1. Introduction
Even though the Bohr–Sommerfeld theory was very successful in predicting some physical results, it was never accepted by physicists as a valid quantum theory in the same class as the Schrödinger theory or the Bargmann–Fock theory. The reason for this was that the original Bohr–Sommerfeld theory did not provide operators of transition between quantum states. The need for such operators in the Bohr–Sommerfeld quantization was already pointed out by Heisenberg [
1]. The aim of this paper is to derive operators of transition between quantum states in the Bohr–Sommerfeld theory, which we call shifting operators, from the first principles of geometric quantization.
The first step of geometric quantization of a symplectic manifold
is called prequantization. It consists of the construction of a complex line bundle
with connection whose curvature form satisfies a prequantization condition relating it to the symplectic form
. A comprehensive study of prequantization, from the point of view of representation theory, was given by Kostant in [
2]. The work of Souriau [
3] was aimed at quantization of physical systems, and studied a circle bundle over phase space. In Souriau’s work, the prequantization condition explicitly involved Planck’s constant
. In [
4], Blattner combined the approaches of Kostant and Souriau by using the complex line bundle with the prequantization condition involving Planck’s constant. Since then, geometric quantization has been an effective tool in quantum theory.
We find it convenient to deal with connection and curvature of complex line bundles using the theory of principal and associated bundles [
5]. In this framework, the prequantization condition reads
where
is the connection 1-form on the principal
-bundle
associated to the complex line bundle
, and
is the multiplicative group of nonzero complex numbers.
The aim of prequantization is to construct a representation of the Poisson algebra
of
on the space of sections of the line bundle
L. Each Hamiltonian vector field
on
P lifts to a unique
-invariant vector field
on
that preserves the principal connection
on
. If the vector field
is complete, then it generates a 1-parameter group
of symplectomorphisms of
. Then, the vector field
is complete and it generates a 1-parameter group
of connection preserving diffeomorphisms of the bundle
, called quantomorphisms, which cover the 1-parameter group
. The term quantomorphism was introduced by Souriau [
3] in the context of
-principal bundles and discussed in detail in his book [
6]. The construction discussed here follows [
7], where the term quantomorphism was not used. In this case,
and
are 1-parameter groups of diffeomorphisms of
P and
, respectively. We refer to
and
as flows of
and
. Since
L is an associated bundle of
, the action
, induces an action
which gives rise to an action on smooth sections
of
L by push forwards,
. Although
may not be defined for all sections
and all
t, its derivative at
is defined for all smooth sections. The prequantization operator
where
is Planck’s constant
divided by
, is a symmetric operator on the Hilbert space
of square integrable sections of
L. The operator
is self adjoint if
is complete.
The whole analysis of prequantization is concerned with
global Hamiltonian vector fields. Since every vector field on
that preserves the symplectic form is locally Hamiltonian, it is of interest to understand how much of prequantization can be extended to this case. In particular, we are interested in the case where the locally Hamiltonian vector field is the vector field
of the integer angle variable
that is defined up to an additive term
where
For a globally Hamiltonian vector field
,
where
is the horizontal transport of section
by parameter
t along integral curves of
Replacing
f by a multivalued function
, defined up to an additive
n, yields the multivalued expression
We observe that, for
, Equation (
3) gives a single valued expression
The shifting operator
is an operator on
which shifts the support of
by
along the integral curve of
. If the vector field
is complete, then
for every
.
Our results provide an answer to Heisenberg’s criticism that in Bohr–Sommerfeld theory there are not enough operators to describe transitions between quantum states [
1].
Superficially, the shifting operator
, see Equation (
5), appears to be a quantization of an angle
. It depends on
and has the factor
considered by Dirac [
8]. However, the factor
, describing the parallel translation by
along integral curves of
makes it nonlocal in the phase space. Therefore,
cannot satisfy local commutation relations with any local quantum variable that is described by a differential operator. Hence, it cannot be the canonical conjugate of the corresponding action operator, or any other operator, which is local in the phase space.
