Bohr-Sommerfeld-Heisenberg Quantization of the Mathematical Pendulum

In this paper we give the Bohr-Sommerfeld-Heisenberg quantization of the mathematical pendulum.


Introduction
The Dirac's formulation of quantum mechanics [7] can be described as a precursor of the theory of C * algebras. Quantum observables are self adjoint operators in a complex vector space of quantum states. In chapter 3 of [8] Dirac represents quantum states as functions on the spectrum of the maximal abelian subalgebra (complete set of commuting observables). Classical Hamiltonian mechanics may be regarded as the limit of quantum mechanics when tends to zero. There are quantum systems without classical analogues.
Quantization is an attempt to find a quantum system corresponding to a given classical system. Since there may be several different approaches, quantization may give inequivalent results. Because quantum observables may be represented as operators on the space of functions on the spectrum of the maximal abelian subalgebra, the usual approach to quantization is to identify a complete set of commuting observables and to study operators on its spectrum.
For a completely integrable Hamiltonian system, Bohr-Sommerfeld quantization [1,12] of the action variables gives rise to a space of quantum states and a complete set of commuting observables acting of this space of states.
Bohr-Sommerfeld theory does not provide operators of transition between the eigenstates of operators corresponding to the actions. These transitions are accounted for by shifting operators. Because the general theory of these operators requires an extension of geometric quantization to locally Hamiltonian vector fields, which is far a field from the topic of this paper, we refer the reader to [6]. However, we do treat a special case relevant to this paper in the appendix. The commutation relations satisfied by the shifting operators are the same as the commutation relations satisfied by formal quantization of the functions e ±iϑ , where ϑ is an angle in the action angle coordinates for the integrable system. Moreover, if ϑ were a single-valued function, then its Hamiltonian vector field X ϑ would generate a local group e tX ϑ of local symplectomorphisms of the phase space preserving the Bohr-Sommerfeld polarization, which would lift to a local group e tZ ϑ of local quantomorphisms. Since the angle ϑ is a multi-valued function, e tZ ϑ is not well defined for t = nh, where h is Planck's constant and n ∈ Z. However, the shifting operators, given by e ±hZ ϑ are well defined and correspond to the operators of multiplication by e ±iϑ . The existence of shifting operators answers Heisenberg's criticism [10] of the Bohr-Sommerfeld theory.
In geometric quantization, a complete set of commuting observables corresponds to a polarization. For a completely integrable Hamiltonian system with a regular foliation by Lagrangian tori, we get Bohr-Sommerfeld theory by choosing a polarization tangent to the tori of the foliation [4]. Taking into account the existence of shifting operators, we obtain a full geometrically based quantum theory. We do not try to compare the results of our quantization scheme with observations. For readers who would like to compare the energy spectra of the Schrodinger and the Bohr-Sommerfeld quantizations of the mathematical pendulum, we provide implicit equations for the energy spectrum in Bohr-Sommerfeld theory. In interesting completely integrable systems [5], the foliation by tori is not regular and we have to take into account the singularities of the polarization to obtain the Bohr-Sommerfeld quantum spectrum.
In this paper we discuss how to treat the singularities in the mathematical pendulum.
2 The classical mathematical pendulum
The Hamiltonian H has two critical points: one at (0, 0) with H(0, 0) = 0 and the other at (0, π) with H(0, π) = 2. These correspond to a stable elliptic and an unstable hyperbolic equilibrium point of X H , respectively.

