Quantomorphisms and Quantized Energy Levels for ... › bitstream › 1807 › ...Quantomorphisms and Quantized Energy Levels for Metaplectic-c Quantization Jennifer Vaughan Doctor

Quantomorphisms and Quantized Energy Levels for Metaplectic-cQuantization

by

Jennifer Vaughan

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics

University of Toronto

c© Copyright 2016 by Jennifer Vaughan

Abstract

Quantomorphisms and Quantized Energy Levels for Metaplectic-c Quantization

Jennifer Vaughan

Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

2016

Metaplectic-c quantization was developed by Robinson and Rawnsley as an alternative to the

classical Kostant-Souriau quantization procedure with half-form correction. This thesis extends

certain properties of Kostant-Souriau quantization to the metaplectic-c context. We show that

the Kostant-Souriau results are replicated or improved upon with metaplectic-c quantization.

We consider two topics: quantomorphisms and quantized energy levels. If a symplectic man-

ifold admits a Kostant-Souriau prequantization circle bundle, then its Poisson algebra is realized

as the space of infinitesimal quantomorphisms of that circle bundle. We present a definition

for a metaplectic-c quantomorphism, and prove that the space of infinitesimal metaplectic-c

quantomorphisms exhibits all of the same properties that are seen in the Kostant-Souriau case.

Next, given a metaplectic-c prequantized symplectic manifold (M,ω) and a function H ∈

C∞(M), we propose a condition under which E, a regular value of H, is a quantized energy

level for the system (M,ω,H). We prove that our definition is dynamically invariant: if two

functions on M share a regular level set, then the quantization condition over that level set

is identical for both functions. We calculate the quantized energy levels for the n-dimensional

harmonic oscillator and the hydrogen atom, and obtain the quantum mechanical predictions in

both cases. Lastly, we generalize the quantization condition to a level set of a family of Poisson-

commuting functions, and show that in the special case of a completely integrable system, it

reduces to a Bohr-Sommerfeld condition.

ii

For David.

iii

Acknowledgements

Many thanks to my advisor, Yael Karshon, for her support, advice and insight over the last

five years.

Portions of this document arose out of conversations with Alejandro Uribe, Richard Cush-

man and Mark Hamilton, and I am grateful to each of them for their time and expertise.

I would not be a mathematician if it weren’t for my father. I would not have gotten through

the past year if it weren’t for my mother. Thank you both.

iv

Contents

1 Introduction 1

2 Background 4

2.1 Symplectic Manifolds and Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Hamiltonian vector fields and the Poisson algebra . . . . . . . . . . . . . . 5

2.1.2 Circle bundles and connection one-forms . . . . . . . . . . . . . . . . . . . 5

2.1.3 Complex line bundles and connections . . . . . . . . . . . . . . . . . . . . 6

2.1.4 Holonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Kostant-Souriau Quantization and the Half-Form Correction . . . . . . . . . . . 7

2.2.1 Prequantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.2 Quantization and the half-form correction . . . . . . . . . . . . . . . . . . 8

2.3 Metaplectic-c Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.1 Definitions on vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.2 Definitions over a symplectic manifold . . . . . . . . . . . . . . . . . . . . 12

3 Metaplectic-c Quantomorphisms 15

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Kostant-Souriau Quantomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.1 Infinitesimal quantomorphisms of a prequantization circle bundle . . . . . 16

3.2.2 The Lie algebra isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.3 An operator representation of C∞(M) . . . . . . . . . . . . . . . . . . . . 20

3.3 Metaplectic-c Quantomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.1 Infinitesimal metaplectic-c quantomorphisms . . . . . . . . . . . . . . . . 22

v

3.3.2 The Lie algebra isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.3 An operator representation of C∞(M) . . . . . . . . . . . . . . . . . . . . 28

3.4 Two Diffeomorphisms That Are Not Quantomorphisms . . . . . . . . . . . . . . 29

4 Quantized Energy Levels and Dynamical Invariance 33

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 Quantized Energy Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2.1 Vector subspaces and quotients . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2.2 Quotients and reduction over a manifold . . . . . . . . . . . . . . . . . . . 36