In our earlier papers [
9,
10,
11,
12], we followed an algebraic analysis, similar to that used by Dirac [
8], supplemented by heuristic guesses about the behaviour of the shifting operators at the points of singularity of the polarization. In particular, we assumed that
vanishes on the states concentrated on a set of limit points of
as
In the present paper, we derive shifting operators in the framework of geometric quantization, and extend our result to cases with a variable rank polarization.
The second stage in geometric quantization consists of the choice of a polarization, which is an involutive complex Lagrangian distribution F on the phase space. Suppose that P is the cotangent bundle space of the configuration space. In this case, the choice of F containing the vertical directions, leads the quantum mechanics of Schrödinger. If F leads to complex analytic structure on P, we have the Bargmann–Fock theory. If F is spanned by the Hamiltonian vector fields of a completely integrable system, we have Bohr–Sommerfeld theory. Each of these theories have specific structure, which is helpful in formulation and solving problems. In the following, we restrict our investigation to the Bohr–Sommerfeld theory in order to emphasize its membership in the class of quantum theories corresponding to different polarization.
A common problem in arising in quantum theories is occurrence of singularities. Usually, one studies the geometric structure of the theory in the language of differential geometry of smooth manifolds, and then investigates the structure of singularities separately. The theory of differential spaces, introduced by Sikorski [
13,
14], is a powerful tool in the study of the geometry of spaces with singularities [
15]. The main singularity encountered here corresponds to the fact that the polarization
F spanned by the Hamiltonian vector fields of a completely integrable system does not have constant rank. This singularity is so well known that we do not have to use the language of differential spaces to get results. It should be noted that the results in [
9,
11] rely on the theory of differential spaces.
In conclusion, it should be mentioned that the scientists, who used visual presentation of the Bohr–Sommerfeld spectra in terms of dots on the space of the action variables, are familiar with handling shifting operators. The line segments joining two dots corresponding to quantum states represent the shifting operators between these states.
To make the paper more accessible to the reader, we have provided an introductory section with a comprehensive review of geometric quantization. Experts may omit this section and proceed directly to the next section on Bohr–Sommerfeld theory.
3. Bohr–Sommerfeld Theory
3.1. Historical Background
Consider the cotangent bundle
of a manifold
Q. Let
be the cotangent bundle projection map. The Liouville 1-form
on
is defined as follows. For each
,
and
,
The exterior derivative of
is the canonical symplectic form
on
.
Let . A Hamiltonian system on with Hamiltonian is completely integrable if there exists a collection of functions , which are integrals of , that is, for , such that for i, . Assume that the functions are independent on a dense open subset of . For each , let be the orbit of the family of Hamiltonian vector fields passing through p. This orbit is the largest connected immersed submanifold in with tangent space equal to . The integral curve of starting at p is contained in . Hence, knowledge of the family of orbits provides information on the evolution of the Hamiltonian system with Hamiltonian .
Bohr–Sommerfeld theory, see [
16,
17], asserts that the quantum states of the completely integrable system
are concentrated on the orbits
, which satisfy the
Bohr–Sommerfeld Condition:For every closed loop , there exists an integer n such thatwhere h is Planck’s constant. This theory applied to the bounded states of the relativistic hydrogen atom yields results that agree exactly with the experimental data [
17]. Attempts to apply Bohr–Sommerfeld theory to the helium atom, which is not completely integrable, failed to provide useful results. In his 1925 paper [
1], Heisenberg criticized Bohr–Sommerfeld theory for not providing transition operators between different states. At present, the Bohr–Sommerfeld theory is remembered by physicists only for its agreement with the quasi-classical limit of Schrödinger theory. Quantum chemists have never stopped using it to describe the spectra of molecules.