Action-angle coordinates
In this subsection we find action-angle coordinates (I, ϑ) for the mathematical pendulum.
First, we introduce the action function I on T * S 1 such that for every connected component C(e) of the energy level H −1 (e), the restriction of I to C(e) is Before giving explicit expressions for I and ϑ we compute the Poisson bracket {I, ϑ} as follows: since ω = dθ and C(e) is parametrized by a periodic integral curve γ of X H of period T = T (e). We reparametrize C(e) using ϑ = 2π T t, which is the angle function. Because the matrix of the symplectic form ω in action angle coordinates is it follows that ω = dI ∧dϑ. Similarly, the Poisson bracket {I, H} is computed as follows: since the curve γ is closed. Thus I is constant on the integral curves of X H . So I is constant on C(e). Consequently, We now give explicit expressions for the action I and the angle ϑ of the mathematical pendulum. There are two cases.
We denote by I 0 the restriction of I to the region P 0 = {(p, α) ∈ T * S 1 | H(p, α) < 2}. Because (0, 0) is a nondegenerate minumum of the Hamiltonian H with minimum value 0, for e near 0 the level set H −1 (e) is diffeomorphic to a circle S 1 and hence is connected. From the Morse isotopy lemma it follows that for every e with 0 < e < 2 the level set H −1 (e) is diffeomorphic to a circle and hence is connected. By definition where e = 1 − cos α ± , which implies that α − = −α + , since cos is an even function. Therefore using the identity cos α = 1 − 2 sin 2 α 2 and the change of variables sin α 2 = sin α + 2 sin ϕ. We check some limiting cases. First when e 2 we obtain lim e 2 I 0 (e) = 8 π . When e 0 we find that I 0 (e) ∼ 4e π π/2 0 cos 2 ϕ dϕ = e, which is what is given by the harmonic oscillator.
We now find the corresponding angle ϑ 0 . By definition where T = T (e) is the period of the motion of the mathematical pendulum on H −1 (e). From Hamilton's equations it follows that Again we check some limiting cases. First, when e 2 we find that T ∞. So ϑ 0 0. Second, when e 0 we get T 4 π/2 0 dϕ = 2π. So ϑ 0 2ϕ = α, which checks with the angle given by the harmonic oscillator.
First we find the restrictions I ± of I to the regions P ± = {(p, α) ∈ T * S 1 | H(p, α) > 2, ± p > 0}. Because (0, π) is a nondegenerate critical point of Morse index 1 of the Hamiltonian H with critical value 2, for e > 2 but near to 2 the level set H −1 (e) is diffeomorphic to the disjoint union of two circles C ± (e). By the Morse isotopy lemma it follows that for all e > 2 the level set H −1 (e) is diffeomorphic to C − (e) C + (e). By definition We check two limiting cases. When e 2, lim e 2 I ± (e) = 4 π π/2 0 cos ϕ dϕ = 4 π , which is one half of the action I(e) at e = 2. This is correct because as We now find the corresponding angle ϑ ± . By definition where T ± = T ± (e) is the period of the motion of the mathematical pendulum on H −1 (e). From Hamilton's equations it follows that Again we check some limiting cases. First, when e 2 we find that T ± ∞. So ϑ ± 0. Second, when e ∞ we get T ± ∼ π √ 2e . So ϑ ∼ 4ϕ = 2α. It follows from the above discussion that the action function I, defined by equation (4) is continuous on [0, ∞). However, I(e) is not smooth at e = 2, see Dullin [9].

Elements of geometric quantization
In this section, we review the elements of geometric quantization applicable to the mathematical pendulum following [11].
Consider a trivial complex line bundle L = C × T * S 1 with projection map ρ : L → T * S 1 : z, (p, α) → (p, α) and trivializing section λ 0 : T * S 1 → L : (p, α) → (1, (p, α)). Define a connection ∇ on L by setting where is Planck's constant divided by 2π and θ = p dα is the canonical 1form on T * S 1 . Since ω = dθ, it follows that the curvature of the connection ∇ is −1 2π ω. We consider the geometric quantization of the mathematical pendulum with respect to the singular polarization D of T * S 1 consisting of all integral curves of the Hamiltonan vector field X H (2) 1 associated to the Hamiltonian function H (1). This means that quantum states of the mathematical pendulum are represented by sections of the prequantization line bundle L that are covariantly constant along D. It should be noted that this representation is not unique. Multiplication of every section by a constant phase factor leads to an equivalent representation. For e / ∈ {0, 2}, the leaves of D are smooth and are topological circles due to the conclusions of subsection 2.2. Moreover, this polarization has singularities consisting of the equilibrium points (0, 0) and (0, π) of X H and two homoclinic orbits of X H , which have (0, π) as a common boundary. Therefore, we are extending geometric quantization to a singular polarization, which leads to the difficulties encountered here.

