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Institute of Biophysics, Bulgarian Academy of Sciences,


Reduction in Stages and Complete Quantization of the MIC-Kepler Problem
Iva lo M. Mladenov and Vasil V. Tsanov

y

4 May, 1998

? Institute

of Biophysics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 21, So a { 1113, Bulgaria
yDepartment

of Mathematics , University of So a, 5 James Bourchier Bld., 1126 So a, Bulgaria

Abstract. The one-parameter deformation family of the standard Kepler problem known as the MIC{Kepler problem is completely quantized using the explicit momentum mapping of the torus actions on some toric manifolds and some equivariant cohomology theory. These manifolds appear as symplectic faces of the system. At any level of the reduction process the geometric quantization scheme produces all relevant quantum{mechanical numbers. Key words: geometric quantization, equivariant cohomoligies AMS subject classi cations: 58F06, 58F07, 81S10
Research partially supported by Bulgarian NSF-Project F-644/96. Research partially supported DFGvproject number 436 BUL 113/96/5.

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1 Introduction
As a rule phase spaces of the most interesting classical Hamiltonian systems are cotangent bundle of smooth con guration manifolds and their quantization does not present serious problems (see Sect.2). However, the rst step { the prequantization { produces only part of the quantum numbers and one should use other devices in order to obtain the spectrum of the complete set of Dirac observables. Here we present a detailed treatment of a concrete dynamical system and show that reduction in stages (either at classical or quantum level) produces the desired information about the spectrum. The system in question is the one-parameter deformation family of the Kepler problem known as MIC{ Kepler problem(see Sect. 4). We should point out that despite geometric quantization concept is thirty years old such treatment is absent even for the standard Kepler problem. A possible explanation of this situation can be traced back to the general fact that one can quantize unambiguously only functions which are polynomials up to a second degree in phase space coordinates while the square of the angular momentum which is fourth degree polynomial does not belongs to this set. On the other side the choice of the momentum as an element of the complete set of observables is dictated by the spherical symmetry of the problem. Another wellknown fact is that the symmetries manifest themselves in the possibilities of separating the variables in the Schrodinger equation in di erent coordinate systems and this is connected with the existence of constants of motion. Simultaneous diagonalization of the Hamiltonian and the third components of momentum and Runge{ Lenz vector corresponds to separation of variables in parabolic coordinates and has been noticed by Bargmann. Working in much more abstract setting we will follow essentially the same idea in order to derive missing quantum numbers. >From mathematical point of view the results which will be presented below follow from quantization of the momentum map associated with free torus actions on some symplectic (toric) manifolds.

On any symplectic manifold (M; !) the symplectic form ! generates a Lie algebraic structure in the space R1(M ) of smooth real-valued functions on M: The problem for nding the representations of R1(M ) was approached for the rst time by Dirac 1] in the case (M R2n ; ! dp ^ dq), and after that has been generalized by Segal 2] for the phase spaces which are cotangent bundles and nally by Kostant 3] and Souriau 4] for arbitrary symplectic manifolds. The starting point is the observation that if we are able to associate with every classical variable a quantum one then the commutator of two quantum variables should represents up to a multiplicative number the Poisson bracket of the classical ones. This part of the programme can be easily realized and nowadays is referred as prequantization. Below we summarize the relevant notion and de nitions. De nition 2.1 The symplectic manifold (M ; !) is pre-quantizable if !=2 ] is in the image of the map 1

2.1 Kostant-Souriau Programme

2 Geometric Quantization

2 2 HCech(M; Z) ! HdeRham (M; R ); (2.1) where ] denotes the de Rham cohomological class. When M is a compact manifold this condition is equivalent to Z 1 ! 2 Z; for every two ? cycle 2 H (M; Z): (2.2) 2 2 and produces the quantization of charge, spin and energy levels of some physical systems. If (M; !) is pre-quantizable, there exists a hermitian line bundle L ! M , whose Chern class is 1 !], equipped with a connection r which curvature form is ?i! and hermitian 2 scalar product h( ; ) that is invariant with respect to the parallel transport 3, 5, 6, 7]. The irreducibility of the representations which is the second stage (quantization) of the programme is achieved by introducing additionally a new structure called polarization. A real polarization over M is a such map that juxtapose to each point m 2 M a real subspace Fm Tm (M ) which is maximally isotropic integrable distribution. Example 2.2 Let Q be a smooth manifold and let T Q be its cotangent bundle. If fpi; qig are the local canonical coordinates in T Q, then an easy check shows that the vector elds @ @ @ X1 = @p ; X2 = @p ; : : : ; Xn = @p 1 2 n de ne a real polarization over T Q which is known as vertical polarization.