4.3 Dynamical Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3.1 Invariance under a diffeomorphism . . . . . . . . . . . . . . . . . . . . . . 40

4.3.2 Regular level set of multiple functions . . . . . . . . . . . . . . . . . . . . 41

4.4 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.4.1 Initial choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.4.2 From Cartesian to symplectic polar coordinates . . . . . . . . . . . . . . . 48

4.4.3 Quantized energy levels of the harmonic oscillator . . . . . . . . . . . . . 55

4.4.4 Example of dynamical invariance . . . . . . . . . . . . . . . . . . . . . . . 58

4.5 Comparison with Kostant-Souriau Quantization . . . . . . . . . . . . . . . . . . . 61

4.5.1 The quantized energy condition and its properties . . . . . . . . . . . . . 61

4.5.2 Lack of half-shift in the harmonic oscillator . . . . . . . . . . . . . . . . . 63

5 The Hydrogen Atom 65

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.1.1 Choices for subsequent calculations . . . . . . . . . . . . . . . . . . . . . . 66

5.2 Quantizing the Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2.2 Metaplectic-c prequantization of (M,ω) . . . . . . . . . . . . . . . . . . . 69

5.3 Ligon-Schaaf Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.3.1 TS3 and the Ligon-Schaaf map . . . . . . . . . . . . . . . . . . . . . . . . 71

5.3.2 Metaplectic-c prequantization for TS3 . . . . . . . . . . . . . . . . . . . . 73

5.3.3 Lifting the Ligon-Schaaf map . . . . . . . . . . . . . . . . . . . . . . . . . 75

vi

5.3.4 Rescaling the Delaunay Hamiltonian . . . . . . . . . . . . . . . . . . . . . 77

5.4 Quantization of a Free Particle on S3 . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.4.1 Orbits of ξK and local coordinates for TS3 . . . . . . . . . . . . . . . . . 78

5.4.2 Change of variables over C . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.4.3 Restrictions to C ⊂ TS3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.4.4 Quantized energy levels for (TS3, ω3,K) . . . . . . . . . . . . . . . . . . . 87

6 Generalization of the Quantized Energy Condition 92

6.1 The Quantization Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.2 Generalized Dynamical Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.3 Completely Integrable Systems and Bohr-Sommerfeld Conditions . . . . . . . . . 98

6.3.1 Kostant-Souriau quantization and Bohr-Sommerfeld conditions . . . . . . 98

6.3.2 Quantized energy condition for k = n . . . . . . . . . . . . . . . . . . . . 99

6.3.3 The n-dimensional harmonic oscillator . . . . . . . . . . . . . . . . . . . . 100

6.3.4 The 2-dimensional hydrogen atom . . . . . . . . . . . . . . . . . . . . . . 103

vii

Chapter 1

Introduction

In physics, the phase space for a classical mechanical system contains position and momentum

coordinates for the component particles. The physically observable quantities for the system

are functions on phase space. The quantum mechanical version of such a system consists of a

Hilbert space of quantum states, and an identification of observables with self-adjoint operators.

Geometric quantization is a branch of symplectic geometry that generalizes the concept

of quantization to abstract symplectic manifolds. Given a symplectic manifold (M,ω) and a

function H ∈ C∞(M), the corresponding Hamiltonian vector field ξH on M has integral curves

that satisfy Hamilton’s equations in local canonical coordinates. Thus we can view (M,ω) as a

classical phase space, and H as a Hamiltonian energy function. To quantize (M,ω), we require

a procedure for constructing a suitable Hilbert space of quantum states, and a Lie algebra

isomorphism from C∞(M), or a subalgebra of it, to operators on those states.

The original formulation of geometric quantization is due to Kostant [14] and Souriau [22].

Their quantization procedure requires that (M,ω) admit a prequantization circle bundle and

a metaplectic structure. In Chapter 2, after we review the necessary elements of symplectic

geometry and principal bundles, we present the Kostant-Souriau quantization procedure with

half-form correction.