Bohr-Sommerfeld conditions
Consider an integral curve γ : R → T * S 1 : t → γ(t) = p(t), α(t) of the Hamiltonian vector field X H . Suppose that e = H(γ(t)) is not 0 or 2. Then, γ is periodic with period T = 0. The cases when e = 0 and e = 2 will be discussed separately.
Let σ : T * S 1 → L be a section of the prequantization line bundle that is covariantly constant along D. Then γ * σ : R → L is a horizontal lift of γ to L. γ * σ is periodic with period T if either the restriction of the connection ∇ to the image of γ has trivial holonomy group, or σ restricted to the image of γ is identically zero.
for some n ∈ Z.
Proof. Consider an integral curve γ : R → T * S 1 : t → γ(t) = p(t), α(t) of the Hamiltonian vector field X H . It satisfies Hamilton's equations (3). Suppose that e = H(γ(t)) ∈ (0, 2). Then, the curve γ is periodic with period T , see (9). Let be a horizontal lift of γ. Then the covariant derivative D dt γ(t) of γ must vanish. Equation (12) implies that Here the + sign corresponds to α ∈ [0, α + ] and the − sign corresponds to α ∈ [α − , 0] = [−α + , 0]. Integrating (14) from α − to α + and using the fact that cos is an even function, we get by equation (6). The horizontal lift γ of the closed curve γ is a closed curve in the line bundle L if and only if z(α + ) = z(α − ). Since ln is a multivalued function and ln 1 = 2πni, it follows that γ is a closed curve in L if and only if we have I| C(e) = n . Equation (13) gives the Bohr-Sommerfeld conditions discussed in the introduction. The action integral is independent of the parametrization of γ within its orientation class. However, the change of orientation of γ would lead to the change from n to −n. Therefore, the Bohr-Sommerfeld condition (13) depends only on the image of γ. In the following, we shall refer to the image an integral curve γ of X H that satisfies equation (13) as a Bohr-Sommerfeld torus. The integer n on the right hand side of equation (13) is called the quantum number of the corresponding Bohr-Sommerfeld torus.
Since integral curves of X H preserve the Hamiltonian H, we may rewrite equation (13) in the form where C(e) is a connected component of the energy level H −1 (e). Thus, Bohr-Sommerfeld conditions (13) impose conditions on the energy. The set of values of the energy allowed by Bohr-Sommerfeld conditions is interpreted as the quantum energy spectrum of the system.
From the discussion preceding theorem 4 it follows that a section σ of the prequantum line bundle, which is covariantly constant along D, has support contained in the union of Bohr-Sommerfeld tori and the energy levels H −1 (0) and H −1 (2). Since H −1 (0) is a critical point, the restriction of σ to H −1 (0) is the value of σ at H −1 (0) which is not restricted by the condition that σ is covariantly constant along D. So we may allow the value e = 0 in equation (15). On the one hand, we consider H −1 (0) as a (singular) Bohr-Sommerfeld torus corresponding to the quantum number n = 0. On the other hand, we assume that the singular level set H −1 (2) is not a (singular) Bohr-Sommerfeld torus.
Since a section of theprequantization line bundle that is covariantly constant along D has its support in the union of Bohr-Sommerfeld tori, which has empty interior, such sections can be smooth only in the sense of distributions. Therefore, we adopt the following definition.
Definition 4.2 A quantum state of the mathematical pendulum is a section σ of the prequantization line bundle ρ, whose support lies in the union of Bohr-Sommerfeld tori such that for each Bohr-Sommerfeld torus C the restriction σ| C of σ to C is a smooth covariantly constant section of ρ| C .
Let H be the space of quantum states of the mathematical pendulum. For each Bohr-Sommerfeld torus C, we choose a non-vanishing smooth covariantly constant section σ of L |C . The family {σ |C } is a basis of H, which we shall refer to as a Bohr-Sommerfeld basis. Give H a hermitian scalar product (· | ·) so that the Bohr-Sommerfeld basis {σ |C } is orthonormal. Thus, we have obtained a vector space structure on the space of states of the mathematical pendulum. Note that this structure is not uniquely determined by the geometry of the classical phase space. We have the freedom of multiplying each basis vector σ |C by a nonzero complex number.
if it is constant on Bohr-Sommerfeld tori. Bohr-Sommerfeld quantization assigns to a quantizable function f a linear operator Q f on H such that, for each Bohr-Sommerfeld torus C Observe that the operators Q f corresponding to Bohr-Sommerfeld quantizable functions f are diagonal in the Bohr-Sommerfeld basis. Since the Bohr-Sommerfeld tori are closed and mutually disjoint, for any function C → λ C on the collection of Bohr-Sommerfeld tori, there exists a function f ∈ C ∞ (T * S 1 ) such that f |C = λ C . Thus, each basis vector σ |C is an eigenvector of the operator Q f corresponding to an eigenvalue λ C .

Structure of the Bohr-Sommerfeld basis
We now study of the structure of the Bohr-Sommerfeld basis {σ| C }.