Example 2.3 The two-dimensional sphere does not allows real polarization because of
the non-existence of non-singular real vector eld on S 2 :

This situation suggests also the generalization of the above notion, namely: A complex polarization over M is a map F which assigns to each point m 2 M a subspace Fm of TmC (M ) which is maximally isotropic integrable distribution, and besides the distribution Dm = Fm \ Fm is of some xed dimension at each point m 2 M: The polarization F is called Kahlerian if Fm \ Fm = 0: For any kind of polarization F the potential of the symplectic form ! (i.e. ! = d ) is called an adapted to it if (X ) = 0 for every X 2 F: The quantum pre-Hilbert space is built up by the polarized sections of L which de nition is as follows: Let M; !; L; r and F be as de ned above. The polarized sections of L form the line bundle

LF = fs 2 Sect(L) ; rX s = 0;

for all X 2 X(M; F )g:

In order to have true Hilbert space we need some measure(or density) which is an element of a second line bundle. This can be introduced if we consider the elements of the cotangent bundle T (M ) that vanish on F and form a subbundle F T C (M ) which is called annihilator of F: By the very de nition of the symplectic form we have that the map

2 F ! i( )! 2 F
2

is an isomorphism of F and F : This means that we can form the line bundle KF = ^nF over M that will be further referred as a canonical bundle of F: If f 1; 2; : : : ; ng is a basis of F , then K! = i( 1)! ^ i( 2 )! ^ : : : ^ i( n)! is a basis in KF and for every g 2 GL(n; C ); (K g )! = detg:K! : Let (M; !) be a symplectic manifold and F is a complex polarization on it. We will say that M is a metaplectic manifold if there exists a line bundle N 1=2 over M such that

N 1=2 N 1=2 = KF :
One can show that (M; !) is metaplectic if and only if the rst Chern class of KF is zero modulo two and this property does not depend on the choice of F . In this case the group H 1(M; Z2) parameterizes the set of \square roots", i.e. the set of all N 1=2 which 1 satisfy the above condition. The sections of NF=2 which are constant along F are called half-forms normal to F: 1 ~ The line bundle Q = LF NF=2 over M is called a quantum line bundle because its sections are considered as elements of the Hilbert space HF . The classical observables which can be quantized directly are those that preserve the polarization F , i.e. ff 2 R1(M ) ; Xf ; F ] F g, where Xf is de ned by the equation i(Xf )! = ? df: If = s , 1 ~ where 2 ?(Q); s 2 ?(LF ); 2 ?(NF=2 ) are sections of the corresponding line bundles, the associated with f quantum operator acts in HF as speci ed below : f^( ) = (?irXf + f )s ? is L(Xf ) : (2.3) Identifying the sections of LF with functions on M (which is possible because LF is a line bundle) the action of f^ in HF can be written in the form f^ = (?iXf ? (Xf ) + f )' ? i' L(Xf ) (2.4) where is the potential one form of !: Actually this explicit formula has found very few applications (cf. Sect. 6 ) as most of the considerations end with checking the consistency of the scheme relying on (2.3). After cotangent bundles and co-adjoint orbits Kahlerian manifolds form another important class of symplectic manifolds. According Darboux theorem all symplectic manifolds (of xed dimension) are locally the same but in practice they appear with some additional geometric structure. It presence in the setting of geometric quantization helps in many cases to answer de nitely the question if the given symplectic manifold (M; !) allows such quantization. A trivial example is provided by even-dimensional complex projective spaces. The well-known fact for these manifolds is that

2.2 Czyz-Hess Scheme

H 2(CP 2n; Z2) = Z2:
3

On the other hand we know that the symplectic manifold (M; !) can be quantized if M is a metaplectic manifold, i.e. H 2(M; Z2) = 0: So, even-dimensional complex projective spaces can not be treated in Kostant-Souriau scheme . On the other hand they appear as orbit manifolds of the odd-dimensional harmonic oscillators which form an important class of dynamical systems . Fortunately this problem can be taken away by slight modi cation of geometric quantization scheme as developed by Czyz 8] and Hess 9] and outlined bellow. De nition 2.4 Let (M; !) be such Kahlerian manifold that q] = 21 !] ? 1 c1 (M ) belongs 2 to the image of : H 2 (M; Z) ! H 2 (M; R ) and q is positive, i.e. q( ) 0 for any positively oriented two-cycle 2 H2(M; R ): The complex line bundle Q whose rst Chern class c1 (Q) is q is called quantum bundle. 1 1 ~ If the bundles LF and NF=2 exist then there exists also the bundle Q = LF NF=2 so ~ ~ that c1(Q) = c1(Q) and therefore Q and Q are equivalent. Among symplectic manifolds the Kahlerian ones are those which possess canonical anti-holomorphic polarization that makes identi cation of quantum states with holomorphic sections quite natural. Now, xing a positive harmonic representative 2 c1(Q) and connection r which curvature is ?2 i we are in position to de ne also and r - invariant hermitian structure h ( ; ) on Q: We recall that, the curvature of the hermitean metric h on the bundle Q satis es i @ @ log h ' ! ? 1 c (M ): 2 2 21 The space of holomorphic sections H 0((M; Q) of Q can be converted into Hilbertian space H if we introduce the scalar product Z 1 < s; t >= h (s ; t) ; ! = 2 ; s; t 2 ?(M; Q) ; n = 2 dimM: M
n(n?1)=2 and where := (?1) ! ^ ! ^ : : : ^ ! is the natural volume form on M: If our n! manifold M is simply-connected the hermitian structure is de ned up to a positive factor and H is de ned up to an isomorphism which depends on the choice of the connection r: The representations are build up following the prequantization recipe in which (L; !) is exchanged for (Q; ! ) i.e. to the classical observable (i.e. a function f on the phase space), there corresponds a quantum operator

