Classical and quantum spherical pendulum

This paper extends the Bohr-Sommerfeld quantization of the spherical pendulum to a full quantum theory. This the first application of geometric quantization to a classical system with monodromy.

classical functions. In his 1925 paper [13], Heisenberg emphasized the importance of operators that provide transitions between different quantum states. Since Bohr-Sommerfeld approach did not provide transition operators, it was abandoned in favour of the matrix mechanics of Born, Jordan and Heisnberg [4], [5] and the wave mechanics of Schrödinger [17]. Dirac incorporated both approaches in his Principles of Quantum Mechanics published in 1930 [11].
In [9], we showed that, if the Bohr-Sommerfeld basis of the space H of quantum states of a completely integrable system has a structure of a global lattice with boundary and the action-angle variables are globally defined in the open dense set of regular points of the energy-momentum map EM, then the lowering operators that take σ n,m to σ n−1,m can be interpreted as quantization of e −iϕ 1 . Similarly, the operators that take σ n,m to σ n,m−1 can be interpreted as quantization of e −iϕ 2 . In this paper, we show that the Bohr-Sommerfeld basis of the space of quantum states of the spherical pendulum has the structure of the global lattice with boundary so that shifting operators are well defined. However, due to the classical monodromy, our actionangle variables fail to satisfy the defining equation ω = dA 1 ∧dϕ 1 +dA 2 ∧dϕ 2 on L −1 (0). Thus, Dirac's quantization conditions allow us to interperate the lowering operators as quantizations of e −iϕ 1 and e −iϕ 2 only in the complement of L −1 (0) in EM −1 (R), where R is the set of regular values of the energy momentum map.

The classical spherical pendulum
In this section we describe the geometry of the classical spherical pendulum. More details can be found in [6, chpt V].

The basic system
We discuss the spherical pendulum as a constrained system. First we give the unconstrained system. Let T * R 3 = R 3 × (R 3 ) * have coordinates (q, p) and symplectic form ω = 3 i=1 dp i ∧ dq i = dθ, where θ = p, dq . Here , is the Euclidean inner on R 3 , which we use to identify T * R 3 with T R 3 . The unconstrained Hamiltonian system (H, T R 3 , ω) has unconstrained Hamiltonian H : T R 3 → R : (q, p) → 1 2 p, p + q, e 3 . Here {e i } 3 i=1 is the standard basis of R 3 . Now constrain the system ( H, T R 3 , ω) to the tangent bundle T S 2 = {(q, p) ∈ T R 3 q, q = 1 & q, p = 0} of the 2-sphere S 2 with symplectic form ω = ω |T S 2 . Again we use the Riemannian metric on S 2 induced from the Euclidean inner product on R 3 to identify the cotangent bundle T * S 2 with the tangent bundle T S 2 . The constrained Hamiltonian is H = H |T S 2 , that is, The classical spherical pendulum is the Hamiltonian system (H, T S 2 , ω).
The integral curves of the Hamiltonian vector field X H of the Hamiltonian H (1) satisfy dq dt = p (2a) dp dt = −e 3 + ( q, e 3 − p, p )q (2b) on T R 3 . Since T S 2 is an invariant manifold of (2a)-(2b), it follows that they define the integral curves of a vector field X H = X H |T S 2 , which governs the motion of the spherical pendulum. A calculation shows that H = H |T S 2 and L = L |T S 2 are constants of motion of the vector field X H .

Reduction of symmetry
The angular momentum L of the unconstrained system ( H, T R 3 , ω) is a constant of motion, because the unconstrained Hamiltonian H is invariant under the S 1 -action Φ : S 1 × T R 3 → T R 3 : s, (q, p) → Φ s (q, p) = (R s q, R s p), where R s =   cos s − sin s 0 sin s cos s 0 0 0 1   . Since R s is a rotation, T S 2 is invariant under Φ s . The infinitesimal generator of the S 1 -action restricted to T S 2 is X L |T S 2 , whose integral curves satisfy So the constrained Hamiltonian H is invariant under the S 1 -action Φ : S 1 × T S 2 → T S 2 : s, (q, p) → Φ s (q, p) = (R s q, R s p).
The invariance of the constrained angular momentum L = L|T S 2 under the S 1 -symmetry Φ s shows that it is an integral of the spherical pendulum. Thus the spherical pendulum is an integrable system (H, L, T S 2 , ω).
Thus the reduced equations of motion on the reduced phase space P arė Note that the function π 2 2 + 2 − π 3 (1 − π 2 1 ), whose zero set defines the reduced phase space P , is a constant of motion of the reduced equations of motion. On R 2 with coordinates (π 1 , π 2 ) the structure matrix W R 2 of the So the reduced equations of motion on R 2 of the reduced system (H , Note that the reduced Hamiltonian H is a constant of motion of the reduced equations of motion.