The energy level
A Bohr-Sommerfeld torus in P 0 can be labelled by its quantum number n = 0, . . . , N , where N is the largest nonnegative integer such that N < I (2). Thus {σ| C } 0 = {σ 0 0 , . . . , σ 0 N }, where the subscript n = 0, . . . , N is the quantum number of the state σ 0 n and the superscript 0 reminds us that σ 0 n lies in H 0 . Similarly a Bohr-Sommerfeld torus in P ± can be labeled by it quantum number m ≥ M , where 2M is the smallest even nonnegative integer greater than or equal to N + 1. In other words, The structure of the Bohr-Sommerfeld set can be used to study the energy spectrum of the mathematical pendulum, which consists of values of e n such that a connected component C(e n ) of the energy level H −1 (e n ) satisfies Bohr-Sommerfeld conditions 1 2π C(en) pdα = n for some integer n. See the discussion following equation (12). The part of energy spectrum contained in the interval interval [0, 2] is simple, and it can be obtained by solving for e n equation where 0 ≤ n ≤ N , and N is the largest positive integer such that e N < 2. We have assumed that e = 2 is not in the energy spectrum of the mathematical pendulum. For e > 2, The part of the energy spectrum contained in the half line [2, ∞) can be obtained by solving for e m the equation where 2m ≥ N + 1 ensures that e m > 2. In this range, each eigenspace is 2-dimensional.

Transitions between quantum states
In this subsection we discuss the transitions between quantum states given by the horizontal arrows in diagram (18). The transitions from σ 0 N to σ ± M given by slanted arrows in diagram (18) involve crossing the energy level 2, where the action I is continuous but not differentiable. This requires understanding the Z 2 symmetry of the mathematical pendulum, which will be treated in the next section.
In diagram (18) transitions involving the right pointing horizontal arrows correspond to the action of an operator b on H such that We refer to b as the raising operator on H. Transitions involving the left pointing horizontal arrows give rise to the lowering operator a such that and Since σ 0 0 is the lowest point in the lattice, we require that Thus in diagram (18) the lowering operator a corresponds to left pointing horizontal arrows, while the raising operator b corresponds to right pointing horizontal arrows. 2 In order to interpret the slanted arrows in the diagram we need to discuss the Z 2 symmetry of the mathematical pendulum. The shifting operators a and b will be constructed by lifting the shifting operator in the Z 2 -reduced quantum system. In the appendix we construct this shifting operator using geometric quantization.
In order to identify the function whose quantization might lead to the operator b we extend Dirac's quantization rule to complex valued functions. Action angle coordinates (I, ϑ) on P 0 ∪P + ∪P − restrict to action angle coordinates (I 0 , ϑ 0 ) on (P 0 , ω| P 0 ) and (I ± , ϑ ± ) on (P ± , ω| P ± ), respectively. The latter action angle coordinates have been computed in section 2.2. They satisfy the Poisson bracket relations {I 0 , ϑ 0 } = −1 and {I ± , ϑ ± } = −1 on P 0 and P ± , respectively. Therefore If we introduce the quantum operator Q e iϑ on H, equations (24) and (25) imply On the other hand, equation (16) yields and Equations (19) and (20) imply that the operator b and the quantized actions satisfy the commutation relations for n = 0, 1, . . . , N − 1 and m = M, M + 1, . . ., respectively. Comparing equations (26) and (29) shows that the raising operator b defined in equations (19) and (20) satisfy the same commutation relations as the operator Q e iϑ .
It is of interest to see to what degree the raising and the lowering operators on H correspond to quantization of classical functions on P = P 0 ∪ P − ∪ P + . Their restrictions to the subspaces H 0 , H − and H + of H with supports in H 0 , H − and H + , respectively, can be related to quantization of the functions e ±iϑ 0 , e ±iϑ − , and e ±iϑ − on P 0 , P + and P − , respectively.

Choose the basic sections of basic sections {σ
follows that the restriction b 0 of the raising operator b to H 0 may be interpreted as the operator Q e iϑ 0 , which sends σ 0 n into e iϑ 0 σ 0 n for 1 ≤ n ≤ N . Similarly, the restriction b ∓ of the raising operator b to H − may be interpreted as the operator Q e iϑ − of multiplication by e iϑ 0,∓ . In other words, b − = Q e iϑ − . Similarly, b + = Q e iϑ + , is the restriction of the raising operator b to H +. On the other hand, the restriction b 0 of the raising operator b to H 0 agrees with e iϑ 0 except on σ 0 n because bσ 0 n is not in H 0 . In the same way, the lowering operators a 0 , a − , and a + may be interpreted in terms of the multiplication operators Q e iϑ 0 , Q e iϑ − , and Q e iϑ + , respectively.

The Z 2 -symmetry
The mathematical pendulum (H, T * S 1 , ω) has a Z 2 -symmetry generated by because the Hamiltonian H (1) and the symplectic form ω = dp ∧ dα are invariant. In more detail, for every The quantized mathematical pendulum (H, T * S 1 , ω) has quantum line bundle is a section, which trivializes the bundle ρ. The bundle space L has a Z 2 -symmetry generated by the mapping The mapping µ covers the Z 2 -symmetry of the mathematical pendulum generated by ζ (30), since ρ • µ = ζ • ρ.