(f ) 2 End H 0(M; Q); (f )s (?irXf + f )s where s 2 H 0(M; Q), and now the vector eld Xf is de ned by:

i(Xf )! = ? df: The only problem here is that ! is not always non-degenerated. More detailed exposition can be found in Czyz 8] and Hess 9].
4

3 Classical and Quantum Reductions
When a Lie group G acts symplectically(canonically) on the phase space (P; !) of the Hamiltonian system (P; !; H ) leaving the Hamiltonian H invariant it generates quite naturally a mapping from P into the dual space g of its Lie algebra g whose components are integrals of motion for the dynamical system. This means that the motion takes place inside a constraint submanifold C P and sometimes possesses gauge degrees of freedom. Passing on a new manifold where they are discarded is known for centuries in mechanics as reduction procedure and its modern formulation given below is due to Marsden and Weinstein 10]. Theorem 3.1 Let (P; !) be a symplectic manifold on which acts canonically the Lie group G, and J : P ! g be the Ad -equivariant momentum mapping of this action. Let us suppose that 2 g is a regular value of J and that the isotropy group G act freely and properly on J ?1 ( ): Then P = J ?1 ( )=G is a symplectic manifold with a symplectic form de ned by ! = i ! where : J ?1 ( ) ! P is the canonical projection and i : P ! P is the embedding. Let H : P ! R is G-invariant Hamiltonian function. The ow induced on P is also a Hamiltonian one with Hamiltonian function H de ned by the relation H =H i : If the Hamiltonian system (M; ! ; H ) allows a symmetry group action commuting with that of G, the reduced system (P ; ! ; H ) keeps this symmetry. A special case of the above theorem which will be of immediate interest in the sequel is the case when P is a cotangent bundle T (M ) of some manifold M on which acts freely and properly the one-parameter Lie group G: Let M ! N = M=G be the induced principal G-bundle and ~ be the connection one-form. The reduced symplectic manifold P is symplectomorphic with T N which symplectic form ! is the sum of the canonical form on T N and a magnetic term N d ~ where N is the canonical projection N : T N ! N 11]. Thus, each Hamiltonian system with symmetry can be treated as dynamical system either on (P; !) or (P ; ! ) and what is more important - there is no formal distinction at classical level between working on the initial or reduced phase space. There are plenty of strong results concerning the quantum mechanical counterpart of this situation which tell us when quantization and reduction are coherent procedures (see 13, 12, 14, 15]). In order to give the reader the avor what to expect in this situation and because we will make use of it we quote the following result: Theorem 3.2 (Guillemin & Sternberg 13]). Let us suppose that the (extended) phase space (P; !) is a compact and quantizable, G is a compact Lie group, 0 2 g is a regular value of J and F is a Kahlerian G-invariant polarization over P: Then, there exists an isomorphism between the G-invariant sections of LF and the sections of the quantum line bundle over the reduced phase space (P0 ; !0): The situation is even more favorite - in the above setting the reduction and (pre)- quantization are interchangeable procedures. 5

4 The MIC{Kepler Problem
=d +

_ The Hamiltonian system (T R3 ; ; H ) , where _ T R3 T (R3 n f0g) f(p; q) 2 R3 R3 ; q 6= 0 g