3.2. Geometric Quantization in a Toric Polarization
To interpret Bohr–Sommerfeld theory in terms of geometric quantization, we consider a set consisting of points where are linearly independent and the orbit of the family of Hamiltonian vector fields on is diffeomorphic to the k torus . We assume that P is a -dimensional smooth manifold and that the set is a quotient manifold of P with smooth projection map . This implies that the symplectic form on restricts to a symplectic form on P, which we denote by . Let D be the distribution on P spanned by the Hamiltonian vector fields . Since for i, , it follows that D is an involutive Lagrangian distribution on . Moreover, is a strongly admissible polarization of .
Since the symplectic form
on
is exact, we may choose a trivial prequantization line bundle
with connection 1-form
. Let
be the restriction of
to
P and let
be the 1-form on
P, which is the restriction of
to
P, that is,
. Then,
is a principal
bundle over
P with projection map
and connection 1-form
. The complex line bundle
associated to the principal bundle
is also trivial. Prequantization of this system is obtained by adapting the results of
Section 2.
Since integral manifolds of the polarization D are k-tori, we have to determine which of them admit nonzero covariantly constant sections of L.
Theorem 4. An integral manifold M of the distribution D admits a section σ of the complex line bundle L, which is nowhere zero when restricted to M, if and only if it satisfies the Bohr–Sommerfeld condition (47). Proof. Suppose that an integral manifold
M of
D admits a nowhere zero section of
. Since
is nowhere zero, it is a section of
. Let
be a loop in
M. For each
, let
be the tangent vector to
at
t. Since
is covariantly constant along
M, Claim 2 applied to the section
gives
for every
and every
. Taking
and
gives
Since
and
, we get
Hence, Equation (
48) is equivalent to
which integrated from 0 to
gives
If
bounds a surface
, then Stokes’ theorem together with Equation (
47) and the quantization condition (
7) yield
because
M is a Lagrangian submanifold of
. Thus,
, which implies that the nowhere zero section
is parallel along
. If
does not bound a surface in
M, but does satisfy the Bohr–Sommerfeld condition
(
47) with
replaced by its pull back
to
P, then
so that
Hence,
and the nowhere zero section
is parallel along
. □
Note that the manifolds
M that satisfy Bohr–Sommerfeld conditions (
47) are
k-dimensional toric submanifolds of
We call them
Bohr–Sommerfeld tori. Since Bohr–Sommerfeld tori have dimension
, there is no non-zero smooth section
that is covariantly constant along
D. However, for each Bohr–Sommerfeld torus
M, Theorem 4 guarantees the existence of a non-zero, covariantly constant along
smooth section
, where
denotes the restriction of
L to
M.
Let
be the set of Bohr–Sommerfeld tori in
P. For each
, there exists a non-zero, covariantly constant along
smooth section
of
L restricted to
M determined up to a factor in
. The direct sum
is the the space of quantum states of the Bohr–Sommerfeld theory. Thus, each Bohr–Sommerfeld torus
M represents a 1-dimensional subspace
of quantum states. Moreover,
if
because Bohr–Sommerfeld tori are mutually disjoint. Hence, the collection
is a basis of
For our toral polarization
, the space of smooth functions on
P that are constant along
F, see Equation (
44), is
, see Lemma A3. For each
, the Hamiltonian vector field
is in
D, that is,
for every basic state
. Hence, the prequantization and quantization operators act on the basic states
by multiplication by
f, that is,
Note that
is a constant because
. For a general quantum state
We see that, for every function , each basic quantum state is an eigenstate of corresponding to the eigenvalue . Since eigenstates corresponding to different eigenvalues of the same symmetric operator are mutually orthogonal, it follows that the basis of is orthogonal. This is the only information we have about scalar product in . Our results do not depend on other details about the scalar product in .