Regular values
In this section we determine the topology of the set of regular values, which lie in the image of the energy-momentum mapping EM : T S 2 → R 2 : (q, p) → H(q, p), L(q, p) = 1 2 p, p + q, e 3 , q 1 p 2 −q 2 p 1 of the spherical pendulum.
First we determine the set of critical values of the energy momentum map. The pair (h, ) is a critical value of EM if and only if the h-level set of the reduced Hamiltonian H , that is, the 2-plane 1 2 π 3 + π 1 = h in R 3 with coordinates (π 1 , π 2 , π 3 ), intersects the reduced space P ⊆ R 3 , defined by π 2 2 + 2 = π 3 (1 − π 2 1 ) with |π 1 | ≤ 1 and π 3 ≥ 0, at a point of multiplicity greater than 1. In other words, the polynomial which is obtained by eliminating π 3 from the defining equation of P , has a multiple root (π 1 , π 2 ) ∈ [−1, 1] × R, that is, 0 = Q(π 1 , π 2 ) and (0, 0) = DQ(π 1 , π 2 ) = − P h, (π 1 ), 2π 2 . Clearly π 2 = 0 and π 1 is a multiple root of Equating the coefficients of like powers of π 1 in the preceding equality gives Eliminating t from (13) gives the following parametrization of the discriminant locus {∆ = 0} is the union of two curves B ± , which go to +∞ as s 0, are reflections in the h-axis of each other, meet at a right angle and end when s = −1, that is, when (h, ) = (−1, 0). Otherwise they do not intersect. When s = 1 we obtain the isolated point (1, 0). The image of the energy momentum mapping EM of the spherical pendulum is the closed subset of R 2 bounded by the curves B ± , which contains the point (1, 0). The set R of regular values in the image of EM is the interior of the image of EM with the point (1, 0) removed. Thus R is diffeomorphic to an open 2-disk with its center deleted and so is not simply connected.  Actually, the argument above gives the following statification of the range of the energy momentum map EM, which is a semialgebraic subset of R 2 .
The orbit map of the S 1 -action Φ on L −1 ( ) is given by ρ : L −1 ( ) ⊆ T S 2 → P ⊆ R 3 : (q, p) → π 1 (q, p), π 2 (q, p), π 3 (q, p) . (14) Suppose that (h, ) ∈ R. Then the h-level set of the reduced Hamiltonian H : P ⊆ R 3 → R : (π 1 , π 2 , π 3 ) → 1 2 π 3 + π 1 is Here π ± 1 are consecutive roots of the polynomial which is diffeomorphic to a circle. From the construction of the S 1 -orbit space P it follows that ρ −1 ( H ) −1 (h) is the total space of an S 1 -bundle Σ with base S 1 = ( H ) −1 (h). Because P is homeomorphic to R 2 , we deduce that ( H ) −1 (h) is contractible to a point in P . Thus Σ is a product bundle, which implies that ρ −1 ( H ) −1 (h) is diffeomorphic to a smooth 2-torus whose fiber over (h, ) ∈ R is the smooth torus T 2 h, . Since every fiber of this mapping is compact, the fibration is locally trivial. Moreover, the bundle π| B ± : EM −1 (B ± ) → B ± is a trivial S 1 bundle over R because each curve B ± is contactible to a point. An integral curve of X L on EM −1 (B ± ), which parametrizes a fiber of the bundle π| B ± is noncontractible curve and is the limit of an integral curve of X L on T 2 h, when (h, ) ∈ R converges to a point on B ± .

Action-angle coordinates
In this section we construct local action-angle coordinates for the spherical pendulum.
The smooth locally trivial 2-torus fibration π : EM −1 (R) → R satisfies the hypotheses of the action-angle coordinate theorem for the integrable Hamiltonian system (H, L, T S 2 , ω) describing the spherical pendulum, see where T 2 is the affine 2-torus R 2 /(2πZ) 2 , such that 1. The symplectic form ω on T S 2 when restricted to U is exact and 2. The actions A 1 and A 2 are smooth functions of H and L on U.

The vector field
, corresponding to the Hamiltonian H = ϕ * (H|U). Moreover, the integral curves of X H satisfy Choose V sufficiently small so that the fibration EM| U is trivial. Then we have the following commutative diagram . Thus the bundle π is a local trivialization of the bundle EM| EM −1 (R) .
We now construct the action functions for the spherical pendulum. According to the proof of the action-angle coordinate theorem in [6], the action functions A i on U are constructed by finding linear combinations on U of the vector fields X H| U and X L| U , whose coefficients are smooth functions on EM(U) = V, which have periodic flow of period 2π when restricted to T 2 h, . Here (h, ) = EM(u) with u ∈ U. Since X L has periodic flow of period 2π on EM −1 (R) ⊆ T S 2 , we may define the second action function on U as To construct the first action function A 1 we need to determine certain functions related to the flows (ϕ H t )| T h, is periodic of period 2π, Γ 1 is a closed curve on T 2 h, . Since X H (u) is nonzero and is transverse to X L (u) for every u ∈ T 2 h, and because (h, ) is a regular value of EM, the integral curve Γ 2 : t → ϕ H t (u), which starts at u ∈ Γ 1 ([0, 2π]), has a unique positive first time T for which ϕ H T (u) ∈ Γ 1 ([0, 2π]). The time T is a smooth function of (h, ) and does not depend on the choice of starting point u in Γ 1 ([0, 2π]), because it is the period of the reduced vector field X H on ( H ) −1 (h) ⊆ P . Let Θ be the smallest positive time it takes the curve Γ : The function Θ does not depend on the choice of the point u and is a smooth function of (h, ).
To find an explicit expression for the push forward A 1 of first action A 1 to the image of the energy momentum mapping, consider the formula where Γ : [0, 2π] → T S 2 is a closed path in T 2 h, starting at (q, p), which is a sum of two paths Γ 1 and Γ 2 on T 2 h, . Suppose that (h, ) ∈ R. Choose the paths Γ 1 and Γ 2 to be and respectively. Here T (h, ) = T (h, )/(2π) and Θ(h, ) = Θ(h, )/(2π). Using the path parameters as integration variables, we get Using (2a) and (3a) we obtain But h = H = 1 2 π 3 + π 1 and dt = 1 π 2 dπ 1 , using (11a). So Therefore, when (h, ) ∈ R, where 2π I(h, ) = 4 Since we may rewrite the right hand side of (16) as Using (19) straightforward calculation shows that We have to show that the flow of the Hamiltonian vector field X A 1 is periodic of period 2π. From (20) it Pulling back the preceding equation by the energy momentum mapping EM gives Applying the map ω (q, p) yields h, is given by The preceding flow is periodic of period 2π because The last equality above follows from the definition of the time T (h, ) of first return and the definition of the rotation number of the flow of X H on T 2 h, . Therefore A 1 is an action function on

see (21) and fact 2.2. However, the flow of X
Thus the flow ϕ A 1 t on T 2 h,0 is well defined and continuous for every −1 < h < 1 or h > 1.
h, . From the definition of the first action function we have Taking the limit as ± 0 gives This completes the proof that A 1 is an action function on EM −1 (R).