; =

3 X

1 2 2 2 H = 2 jpj2 ? =r + 2=2r2; jqj2 = q1 + q2 + q3 = r2; ; 2 R; > 0; is known as the MIC{Kepler problem 16, 17]. Using more or less standard physical terminology, the problem consists in studying the motion of charged particle in a eld which is ~ a superposition of a magnetic monopole eld B = ? ~=jqj3 and the elds generated by q the Newtonian potential ? =r and centrifugal potential 2 =2r2. We will see that the energy level submanifolds H ?1(E ) for negative values of the energy are lled up with closed orbits. This hints a presence of \hidden" symmetry and \accidental" degeneracy of the _ spectrum. Actually, the \hidden" symmetry of the Hamiltonian system (T R3; ; H ) is SO(4) generated by the constants of motion ~ q ~ q ~ ~ p q p L = ~ p + ~=r ; A = (L ~ + ~=r)= ?H ; which have interpretation of a \total angular momentum" and generalized Runge{Lenz vector. The names are borrowed by the standard Kepler problem which can be viewed as a special \point" of this one-parameter deformation family. The classical Kepler problem ( = 0) was geometrically quantized by Simms 18] and Mladenov & Tsanov 19]. Here we will apply the geometric quantization to extended and the reduced phase spaces of the _ Hamiltonian system (T R3; ; H ) which results in coinciding spectra. We will present them as Theorem 4.1 (Mladenov & Tsanov 17]). The discrete spectrum (bound states) of the MIC{Kepler problem ( - xed, - xed and quantized) consists of energy levels :

j =1

pj dqj ;

=?

=(2jqj3)

3 X

i;j;k=1

ijk qi dqj ^ dqk ;

(4.1)

EN = ? 2=2N 2 ;

N = j j + 1; j j + 2; j j + 3 : : :

The magnetic charge can take the values = 0; 1 ; 1; 3 ; 2 : : : 2 2 and the multiplicity of the energy level EN is :

m(EN ) = N 2 ? 2:

6

_ The Hamiltonian system (T R4 ; ; H ), where : _ T R4 = T (R4 n f0g) = f(y; x) 2 R4 R4 ; x 6= 0 g; = dy ^ dx = and
?

5 Conformal Kepler Problem
4 X

j =1

dyj ^ dxj

(5.1)

H = jyj2 ? 8 =8jxj2 ; ? xed constant is known as a conformal Kepler problem 20]. Let us introduce additionally two other _ Hamiltonian functions on the phase space (T R4 ; ), namely, that one of the Harmonic oscillator : K = jyj2 + 2 jxj2 =2 ;
?

? an arbitrary positive constant;

and that one of the \momentum" : 1 M = 2 (x1 y2 ? x2 y1 + x3 y4 ? x4 y3): p Lemma 5.1 Let E < 0 and = ?8E: Then

H ?1(E ) = K ?1(4 )
and the ows de ned by the Hamiltonians H and K coincide on these hypersurfaces up to re-parameterization.

Proof. Taking into account the above de nition it is obvious that we have 4jxj2(H + 2=8) = K ? 4 ;
which proves the rst statement. Further on H and K will be denoted by H and K . In order to prove the second one we need only to notice that the Hamiltonian vector elds XH and XK when restricted to energy level submanifolds H ?1(E ) = K ?1(4 ) are related as follows : 4jxj2XH = XK ; and so the proof of the lemma is complete. _ The complex coordinates on (T R4 ; ) written below depend on the same arbitrary positive constant chosen above

z1 = (x1 + ix2 ) ? i(y1 + iy2); z3 = (x1 ? ix2 ) ? i(y1 ? iy2);
7

z2 = (x3 + ix4 ) ? i(y3 + iy4); z4 = (x3 ? ix4 ) ? i(y3 ? iy4 ):

(5.2)

_ In these coordinates T R4

n D, where D = fz 2 C 4 ; z1 = ?z3 ; z2 = ?z4 g;
C4

and the symplectic form turns out to be (up to a multiplicative constant) the standard Kahler form on C 4 4 X dzj ^ dzj : = 4i dz ^ dz = 4i Finally, the hamiltonian functions K and M can be written in these coordinates as
j =1

K = (jz1 j2 + jz2j2 + jz3 j2 + jz4 j2)=4
and

(5.3) (5.4)

M = (jz1j2 + jz2 j2 ? jz3j2 ? jz4j2 )=8 :

It should be noted that these hamiltonians and the symplectic form are well de ned over the manifold _ _ C 4 = C 4 n f0g T R4 : Let Kt ; Mt denote the ows of the Hamiltonian systems (C_ 4 ; ; K ), (C_ 4 ; ; M ): Lemma 5.2 For every z 2 C_ 4 and s; t 2 R; the corresponding ows are:

Kt z = ( ei t z1 ; ei t z2 ; ei t z3 ; ei t z4 ); Msz = (eis=2 z1 ; eis=2z2 ; e?is=2z3 ; e?is=2z4 ):

(5.5) (5.6)

In particular, the ows of all three Hamiltonians H; K and M commute where de ned. Proof. The explicit expressions for the ows Kt ; Ms are obtained by direct calculations. The last assertion follows from these expressions and Lemma 5.1. In view of the lemma that have been just proved, the ow Ms de nes a symplectic U (1)action over C_ 4 : The \momentum" for this action is M itself. Let us remark that the set _ D and consequently its complementary set T R4 are invariant under this U (1)-action. Through every point there pass just one orbit and the Hamiltonian function H invariant _ on these orbits. All this means that the Hamiltonian system (T R4 ; ; H ) can be reduced with respect to U (1): The result of this reduction is summarized in the following lemma : Lemma 5.3 20, 17] Let 2 R be the value of the momentum map of the lifted Hopf _ action on T R3 : Then _ M ?1 ( )=U (1) T R3 and when reduced and H produce and H , i.e. one ends with the MIC{Kepler problem. 8