The metaplectic-c group is a circle extension of the symplectic group. Metaplectic-c quan-

tization, which was developed by Hess [13] and Robinson and Rawnsley [17], with earlier work

by Czyz [5]. It is a variant of Kostant-Souriau quantization in which the prequantization circle

bundle and metaplectic structure are replaced by a single object called a metaplectic-c prequan-

1

Chapter 1. Introduction 2

tization. Hess, Robinson and Rawnsley proved that metaplectic-c quantization can be applied to

all systems that admit Kostant-Souriau quantizations, and to some where the Kostant-Souriau

process fails. In the final section of Chapter 2, we give a detailed description of a metaplectic-c

prequantization and its properties.

The objective of this document is to explore some of the ways in which metaplectic-c quan-

tization replicates or improves upon known results for Kostant-Souriau quantization. We begin

in Chapter 3 by examining the concept of a quantomorphism: that is, an isomorphism of pre-

quantization bundles that preserves all of their structures. In the context of a Kostant-Souriau

prequantization circle bundle for (M,ω), it is known that the Lie algebra of infinitesimal quan-

tomorphisms is isomorphic to C∞(M). We formulate a definition of a metaplectic-c quanto-

morphism, and prove that the space of infinitesimal metaplectic-c quantomorphisms is again

isomorphic to C∞(M).

The remainder of the document focuses on the concept of a quantized energy level. In

quantum mechanics, the energy spectrum for a spatially confined particle is discrete: only

certain energy levels are permitted. Various interpretations of this phenomenon can be found

in the literature for different forms of geometric quantization. Given (M,ω) and a function

H ∈ C∞(M), we propose a new definition for a quantized energy level E of H in the case where

(M,ω) admits a metaplectic-c prequantization. Our definition is evaluated over a regular level

set of H, and it does not require either symplectic reduction or a choice of polarization.

We present our definition in Chapter 4, and show that its properties compare favourably

with others that have been studied. Our main result, Theorem 4.3.5, states that if two functions

H1, H2 ∈ C∞(M) are such that H−11 (E1) = H−1

2 (E2) for regular values E1, E2, then E1 is a

quantized energy level for (M,ω,H1) if and only if E2 is a quantized energy level for (M,ω,H2).

That is, our condition is a geometric property of the level set, and does not depend on the

dynamics of a specific choice of H. As such, we refer to Theorem 4.3.5 as the dynamical

invariance theorem. Also in Chapter 4, we demonstrate a computational technique for lifting

a local change of coordinates on M to the level of metaplectic-c prequantizations, and use this

technique to evaluate the quantized energy levels for the n-dimensional harmonic oscillator.

In Chapter 5, we calculate the quantized energy levels of the hydrogen atom, using the

physical model that is identical to the Kepler problem. This calculation is more technically de-

Chapter 1. Introduction 3

manding than that for the harmonic oscillator. We use the Ligon-Schaaf regularization map to

transport the problem to TS3, and apply a further transformation to relate the negative quan-

tized energy levels of the hydrogen atom to the positive quantized energies of a free particle on

S3. The latter step relies on the dynamical invariance property. We show that the metaplectic-c

quantized energy levels agree with the physical prediction from quantum mechanics.

Lastly, in Chapter 6, we show that our quantized energy condition generalizes from one

function on M to a family of Poisson-commuting functions. After proving a generalized version

of the dynamical invariance theorem, we consider the special case of a completely integrable

system, where the size of the Poisson-commuting family is maximal. In that case, the quan-

tized energy condition simplifies to a Bohr-Sommerfeld condition. Thus our quantized energy

condition provides a framework that encompasses both the quantized energy levels of a single

function H and the Bohr-Sommerfeld leaves of the real polarization generated by a completely

integrable system.

Significant portions of this document have already been published or posted on the arXiv.

Chapter 3 appears in [23], Chapter 4 appears in [24], and Chapter 5 appears in [25]. Portions of

the abstract, Chapter 2 and this introduction are amalgams of material from all three papers.

Minor truncations and edits have been performed for the sake of internal consistency and to

eliminate redundancy. Additional background material has been added to Chapter 2, and the

content of Chapter 6 does not yet appear elsewhere.

