Generalizations of a method for constructing first integrals of a class of natural Hamiltonians and some remarks about quantization
aa r X i v : . [ n li n . S I] F e b Generalizations of a method for constructing firstintegrals of a class of natural Hamiltonians andsome remarks about quantization
Claudia ChanuDipartimento di Matematica e Applicazioni,Universit`a di Milano Bicocca. Milano, via Cozzi 53, Italia.Luca Degiovanni, Giovanni RastelliFormerly at Dipartimento di Matematica,Universit`a di Torino. Torino, via Carlo Alberto 10, Italia.e-mail: [email protected]@[email protected] 18, 2018
Abstract
In previous papers we determined necessary and sufficient conditionsfor the existence of a class of natural Hamiltonians with non-trivial firstintegrals of arbitrarily high degree in the momenta. Such Hamiltoni-ans were characterized as (n+1)-dimensional extensions of n-dimensionalHamiltonians on constant-curvature (pseudo-)Riemannian manifolds Q.In this paper, we generalize that approach in various directions, we ob-tain an explicit expression for the first integrals, holding on the moregeneral case of Hamiltonians on Poisson manifolds, and show how theconstruction of above is made possible by the existence on Q of particu-lar conformal Killing tensors or, equivalently, particular conformal mastersymmetries of the geodesic equations. Finally, we consider the problemof Laplace-Beltrami quantization of these first integrals when they are ofsecond-degree.
In recent years, several progresses have been done in the field of integrable andsuperintegrable Hamiltonian systems, both classical and quantum, by the in-troduction of new techniques for the study of higher-degree polynomial firstintegrals and higher-order symmetry operators. After researches exposed in [4],17] and [10] is now possible to explicitly build and analyze Hamiltonian systemspossessing symmetries of arbitrarily-high degree. For a more detailed introduc-tion see the contribution to the QTS 7 proceedings written by W. Miller Jr.In several papers ([4], [5], [6]) we developed the analysis of a class of systemswhich, in dimension two, are a subset of the celebrated Tremblay-Turbiner-Winternitz (TTW) systems and are strictly related with the Jacobi-Calogeroand Wolfes three-body systems [4], [6]. In [5] we generalized these systems tohigher-dimensions by introducing a ( n + 1)-dimensional extension H of a given n -dimensional natural Hamiltonian L . We obtained necessary and sufficientconditions for the existence of a first integral of H in a particular form, onenecessary condition being the constant curvature of the configuration manifoldon which L is defined (for superintegrable systems with higher-degree first in-tegrals on constant curvature manifolds see also [8]). The first integral of H ,which is independent from those of L , is polynomial in the momenta and can beexplicitly constructed through a differential operator. In the present paper, wegeneralize the analysis done in [5] in several directions. In Sec. 2 we extend theconstruction to non natural Hamiltonians on a general Poisson manifolds andobtain, also in this case, an explicit expression for the polynomial first integral.In Sec. 3 we restrict ourselves to cotangent bundles of (pseudo-)Riemannianmanifolds and consider a wider class of higher-degree first integrals, we provethat a necessary condition for their existence is the presence of a particular classof conformal Killing tensors or, equivalently, of conformal master symmetries ofthe geodesic equations; we end the section with an example showing how themethod can provide several independent first integrals of degree m . In Sec. 4 wecharacterize our construction in an invariant way and determine necessary andsufficient conditions for the constant curvature or conformal flatness of the con-figuration manifold of H , conditions employed in Sec. 5, where the quantizationof the second-degree first integrals obtained by our method is considered. Let us consider a Poisson manifold M and a one-dimensional manifold N . Forany Hamiltonian function L ∈ F ( M ) with Hamiltonian vector field X L , weconsider its extension on ˜ M = T ∗ N × M given by the Hamiltonian H = 12 p u + α ( u ) L + β ( u ) (1)where ( p u , u ) are canonical coordinates on T ∗ N and α ( u ) = 0. The Hamiltonianflow of (1) is X H = p u ∂∂u − ( ˙ αL + ˙ β ) ∂∂p u + αX L , where dots denotes total derivative w.r.t. the (single) variable u .It is immediate to see that any first integral of L is also a constant of motionof H , when considered as a function on ˜ M . We recall that a function F is afirst integral of H if and only if X H F = { H, F } = 0.In [5] we determined on L , α and β necessary and sufficient conditions forthe existence of two functions γ ∈ F ( N ) and G ∈ F ( M ) such that, given thedifferential operator U = p u + γ ( u ) X L , (2)2he function U m ( G ) obtained applying m = 0 times U to G is a non trivialadditional first integral for H .In particular, if L is a natural Hamiltonian on the cotangent bundle of a(pseudo-)Riemannian manifold ( Q, g ) L = 12 g ij p i p j + V and α is assumed to be not constant, an integral of the form U m ( G ) exists, with G not dependent on the momenta, if and only if G satisfy for some constant c = 0 the equations: ∇ i ∇ j G + mcg ij G = 0 , (3) ∇ i V ∇ i G = 2 mcV G, (4)which are equivalent to {∇ i Gp i , L } = 2 mcGL, meaning that ∇ i Gp i is a conformal first integral of L .If a solution of the previous equations exists, then the extended Hamiltonian(1) and the differential operator (2) take the form H = 12 p u + mcS κ ( cu + u ) L (5) U = p u + 1 T κ ( cu + u ) X L , (6)where the trigonometric tagged functions (see [3, 9]) are employed S κ ( x ) = sin √ κx √ κ κ > x κ = 0 sinh √ | κ | x √ | κ | κ < T κ ( x ) = tan √ κx √ κ κ > x κ = 0 tanh √ | κ | x √ | κ | κ < Proposition 1.