Properties of the functions Θ, T and A 1
We just state the principal analytic properties of the functions Θ (18b) and T (18a). Their proofs may be found in [6, chpt V and exercises].

Fact 2.2
1. On R \ { = 0} the function Θ is real analytic and odd in , that is, 3. The function Θ : R ⊆ R 2 → R is multivalued. Along any positively oriented closed curve, which generates the fundamental group of R, its value decreases by 1. 4. The function T : R ⊆ R 2 → R ≥0 is real analytic and even in , We now determine the limiting values of the functions T , Θ, and A 1 (19) as (h, ) ∈ R converges to (h(s), (s)) ∈ ∂R for some s ∈ [−1, 0).
Proof. Consider C ∨ , the extended complex plane, which is cut along the real axis between x − and x + and again between x 0 and ∞. Here . Let C be a positively oriented closed curve in C ∨ , which crosses the Re z axis twice: once in (−1, x − ) and once in (x + , 1). Note that the complex square root is negative just above the cut [x − , x + ]. In the limit as We can write the integral I(h, dx as the complex integral Thus when (h, ) converges to (h(s), (s)) the integral (26) becomes Therefore when (h(s), (s)) ∈ ∂R ∩ { ≥ 0} using (16) we get A similar argument shows that This confirms a result in [16].

Additional properties of A 1
The next proposition gives some addtional properties of the first action function A 1 (19).
which is strictly decreasing when < 0, is strictly increasing when > 0, and has a positive minimum value h * (a) at = 0.
Proof. First we show that I(h, where C and the complex square root are chosen as in the proof of (25a)-(25c). Thinking of h and as complex variables we obtain and similarly ∂I ∂ = 0. Therefore locally I is a complex analytic function. Restricting h and to be real variables shows that locally I is a real analytic function on R. Clearly, I(h, − ) = I(h, ). To show that I is single-valued, consider the positively oriented rectangular path Γ ε in R, which consecutively joins the vertices (h, ε), (h + 1, ε), (h + 1, −ε), (h, −ε), and (h, ε), where ε is chosen sufficiently small and positive so that Γ ε lies in R and −1 < h < 1.
This shows that I is single valued on R, because Γ ε generates the fundamental group of R.
That A 1 is locally a real analytic function on R\{ = 0} follows from (17) and the fact that T , ϑ, and I are locally real analytic functions on (17), because T and I are continuous there. From (20) we see that Thus the assertions about ∂A 1 ∂ follow from the properties of the function Θ.
1. The fact that A 1 has a real analytic extension to ∂R follows from the proof in proposition 2 Thus A 1 has a continuous extension to R.
Making the successive changes of variables Evaluating K we get To see this we compute since the integrand is positive. So the graph of A 1 | R∩{ =± (s 0 )} is strictly convex. Because of convexity, the function A 1 | R∩{ =± (s 0 )} is proper. Thus if its image were compact, then so would be its domain. But this is a contradiction, since its domain From (20) and the fact that T > 0 on R, it follows that we have ∂A 1 ∂h > 0 on R ∩ { = 0}. Therefore by the implicit function theorem, near Differentiating (30) with respect to and then evaluating the result at (h 0 , 0 ) gives using (20). A a has a jump discontinuity at R ∩ { = 0}, because Thus a connected component of A −1 1 (a) in R is the graph of a piecewise real analytic function of . Suppose that the domain of A a is the compact interval [ * , * ]. If * < 0, then the function A a would have a minimum value h * at * , since it is strictly decreasing in we have A a ( * ) < 0. Therefore by the implicit function theorem, there is an 0 > > * close to * such that A a is defined. But this contradicts the hypothesis that [ * , * ] is the domain of A a . Therefore * > 0. Then the function A a would have a maximum value h * at * , since it is strictly Therefore by the implicit function theorem, there is an > * close to * such that A a is defined. But this contradicts the hypothesis that [ * , * ] is the domain of A a . Hence * does not exist. A similar argument shows that * does not exist. Thus the domain of A a is (−∞, ∞). This implies that the Suppose that a = 4/π. Then the line segments {(h, − (s)) ∈ R s ∈ [−1, 0)} and {(h, (s)) ∈ R s ∈ [−1, 0)}, which are parallel to the h-axis, each intersect the graph of A a : (−∞, ∞) −→ [h * (a), ∞) exactly once, since A 1 | R∩{ =± (s)} is strictly increasing and has range (h(s), ∞). Thus the alevel set of A 1 with a > 0 and = 4/π is connected and is the graph of a piecewise real analytic function, whose graph intersects the h-axis in R exactly once at (h * (a), 0).