Chapter 2

Background

Our starting point is a symplectic manifold (M2n, ω): that is, a 2n-dimensional manifold

equipped with a closed, nondegenerate two-form. We think of this object as a classical me-

chanical phase space. Classical observables are elements of C∞(M), the smooth real-valued

functions on M . One objective of geometric quantization is to build a Hermitian line bundle

over M and a Lie algebra homomorphism from C∞(M) to operators on sections of the bundle,

thereby reproducing the transition from classical to quantum mechanics in which observables

become operators on a Hilbert space of quantum states. In Section 2.1, we review the basic facts

that we will require concerning symplectic manifolds, circle bundles and complex line bundles,

and connections.

In Section 2.2, we describe the Kostant-Souriau quantization procedure. We begin with the

prequantization stage, in which (M,ω) is required to admit a prequantization line bundle (L,∇).

Then we introduce a metaplectic structure and a choice of polarization F , and sketch how to

construct the complex line bundle of half-forms ∧1/2F . We conclude with the quantization

stage, in which the functions in C∞(M) whose flows preserve the polarization are mapped to

operators on polarized sections of L⊗ ∧1/2F .

Finally, in Section 2.3, we summarize the construction of a metaplectic-c prequantization,

as developed by Hess [13] and Robinson and Rawnsley [17]. A metaplectic-c prequantization

is an object that performs the functions of a prequantization line bundle and a metaplectic

structure simultaneously. As we will see in later chapters, certain features of Kostant-Souriau

quantization with half-form correction can be reproduced by a metaplectic-c prequantization

4

Chapter 2. Background 5

alone, without requiring a choice of polarization. Since our work focuses on the prequantization

stage of Robinson and Rawnsley’s procedure, we omit a detailed discussion of the quantiza-

tion stage. Some elements of it are presented in Section 3.3.3 in the context of metaplectic-c

quantomorphisms.

Some global remarks concerning notation: for any vector field ξ, the Lie derivative with

respect to ξ is written Lξ. The space of smooth vector fields on a manifold P is denoted by

X (P ). Given a smooth map F : P → M and a vector field ξ ∈ X (P ), we write F∗ξ for the

pushforward of ξ only if the result is a well-defined vector field on M . If P is a bundle over

M , Γ(P ) denotes the space of smooth sections of P , where the base is always taken to be the

symplectic manifold M . Planck’s constant will only appear in the form ~.

2.1 Symplectic Manifolds and Bundles

2.1.1 Hamiltonian vector fields and the Poisson algebra

Let (M,ω) be a symplectic manifold. Given f ∈ C∞(M), define its Hamiltonian vector field

ξf ∈ X (M) by

ξfyω = df.

Define the Poisson bracket on C∞(M) by

f, g = ξfg = −ω(ξf , ξg), ∀f, g ∈ C∞(M).

A standard calculation establishes that for all f, g ∈ C∞(M),

[ξf , ξg] = ξf,g.

2.1.2 Circle bundles and connection one-forms

Let Yp−→ M be a right principal U(1) bundle over M . For any θ ∈ u(1), the Lie algebra of

U(1), let ∂θ be the vector field on Y given by

∂θ(y) =d

dt

∣∣∣∣t=0

y · exp(tθ), ∀y ∈ Y.


The vector field ∂θ is called the vector field generated by the infinitesimal action of θ ∈ u(1).

A u(1)-valued one-form γ on Y is called a connection one-form if γ is invariant under the

right principal action, and for all θ ∈ u(1), γ(∂θ) = θ. If Y is equipped with a connection

one-form γ, then there is a two-form $ on M , called the curvature of γ, such that dγ = p∗$.

For any ξ ∈ X (M), let ξ be the lift of ξ to Y that is horizontal with respect to γ. That

is, p∗ξ = ξ and γ(ξ) = 0. For any θ ∈ u(1), note that p∗∂θ = 0, which implies that p∗[ξ, ∂θ] =

[p∗ξ, p∗∂θ] = 0 and γ([ξ, ∂θ]) = −(p∗$)(ξ, ∂θ) = 0. Therefore [ξ, ∂θ] = 0 for all θ.