3.3. Shifting Operators
3.3.1. The Simplest Case
We begin by assuming that
with canonical coordinates
where, for each
,
is the canonical angular coordinate on the
torus and
is the conjugate momentum. The symplectic form is
In this case, action–angle coordinates
are obtained by rescaling the canonical coordinates so that, for every
we have
and
. Moreover, the rescaled angle coordinate
is interpreted as a multi-valued real function, the symplectic form
and the toric polarization of
is given by
In terms of action–angle coordinates, the Bohr–Sommerfeld tori in
are given by equation
where
. For each
, we denote by
the corresponding Bohr–Sommerfeld torus in
. If
is the connection form in the principal line bundle
, then sections
form a basis in the space
of quantum states.
For each
the vector field
is transverse to
D and
, so that
is the Hamiltonian vector field of
. In the following, we write
to describe the actual vector field
without referring to its relation to the action angle coordinates
. Equation (
36) in
Section 2.1, for
is multi-valued because the phase factor is multi-valued, and
Claim 5. If , thenis well defined. Proof. For every
, consider an open interval
in
such that
. Let
Since the action–angle coordinates
are continuous,
W is an open subset of
P. Let
be a unique representative of
with values in
. With this notation,
The restriction to
W of the vector field
is the genuinely Hamiltonian vector field of
, namely,
The vector field
is well defined. Equation (
36) yields
. Hence,
If we make another choice of intervals
in
such that
and let
. Then,
with values in
differs from
by an integer, so that
, and, in
, we have
Moreover,
, so that
Since we can cover
P by open contractible sets defined in Equation (
57), we conclude that
is well defined by Equation (
56) and depends only on the vector field
□
Consequently, there exists a connection preserving automorphism
such that, if
, where
is given by Equation (
57), then
Claim 6. The connection preserving automorphism , defined by Equation (62) depends only on the vector field and not the original choice of the action–angle coordinates.
Proof. If
is another set of action–angle coordinates then
where the matrices
and
lie in
and
. In the new coordinates,
Clearly,
To compare the phase factor entering Equation (
55), we consider an open contractible set
. As before, for each
choose a single-valued representative
of
. Then,
where each
is an integer and thus
is also an integer. Hence,
where
are integers. Since
l is constant,
Therefore,
which shows that the automorphism
depends on the vector field
and
not on the action angle coordinates in which it is computed. □
Claim 7. For each , the symplectomorphism where h is Planck’s constant, preserves the set of Bohr–Sommerfeld tori in P.
Proof. Since
is complete,
is a 1-parameter group of symplectomorphisms of
. Hence,
is well defined. By Equation (
52),
for every Bohr–Sommerfeld torus
, where
.
Since
,
This implies that, for every
,
and
. Therefore, if
, then
, if
, and
if
. This implies that
is a Bohr–Sommerfeld torus. □
We denote by
the action of
on
L. The automorphism
acts on sections of
L by pull back and push forward, namely,
Since
is a connection preserving automorphism, it follows that, if
satisfies the Bohr–Sommerfeld conditions, then
and
also satisfy the Bohr–Sommerfeld conditions. In other words,
and
preserve the space
of quantum states. The
shifting operators and
corresponding to
are the restrictions to
of
and
respectively. For every
Equations (
53) and (
56) yield
For each
,
. In addition, the operators
,
for
, generate an abelian group
of linear transformations of
into itself, which acts transitively on the space of one-dimensional subspaces of
.
Given a non-zero section
supported on a Bohr–Sommerfeld torus, the family of sections
is a linear basis of
invariant under the action of
Since
is abelian, there exists a positive, definite Hermitian scalar product
on
, which is invariant under the action of
, and such that the basis in (
71) is orthonormal. It is defined up to a constant positive factor. The completion of
with respect to this scalar product yields a Hilbert space
of quantum states in the Bohr–Sommerfeld quantization of
. Elements of
extend to unitary operators on
.
3.3.2. General Case of Toral Polarization
Hilbert Space and Operators
Let
be a symplectic manifold with toroidal polarization
D and a covering by domains of action–angle coordinates. If
U and
are the domain of the angle-action coordinates
and
, respectively, and
then in
we have
where the matrices
and
lie in
and
.