Fact 2.4 The action map of the spherical pendulum is
is a homeomorphism of R onto R >0 × R \ {(1, 0)}, which is a real analytic diffeomorphism on R \ { = 0}. This homeomorphism extends to a homeo- The second equality follows from (20). Since T > 0 on R, the action map

The period lattice and its degeneration
Here we discuss the period lattice of the 2-torus T 2 h, = EM −1 (h, ) when (h, ) ∈ R and study its degeneration as (h, ) converges to a boundary point of R.
For (h, ) ∈ R consider the R 2 -action on the 2-torus T 2 h, defined by Then Ψ is locally transitive at p ∈ T 2 h, because T p T 2 h, = span{X H (p), X L (p)}. Since T 2 h, is connected, it follows that the action Ψ is transitive on T 2 h, . Let P h, = {(t 1 , t 2 ) ∈ R 2 Ψ (t 1 ,t 2 ) (p) = p} be the isotropy group of the R 2 -action Ψ at p. Then T 2 h, = R 2 /P ,h . Because R 2 is abelian, the isotropy group P h, does not depend on the choice of p. Since P h, is closed subgroup of the Lie group R 2 , it is a Lie group. Suppose that dim P h, ≥ 1. Then P h, has a one parameter subgroup, namely, s → (a s, b s) for some (a, b) ∈ R 2 \ {(0, 0)}. So p = ϕ H a s • ϕ L b s (p), which implies that But X H (p) and X L (p) are linearly independent in T p T 2 h, . Hence (34) implies that a = b = 0, which contradicts the definition of a and b. Thus P h, is a 0-dimensional Lie group and hence is discrete. This shows that P h, is a Z-lattice, called the period lattice. By definition of the functions T and Θ, we see that the vectors T 2π , which to each (h, ) ∈ R assigns the basis of P h, , is locally a real analytic matrix valued function.
We now look at what happens to P h, as (h, ) ∈ R converges to a point (h(s), (s)) ∈ ∂R for some s ∈ [−1, 0). Let p h, ∈ T 2 h, and suppose that p h, converges to p s ∈ EM −1 ((h(s), (s)) = S 1 s . In fact p s is a relative equilibrium of the Hamiltonian vector field X H on (T S 2 , ω). In other words, the integral curve of X H starting at p s is a 1-parameter subgroup of the S 1 symmetry generated by the angular momentum Hamiltonian vector field X L . So X H (p s ) and X L (p s ) are linearly dependent. Thus the isotropy group P h, at p h, of the R 2 -action (33) degenerates to the isotropy group P s = P h(s), (s) of the R-action which is generated by (0, 2π).
Another way to see that this degeneration is to use the action functions A 1 = π * (A 1 ) and A 2 = π * (A 2 ) on EM −1 (R * ), where R * = R\{(1, 0)}. Here Consider the R 2 -action The period lattice P q , which is the isotropy group of the action Ψ at q, is generated by {(2π, 0), (0, 2π)}, when q ∈ EM −1 (R), because the flows of the vector fields X A 1 and X A 2 are periodic of period 2π. When q ∈ EM −1 (∂R), the period lattice P q is generated by {(0, 2π)}, because X A 1 (q) = 0, while X A 2 has periodic flow of period 2π on EM −1 (∂R).

Monodromy
In this subsection we show that the spherical pendulum has monodromy by looking at the variation of the period lattice along a homotopically nontrivial loop in R.
Let Γ : [0, 1] → R : t → Γ(t) = h(t), (t) be a smooth closed curve in R, which encircles the point (1, 0) and generates the fundamental group of R. We transport the period lattice along Γ, namely we look at the curve 2π .
We note that In other words, after transporting the period lattice P Γ(0) along Γ, its initial basis { T (Γ(0)), −Θ(Γ(0)) t , 0, 2π t } at Γ(0) becomes the final basis . The matrix of this linear transformation with respect to the initial basis is M = 1 0 1 1 . Thinking of the 2-torus T 2 Γ(t) associated to the period lattice P Γ(t) , we have a smooth bundle of 2-tori over Γ whose fiber over Γ(t) is T 2 Γ(t) . The gluing map of the fibers over the end points Γ(0) and Γ(1) is the linear map of R 2 into itself with matrix M. Since M ∈ Gl(2, Z), it maps the lattice (2πZ) 2 into itself. Thus M is a diffeomorphism of the affine 2-torus T 2 = R 2 /(2πZ) 2 into itself. Recalling that action-angle coordinates identify T 2 h, with T 2 , we see that M is the monodromy map of the 2-torus bundle over Γ. Different choices of the action functions or of the closed curve in the homotopy class of Γ lead to a 2-torus bundle, which is isomorphic to the one constructed above. The isomorphism class of the new bundle is determined by the conjugacy class of the monodromy map M in Sl(2, Z). Consequently, the spherical pendulum has no global action-angle coordinates.

Quantum spherical pendulum
We begin with a brief review of the elements of geometric quantization, which we use here. For details see [20] 3.1 Elements of geometric quantization

The prequantization line bundle
The first step in geometric quantization of the symplectic manifold (P, ω) is the construction of a complex line bundle λ : L → P with connection ∇ such that the curvature of ∇ is − 1 2π ω, where is Planck's constant divided by 2π. 3 For the spherical pendulum P = T * S 2 , and ω = dθ, where θ is the Liouville form of the cotangent bundle of the sphere. Hence, L is a trivial bundle, and we can introduce a global trivializing section σ 0 : P → L such that, for every vector field X on P , the covariant derivative of σ 0 in direction X is Moreover, we introduce a Hermitian form ·, · on L such that σ 0 , σ 0 = 1.
The choice of the trivializing section σ 0 : P → L gives rise to an identification Under this identification, every section σ of L corresponds to a complexvalued function ψ on P such that σ = ψσ 0 .
If ψ(Γ(t)) = 0, we divide by ψ(Γ(t)) and integrate to get which is equivalent to We have proved the following version of the Bohr-Sommerfeld conditions. Theorem 3.1 If Γ is a closed curve in P and σ is a section of L which is covariantly constant along Γ, then the pull-back of σ to Γ is identically zero unless for some integer n.