Besides, if one reduce the constants of motion of the conformal Kepler problem : M1 = (z1z2 + z2 z1 ? z3 z4 ? z4 z3)=8 A1 = (z1z2 + z2 z1 + z3 z4 + z4z3 )=8 M2 = (z1z2 ? z2 z1 + z3 z4 ? z4 z3)=8 i A2 = (z1 z2 ? z2 z1 ? z3 z4 + z4 z3)=8 i (5.7) M3 = (jz1j2 ? jz2j2 ? jz3 j2 + jz4j2)=8 A3 = (jz1j2 ? jz2j2 + jz3j2 ? jz4j2)=8 ~ ~ one gets the momentum L and the generalized Runge{Lenz vector A which are constants of the motion for the MIC{Kepler problem.

6 Quantization of the Extended Phase Space

@ z1 @ z2 @ z3 @ z4 i zdz of : The Hilbert space consists of \wave functions" of adapted potential = ? 4 the form = ' where ' is holomorphic and = (dz1 ^ dz2 ^ dz3 ^ dz4)1=2 : Essentially Dirac's idea concerning quantization in the presence of constraints that are not eliminated at the classical level is that they should be enforced at the quantum one. In our case the constraints K = 4 and M = select the energy-momentum manifold EM( ; ) and therefore the acceptable quantum states are those that belong to the subspace HJ of H de ned below : ^ ^ HJ = f 2 H ; K = 4 ; M = g: Taking into account all of the above and formula (2.4) we write down the quantized version of our operators as
^ K = = (z1 @ + z2 @ + z3 @ + z4 @ + 2)' @z1 @z2 @z3 @z4 (N + 2) = 4 9

are called energy-momentum manifolds. _ EM( ; ) = f(y; x) 2 T R4 ; K = 4 ; M = g: Underp reduction EM( ; ) falls down (via ) over the energy hypersurface H = ? 2=8 ( = ?8E ) of the MIC{Kepler problem. As a set EM( ; ) is not empty if and satisfy the condition jj 2 : In this section we suppose (by the reasons that will be clari ed in the next one) that we have strong inequality j j < 2 : Following 21] (see also 22]- 23]) we will change our _ _ viewpoint and will consider (T R4 ; ) as an \extension" of (T R3 ; ): We will prove Theorem 4.1 working with the complex coordinates de ned in (5.2) , the polarization F \spanned" by the anti-holomorphic directions f @ ; @ ; @ ; @ g and

De nition 6.1 The level hypersurfaces of the map J : C 4 nD ! R2, J (z) = (K (z); M (z))

;

N = 0; 1; 2; : : :

1 @ @ @ @ ^ M = 2 (z1 @z + z2 @z ? z3 @z ? z4 @z )' = 1 2 3 4 where ' is a homogeneous monom of degree N in z1; z2 ; z3 and z4 : Introducing N = N =2 + 1 and solving 2N ?8E = 4 ; we obtain the energy spectrum EN = ? 2 =2N 2 as well n1 + n2 + n3 + n4 = 2N ? 2; n1 + n2 ? n3 ? n4 = 2 , ni

and

;

p

0; i = 1; 2; 3; 4:

The last constraint relation is equivalent to Dirac's quantization of the magnetic charge : 1 = 0; 2 ; 1; 3 ; 2 : : : 2 Besides we get : n1 + n2 = N + ? 1 = N1 = 0; 1; 2; ::: and n3 + n4 = N ? ? 1 = N2 = 0; 1; 2; ::: which combined tell us that the possible values of N are given by the formula N = j j +1 , j j + 2 , j j + 3 ,. . . In order to nd the degeneracies m(EN ) one should notice that ' can be represented as a product '1 (z1; z2 ):'2(z3 ; z4) of homogeneous monomials of degree N1 and N2 respectively. So, the dimension of the Hilbert space H ;N is :

m(EN ) = dimH ;N = (N1 + 1)(N2 + 1) = N 2 ? 2;
and this ends the proof of the theorem.
resentation ( N1 ; N2 ) = ( 2 ; 2 ) of the global symmetry group of the MIC{Kepler 2 2 problem Spin(4) = SU (2) SU (2) ( - half-integer) or SO(4) ( -integer).

Remark 6.2 The Hilbert +space NH ;N1 is the carrier space for the unitary irreducible repN ?1 ? ?