The Z 2 -symmetric quantum system
Consider the Z 2 -symmetry on L generated by the mapping µ (32). The Z 2 -symmetric quantized system is the mathematical pendulum with Z 2symmetry generated by ζ (30) and quantum line bundle ρ (31) having the Z 2 -symmetry generated by µ (32).
Let Γ(ρ) be the vector space of smooth sections of the bundle ρ.
Thus the operator P 0,± is bijective. Since P 2 0,± = id H ± , it follows that P −1 0 = P 0 and P −1 ± = P ∓ . We say that a quantum state σ| C of the Z 2 -quantized mathematical pendulum (H, T * S 1 , ω) with quantum line bundle ρ is even if it is covariantly constant even section σ| C of ρ, that is, P(σ| C ) = σ| C . Let H even be the vector space spanned by the even quantum states. σ| C is an odd quantum state if it is covariantly constant odd section σ| C of ρ, that is, P(σ| C ) = −σ| C . Let H odd be the vector space spanned by the odd quantum states.
Theorem 6.1.4 Let σ| C be a nonzero even or odd quantum state in H even ∩ H 0 or H odd ∩ H 0 , respectively. Then its quantum number is even or odd, respectively.
Note that z| C is nowhere vanishing, for if it vanished at some point in C then it would be identically zero on C, since σ| C is covariantly constant. But this contradicts our hypothesis. Because σ| C ∈ H 0 by hypothesis, it follows that C = C(e) = (H × ) −1 (e), where H × = H|T × S 1 and 0 < e < 2.
Corollary 6.1.5 Let σ| C be a nonzero even or odd quantum state in H even ∩ H 0 or H odd ∩H 0 , respectively. Then the quantum number of σ| C is the number of Z 2 -orbits of ε µ (32) on the image of the horizontal lift of the curve γ, which parametrizes C.
It follows from lemma 6.1.2 that the operators P ± enable us to go from H + to H − and back. Thus they play the role of shifting operators. In particular {P + σ + m } m≥M is a basis of H − . Because the parity operator P induces a Z 2 -symmetry on the Hilbert space H, by averaging the given inner product, we may assume that the parity operator P preserves the new inner product on H. In order to simplify the presentations we choose the orthonormal bases {σ ± m } of H ± so that P ± σ ± m = σ ∓ m . In order to construct operators relating H 0 to H ± we need to show that reduction of the Z 2 -symmetry of the mathematical pendulum gives rise to the quantized Z 2 -reduced mathematical pendulum.

Z 2 -quantization and reduction
In this subsection discuss quantization of the Z 2 -reduced mathematical pendulum.