Consider a complete locally Hamiltonian vector field
X on
such that, for each angle-action coordinates
with domain
U,
for some
. Equation (
72) shows that in
, we have
where
, for
As in the preceding section, Equation (
36) with
, which is multi-valued, gives
which is multivalued, because the phase factor is multi-valued. As before, if we set
, we would get a single-valued expression
because
. This would work along all integral curves
for
which are contained in
U.
Now, consider the case when, for
,
and there exists
such that
, where
U and
are domains of action–angle variables
and
respectively. Moreover, assume that
for
and
for
Using the multi-index notation, for
, we write
Let
W be a neighborhood of
in
P such that
and
are contractible. For each
let
be a single-valued representative of
as in the proof of Claim 5. Similarly, we denote by
a single-valued representative of
. Equation (
73) shows that in
, the functions
and
are local Hamiltonians of the vector field
X and are constant along the integral curve of
. Hence, we have to make the choice of representatives
and
so that
With this choice,
, and
is well defined. It does not depend on the choice of the intermediate point
in
.
In the case when
, action–angle coordinate charts with domains
are needed to reach
from
; we choose
and end with
. At each intermediate point
we repeat the the argument of the preceding paragraph. We conclude that there is a connection preserving automorphism
well defined by the procedure given here, and it depends only on the complete locally Hamiltonian vector field
X satisfying condition (
73). The automorphism
of the principal bundle
leads to an automorphism
of the associated line bundle
L. As in Equation (
69), the shifting operators corresponded to the complete locally Hamiltonian vector field
X are
In absence of monodromy, if we have
k independent, complete, locally Hamiltonian vector fields
on
that satisfy the conditions leading to Equation (
73), then the operators
,
for
generate an abelian group
of linear transformations of
. If the local lattice
of Bohr–Sommerfeld tori is regular, then
acts transitively on the space of one-dimensional subspaces of
. This enables us to construct an
-invariant Hermitian scalar product on
, which is unique up to an arbitrary positive constant. The completion of
with respect to this scalar product yields a Hilbert space
of quantum states in the Bohr–Sommerfeld quantization of
.
Local Lattice Structure
The above discussion does not address the question of labeling the basic sections in by the quantum numbers associated to the Bohr–Sommerfeld k-torus , the support of .
These quantum numbers
do depend on the choice of action angle coordinates. If
is another choice of action angle coordinates in the trivializing chart
, where
, then the quantum numbers
of
T in
coordinates are related to the quantum numbers
of
T in
coordinates by a matrix
such that
, because by Claim A2 in
Appendix A on
the action coordinates
is related to the action coordinate
j by a constant matrix
. Let
. Then,
is the
local lattice structure of the Bohr–Sommerfeld tori
, which lie in the action angle chart
. If
and
are action angle charts, then the set of Bohr–Sommerfeld tori in
are
compatible. More precisely, on
the local lattices
and
are compatible if there is a matrix
such that
. Let
be a
good covering of
P, that is, every finite intersection of elements of
is either contractible or empty, such that for each
we have a trivializing chart
for action angle coordinates for the toral bundle
. Then,
is a collection of pairwise compatible local lattice structures for the collection
of Bohr–Sommerfeld tori on
P. We say that
has a
local lattice structure.
The next result shows how the operator
of
Section 3.3 affects the quantum numbers of the Bohr–Sommerfeld torus
.
Claim 8. Let be a chart in for action angle coordinates . For every Bohr–Sommerfeld torus in U with quantum numbers , the torus is also a Bohr–Sommerfeld torus , where .