The wave functions in H = H ;N are labeled uniquely by four quantum numbers which ^ ^ ^ ^ are the eigenvalues of the complete set of commuting operators M ( ); H (N ); M3 (m); A3 (`); where 1 @ @ @ @ ^ M3 = 2 (z1 @z ? z2 @z ? z3 @z + z4 @z ) = m (6.1) 1 2 3 4 and @ @ @ @ ^ (6.2) A3 = 1 (z1 @z ? z2 @z + z3 @z ? z4 @z ) = ` : 2 1 2 3 4 10

Looking at (6.1) and (6.2) one can conclude immediately that m and l can take either integer or half-integer values. It is to be noted also that this result is closely related with convexity theorem of the torus actions on symplectic manifolds 23]. Indeed, let us consider the ows U& ; V generated by M3 and A3 ,

U& z = (ei&=2 z1 ; e?i&=2 z2 ; e?i&=2 z3; ei&=2 z4 )

(6.3)

V z = (ei =2 z1; e?i =2z2 ; ei =2 z3 ; e?i =2 z4 ) (6.4) in conjunction with Kt and Ms. Doing so we realize that we have at disposition an action of the four-torus T 4 on our symplectic manifold (C_ 4 ; ). Introducing new \time" variables s & 1 = t+ + + 2 2 2 s & 2 = t+ ? ? 2 2 2 (6.5) s?&+ 3 = t? 2 2 2 s & 4 = t? + ? 2 2 2
this action takes the form ( ; z) = (ei 1 z1 ; ei 2 z2 ; ei 3 z3 ; ei 4 z4 ) (6.6) and the associated moment J is readily given by J (z) = 81 (jz1j2 ; jz2j2; jz3j2; jz4j2) (6.7) which makes obvious that the image set is convex. Besides, our representation space is spanned by the homogeneous polynomials of degree N in the variables z1 ; z2; z3 ; z4 on which the torus element g = (ei 1 ; ei 2 ; ei 3 ; ei 4 ) is represented by the transformation
X

anzn !

X

an ei :nzn:

(6.8)

The multi-indices n = (n1; n2; n3 ; n4) which appear above obey n1 + n2 + n3 + n4 = N and provide labels for the irreducible multiplicity-free representations N of the torus T 4 :

7 Quantization of the Orbit Manifolds
In Section 5 we have established that the energy level submanifolds consist entirely of closed orbits. This allows H ?1(E ) to be factorized with respect to dynamical ow and the so obtained manifold H ?1(E )=U (1) = O (E ) is known as an orbit manifold. Its complete description as a symplectic manifold is given below : 11

p Theorem 7.1 (Mladenov & Tsanov 17]). Let E < 0 and = ?8E: Then : i) if j j < 2 O (E ) = P 1 P 1 ii) if j j = 2 O (E ) = P 1 iii) if j j > 2 H ?1(E ) ;
The reduced symplectic form over P 1 P 1 is : (E ) = 2 (2 + ) !1 + 2 (2 ? where )!
2

(7.1)

dj (7.2) !j = 2i (1 +^ d2)j2 ; j = 1; 2 jj and ( 1; 2) are whichever non-homogeneous coordinates on P 1 P 1: The symplectic form over P 1 in item ii) is the respective non-zero component of O (E ) (depending on the sign of ). This theorem reduces quantization of the MIC{Kepler problem to geometric quantization of the compact Kahler manifolds P 1 P 1 and P 1. The proof is based on the following lemma: Lemma 7.2 O (E ) = J ?1(4 ; )=U (1) U (1):

Proof. When 6= 0 the orbits of the Hamiltonian H coincides with that of K described _ by Lemma 5.1 . In particular, none of them belongs to C_ 4 n T R4 = D nf0g and therefore we have one-to-one correspondence between the orbits of the MIC{Kepler problem on the energy hypersurface H = E and the orbits of the torus action on J ?1 (4 ; ) described in Lemma 5.2 and this implies that the orbit spaces are identical. What remains to be done in order to prove Theorem 7.1 is to describe properly J ?1 (4 ; ): For that purpose we remark that the system of equations K = 4 ; M = is equivalent to the system
jz1 j2 + jz2j2 = 4(2 + );
so we can conclude that

jz3 j2 + jz4 j2 = 4(2 ? )

J ?1(4

; )=:

8 <

S 3 S 3 when j j < 2 , S3 when j j = 2 ; ; when j j > 2 :

The projection p : S 3 S 3 ! P 1 P 1 is de ned through the Hopf's map of the corresponding factors p(z1 ; z2; z3 ; z4) = ( z1 : z2]; z3 : z4]); where z1 : z2 ]; z3 : z4 ] are the homogeneous coordinates over P 1 P 1: In accordance with Lemma 7.2 the projection p is just the factor-map