Let H be the extension (1) of the Hamiltonian L on the Poissonmanifold ˜ M , let U the differential operator (2) and G ∈ F ( M ) a function suchthat X L ( G ) = 0 . Then, U m ( G ) is a first integral for H if and only if G satisfies X L ( G ) + 2 m ( cL + L ) G = 0 c, L ∈ R . (7) and α , β and γ satisfy α = − m ˙ γ, (8) β = mL γ + β , β ∈ R , (9)¨ γ + 2 cγ ˙ γ = 0 . (10) Proof.
In [5] it is proved that we have that X H U m ( G ) = 0 for a function G ∈ F ( M ) if and only if L , α , β satisfy( m ˙ γ + α ) X L ( G ) = 0 , (11) αγX L ( G ) − m ( ˙ αL + ˙ β ) G = 0 . (12)3ecause X L ( G ) = 0, from (11) it follows that α = − m ˙ γ (13)and condition (12) becomes ˙ γγ X L ( G ) G = m ¨ γL − ˙ β. Since ˙ γ = − α/m = 0, we get X L ( G ) G = m ¨ γγ ˙ γ L − ˙ βγ ˙ γ , (14)which derived with respect to u gives ddu (cid:18) ¨ γγ ˙ γ (cid:19) L = ddu ˙ βmγ ˙ γ ! . But L is a non-constant function on M , hence the functions ¨ γ and ˙ β must beboth proportional to γ ˙ γ :¨ γ = − cγ ˙ γ = − c ddu (cid:0) γ (cid:1) , ˙ β = 2 mL γ ˙ γ = mL ddu (cid:0) γ (cid:1) . By integrating and substituting in (14), we obtain conditions (7) and (9).
Remark 1. If X L ( G ) = 0 we trivially have U m ( G ) = p mu , which is a firstintegral of H only if α and β are constant. Hence, it is a constant of motionfunctionally dependent on L and H . Remark 2.
The equation (7) is obviously equivalent to { L, { L, G }} = − m ( cL + L ) G ;this condition can be interpreted in terms of master symmetries: the Hamilto-nian vector field X G is a master symmetry for the Hamiltonian vector field X L on the hypersurfaces L = 0 or G = 0. Further remarks about the special casewhen L is a natural Hamiltonian are at the end of Sec. 3.By integrating the equations for α , β and γ in Proposition 1 the explicitexpression for the extended Hamiltonian H and the differential operator U canbe found. From equation (7) we have [5] Theorem 2.
Let H be the extension (1) of the Hamiltonian L on the Poissonmanifold ˜ M , let U the differential operator (2) and G ∈ F ( M ) a function satis-fying X L ( G ) = 0 and (7). Then, U m G is a first integral of H if and only if H and U are in either one of the two following forms characterized by the value of c in (7) ) for c = 0 H = 12 p u + mcS κ ( cu + u ) ( L + V ) + W , (15) U = p u + 1 T κ ( cu + u ) X L , ii) for c = 0 H = 12 p u + mA ( L + V ) + B ( u + u ) , (16) U = p u − A ( u + u ) X L , with κ, V , W ∈ R , B = mL A and A = 0 .Proof. By Proposition 1, α , β , γ must satisfy (8), (9), (10). In the case c = 0equation (10) becomes ˙ γ + c ( γ + κ ) = 0, whose solution is γ = 1 T κ ( cu + u ) . Hence, α = mcS κ ( cu + u ) ,β = mcV S κ ( cu + u ) + W , with V = L /c and W = β − mκL . In the case c = 0, equation (10) gives˙ γ + A = 0 with A = 0 in order to avoid α = 0. Hence, α = mA,β = mAV + B ( u + u ) ,γ = − A ( u + u ) , where V is now an arbitrary constant. Remark 3.