Prequantization operators
For each f ∈ C ∞ (P ), the Hamiltonian vector field X f of f generates a local 1-parameter group exp tX f of local diffeomorphisms of P that preserve the symplectic form ω. There is a unique lift of X f to a vector field X f on the prequantization line bundle L such that exp t X f preserves the connection on L [20]. The prequantization operator P f associated to f is given by for every σ ∈ S ∞ (L). Direct computation gives where Proof. Therefore, We refer to the map as the prequantization map. Equation (39) implies that the map f → 1 i P f is a representation of the Poisson algebra of C ∞ (P ) on the space S ∞ (L), which we call the prequantization representation.

Polarization
In Dirac's formulation of quantum mechanics, the space of quantum states of the spherical pendulum consists of functions on the spectrum of the complete set of commuting observables. This idea can be identified with the modern theory of representations of C * algebras. In the framework of geometric quantization, it gave rise to the notion of polarization of (P, ω), which is given by a complex involutive Lagrangian distribution F on the phase space P . For P = T * S 2 , the choice of F determines the representation of the quantum theory of the spherical pendulum.
Let F be a complex involutive Lagrangian distribution F on P . In other words, F is a complex distribution on P such that dim C F = 2n, where dim P = 2n. Moreover, ω(u, v) = 0 for every pair of vectors u, v ∈ F attached at the same point in P . In the following we also allow for polarizations with singularities, that is, smooth distributions that are Lagrangian on an open dense subset of P. Quantization in the F -representation leads to the space of quantum states In other words, S F (L) is the space of sections of L that are covariantly constant along F . For each f ∈ C ∞ (P ), the Hamiltonian vector field X f of f lifts to a unique vector field on the prequantization line bundle that preserves the connection. Hence, if f ∈ C ∞ F (P ), it follows that P f action leaves S F (L) invariant.
Definition 3.2 Direct quantization in the F -representation is given by restricting the domain of the prequantization map to C ∞ F (P ) × S F (L) and its codomain to S F (L). In other words, Quantization in the F -representation of functions that are not in C ∞ F (P ) requires additional assumptions. Definition 3.3 A polarization of (P, ω) is real, if it is a complexification of a (real) involutive Lagrangian distribution. In other words, where D is an involutive Lagrangian distribution on P .
In the following we assume Condition D is locally spanned by Hamiltonian vector fields.
This condition allows for a generalization to polarizations with singularity.

Schrödinger quantization
Schrödinger quantization of the spherical pendulum corresponds to the real polarization tangent to fibres of the cotangent bundle projection π : T * S 2 → S 2 . In other words, the Schrödinger polarization is ker T π ⊗ C, where For every p ∈ T * S 2 and w ∈ T p (T * S 2 ), the evaluation of the Liouville form θ on w equals the evaluation of p on T π(w), that is, θ(w) = p(T π(w)).
Hence, θ vanishes on ker T π, which implies that the extension of θ to the complexification of ker T π vanishes. Equation (35) implies that the trivializing section σ 0 of L is covariantly constant along ker T π. Thus, every section σ of L that is covariantly constant along ker T π is of the form σ = ψσ 0 , where ψ is a complex-valued function on T * S 2 that is constant along ker T π. However, functions on T * S 2 that are constant along ker T π are pull-backs by π : T * S 2 → S 2 of functions on S 2 . Thus, Equation (43) shows that we may identify the space S ker T π (L) of sections of L that are covariantly constant along ker T π with the space of smooth complex-valued functions on S 2 . The space C ∞ ker T π (T * S 2 ) of functions on T * S 2 that are directly quantizable in terms of the Schrödinger polarization ker T π consists of functions such that their Hamiltonian vector fields preserve ker T π. It contains pullbacks by π of smooth functions on S 2 . Moreover, for every vector field X on S 2 , its natural extension ofX to T * S 2 is a Hamiltonian vector field with Hamiltonian θ(X) and it preserves ker T π. It can be easily verified that where X(S 2 ) denotes the space of smooth vector fields on S 2 . It follows from equations (41) and (38) that, for everyf ∈ C ∞ (S 2 ), X ∈ X(S 2 ) and Ψ ∈ C ∞ (S 2 ) ⊗ C, and Schrödinger quantization of functions on T * S 2 , which can be expressed as polynomials in π * f and θ(X), assigns the corresponding polynomial in Q π * f and Q θ(X) . In this case, the result depends on the ordering of the factors. It is usually postulated that the quantum Hamiltonian is where ∆ denotes the Laplace-Beltrami operator on S 2 . There are several derivations of this result, but each of them requires additional assumptions, [1], [20], [23] It is not a direct consequence of prequantization and polarization. An additional assumption of the Schrödinger theory is that the scalar product of quantum states (π * Ψ 1 )σ 0 and (π * Ψ 2 )σ 0 is given by where dµ is the area form of S 2 . The demand that quantum observables are given by self-adjoint operators, requires that the operators in equation (46) should be symmetrized. If the scalar product is introduced in geometric quantization in terms of half-forms, then the operators corresponding to equation (46) are symmetric [20].