J ?1 (4 ; ) ! J ?1 (4 ; )=(U (1) U (1))
12

In this way item i) of the Theorem 7.1 is proven. It is obvious that the restriction of p on the non-trivial factor gives the map we are needed in order to prove ii): Finally, item iii) is a trivial statement and what else has to be done is to compute the reduced symplectic form. In the non-homogeneous coordinates ( 1; 2) = (z2 =z1; z4 =z3) over P 1 P 1; p is simply

p(z1 ; z2; z3 ; z4) = ( 1; 2): Referring to Lemma 7.2 we can write
(E ) = jS3 S3 ; where S 3 S 3 are spheres de ned above. An easy check in coordinates shows that this is true which means that Theorem 7.1 is also proved.

p

if

De nition 7.3 The line bundle L over the compact Kahler manifold X is called positive,
Z

c1(L) 0; for every positively oriented cycle 2 H2(X; Z):

For this type of bundles H 0(X; O(L)) 6= 0:

Theorem 7.4 (see Gri ths & Harris 24]). The group H 2(P 1 P 1 ) = Z
by !1 ]; !2 ] and

Z is generated

1 1 c1(NF=2 ) = ? 2 c1(P 1 P 1) = ? ( !1] + !2]): In view of the prequantization condition (2.2) we have 1 (E ) = N ! + N ! ; N ; N 2 Z; 1 1 2 2 1 2 2 which means that 2 + = N1 as well 2 ? = 1 (N1 ? N2) ; 2 13 = N2 ; = 4 (N1 + N2 ) :

Introducing N = 1 (N1 + N2 ), we get immediately N1 = N + ; N1 = N ? as well 2 the energy spectrum of the MIC{Kepler problem EN = ? 2 =2N 2: The Hilbert space H 0(P 1 P 1 ; QN ) is non-trivial if the rst Chern class of the line bundle QN ! P 1 P 1

c1(QN ) = (N1 ? 1) !1] + (N2 ? 1) !2]
is positive, i.e. N1; N2 1 and N j j+1: Finally, the degeneracies m(EN ) of the energy levels EN which coincide with dimensionalities of the spaces of holomorphic sections of quantum line bundles QN are calculated by Riemann{Roch{Hirzebruch theorem:

m(EN ) = dimH 0 (O (E ); QN ) = N1 N2 = N 2 ?

2

:

The expression for the third component of the Runge{Lenz vector is actually the momentum mapping of the circular action around vertical axes of the spheres. If we x its value to be ` then the momentum manifold j 2 j 2 N1 1 + 1jj j2 + N2 1 + 2jj j2 = N1 + N2 ? ` = N ? ` (7.3) 2 2 1 2 is either the sphere S 3 when N ? ` > 0, four points when N ? ` = 0 or the empty set in the case N ? ` < 0. This can be seen quite easily if we introduce the following set of coordinates N1 1=2 ; N2 1=2 (7.4) 1= 1 2= 2 1 + j 1 j2 1 + j 2j2 in which (7.3) becomes obviously

Remark 7.5 The observables M3 and A3 in the complete set which survive under reduction can be expressed in the nonhomogeneous coordinates ( 1 ; 2 ) over O (EN ) as follows: 2 2 M3 ;N = N1 1 ? jj 1jj2 ? N2 1 ? jj 2jj2 2 1+ 1 2 1+ 2 ? 2 ? 2 A3 ;N = N1 1 + jj 1jj2 + N2 1 + jj 2jj2 : 2 1 1 2 1 2

j 1j2 + j 2j2 = N ? `:

(7.5)

In the rst of the above listed cases we have a free action of SO(2) on J ?1(`) and therefore we can factorize it. The reduced manifold is topologically the sphere S 2 and the reduced symplectic form is !` = 2 (N ? `) ; (7.6) where is the form (7.2) written in any of the non-homogeneous coordinates on the projective line 1 : 2]: Now the quantization condition reads (N ? `) ? = k ; k 0 14 (7.7)