The constants u , V and W are not essential. Indeed, H and L are defined up to additive constant W and V while u can be eliminated by atranslation of u . In the case c = 0, the choice V = W = 0 gives the expressions(5) and (6) for H and U obtained in [5]. Moreover, by including the constant L in the Hamiltonian L , the condition (7) assumes the simpler form X L ( G ) + 2 mcLG = 0 . Once L and G satisfy condition (7) the first integrals U m ( G ) are explicitlydetermined for any G : M −→ R . Theorem 3.
Under the hypothesis of Proposition 1 the functions U m G can beexplicitely written as U m G = P m G + D m X L G, (17)5 here P m = [ m/ X k =0 (cid:18) m k (cid:19) γ k p m − ku ( − m ( cL + L )) k ,D m = [ m/ − X k =0 (cid:18) m k + 1 (cid:19) γ k +1 p m − k − u ( − m ( cL + L )) k , m > , where [ · ] denotes the integer part and D = γ .Proof. From equation (7) it follows that for all k ∈ N we have X k +1 L G = ( − m ( cL + L )) k X L G, X kL G = ( − m ( cL + L )) k G. (18)By expanding U m using the binomial formula U m G = ( p u + γX L ) m = m X k =0 (cid:18) mk (cid:19) p ku ( γX L ) m − k , and separating even and odd terms in k , by taking in account relations (18) weget equation (17).The setting described in the previous section can be further generalized asfollows. Let X L be a Hamiltonian vector field on a Poisson manifold ˜ M , let on˜ M X H = Y + f X L , for a vector field Y and U = f + f X L , where f i : ˜ M → R . Following the same proof procedure as in [5] we get Proposition 4. If X L ( f i ) = 0 and [ Y, X L ] = 0 then X H U m ( G ) = 0 , i.e. U m ( G ) is a first integral of H , if and only if (cid:0) f Y + ( mY ( f ) + f f ) X L + f X L Y + f f X L (cid:1) ( G ) = − mY ( f ) G. (19) Proof. If X L ( f i ) = 0 and [ Y, X L ] = 0,then { H, L } = 0 , [ X H , U ] = Y ( f ) + Y ( f ) X L , [[ X H , U ] , U ] = 0 . Thus, X H U m = U m − ( m [ X H , U ] + U X H ) == U m − (cid:0) mY ( f ) + f Y + ( mY ( f ) + f f ) X L + f X L Y + f f X L (cid:1) . and the thesis follows.The analysis of such a generalization will not be considered here.6 Extensions of a natural Hamiltonian
In the following sections we will assume that L is a natural n -dimensional Hamil-tonian on M = T ∗ Q for a (pseudo-)riemannian manifold ( Q, g ): L = 12 g ij ( q h ) p i p j + V ( q h ) , (20)where g ij are the contravariant components of the metric tensor and V a scalarpotential. This assumption, together with the hypothesis that G is polynomialof degree d in the momenta ( p i ), allows us to expand condition (7) into anequality of two polynomials in ( p i ) of degree d +2 that can be splitted into severaldifferential conditions involving the metric, the potential and the coefficients of G . Indeed, being L a natural Hamiltonian, we have (in [5] the equation for X L was mistyped, however, this does not affects any of the results of the paper, X L = p i ∇ i − ∇ i V ∂∂p i ,X L = p i p j ∇ i ∇ j − ∇ i V ∇ i − p j ∇ i V ∇ j ∂∂p i − p i ∇ i ∇ j V ∂∂p j + ∇ i V ∇ j V ∂ ∂p i ∂p j . In [5] we dealt with the case c = 0, d = 0, i.e. G independent of momenta,obtaining the conditions (3) and (4). The maximal dimension of the space ofsolutions of equation (3) is n + 1 and it is achieved only if the metric g on Q has constant curvature. We call complete the solutions G of (3) satisfying thisintegrability condition (see [5]).In the following, we analyze in details the d = 1 case ( G linear in themomenta), in order to show how the procedure works. Proposition 5.