Bohr-Sommerfeld spectrum
Bohr-Sommerfeld quantization corresponds to the polarization ker T EM⊗C tangent to the fibres of the energy-momentum map EM : T * S 2 → R 2 . The range of the energy momentum map is stratified, with open dense stratum given by the set R of regular values. There are two one dimensional strata B + and B − corresponding to the minimum of energy for a positive or negative value of the angular momentum, respectively, and two singular points (−1, 0) and (1, 0). As before, quantum states are sections of the prequantization line bundle that are covariantly constant along the polarization. Since fibres of the energy momentum map are compact, values (h, l) of the energy and the angular momentum that are in supports of sections of L, which are covariantly constant along the polarization ker T EM ⊗ C, are restricted by the Bohr-Sommerfeld conditions; see Theorem 3.1.2. These conditions can be easily described in terms of the action variables A 1 , A 2 , where A 1 is a continuous function of the Hamiltonian H and the angular momentum L, and A 2 = L, see proposition 2.4. Recall that A 1 is a continuous function on T * S 2 . Moreover, (A 1 )| EM −1 (1,0) = 4/π, (A 1 )| EM −1 (−1,0) = 0, and (A 1 )| EM −1 (B ± ) = 0.
In order to avoid excessive notation, for (h, ) in the range of EM, we write A 1 (h, ) for the value at (h, ) of the push-forward of A 1 by the energy momentum map EM. In other words,

Definition 3.4
The Bohr-Sommerfeld energy-momentum spectrum of the quantum spherical pendulum is the set S of (h, ) in the image of EM that satisfy the Bohr-Sommerfeld conditions and It remains to consider the singular point (1, 0). Since A 1 (1, 0) = 4/π and A 2 (1, 0) = 0, equation (49) gives 4/π = n , which implies that Planck's constant 2π = 8/n. This is a very strong condition on Planck's constant, unlikely to be satisfied in physics. Therefore, we assume that the Bohr-Sommerfeld conditions are not satisfied by the unstable equilibrium point (1, 0).

Conclusion
The Bohr-Sommerfeld energy-momentum spectrum S is the range of a map In other words, Definition 3.5 In physics the pairs (n, m) ∈ Z 2 such that n ≥ 0 are called quantum numbers of the spherical pendulum.
For every pair (n, m) of quantum numbers, there exists a non-zero covariantly constant section σ n,m of L restricted the fibre EM −1 (h(n, m), m ). The family of sections B = σ n,m (n, m) ∈ Z 2 and n ≥ 0 (51) forms a basis in the space of quantum states of the Bohr-Sommerfeld theory. We consider a Hilbert space H of quantum states in which B is an orthonormal basis.
- Figure 3. The Bohr-Sommerfeld quantum states of the spherical pendulum in R.
According to the general principle of geometric quantization, the space C ∞ ker T EM (T * S 2 ) of function that are directly quantizable in the Bohr-Sommerfeld theory consists of smooth real-valued functions f on T * S 2 such that the Hamiltonian vector field X f preserves the polarization ker T EM. Proof. Locally, we express f as a function of local angle action coordinates (A 1 , A 2 , ϕ 1 , ϕ 2 ) and write f =f (A 1 , A 2 , ϕ 1 , ϕ 2 ). Therefore, Since X A 1 and X A 2 are linear combiations of X H and X L , the assumption that [X H , X f ] and [X L , X f ] are linear combinations of X H and X L implies Integrating the preceding equation gives Taking into account ∂ 2f ∂ϕ 1 ∂ϕ 1 = 0 we get thatf 1 is a linear function of ϕ 1 .
Similarly, ∂ 2f ∂ϕ 2 ∂ϕ 2 = 0 implies thatf 2 is a linear function of ϕ 2 . However, linear functions of angles are not single-valued. By assumption,f is singlevalued. Therefore,f is independent of ϕ 1 and ϕ 2 , and herefore, Since A 1 and A 2 ar functions of H and L, it follows that f restricted to the domain of a local action-angle coordinates (A 1 , A 2 , ϕ 1 , ϕ 2 ) is a function of the restrictions of H and L to the same domain.
Domains of local action angle coordinates cover the open set EM −1 (R) ⊂ T * S 2 , where R is the set of regular values of the energy momentum map. Therefore, f restricted to EM −1 (R) is a function of H and L restricted to EM −1 (R). In other words, f restricted to EM −1 (R) is constant along fibres of the restriction to R of EM : T * S 2 → R 2 . Since EM −1 (R) is dense in T * S 2 and f is continuous, it follows that f is constant on fibres of EM : T * S 2 → R ⊆ R 2 . Therefore, there exists a functionf : R = image EM → R such that f =f (H, L).
Corollary 3.5 The space of functions whose Hamiltonian vector fields preserve ker T EM is Equations (41) and (38) imply that the Bohr-Sommerfeld quantization assigns to a functionf (H, L) ∈ C ∞ ker T EM (T * S 2 ) an operator Qf (H,L) on H such that operator for every σ (h, ) ∈ B. It should be emphasized that C ∞ ker T EM (T * S 2 ) does not contain any function f for which the operator Q f is not diagonal in the Bohr-Sommerfeld basis B.

Shifting operators
The weakness of the Bohr-Sommerfeld theory is that it does not provide operators corresponding to transitions between different states. Of course, there are such operators acting on H. Since the Bohr-Sommerfeld basis B in H has a lattice structure given by equation (51), there exist shifting operators a 1 and a 2 on H such that a 1 σ n,m = σ n−1,m for n ≥ 0 0 for n = 0, a 2 σ n,m = σ n,m−1 .