Proof. For simplicity, we assume that
. Suppose that the image of the curve
lies in
, where
. For
and
we have
and
. Since
has action angle coordinates
in
U, the point
has action angle coordinates
. In particular, the point
has action angle coordinates
. Thus,
and
for
. Since
T is the Bohr–Sommerfeld torus
, we have
. Then,
Thus, the torus
is a Bohr–Sommerfeld torus
with
.
Now, consider the case when the image of the curve
is not contained in
V. This means that
, where
, does not contain the torus
T. Since
is a 1-parameter group of symplectomorphisms of
, for every
, the functions
, with
and
are action angle coordinates on
. Choose
so that
. Suppose that
, where
. Observe that for
the action angle coordinates
in
U satisfy
Hence,
and
because
is constant. Moreover,
Similarly,
because
T is a Bohr–Sommerfeld torus
with quantum numbers
. Thus,
is a Bohr–Sommerfeld torus corresponding to the quantum numbers
. This argument may be extended to cover the case where
for any positive integer
k and
. □
3.4. Singularity of Toral Polarization in Completely Integrable
Hamiltonian Systems
A completely integrable Hamiltonian system on a symplectic manifold of dimension is given by k functions which Poisson commute with each other, and are independent on the open dense subset of We assume that, for every and each , the maximal integral curve of through x is periodic with period The complement of in P is the set of singular points of the real polarization of
Applying the arguments of
Section 3.1 and the beginning of
Section 3.2, we obtain the set
of Bohr–Sommerfeld tori
M in
P. Each
M is an integral manifold of
D, which admits a covariantly constant section
. The section
is determined up to a non-zero constant. The direct sum
is the space of quantum states of the Bohr–Sommerfeld theory. Each Bohr–Sommerfeld torus
M represents a one-dimensional subspace of quantum states. The collection
is a basis of
, and
Let
be the set of the Bohr–Sommerfeld tori in
. Then,
is the space of quantum states of the system, which are described by the Bohr–Sommerfeld quantization of
. The collection
is a basis of
, and
for every
The restriction
of
D to
is a toral polarization of
discussed earlier. The functions
, which define the system, give rise to action–angle coordinates
on
, where for each
and
is the multivalued angle coordinate corresponding to
. Since we deal with the single set of action–angle coordinates, most of the analysis of
Section 3.3.1 applies to this problem. As in
Section 3.3.2, Equation (
54), for
we introduce the notation
Each
is a locally Hamiltonian vector field on
. However, since
, we cannot assume that the vector field
is complete.
In terms of action–angle coordinates
on
the Bohr–Sommerfeld tori in
are given by equation
where
. For
,
denotes the Bohr–Sommerfeld torus in
corresponding to the eigenvalue
of
If
is not in the spectrum of
, then
In a trivialization
of the complex line bundle
L restricted to
, for each
we can choose
form a basis in the space
of quantum states in
.
Claim 5 implies the following
Corollary 1. If, for every and each Planck constant is in the domain of the maximal integral curve of starting at x, then is well defined.
Under the assumptions of Corollary 1, we may follow the arguments of
Section 3.3.1 leading to Equation (
70). Applied to the case under consideration, it may be rewritten as follows. For every
such that
one has
if
, and
if
.
It remains to extend the action of and given above to all states in This involves a study of the integral curves of on P, which originate or end at points in the singular set .
Suppose we manage to extend the action of the shifting operators to all states in
Monodromy occurs when, there exist loops in the local lattice of Bohr–Sommerfeld tori such that for some
the mapping
need not be the identity on
. In this case shifting operators are multivalued, and there exists a phase factor
such that
Given a non-zero section
supported on a Bohr–Sommerfeld torus
Any maximal family
of sections in
, such that no two sections in
B are supported on the same Bohr–Sommerfeld torus, is a linear base of
. We can define a scalar product
on
as follows. First, assume that basic sections supported on different Bohr–Sommerfeld tori are perpendicular to each other. Then, assume that for every
,
This definition works even in the presence of monodromy. The completion of
with respect to this scalar product yields a Hilbert space
of quantum states in the Bohr–Sommerfeld quantization of the completely integrable Hamiltonian system under consideration.