2.1.3 Complex line bundles and connections

Now let L→M be a complex line bundle over M . A connection ∇ on L allows us to take the

derivative of a section s of L in the direction of a vector field ξ on M . It can be viewed as a

map from sections of L to L-valued one-forms on M in the sense that, if s is a section of L and

ξ is a vector field on M , then at every point m ∈ M , ∇s acts on the vector ξ(m) to yield a

value in Lm, which we interpret as the derivative of s in the direction of that vector. Formally,

a connection on L is a map

∇ : Γ(L)→ Γ(L)⊗ Ω1(M)

such that, for all r, s ∈ Γ(L), all ξ ∈ X (M), and all f ∈ C∞(M),

∇ξ(r + s) = ∇ξr +∇ξs,

∇ξ(fs) = f∇ξs+ (ξf)s.

If (Y, γ)p−→M is a circle bundle with connection one-form over M , then it has an associated

Hermitian line bundle with connection (L,∇) over M . The line bundle L is given by L =

Y ×U(1)C, with Hermitian structure induced from the usual Hermitian inner product on C. We

write an element of L as an equivalence class [y, z] with y ∈ Y and z ∈ C. The connection ∇

on L is constructed from the connection one-form γ through the following process.

Given any s ∈ Γ(L), define the map s : Y → C so that [y, s(y)] = s(p(y)) for all y ∈ Y .

Then s has the equivariance property

s(y · λ) = λ−1s(y), ∀ y ∈ Y, λ ∈ U(1).


Conversely, any map s : Y → C with the above equivariance property can be used to construct

a section s of L by setting s(m) = [y, s(y)] for all m ∈M and any y ∈ Y such that p(y) = m.

Let ξ ∈ X (M) be given, and let ξ be its horizontal lift to Y . If s : Y → C is an equivariant

map, then so is ξs. This follows from the fact that [ξ, ∂θ] = 0 for all θ ∈ u(1). Define the

connection ∇ on L so that for any ξ ∈ X (M) and s ∈ Γ(L), ∇ξs is the section of L that

satisfies

∇ξs = ξs.

2.1.4 Holonomy

Let (Y, γ) be a circle bundle with connection one-form over M . Let u : R→M be a path in M

such that u(t+1) = u(t) for all t. Given a starting point y0 ∈ Yu(0), there is a unique lift of u(t)

to a path u(t) in Y such that u(0) = y0 and every tangent vector ˙u(t) satisfies γ(

˙u(t))

= 0.

Such a lift is called horizontal with respect to γ. Since u(t) is a lift of u(t), u(1) ∈ Yu(0). The

resulting map y0 7→ u(1) is automorphism of the fiber Yu(0), called the holonomy1 of γ over

u(t). If the horizontal lift is itself a closed loop, then γ is said to have trivial holonomy over

u(t).

There is an equivalent formulation of holonomy on the corresponding line bundle with

connection (L,∇). Let C ⊂M be the image of u in M , and let u(t) be a lift of u(t) to L. The

image of u(t) can be thought of as a section s of L, defined only over C. The lift u(t) is called

horizontal if ∇u(t)s = 0 for all t. Given a starting point u(0) in the fiber Lu(0), the condition of

being horizontal uniquely determines the rest of the lift. As before, u(1) ∈ Lu(0), so we obtain

a linear map u(0) 7→ u(1) from the fiber Lu(0) to itself, called the holonomy of ∇ over u(t).

2.2 Kostant-Souriau Quantization and the Half-Form Correc-

tion

Geometric quantization in its original form was developed in the 1960’s by Kostant [14] and

Souriau [22]. The half-form correction was added by Blattner, Kostant and Sternberg [2]. The

following overview is based on the detailed treatments that can be found in [11, 20, 26].

1This material is standard, but our main source is the exposition in [12].


2.2.1 Prequantization

Let (M,ω) be a symplectic manifold. In the prequantization stage, we require (M,ω) to admit

a prequantization circle bundle.

Definition 2.2.1. A prequantization circle bundle for (M,ω) is a right principal U(1)

bundle Yp−→M , together with a connection one-form γ on Y satisfying dγ = 1

i~p∗ω.

If (M,ω) admits a prequantization circle bundle, then the associated Hermitian line bundle

with connection (L,∇) is called a prequantization line bundle. The manifold (M,ω) admits a

prequantization circle bundle (equivalently, a prequantization line bundle) if and only if the

cohomology class[

12π~ω

]∈ H2(M,R) is integral.