from which follows that the maximal value of ` is N ? 1: Using Riemann{Roch theorem one can easily nd that the number of the global holomorphic sections of the reduced quantum bundle Lk over the sphere S 2 is k + 1 = N ? ` . Introducing := 1= 2 the last function M3 ;N from the complete set of observables can be written as 2 (N ? `) 1 ? j j2 + (7.8) 1+j j while the corresponding \quantum" operator is ?2 @@ + N ? ` ? 1 + : (7.9) The spectrum of this operator in ?(S 2; O(Lk )) consists of equidistant of step two integer or half-integer eigenvalues m 2 ?k + ; ?k + + 2; : : : ; ?k + ? 2; k + ]: At classical level (7.8) is just the momentum map of the circle action around the third axe of the sphere S 2 (so we can forget the additive constant ) and if jmj < N ? ` this action is free. This means that the inverse image of the momentum map is a circle and after reduction we end with a point as reduced phase space. The representation space associated with this point is one-dimensional as the only SO(2)-invariant section which descends from S 2 is the constant section. This can be seen also in another way if we remember that since the very rst days of quantum mechanics there are attempts to associate the volume in the phase space with the number of the pure states. This was proven to be assymptocally true (up to universal factor) by Heckman on the basis of the Duistermaat{Heckman exact stationary phase formula 25]. The main ingredient of this formula is the set of xed points of the action which till now were not taken into account in our considerations. The S 1 action on S 2 was treated in 26] and the result (in our notation) is : 8 < 1 if jmj < N ? `; a point 2)= vol(Sm : (7.10) 0 if jmj N ? `; empty: The S 1 diagonal action on S 2 S 2 which has four xed points mentioned above is studied by Wu 27] and in that case 8 < 2 (N ? `) if N ? ` > 0; ? vol (S 2 S 2 )` = : (7.11) 0 if N ? ` 0: Finally the volume of the orbit manifold O (N ) is 4 2N1N2 and all this coincides with the results we have obtained before. In view of the complete coherence of the results obtained at all level of consideration, starting with the extended and ending with a point, we can conclude that the reduction{ quantization technique is the right and straightforward formalism for the treatment of systems with high symmetries. The missing quantum numbers can be derived by quantizing various symplectic manifolds which appear at di erent stages of the reduction procedure. 15

References
1] P. Dirac, Quantum mechanics and a preliminary investigation of the hydrogen atom, Proc. Roy. Soc. A, 110 (1926) 561-579 2] I. E. Segal, Quantization of nonlinear systems, J. Math. Phys. 1 (1960) 468-488 3] B. Kostant, Quantization and unitary representations, Lect. Notes Math., vol. 170 (1970) 87-208 4] J.-M. Souriau, Structure des Systemes Dynamics , Dunod, Paris, 1970 5] D. Simms and N. Woodhouse, Lectures on Geometric Quantization, Lect. Notes Phys., vol. 53, Springer, New York, 1976 6] P. Robinson and R. Rawnsley, The metaplectic representations, Mpc structures and geometric quantization, Memoirs AMS vol.410, 1989 7] J. Sniatycki, Geometric Quantization and Quantum Mechanics, Springer, Berlin, 1980 8] J. Czyz, On Geometric quantization and its connections with the Maslov theory, Rep. Math. Phys., 15(1979) 57-97 9] H. Hess, On geometric quantization scheme generalizing those of Kostant{Souriau and Czyz, Lect. Notes Phys. vol. 139 (1981) 1-35 10] J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys., 5 (1974) 121-130 11] M. Kummer, On the construction of reduced phase space of a Hamiltonian system with symmetry, Indiana Univ. Math. J., 30 (1981) 281-291 12] J. Sniatycki, Constraints and quantization, Lect. Notes Math., vol. 1037 (1983) 301334 13] V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. math., 67 (1982) 515-538 14] M. Puta, On the reduced phase space of a cotangent bundle, Lett. Math. Phys, 8 (1986) 189-194 15] M. Gotay, Constraints, reduction and quantization, J. Math. Phys., 27 (1986) 20512066 16] H. McIntosh and A. Cisneros, Degeneracy in the presence of a magnetic monopole J. Math. Phys., 11 (1970) 896-916 16

17] I. Mladenov and V. Tsanov, Geometric quantization of the MIC{Kepler problem, J.Physics A : Math. & Gen., 20 (1987) 5865-5871 18] D. Simms, Geometric quantization of the energy levels in the Kepler problem, Symposia Mathematica, vol XIV (1974) 125-137 19] I. Mladenov and V. Tsanov, Geometric quantization of the multidimensional Kepler problem , J. Geom. Phys., 2 (1985) 17-24 20] T. Iwai and Y. Uwano, The four-dimensional conformal Kepler problem reduces to three-dimensional Kepler problem with a centrifugal potential and Dirac monopole eld. Classical theory, J. Math. Phys., 27 (1986) 1523-1529 21] I. Mladenov, Geometric quantization of the MIC{Kepler problem via extension of the phase space, Annales de l'Institute Henri Poincare, 50 (1989) 219-227 22] V. Guillemin and S. Sternberg, Variations on the theme by Kepler, AMS Colloquium Publications, vol. 42, AMS, Providence, RI, 1990 23] M. Audin, The Topology of the Torus Actions on Symplectic Manifolds, Birkhauser, 1991 24] Ph. Gri ths and J.Harris, Principles of Algebraic Geometry , Wiley, New York, 1978 25] J. Duistermaat and G. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. math., 69 (1982) 259-268 ; Addendum, ibid. 72 (1983) 153-158 26] E. Witten, Two dimensional gauge theories revisited, J. Geom.& Phys. 9 (1992) 303-368 27] S. Wu, An integration formula for the square of moment maps of circle actions, Lett. Math. Phys. 29 (1993) 311-328

17


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