Reduction of the Z 2 -symmetry
Here we reduce the Z 2 -symmetry of the mathematical pendulum (H, T * S 1 , ω) generated by ζ (30).
First we determine the reduced phase space P , which is the space T * S 1 /Z 2 of orbits of the Z 2 -symmetry on T * S 1 . To start with we use cut and paste geometric methods to construct the Z 2 -orbit space. Recall that a connected subset ∆ of T * S 1 is a fundamental domain for the Z 2 -symmetry generated by ζ (30), if it contains exactly one point of each Z 2 -orbit in T * S 1 .
Look at the closure ∆ of ∆ in T * S 1 and identify the points on the boundary of ∆, which lie on the same Z 2 -orbit. The resulting space is a model for the orbit space T * S 1 /Z 2 .
We now give another construction for the reduced phase space using invariant theory and the concept of a differential space, see [2]. The algebra of real analytic functions on T * S 1 , which are invariant under the symmetry group Z 2 , is generated by These invariant functions are subject to the relation which defines the Z 2 -orbit space P = T * S 1 /Z 2 as a semialgebraic variety in R 3 with coordinates τ = (τ 1 , τ 2 , τ 3 ). We say that a function f on P is smooth if there is a smooth function F on R 3 such that f = F | P . Let t1 t2 t3 Figure 2. The Z2-reduced space in R 3 .
C ∞ ( P ) be the space of smooth functions on P . Then P , C ∞ ( P ) is a locally compact subcartesian differential space, because P is a semialgebraic.
Next we construct the reduced Hamiltonian. Since the Hamiltonian H of the mathematical pendulum is invariant under the Z 2 -symmetry, it induces a smooth function H on P given by restricting the smooth function τ 3 : R 3 → R : τ → τ 3 to P .
In order to have dynamics on P , we first need a Poisson bracket { , } R 3 on C ∞ (R 3 ). A calculation using the Poisson bracket { , } on P shows that Here , is the Euclidean inner product on R 3 and × is the vector product.
Because of (41), the defining function C (40) of P is a Casimir in the Poisson Hence, the collection I of all smooth functions on R 3 , which vanish identically on P , is a Poisson ideal in A.
Consequently, the Poisson bracket { , } P is well defined and Consider the derivation −ad τ 3 on the Poisson algebra A. This derivation gives rise to the Z 2 -reduced Hamiltonian vector field −ad H on the locally compact subcartesian differential space ( P , C ∞ ( P )) associated to the Z 2reduced Hamiltonian H. To see this note that on R 3 the integral curves of −ad τ 3 satisfyτ Because C is a Casimir of the Poisson algebra A, we obtain 0 = {C, τ 3 } R 3 . In other words, C is an integral of −ad τ 3 . A calculation shows that −ad τ 3 leaves the sets C −1 (0), {τ 3 + τ 1 − 1 = 0}, and {τ 1 = ±1} invariant. Thus the Z 2 -reduced space P is invariant under the flow of −ad τ 3 . Consequently, the Z 2 -reduced Hamiltonian vector field −ad H , where H = τ 3 | P , is defined on P . Because the Hamiltonian vector field X H of the mathematical pendulum is complete, the reduced vector field −ad H is complete. Its flow ϕ H t is a 1-parameter group of diffeomorphisms of P . In fact, for p ∈ H −1 (e) the closure of the integral curve t → ϕ H t ( p) is a connected component of the level set H −1 (e), since a level set of the reduced Hamiltonian H is compact.
We now compute the Z 2 -reduced actions. On Consequently, the reduced action when 0 < e < 2 or e > 2. We now calculate the integral in (44). First we consider the case when 0 < e < 2. Letting u 2 = τ 1 − (1 − e), u = √ ev, and then v = cos ϕ, we get successively Next we treat the case when e > 2. Letting τ 1 = cos ϑ and ϑ = 2ϕ successively, we get Note that the Z 2 -reduced action I × (45) and (46) is one half the original action I × = I| T × S 1 (8) and (10). Moreover, the function e → I × (e) is continuous at e = 2. Dullin [9] shows, I × has a logarithm term in its series expansion in e − 2, which shows I × is not differentiable at e = 2.
The corresponding Z 2 -reduced angle ϑ × is where t ∈ [max(1 − e, −1), 1] and In order to simplify the notation, in the following we will use I for the reduced action on the reduced space P and ϑ for the reduced angle, which is not defined at the singular points p 0 and p 2 of P . Note that I × = I| P × and ϑ × = ϑ| P × .

Reduction of the Z 2 -quantum symmetry
In this subsubsection we reduce the Z 2 -symmetry of the quantized mathematical pendulum. In other words, we reduce the Z 2 -action on L ⊆ C × R 2 generated by the mapping µ (32). We use invariant theory.
We need to find a connection ∇ × on the smooth sections of the bundle ρ × , which is related to the original connection ∇ × on P × . On smooth sections of the bundle ρ × we have a connection, whose covariant derivative ∇ × X in the direction of the smooth vector field X on T × S 1 acts on the section Suppose that X is a smooth vector field on P × , which is π × -related to the vector field X, that is, T π × X = X • π × . On the line bundle ρ × with trivializing section In other words, Proof. Equation (50) follows because by definition Next we determine the quantization rules for the Z 2 -reduced Hamiltonian system ( H × = H| P × , P × , ω × ) with quantum line bundle ρ × and trivializing section λ 0 . The mapping P × → T P × : p → span{X H × (p)} defines a smooth Lagrangian distribution D on the symplectic manifold ( P × , ω × ), which is a polarization of ( P × , ω × ). A leaf of D is a connected component of a level set ( H × ) −1 (e) of the Z 2 -reduced Hamiltonian H × on P × , which is a smooth S 1 when e ∈ (0, 2) ∪ (2, ∞).
Let γ : R → P × be an integral curve of X H × of energy e ∈ (0, 2) ∪ (2, ∞). Then γ is periodic of primitive period T = T (e) > 0. Also γ parametrizes a connected component C × (e) of the smooth level set ( H × ) −1 (e). Parallel transport the section λ of the C × -bundle ρ × along γ using the connection ∇ × . Then at every point γ(t) in ( H × ) −1 (e) we have where F (t) = f (γ(t)). For equation (51) to have a nonvanishing solution the holonomy A( T ) of the connection ∇ × along γ must equal 1, because In other words, when e ∈ (0, 2) ∪ (2, ∞) the quantization rule for the (Z 2 , ·)reduced quantized Hamiltonian system ( H × , P × , ω × ) with quantum bundle where I × is the action (44) of the Z 2 -reduced mathematical pendulum. Since the image under π × of a Bohr-Sommerfeld torus of the mathematical pendulum is a Bohr-Sommerfeld torus of the Z 2 -reduced mathematical pendulum.
For every positive integer k, let σ k be a section of the line bundle ρ × , which is supported and covariantly constant on the level set ( I × ) −1 (k ). As before we add the quantum number 0, which corresponds to a section supported on the singular Bohr-Sommerfeld torus corresponding to the singular point (1, 0, 0) of P . The collection { σ k } k∈Z ≥0 is an orthonormal basis of the space H of quantum states of the Z 2 -reduced mathematical pendulum.
Since the quantum states { σ k } k∈Z ≥0 are ordered by increasing k, there exist shifting operators a and b such that b σ k = σ k+1 , for k ≥ 0 a σ k = σ k−1 , for k > 0 and a σ 0 = 0. (53) Because the local lattice structure of the set of Bohr-Sommerfeld tori on P is linear, the shifting operators a and b are also well defined across the singularitiy at the reduced energy value e = 2. As before, the operators a and b satisfy the same commutation relations as the quantum operators Q e −i ϑ and Q e i ϑ , respectively.