Example: The 2-d Harmonic Oscillator
We consider the harmonic oscillator with 2 degrees of freedom, see [
9]. Its configuration space is
with coordinates
. Its phase space
has coordinates
with symplectic form
. The 2-d harmonic oscillator is completely integrable with integrals the Hamiltonian
with
and the angular momentum
with
.
The change of variables
is symplectic, that is,
, preserves the diagonal form of the Hamiltonian
, and diagonalizes the angular momentum
. The functions
are action variables for the two-dimensional harmonic oscillator. The corresponding angle variables are
and
, respectively. In the variables
the symplectic form
is
. The rescaled action angle coordinates
, used previously, are given by
The Bohr–Sommerfeld tori
are parameterized by two integers
The corresponding basic sections are
see Equation (
80). Equations (
83) yield
Hence, the quantum operators
and
act on
as follows.
where
and
The regular part of
is
The singular part of
consists of three strata
is the origin of
, while
and
are cylinders parameterized by
and
respectively.
As before, for
we consider the locally Hamiltonian vector fields
The conditions of Corollary 1 are satisfied. Hence, in
, we get shifting operators
Next, we have to consider limits as integral curves of
. Note that the integral curve
of
originating at
after time
reaches
. Moreover, the integral curve
of
originating at
for
after time
reaches
and after time
reaches the origin
Similarly, the integral curve
of
originating at
after time
reaches
and after time
it reaches
for every
. This argument also applies to
. It enlarges the above table of shifting operators as follows.
Since
is unbounded as
, it is not possible to discuss integral curves of
starting at points in
. However, for
,
Thus,
shifts in the opposite direction to
. Similarly,
shifts in the opposite direction to
It is natural to extend these relations to the boundary and assume that
The actions of the lowering operators
on states
and
on states
not defined, but they never occur in the theory. Therefore, we may assume that
3.5. Monodromy
Suppose that
is a good covering of
P such that for every
the chart
is the domain of a local trivialization of the toral bundle
, associated to the fibrating toral polarization
D of
P, given by the local action angle coordinates
with
. We suppose that the set
of Bohr–Sommerfeld tori on
P has the local lattice structure
of
Section 3.3.
Let p and and let be a smooth curve joining p to . We can choose a finite good subcovering of such that , where and . Using the fact that the local lattices are compatible, we can extend the local action functions on to a local action function on . Thus, using the connection (see Corollary A1), we may parallel transport a Bohr–Sommerfeld torus along the curve to a Bohr–Sommerfeld torus (see Claim 7). The action function at , in general depends on the path . If the holonomy group of the connection on the bundle consists only of the identity element in , then this extension process does not depend on the path . Thus, we have shown
Claim 9. If D is a fibrating toral polarization of with fibration and B is simply connected, then there are global action angle coordinates on P and the Bohr–Sommerfeld tori have a unique quantum number . Thus, the local lattice structure of is the lattice .
If the holonomy of the connection
on
P is not the identity element, then the set
of Bohr–Sommerfeld tori is not a lattice and it is not possible to assign a global labeling by quantum numbers to all the tori in
. This difficulty in assigning quantum numbers to Bohr–Sommerfeld tori has been known to chemists since the early 1920s. Modern papers illustrating it can be found in [
18,
19]. We give a concrete example where the connection
has nontrivial holonomy, namely, the spherical pendulum.