Let be G = λ l ( q i ) p l + W ( q i ) . Then, U m G is a first integral of H if and only if ∇ ( i ∇ j λ l ) + mcg ( ij λ l ) = 0 , (21) ∇ i ∇ j W + mcg ij W = 0 , (22) ∇ i V ( ∇ i λ l + 2 ∇ l λ i ) + λ i ∇ l ∇ i V − mλ l ( cV + L ) = 0 , (23) ∇ i V ∇ i W − m ( cV + L ) W = 0 , (24) Proof.
For G linear in the momenta we have X L G = p i p l ∇ i λ l − ∇ i V λ i + p i ∇ i W,X L G = p i p j p l ∇ i ∇ j λ l − p l ( ∇ i V ( ∇ i λ l + 2 ∇ l λ i ) + λ i ∇ l ∇ i V ) ++ p i p j ∇ i ∇ j W − ∇ i V ∇ i W, and condition (7) holds if and only if p i p j p l ( ∇ i ∇ j λ l + mcg ij λ l ) + p i p j ( ∇ i ∇ j W + mcg ij W ) − p l ( ∇ i V ( ∇ i λ l + 2 ∇ l λ i ) + λ i ∇ l ∇ i V − mλ l ( cV + L ))+2 m ( cV + L ) W − ∇ i V ∇ i W + = 0 , which is equivalent to eqs. (21, 22, 23, 24).7 emark 4. The coefficients of terms with even and odd degree in the momentaare involved in different equations: eq.s (22) and (24) contain the ones of a G independent of p i . Hence, for λ i = 0 we recover the d = 0 case: (22) and(24) are the expansion in coordinates of (3) and (4) for G = W . For W = 0the compatible potentials V have to satisfy both conditions (24) and (23), thusit is impossible to get new potentials other than those compatible with a G independent of the momenta i.e., satisfying conditions (22–24).From (22) one can derive (see [5]) integrability conditions for W ( R hijk − mc ( g hj g ik − g hk g ij )) ∇ h ln W = 0 . (25)If these equations are identically satisfied we have complete integrability whichis equivalent to constant curvature of Q , otherwise W must satisfy all equations(22) and (25). For example, when Q has dimension two, we have from (25)( R − mc det( g ij )) ∇ ln W = 0 , and ( R − mc det( g ij )) ∇ ln W = 0 . Therefore, because of the symmetries of the Riemann tensor, we have
Theorem 6. If Q has dimension , then equations (22) admit non-constantsolutions W only if Q has constant curvature. For each l , the integrability conditions of (21) are weaker than those forthe Hessian equation for G ( q i ) (3) and therefore the curvature of Q could benon-constant.We give two examples in order to illustrate the Proposition 5. Example 1.
As shown in [5] and recalled above, when Q has constant curvature,equation (3), or equivalently equation (22), admits a solution depending on n + 1 real parameters ( a i ). Let G i be a solution determinated by the choiceof a particular set of the ( a i ), let us assume that G i = G j . It is then naturalto consider the relations between U m G i and U m G j and see if some choice ofthe parameters can provide new independent first integrals of the system. Forexample, let L be the natural Hamiltonian on the constant curvature manifold Q = S with ( q = θ, q = φ ) L = 12 ( p θ + 1sin θ p φ ) + V. (26)A complete solution of a 0th degree G ( θ, φ, a , a , a ) has been computed in [5] G = ( a sin φ + a cos φ ) sin θ + a cos θ. (27)and for a = 0, the integration of equation (4) – or equivalently (24) – gives V = 1cos θ F (( a sin φ − a cos φ ) tan θ ) . For different sets of the parameters ( a k ), U m G i and U m G j are no longer simul-taneously first integrals of H unless if F = F = constant , and therefore V = F cos θ . (28)8n this case, let be G = G ( a = 1 , a = 0 , a = 0) and G = G ( a = 0 , a =1 , a = 0). Hence, for any extension of L of the form (1) with α given by (8),the five functions L = L , L = p = p φ , H , U m G and U m G are functionallyindependent first integrals of H . For m = 2, recalling that c = K/m , thecurvature of Q = S is K = 1 and choosing for the other free parameters of α the values κ = 0 and u = 0, we have α = u , and U G and U G are U G = (sin φ sin θ ) (cid:18) p u − p θ u − F u cos θ (cid:19) + p θ p u u cos θ sin φ + p φ p u u cos φ sin θ − p φ u sin φ sin θ ,U G = (cos θ + sin θ − cos θ ) cos φ sin θ (cid:18) p u − p θ u − F u cos θ (cid:19) + p θ p u u cos θ cos φ − p φ p u u sin φ sin θ − p φ u cos φ sin θ . Example 2.