Assume that (M,ω) admits a prequantization circle bundle (Y, γ), with corresponding pre-

quantization line bundle (L,∇). One of the goals of the prequantization process is to produce

a representation r : C∞(M) → End Γ(L). To be consistent with quantum mechanics in the

case of a physically realizable system, the map r is required to satisfy the following axioms,

which are based on an analysis by Dirac [6] on the relationship between classical and quantum

mechanical observables.

(1) r(1) is the identity map on Γ(L),

(2) for all f, g ∈ C∞(M), [r(f), r(g)] = i~r(f, g) (up to sign convention).

In the context of Kostant-Souriau prequantization, a suitable map is given by

r(f) = i~∇ξf + f, ∀f ∈ C∞(M).

The fact that this map satisfies condition (2) can be verified by direct computation; as we will

show in Section 3.2.3, it can also be viewed as a consequence of the Lie algebra isomorphism

between C∞(M) and the space of infinitesimal quantomorphisms of (Y, γ).

2.2.2 Quantization and the half-form correction

The phase space for a classical mechanical system consists of all possible combinations of posi-

tion and momentum coordinates. However, the wave functions for the corresponding quantum


system depend only on the position coordinates (or, more generally, on a complete set of

commuting observables). This observation motivates the introduction of a structure called a

polarization.

Assume that (M,ω) admits a prequantization line bundle (L,∇). A polarization2 F is an

involutive Lagrangian subbundle of the complexified tangent bundle TMC such that dim(Fm ∩

Fm) is constant over all m ∈ M . Let a choice of F be fixed. The canonical bundle for F ,

denoted by KF , is the top exterior power of the annihilator of F in (T ∗M)C. The half-form

bundle, ∧1/2F , is a choice of square root of KF , when such a square root exists. A half-form is

a section of ∧1/2F over M .

To guarantee that the half-form bundle ∧1/2F exists, we require that (M,ω) admit a meta-

plectic structure, which is a lifting of the structure group for (M,ω) from the symplectic

group to its unique connected double cover, the metaplectic group. The manifold (M,ω) ad-

mits such a lifting if and only if the first Chern class c1(TM) ∈ H2(M,Z) is even: that is,

12c1(TM) ∈ H2(M,Z). When this condition is satisfied, the equivalence classes of metaplectic

structures for (M,ω) are in one-to-one correspondence with H1(M,Z2).

A metaplectic structure on (M,ω) induces a metalinear structure on the polarization F , from

which the half-form bundle ∧1/2F can be constructed. From KF , the half-form bundle inherits

a partial connection ∇ζ , defined for all ζ ∈ X (M) such that ζ is parallel to the polarization

F . It also inherits a partial Lie derivative Lξ, defined for all ξ ∈ X (M) such that the flow of ξ

preserves F .

Let C∞F (M) denote the subalgebra of C∞(M) consisting of functions whose Hamiltonian

flows preserve the polarization F . The Kostant-Souriau representation with half-form correc-

tion, rF : C∞F (M)→ End Γ(L⊗ ∧1/2F ), is defined by

rF (f) = r(f)⊗ I + I ⊗ Lξf , ∀f ∈ C∞F (M).

This map is a Lie algebra homomorphism.

A section s⊗ ν ∈ Γ(L⊗∧1/2F ) is called polarized if ∇ζs = 0 and ∇ζν = 0 for all ζ ∈ X (M)

such that ζ is parallel to F . Let ΓF (L ⊗ ∧1/2F ) denote the space of polarized sections. The

2Other technical constraints, such as positivity, may be imposed on F , but since we will not be performingany explicit computations with polarizations, we will not address these details.


operators rF (f) restrict to operators on ΓF (L⊗ ∧1/2F ). The polarized sections with compact

support form the pre-Hilbert space of quantum states, where the Hermitian inner product

arises from the Hermitian structure on L, and a pairing from half-forms to densities. This

completes the construction of the Lie algebra representation and the Hilbert space as suggested

by quantum mechanics.