Example: Spherical Pendulum
The spherical pendulum is a completely integrable Hamiltonian system
, where
is the cotangent bundle of the 2-sphere
with
the Euclidean inner product on
, see [
20]. The Hamiltonian is
where
and the
-component of angular momentum is
The energy momentum map of the spherical pendulum is
Here,
is the closure in
of the set
R of regular values of the integral map
. The point
is an isolated critical value of
. Thus, the set
R has the homotopy type of
and is
not simply connected. Every fiber of
over a point
is a smooth 2-torus
, see chapter V of [
21]. At every point of
there are local action angle coordinates
. The actions are
and
. Here,
where
and
, and
while the angles are
and
, where
and
where
t is the time parameter of the integral curves of the vector field
on the 2-torus
, which are periodic of period
, see
Section 2.4 of [
20]. The action map
is a homeomorphism of
onto
, which is a real analytic diffeomorphism of
onto
, see Fact 2.4 in [
20].
For every
, the Bohr–Sommerfeld tori are
The fibers of
corresponding to the dark points in
Figure 1 are the Bohr–Sommerfeld tori.
The basic sections of the quantum line bundle
are
The family of sections
forms a basis of quantum states of the Bohr–Sommerfeld theory of the spherical pendulum. Let
be the Hilbert space of quantum states for
which
is an orthogonal basis. The Bohr–Sommerfeld energy momentum spectrum
of the spherical pendulum is the range of the map
are the quantum numbers of the spherical pendulum.
In terms of actions
and
, we may write
. Hence, the quantum operators
and
act on the basic sections
as follows
and
The regular part of
is
The singular part of
consists of six strata:
The stratum
is the point
; while the stratum
is the point
. The stratum
is the subset of
, where
and
, which is a cylinder parameterized by
; while
is the subset where
and
, which is a cylinder parameterized by
. The stratum
is the subset of
where
and
, which is a cylinder parameterized by
; while
is the subset where
and
, which is a cylinder parameterized by
.
The conditions of Corollary 1 are satisfied. For
let
. In the regular stratum
we get the shifting operators
Arguing as in the example of the 2-d harmonic oscillator, we can extend the above relations to
In addition, we may assume that
Since the are no global action angle coordinates, the action function
on
R is multi-valued. After encircling the point
, the quantum number of the Bohr–Sommerfeld torus represented by the upper right hand vertex of the rectangle on the
h-axis, see
Figure 2, becomes the quantum number of the upper right hand vertex of the parallelogram formed by applying
to the original rectangle, which is the transpose of the monodromy matrix
M of the spherical pendulum.
The holonomy of the connection is called the monodromy of the fibrating toral polarization D on with fibration .
Corollary 2. Let be the universal covering space of B with covering map . The monodromy map M, which is a nonidentity element holonomy group of the connection on the bundle ρ sends one sheet of the universal covering space to another sheet.
Proof. Since the universal covering space of B is simply connected and we can pull back the symplectic manifold and the fibrating toral distribution D by the universal covering map to a symplectic manifold and a fibrating toral distribution with associated fibration . The connection on the bundle pulls back to a connection on the bundle . Let be a closed curve on B and let M be the holonomy of the connection on B along . Then, lifts to a curve on , which covers , that is, . Thus, parallel transport of a k-torus , which is an integral manifold of the distribution , along the curve gives a linear map M of the lattice defining the k-torus . The map M is the same as the linear map M of into itself given by parallel transporting T, using the connection , along the closed on B because the connection is the pull back of the connection by the covering map . The closed curve in B represents an element of the fundamental group of B, which acts as a covering transformation on the universal covering space that permutes the sheets (= fibers) of the universal covering map . □
In the spherical pendulum, the universal covering space
of
is
. If we cut
R by the line segment
, then
is simply connected and hence represents one sheet of the universal covering map of
R. For more details on the universal covering map, see [
22]. The curve chosen in the example has holonomy
. It gives a map of
into itself, which sends
to the adjacent sheet of the covering map. Thus, we have a rule how the labelling of the Bohr–Sommerfeld torus
, corresponding to
, changes when we go to an adjacent sheet, which covers
, namely, we apply the matrix
M to the integer vector
. Since our chosen curve generates the fundamental group of
, we know what the quantum numbers of Bohr–Sommerfeld are for any closed curve in
, which encircles the origin.