We can use a complete solution G ( q i , a k ) of (3) in order to con-struct solutions λ i of (21). Namely, we can choose λ i = G i , i = 1 , . . . , n where G i denotes any particular solution of (3). We remark that it is not necessarythat G i = G j for i = j , or G i = 0 for all i . By substituting the λ i into (23), theequations become n second-order PDE in V whose solutions provide examplesof compatible potentials. For instance, let us consider again L given by (26)on Q = S . We can choose for λ i the particular values λ = cos( θ ), λ = 0 of(27) as coefficients for a linear homogeneous G . Then, equations (23) can beintegrated yielding, V = c + c sin θ cos θ , which, for c = 0 does not satisfies (4) with G given by (27), hence, for thispotential the construction of U m G is possible only when G depends on themomenta. For the different choice of λ i , λ = 0, λ = cos θ , the integration of(23) gives V = c sin θ , which is compatible with G given by (27) for a = a = 0. The expressions of U m G can be computed by using (17). Remark 5.
By considering the functions λ i as the components of a vector field Λ , equation (21) can be written as[ g , [ g , Λ ]] = − mc Λ ⊙ g , (29)where [ · , · ] are the Schouten-Nijenhuis brackets and ⊙ denotes symmetrizedtensor product. This means that [ g , Λ ] is a particular kind of conformal Killingtensor, or, equivalently, that Λ is a particular conformal master symmetry ofthe geodesic equations, where the conformal factor is a constant multiple of Λ ,instead of an arbitrary vector field. In a similar way, for G polynomial in themomenta of degree k with highest degree term given by λ i ...i k p i . . . p i k , it isstraightforward to show that a necessary condition for U m G to be first integralof H is still of the form (29), where now Λ is a k -tensor field. In the 0-th degreecase G = W ( q i ) eq. (29) becomes [ g , ∇ W ] = − mcW g . efinition 1. We call self-conformal ( s-conformal in short) the ( k + 1) -orderconformal Killing tensor field [ g , Λ] such that [ g , [ g , Λ ]] = C g ⊙ Λ ,C ∈ R , is satisfied. In this case, the k -order tensor Λ is said to be a s-conformalmaster symmetry of the geodesic equations of g . In the case of C = 0, i.e. c = 0, s-conformal Killing tensors and mastersymmetries become the usual Killing tensors and master symmetries. Theorem 7.
Let G be a k -degree polynomial of degree k in the momenta. Anecessary condition for U m G to be first integral of H is that the tensor Λ ofcomponents λ i ...i k given by the coefficients of the highest-degree term of G is aself-conformal master symmetry of the geodesic equations of g or, equivalently,that [ g , Λ ] is a self-conformal tensor field of g with C = − mc . The existence of a complete solution introduced in [5] and recalled abovecan be restated as follows
Corollary 8.
Equation (3) admits a complete solution G = W ( q i ) if and onlyif the dimension of the space of the s-conformal Killing vectors ∇ G , with C = − mc , is maximal and equal to n + 1 . We show under which geometrical conditions a ( n + 1)-dimensional naturalHamiltonian can be written as the extension (15) of a natural Hamiltonian L .Let us consider a natural Hamiltonian H = 12 ˜ g ab p a p b + ˜ V (30)on a ( n + 1)-dimensional Riemannian manifold ( ˜ Q, ˜ g ) and let X be a conformalKilling vector of ˜ g , that is a vector field satisfying[ X, ˜ g ] = L X ˜ g = φ ˜ g , where φ is a function on ˜ Q and [ · , · ] are the Schouten-Nijenhuis brackets. Wedenote by X ♭ the corresponding 1-form obtained by lowering the indices bymeans of the metric tensor ˜ g . Theorem 9.