(52)
Proposition 3.6 The quantized actions and the shifting operators satisfy the commutation relations for j, k = 1, 2.
Since B = {σ n,m | (n, m) ∈ Z 2 , n ≥ 0} is an orthonormal basis in H, the adjoints a † 1 and a † 2 of the shifting operators are given by for all m and n ≥ 0. Taking the adjoints of equations (53), we get It follows from the definition, equations (52), that the operators a 1 and a 2 are analogues of the lowering operators in the Fock space formulation of field theory and the states σ 0,n are ground states for the operator a 1 , [?]. Similarly, the operators a † 1 and a † 2 are analogues of the raising operators.

Quantization of angles
We want to interpret the shifting operators as a 1 and a 2 as quantum operators corresponding to functions f 1 and f 2 on the phase space T * S 2 of the spherical pendulum. In other words, we want to make an identification Since the shifting operators are not self-adjoint, we cannot expect functions f 1 and f 2 to be real-valued. This means that we have to extend the Dirac quantization condition to complex-valued functions. Recall, that the action A 1 is smooth on the complement for j, k = 1, 2.
Proof. By definition, the angle action variables satisfy the Poisson commutation relations where j, k = 1, 2. Hence, as required.
Comparing equations (52) and (58) we see that that the identification (56) satisfies the Dirac quantization condition (57) for f 1 = e −iϕ 1 and f 2 = e −iϕ 2 in the open dense subset EM −1 (R) \ L −1 (0) of T * S 2 , where R is the regular stratum of the range of the energy momentum map EM : T * S 2 → R 2 . In other words, for every (n, m) ∈ Z 2 with n > 0 and m = 0, we may set a 1 σ n,m = Q e −iϕ 1 σ n,m and a 2 σ n,m = Q e −iϕ 2 σ n,m .
Since e −iϕ 1 and e −iϕ 2 are well defined on R ∩ L −1 (0), we can extend the identification (59) to R ∩ L −1 (0) and write a 1 σ n,0 = Q e −iϕ 1 σ n,0 and a 2 σ n,0 = Q e −iϕ 2 σ n,0 , where n > 0. However, on EM −1 (R)\L −1 (0), the Poisson brackets involving A 1 are not defined because dA 1 is not defined there. Hence, the right hand side of equation (57) is not independently defined. This is a manifestation of the presence of monodromy in the spherical pendulum, which will be discussed in Section 3.4.4.

Boundary conditions
It remains to extend operators Q e −iϕ 1 and Q e −iϕ 2 to quantum states σ 0,m supported on the boundary of EM −1 (R). This is analogous to extending Schrödinger quantization to the cotangent bundle of a manifold with boundary and corners.
The angle functions are not globally defined. In particular, the functions e −iϕ 1 and e −iϕ 2 are not defined at the singular points (0, −1) and (0, 1). Moreover, e −iϕ 1 is not defined when A 1 = 0.
In order to extend e −iϕ 1 to a globally defined function, we choose a smooth function χ 1 (h, l) on R = image EM, which is identically 1 on a neighbourhood of R ∩ S in R, where S is the Bohr-Sommerfeld energy spectrum (50), and vanishes to infinite order on the boundary ∂R of R. The product f 1 = χ 1 (L, H)e −iϕ 1 is a globally defined function on T * S 2 that vanishes to infinite order on EM −1 (∂R) and satisfies the Poisson bracket relations for j = 1, 2. Since f 1 = χ 1 (L, H)e −iϕ 1 vanishes to infinite order on ∂R ∩ S and derivatives of χ 1 vanish to infinite order on S, it follows that we can make the identification for all (n, m) ∈ Z 2 with n > 0 and m = 0. On the other hand, f 1 = χ 1 (L, H)e −iϕ 1 vanishes to infinite order on ∂R ∩ S. Hence, we may set This identification is independent of the choice of χ 1 satisfying the required conditions. In order to keep the notation simple, in the following we omit χ 1 and write Q e −iϕ 1 σ n,m = a 1 σ n,m for all (n, m) ∈ Z 2 with n ≥ 0 and m = 0. It remains to consider the action of The function e −iϕ 2 is defined on EM −1 (∂R \ {(−1, 0), (1, 0)}. As before, we can extend e −iϕ 2 to the whole of T * S 2 by multiplying e −iϕ 2 by an appropriate function χ 2 of H and L. By assumption (1, 0) / ∈ S, so vanishing of χ 2 at (1, 0) does not affect the identification of a 2 with Q e −iϕ 2 . However, the point (−1, 0) ∈ S, corresponding to (n, m) = (0, 0), is a corner of the range of EM and ϕ 2 is not defined on EM −1 (−1, 0). Therefore, the identificalion is an essential extension of the definition of Q e −iϕ 2 , which does not follow from the Dirac quantization conditions. Making this identification, we write Q e −iϕ 2 σ n,m = a 2 σ n,m for all (n, m) ∈ Z 2 with n ≥ 0.
Having identified shifting operators a 1 and a 2 with quantizations of e −iϕ 1 and e −iϕ 2 , respectively, we observe that the adjoint operators a † k may be identified with quantization of e iϕ k . In other words, for k = 1, 2. Moreover, as in the Schrödinger theory, to a function f on T * S 2 , which can be expressed as a polynomial in A 1 , A 2 , e −iϕ 1 and e −iϕ 2 , we can assign the corresponding polynomial in Q A 1 , Q A 2 , Q e −iϕ 1 , and Q e −iϕ 2 . In this case, the result depends on the ordering of the factors.