Lifting the shifting operators
In this subsubsection we use the isomorphism R : H even → H to lift the shifting operators on H to shifting operators on H even .
Recall that 2M = N + 2, if N is even To define shifting operators on H even recall that equation (53) defines the shifting operators b and a on H. We may lift the shifting operator a to the shifting operator a even on H even by setting and lift the shifting operator b to the shifting operator b even on H even by setting If k + 1 ≤ K, then Rb even σ 0 2k = bR σ 0 2k = b σ k = σ k+1 = R σ 0 2k+2 .
So b even σ 0 2k = σ 0 2k+2 , because R is injective. Since n = 2k, it follows that b even raises the quantum number n by 2, provided that n + 1 < N . Hence σ 0 n+2 = bσ 0 n+1 = bbσ 0 n = b even σ 0 n . A similar argument shows that b even raises the quantum number m ≥ M by 1 and that b even . Analogous results can be obtained for the lowering operator a even . In particular, if k ≤ K, then This implies that a even σ 0

Crossing the singularity
The operators a even and b even , defined in equation (55a) and (55b), respectively allow for shifting quantum states which cross the singular level set H −1 (2). In order to write this out explicitly, we need to consider the cases when N is even or odd seperately.
We look at the operators a even and b even when N = 2K, M = 1 2 (N +2) = K + 1, and σ 0 N = σ 0 2K ∈ H even ∩ H 0 together with In this case equation (55b) yields Since R : H even → H is injective, we get b even σ 0 Let pr ± : which represents the transition given by the right pointing top and bottom slanted arrows in diagram (18) when N is even. Similarly, which implies Next we look at the operators a even and b even when N = 2K +1 and M = K + 1 and σ 0 N = σ 0 2K+1 ∈ H odd ∩ H 0 . Then by theorem 8 we have σ 0 Therefore, in order to cross from σ 0 N to σ + M + σ − M , we first go to σ 0 N −1 and then to σ Hence for odd N , we have which represents the right pointing top and bottom slanted arrows in diagram (18) when N is odd. Similarly, Let Using the injection mapping ι ± and equations (59) and (62) when N = 2M − 2 we have a even ± (σ ± M ) = a even ι ± (σ ± M ) = a even (σ + The operator a even + represents the transition given by the left pointing top slanting arrow in diagram (18); while the operator a even − represents the transition given by the left pointing bottom slanting arrow in the diagram.

