spaces can be found in [BuO06]. This article alsocontains some discussion of quantum ergodicity questions for vector bundles.As advertised [Zel96] the language of states and ergodicity of states over C ∗ -dynamicalsystems is the appropriate one to describe the high energy limit or quantum ergodicity ofquantum systems. Its application to quantum systems with spin or more mathematicallyto geometric operators acting on vector bundles naturally leads to associated bundle con-structions over the frame bundle. The underlying dynamics being the frame flow. Quantumergodicity questions translate into questions about the frame flow. In particular ergodicity ofthe frame flow has strong implications for quantum systems with spin. It implies quantumergodicity on certain natural subspaces that can be found more or less constructively fromour method. Finally we would like to mention that the frame flow was already consideredby Arnold in [Arn61]. In negative curvature, it was studied by Brin, together with Gromov,Karcher and Pesin, in a series of papers [BrP74, Br75, Br76, BrG80, Br82, BrK84] inde-pendently of any connection to spectral theory for operators on vector bundles or quantumergodicity questions. Many examples of manifolds with ergodic frame flow are known (forexample manifolds of constant negative curvature to name only the simplest ones) and muchprogress has been made towards its understanding. We would like to refer the reader to theabove mentioned literature on frame flows for further details. Acknowledgements.
The author would like to thank the CRM Montreal and the Analysislab in Montreal kind hospitality during his stay in summer 2008
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