Quantum monodromy
In this subsection we will discuss 1. the definition of quantum monodromy; 2. show that the quantum spherical pendulum has quantum monodromy; 3. read off the classical monodromy of the spherical pendulum from the joint spectrum of its Bohr-Sommerfeld-Heisenberg quantization.
We begin by defining quantum monodromy. Our discussion leans heavily on the treatment of Vu Ngoc [22]. Let B be an open 2-disk in R 2 , which is contained in R and is centered at c ∈ R. The intersection B ∩ S of the Bohr-Sommerfeld spectrum S with B is a local lattice, because the image of B ∩ S under the homeomorphism given by the action map A (32) is the intersection of the open subset A(B) with the standard lattice 2π Z 2 . We call the pair (B, A| B ) a local chart at c for the Bohr-Sommerfeld spectrum S. Let α∈I B α be an open covering of R by 2-disks each centered at c α . Suppose that (B α , A| Bα ) and (B β , A| B β ) are local charts for S and that c α ∈ B α ∩ B β . From the construction of action angle coordinates in section 2.4, it follows that the chart transition map is locally constant. Let τ : L → R be the Z 2 -bundle over R with local trivialization given by the top row of the commutative diagram where the left vertical arrow is the restriction of the bundle projection map to τ −1 α (U α ) and the left vertical arrow is the projection map on the first factor. The bottom horizontal map is the identity mapping. More explicitly, the image of the fiber L c over the point c ∈ B α under the trivialization mapping τ α is c, The local transition maps for the bundle L are given by The isomorphism class 4 of the bundle L is the quantum monodromy of the Bohr-Sommerfeld spectrum S ⊆ R.
Action-angle coordinates provide a local trivialization of the bundle π : EM −1 (R) → R. From their construction it follows that the bundle L → R is isomorphic to the bundle P → R of period lattices. But the bundle P is not trivial, since the spherical pendulum has classical monodromy. Thus the Z 2 -lattice bundle L is not trivial. In other words, the quantum monodromy of the spherical pendulum is nontrivial.
Claim 3.7 For each 1 ≤ i ≤ there is a finite shortest sequence a i of shifting operators, each member of which is taken from {a 1 , a 2 , a † 1 , a † 2 }, that shifts the vertex (n i , m i ) of the spectral quadrilateral Q n i ,m i to the vertex (n i+1 , m i+1 ) of Q n 1 +1,m i +1 .
Corollary 3.8 For each 1 ≤ i ≤ the image of Q n i ,m i under the operator a i is Q n 1 +1,m i +1 .
The lower left hand corner of the spectral quadrilateral Q n,m is given by the spectral values corresponding to the edge joining (n, m) to (n + 1, m) and the spectral values correspongind to the edge joining (n, m) to (n, m + 1). Because the quantum spectral values are determined by the intersection of level sets of the action function A 1 with a level set of the action function using (20) and the fact that A 2 (h, ) = .
We now look at the variation of DA along the lower left hand vertices occuring on the polygonal path Γ in R. On the one hand because T is a continuous function on R, its variation along Γ is 0. On the other hand, from fact 2.2 the variation of − Θ along Γ is 1. Thus the variation of DA along Γ is 0 0 1 0 . Since the column vectors of DA(p n,m ) form a basis of the period lattice of the 2-torus T 2 pn,m , the monodromy matrix of the classical spherical pendulum along Γ is the sum of the identity matrix 1 0

Concluding remarks
• We have extended the Bohr-Sommerfeld quantization of the spherical pendulum to a full quantum theory, which we call the Bohr-Sommerfeld-Heisenberg (BSH) quantization. Our approach leads to a matrix formulation of Born, Jordan and Heisenberg [4], [5] with quantum operators expressed as matrices in the Bohr-Sommerfeld basis. According to Mehra and Rechenberg [15, p.265] the connection between quantum mechanics and the Bohr-Sommerfeld theory of multiply periodic systems "seems to have been lost completely in the matrix approach" until it was re-established by Wentzel [24].
• We had an advantage of being able to rely on the guiding principle of geometric quantization for an open dense subset of the phase space in which all our constructions were regular. In treating nowhere dense sets of singular points, we followed Dirac's Principles of Quantum Mechanics [11] • In our presentation, we have included the geometric quantization setting of the Schrödinger theory. It shows that the main difference between the BSH theory and the Schrodinger theory is the choice of polarization. For a completely integrable system, the energy momentum map is regular in an open dense subset of the phase space. Hence in the BSH quantization, we have to deal with polarization with singularities on the boundary of that set. This leads to difficulties analogous to those that appear in formulating the Schrodinger theory on a singular space.
• In the case of the spherical pendulum, the Bohr-Sommerfeld energy spectrum differs from the Schrödinger energy spectrum. In the quasiclassical limit of close to zero, the Bohr-Sommerfeld spectrum and the Schrödinger spectrum of Q A 1 differ by 1 2 . • In physics, Planck's constant 2π is approximately 6.626×10 −34 joules.
However, in the process of quantization, is treated as a parameter.
In the Schrödinger quantization, the reperesentation space is independent of , while the quantum operators depend on explicitly. In the BSH quantization, presented here, the representation space H depends on because it is defined in terms of a basis B consisting eigensections supported on fibres of the energy-momentum map that satisfy the Bohr-Sommerfeld conditions. For every fibre of the energy-momentum map, there exists a value of , treated as a parameter, for which this fibre satisfies Bohr-Sommerfeld conditions.
• For every = 4/πn, where n ∈ N, our construction gives well defined shifting operators a 1 , a 2 and their adjoints a † 1 , a † 2 on the representation space H. It is a consequence monodromy that the interpretation of the shifting operators a 1 , a 2 as quantization of e −iϕ 1 and e −iϕ 2 fails to be global even on the set of regular values of the energy momentum map.