If on ˜ Q there exists a conformal Killing vector field X with con-formal factor φ such that dX ♭ ∧ X ♭ = 0 , (31) dφ ∧ X ♭ = 0 , (32) d k X k ∧ X ♭ = 0 , (33) X ( ˜ V ) = − φ ˜ V , (34)˜ R ( X ) = kX, k ∈ R , (35)10 here ˜ R is the Ricci tensor of the Riemannian manifold, then, there exist on ˜ Q coordinates ( u, q i ) such that ∂ u coincides up to a rescaling with X and thenatural Hamiltonian (30) has the form (15).Proof. Condition (31) means that X is normal i.e., orthogonally integrable: lo-cally there exists a foliation of n dimensional diffeomorphic manifolds Q suchthat T P Q = X ⊥ = { v ∈ T ˜ Q | ˜ g ( v, X ) = 0 } ; it follows that there exists a coordi-nate system ( q = u, q i ) for i = 1 , . . . , n such that ∂ u is parallel to X and thecomponents ˜ g i vanish for all i = 1 , . . . , n . Furthermore, by (32) the conformalfactor φ is constant on the leaves Q ( v ( φ ) = 0 for all v ∈ X ⊥ ); thus, φ dependsonly on u . By expanding the condition that X = F ( q a ) ∂ u is a conformal Killingvector (cid:8) ˜ g ( q a ) p u + ˜ g ij ( q a ) p i p j , F ( q a ) p u (cid:9) = φ ( u ) (cid:0) ˜ g ( q a ) p u + ˜ g ij ( q a ) p i p j (cid:1) we get the equations˜ g (2 ∂ u F − φ ) − F ∂ u ˜ g = 0 (36)˜ g hj ∂ h F = 0 j = 1 , . . . , n (37)˜ g ij φ + F ∂ u ˜ g ij = 0 i, j = 1 , . . . , n (38)By (37), we have F = F ( u ), hence due to (38) we get that ∂ u ln ˜ g ij is a functionof u , the same function for all i, j . Thus, without loss of generality we can assume˜ g ij = g ij ( q h ) α ( u ). Moreover, Eq. (36) implies that, up to a rescaling of u , ˜ g is independent of u . By imposing X ( ˜ V ) = − φV , we obtain ∂ u ln ˜ V = − φ/F ,that means ˜ V = α ( u ) V ( q h ), thus we get H = 12 g ( q h ) p u + α ( u ) (cid:18) g ij ( q h ) p i p j + V ( q h ) (cid:19) . Finally, condition (33) means that the norm of X is constant on Q , that is F ( u ) g ( q i ) is independent of ( q i ). This shows that up to a rescaling and achange of sign of H we can assume g = 1 and in the coordinate system ( u, q i )(30) has the required form (15). By computing again the Poisson bracket, weget relations between φ , α and F : α = k ( F ) − and φ = 2 ˙ F with k a realnot vanishing constant. When X is a proper conformal Killing vector, we canassume that α is proportional to F ( u ) − . The covariant components of the Riccitensor of ˜ Q are given in Lemma 10, in particular we have for i = 1 , . . . , n ˜ R = n ¨ FF , ˜ R i = 0 . Hence, X = F ( u ) ∂ u is an eigenvector of the Ricci tensor with eigenvalue ρ = n ¨ FF ,which is constant if and only if F is proportional to S κ ( cu + u ). Remark 6. If φ = 0 (i.e., X is a Killing vector), then α and F are necessarilyconstant and this gives the geodesic term of the case c = 0, but equation (34)does not characterize the potential of the Hamiltonian (16). Remark 7.
It is straightforward to check that for a Hamiltonian of the form H = p u + F − ( u ) L with L a natural n -dimensional Hamiltonian, X = F ∂ u isa CKV with conformal factor φ = 2 ˙ F such that X ( F − ( u ) V ) = − φ ( F − ( u ) V ).Hence, conditions of the above theorem are necessary for having an extendedHamiltonian of our form. 11e want now to study the geometric properties of the metric ˜ g obtained byan extension of a metric g , in particular when g is of constant curvature.In the following, we assume α ( u ) = f − in order to simplify computations.In particular, f is allowed to be pure imaginary. Lemma 10.
Let ( g ij ) be the components of a n -dimensional metric on Q inthe coordinates ( q i ) . We consider the ( n + 1) -dimensional metric on ˜ Q havingcomponents ˜ g ab ( a, b = 0 , . . . n, i, j = 1 , . . . n ) with respect to coordinates ( q = u, q i ) defined as follows ˜ g ab = a = b = 0 , a = 0 , b = 0 ,f ( u ) g ij ( q h ) a = i, b = j. (39) Then, the relations between the covariant components of the Riemann tensorsassociated with ˜ g and g are for all h, i, j, k = 1 , . . . , n ˜ R hjkl = f R hjkl − ˙ f f (˜ g hk ˜ g jl − ˜ g hl ˜ g jk ) , (40)˜ R jkl = 0 , (41)˜ R j l = − ¨ ff ˜ g jl . (42) Moreover, the covariant components of the Ricci tensors R ij and ˜ R ab of the twometrics are related, for all h, i, j, k = 1 , . . . , n , by ˜ R = n ¨ ff , (43)˜ R i = 0 , (44)˜ R ij = R ij + (cid:16) f ¨ f + ( n −
1) ˙ f (cid:17) f − ˜ g ij , (45) and the relation between the Ricci scalars R and ˜ R is ˜ R = Rf + n f ¨ f + ( n −
1) ˙ f f , (46) where ˙ f and ¨ f denote the first and second derivative w.r.t. u of f ( u ) . Expressions (40), (45), and (46) become simpler when Q is of constant cur-vature, while the other formulas remain unchanged. Lemma 11.