Appendix: construction of the lowering operator
In this appendix we construct the lowering operator a for the quantized Z 2 -reduced system on T × S 1 .
The action integral of Bohr-Sommerfeld quantization is I = 2π 0 p dϑ = 2π p. The variable conjugate to I is θ = ϑ/2π, since ω = dI ∧ dθ. Let (I, θ) be coordinates on T * S 1 = R × S 1 with S 1 = R/Z and let ω = dI ∧ dθ be the symplectic form on T * S 1 . The vector field X = − ∂ ∂I = − 1 2π ∂ ∂p on T * S 1 is locally Hamiltonian, since In what follows we find a quantomorphism Φ h of (L, λ), which covers e hX . In other words, Φ h is a diffeomorphism of L into itself such that Φ * h λ = λ and π • Φ h = e hX • π. Here is the line bundle, associated to the C × principal bundle π × : L × → T * S 1 , with connection 1-form λ = 1 2πi dz − 1 h Idθ. The vector field X is integral, that is, 1) there is an good covering U = {U i } i∈I of T * S 1 by open sets U i , i ∈ I, where every finite intersection of elements of U is either empty or contractible; 2) for every U i , U j ∈ U such that U i ∩ U j = ∅ we have θ| U i − θ| U j is an integer on U i ∩ U j . The local Hamiltonian functions θ| U i for i ∈ I piece together to give a smooth mapping [θ] : T * S 1 → S 1 = R/Z, which is the "coordinate" θ, that is, Consider the vector field Z on L × , whose flow is The flow of Z preserves the connection 1-form β, since for every (b, I, θ)) ∈ L × we have We have where Y θ/h (b, I, θ) = d dt t=0 (b e 2πi tθ/h , I, θ) is a vector field on (L × , β), whose flow is e tY θ/h (b, I, θ) = (b e 2πi tθ/h , I, θ), and liftX is a vector field on (L × , β), which is the horizontal lift of the vector field X, that is, liftX(b, I, θ) ∈ ker β(b, I, θ) for every (b, I, θ) ∈ L × . The vector fields liftX and X are π ×related, that is, Note that the flows of the vector fields Y θ/h and liftX commute.
We now look at the universal covering space (T * R, ω) of (T * S 1 , ω) with coordinates (p, q) and symplectic form ω = dp ∧ dq. The universal covering map is given by κ : T * R → T * S 1 : (p, q) → ( 1 2π I, θ) = (p, q mod Z), since R → S 1 = R/Z : q → q mod Z is the smooth universal covering map of S 1 . Pull the local Hamiltonian vector field X on T * S 1 back by the covering map κ to a vector field X on T * R, which is κ-related to X, that is, T (p,q) κ X(p, q) = X κ(p, q) for every (p, q) ∈ T * R. The integral vector field X is the Hamiltonian vector field X q+c = − ∂ ∂p associated to the Hamiltonian function q : T * R → R : (p, q) → q. The constant c can be choosen to be 0, because the smooth mappings [q] : T * R → S 1 = R/Z : (p, q) → q mod Z and κ * [θ] : T * R → S 1 : (q, p) → [θ] κ(p, q) are equal, namely, [q] = κ * [θ].
The C × bundle π × : L × → T * S 1 pulls back under the covering map κ to the C × bundle π × : L × → T * R : (b, p, q) → (p, q) with connection 1-form The flow e tZ on L × lifts to the 1-parameter group of diffeomorphisms e t Z : L × → L × : (b, p, q) → b e −2πi tq/h , p − t, q , which preserves β. We have e t Z = e −t Y q/h • e t liftXq , where Y q/h is the vector field on L × , whose value at (b, p, q) is d dt t=0 (b e −2πi tq/h , p, q). Its flow is given by e tY q/h (b, p, q) = (b e 2πi tq/h , p, q). Also liftX q is a vector field on L × , which is the horizontal lift of X q using the connection 1-form β on L × . The flow of liftX q is e t liftXq (b, p, q) = (b, p − t, q). Note that the flows e tY q/h and e t liftXq commute. Since L × = κ * L × , there is a smooth mapping κ × : L × → L × : (b, p, q) → (b, p, q mod Z) = (b, 1 2π I, θ), which covers κ, that is, π × • κ × = κ • π × . The flows e t liftXq and e t liftX are κ × -related, that is κ × • e t liftXq = e t liftX • κ × .
The locally Hamiltonian vector field X on T * S 1 with flow e tX : T * S 1 → T * S 1 : (I, θ) → (I − t, θ) lifts to a vector field X on L, whose flow is e t X : L → L : (z, I, θ) → (z, I − t, θ). The map µ : L = C × T * S 1 → L × = C × × T * S 1 : (z, I, θ) → (e z , I, θ) = (b, I, θ) (72) is smooth and µ * β = λ. Moreover, we have µ • e t X = e t liftX • µ. Instead of e t X we will write e tX . The mapping µ intertwines the (right) action of C × on L × with the action of C × on L, namely, µ(b z, I, θ) = µ(b, I, θ)(b ) −1 for every b, b ∈ C × and every (I, θ) ∈ T * S 1 . From these remarks it follows that the operator Φ × h = e −2πi [θ] • ( e h liftX ) * on smooth sections of the line bundle π × : L × → T * S 1 becomes the operator Φ h = e 2πi [θ] ( e h liftX ) * on smooth sections of the line bundle π : L → T * S 1 . Clearly, Φ h covers e hX , that is, π • Φ h = e hX • π, and preserves the connection 1-form λ. So Φ h is the desired lowering operator a. We note that Φ −h is the raising operator b.