Under the hypotheses of Lemma 10 with n > , if g is a metric ofconstant curvature K , then the non zero covariant components of the Riemanntensor ˜ R associated with ˜ g are, for all h, i, j, k = 1 , . . . , n ˜ R hjkl = K − ˙ f f (˜ g hk ˜ g jl − ˜ g hl ˜ g jk ) , (47) Moreover, the covariant components of the Ricci tensor ˜ R ij and the Ricci scalar ˜ R are, for i, j = 1 , . . . , n , ˜ R ij = (cid:16) f ¨ f + ( n − f − K ) (cid:17) f − ˜ g ij , (48)˜ R = n f ¨ f + ( n − f − K ) f . (49)12 heorem 12. Let ( Q, g ) be a n -dimensional Riemannian manifold of constantcurvature K = mc and ( ˜ Q, ˜ g ) the extended manifold with metric (39), thereforei) the metric ˜ g is of constant curvature if and only if either n = 1 or m = 1 or K = c = ˙ f = 0 ,ii) the metric ˜ g is conformally flat if and only if either n > or ˜ g is ofconstant curvature.Proof. For n = 1 the extended metric is, up to a rescaling of q ,˜ g ab = (cid:18) f (cid:19) , which is of constant curvature if and only if ¨ f is proportional to f which istrue if f is any trigonometric tagged function. For n ≥
2, due to the Bianchiidentities, the metric is of constant curvature if the ratios˜ R abcd / (˜ g ac ˜ g bd − ˜ g ad ˜ g bc )are independent of ( a, b, c, d ), that is by (47), (41), and (42)¨ f f + K − ˙ f = 0 , (50)which for c = 0, i.e. f = S κ ( Km u + u ) K , becomes K ( m − m = 0 , which holds only for m = 1 or for K = c = 0, when f is constant (see Theorem2) and (50) holds.For n = 2 the three-dimensional extended metric ˜ g is conformally flat if andonly if the Weyl-Schouten tensor˜ R abc = ˜ ∇ c ˜ R ab − ˜ ∇ ˜ R ac + 12 n (cid:16) ˜ g ac ˜ ∇ b ˜ R − ˜ g ab ˜ ∇ c ˜ R (cid:17) , where ˜ ∇ denotes the covariant derivative w.r.t. ˜ g , vanishes. By applying theformulas derived in Lemma 11 we have that the only non vanishing componentsof ˜ R abc are, for i, k = 1 , R i k = ˙ ff ˜ g ik ( ¨ f f + K − ˙ f ) , which, as shown above, vanish only for m = 1 or in the case when 0 = K = c and f is constant. For n > n + 1)-dimensional extended metric ˜ g isconformally flat if and only if the Weyl tensor¯ C abcd = ˜ R abcd + 1 n − (cid:16) ˜ g ac ˜ R bd − ˜ g ad ˜ R bc + ˜ g bd ˜ R ac − ˜ g bc ˜ R ad (cid:17) ++ ˜ Rn ( n −
1) (˜ g ad ˜ g bc − ˜ g ac ˜ g bd ) . vanishes and, by applying Lemma 11, this is true for all manifold Q of constantcurvature. 13 Quantization
We consider here quantization for the case m ≤ m = 1, it iswell known how to associate a first order symmetry operator with any constantof motion linear in the momenta. In [2] the quantization of quadratic in themomenta first integrals of natural Hamiltonian functions has been analyzed andwe recall here the results relevant for our case.Let ˆ H be the Hamiltonian operator associated with the Hamiltonian H = g ij p i p j + V , we haveˆ H = − ~ ∇ i ( g ij ∇ j ) + V = − ~ V, where ∆ is the Laplace-Beltrami operator. Let T = T ij p i p j + V T be a firstintegral of H , let ˆ T = − ~ ∇ i ( T ij ∇ j ) + V T . (51)We have (Proposition 2.5 of [2]) Proposition 13.
Let be { H, T } = 0 , then [ ˆ H, ˆ T ] = 0 if and only if δC = δ ( T R − RT ) = 0 , (52) where R is the Ricci tensor, T and R are considered as endomorphisms onvectors and one-forms and ( δA ) ij...k = ∇ r A rij...k , is the divergence operator for skew-symmetric tensor fields A . For our purposes we need to apply (52) to the Ricci tensor of the extendedmetric and to the constant of the motion T = U G . By assuming constantthe curvature K of Q , the components of ˜ R ab are given by inserting f = K S κ ( Km u + u ) or f = mA in Lemmas 10 and 11; the covariant componentsof the Ricci tensor are given respectively by˜ R = − n κK m , ˜ R i = 0 , ˜ R ij = K m nκ + ( n − m − T k ( Km u + u )) ! g ij , for K = 0 and ˜ R ab = 0 for K = 0.In order to make computations easier, we remark that for A , B two-tensorson a Riemannian manifold ( ˜ Q, ˜ g ) we have( AB − BA ) ac = A ab B bc − B ab A bc = A ad B cd − g ad g ec B db A be . (53) Lemma 14.
For any symmetric tensor T ij the (1,1) components of C = T ˜ R − ˜ RT , where ˜ R is the Ricci tensor of ˜ g , are C = 0 ,C i = T i W,C i = − ˜ g ij T j W,C ij = 0 , here W = ( n −
1) ¨ f f − ˙ f + Kf . (54) Remark 8.
We immediately have that if W = 0 then C = 0 and, by Proposition13 , { H, T } = 0 implies [ ˆ H, ˆ T ] = 0. However, by Theorem 12, W = 0 if andonly if either n = 1, m = 1 or f is constant, i.e., if and only if ˜ g is of constantcurvature. Theorem 15.
For m = 2 , { H, T } = 0 implies [ ˆ H, ˆ T ] = 0 if and only if ˜ g is ofconstant curvature i.e., if and only if n = 1 or f is constant.Proof. If K = 0, and therefore c = 0 and f is constant, then W = 0. Otherwise,when K = 0 and c = 0, by computing T = U G and by applying Proposition 3we get T = G,T i = γ ∇ i G,T ij = − K γ Gg ij , where γ is given by γ = ( T κ ( Km u + u )) − , as proved in Theorem 2. A straightforward computation gives δC = γW (cid:0) g il ∂ il G + ∂ l G ( ∂ i g il + g il ∂ i ln √ g ) (cid:1) == γW ∆ G = − γnKW G,δC i = f ∂ i G ddu ( γf W ) , where g = det( g ij ). By inserting the expressions of γ and of f = S κ ( Km u + u ) K we have that there are no non-trivial ( G = const. ) solutions to δC = 0 otherthan those such that W = 0, that is, after Remark 8, when n = 1 or ˜ Q is ofconstant curvature.In a recent paper [1], where particular conformally flat, non-constant cur-vature manifolds are considered, it is shown that even if the Laplace-Beltramiquantization of some first integrals of the Hamiltonian fails, their quantizationis somehow made possible by considering the conformal Schr¨odinger operatorinstead of the standard (Laplace-Beltrami) one. The conformal Schr¨odinger op-erator is related to the standard one by an additional term proportional to thescalar curvature. In Theorem 12, we proved that our extended Hamiltoniansfor n > Q, ˜ g ) and by ∆ theLaplace-Beltrami operator of the constant curvature manifold ( Q, g ), a directcalculation shows that ˜∆ = ∂ u + n ˙ ff ∂ u + Kf ∆ , (55)and [ ˜∆ , ∆] = 0 . Therefore, beingˆ H = − ~ ∂ u + n ˙ ff ∂ u ) + Kf ˆ L, L = − ~ V, we have Proposition 16. ˆ L is a symmetry operator of ˆ H : [ ˆ H, ˆ L ] = 0 . Since ˆ H and ˆ L have common eigenfunctions, from ˆ Hψ = µψ and ˆ Lψ = λψ we obtain for the eigenfunction of ˆ H the following characterization Proposition 17.
The function ψ ( u, q i ) is an eigenfunction of ˆ H if and only if ψ is an eigenfunction of ˆ L and − ~ ∂ u ψ + n ˙ ff ∂ u ψ ) + (cid:18) Kλf − µ (cid:19) ψ = 0 . (56) Acknowledgements
This research was partially supported (C.C.) by the European program “Dotericercatori” (F.S.E. and Regione Lombardia).
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