2.3 Metaplectic-c Quantization

The Kostant-Souriau quantization procedure imposes two conditions on (M,ω): namely, that it

admit a prequantization circle bundle and a metaplectic structure. Each of these requirements

comes with a cohomological condition to be satisfied. The motivation behind metaplectic-c

quantization is to replace the prequantization circle bundle and metaplectic structure with a

single bundle that can play both roles simultaneously.

The material in this section is a summary of results originally presented by Hess [13] and

Robinson and Rawnsley [17]. More detail, including proofs of the properties that we state,

can be found in [13, 17]. A similar structure, called a spin-c prequantization, was studied in

[3, 8, 9, 10].

2.3.1 Definitions on vector spaces

Let V be an n-dimensional complex vector space with Hermitian inner product 〈·, ·〉. If we view

V as a 2n-dimensional real vector space, then the action of the scalar i ∈ C becomes the real

automorphism J : V → V . Define the real bilinear form Ω on V by

Ω(v, w) = Im 〈v, w〉 , ∀v, w ∈ V.

Then (V,Ω) is a 2n-dimensional symplectic vector space. The Hermitian and symplectic struc-

tures are compatible in the sense that

〈v, w〉 = Ω(Jv,w) + iΩ(v, w), ∀v, w ∈ V.

The symplectic group Sp(V ) is the group of real automorphisms g : V → V such that


Ω(gv, gw) = Ω(v, w) for all v, w ∈ V . Since the fundamental group for Sp(V ) is Z, Sp(V )

has a unique connected double cover called the metaplectic group, denoted by Mp(V ). Let

Mp(V )σ−→ Sp(V ) denote the covering map. The metaplectic-c group Mpc(V ) is defined to be

Mpc(V ) = Mp(V )×Z2 U(1),

where Z2 ⊂ U(1) is the usual subgroup 1,−1, and where Z2 ⊂ Mp(V ) consists of the two

preimages of I ∈ Sp(V ) under the covering map σ.

By construction, Mpc(V ) contains U(1) and Mp(V ) as subgroups. The inclusion of each

subgroup into Mpc(V ) yields a short exact sequence and a group homomorphism on Mpc(V ).

One such sequence is

1→ U(1)→ Mpc(V )σ−→ Sp(V )→ 1,

where the group homomorphism σ is called the projection map, and its restriction to Mp(V ) ⊂

Mpc(V ) is the double covering map. The other is

1→ Mp(V )→ Mpc(V )η−→ U(1)→ 1,

where the group homomorphism η is called the determinant map, and it has the property that

for any λ ∈ U(1) ⊂ Mpc(V ), η(λ) = λ2. At the level of Lie algebras, the map σ∗ ⊕ 12η∗ yields

the identification

mpc(V ) = sp(V )⊕ u(1).

Given g ∈ Sp(V ), let

Cg =1

2(g − JgJ).

By construction, Cg is an R-linear map that commutes with J , so it is a complex linear trans-

formation of V . The determinant of Cg as a complex transformation is written DetCCg. It can

be shown that Cg is always invertible, so DetCCg is a nonzero complex number.

A useful embedding of Mpc(V ) into Sp(V )×C \ 0 can be defined as follows. An element

a ∈ Mpc(V ) is mapped to the pair (g, µ) ∈ Sp(V )×C\0, where g ∈ Sp(V ) satisfies σ(a) = g,

and µ ∈ C \ 0 satisfies η(a) = µ2DetCCg. The latter condition defines µ up to a sign, and


we adopt the convention that the parameters of I ∈ Mpc(V ) are (I, 1). This choice uniquely

determines the sign of µ for all a ∈ Mpc(V ). Following [17], we refer to (g, µ) as the parameters

of a.

We note two properties of this parametrization, both of which will be useful in later chapters.

First, if a ∈ Mp(V ) ⊂ Mpc(V ), then η(a) = 1, and so the parameters of a are (g, µ) where

σ(a) = g and µ2DetCCg = 1. Second, if λ ∈ U(1) ⊂ Mpc(V ), then the parameters of λ are

(I, λ), and for any other element a ∈ Mpc(V ) with parameters (g, µ), the parameters of the

product aλ are (g, µλ).

2.3.2 Definitions over a symplectic manifold

Let (M,ω) be a 2n-dimensional symplectic manifold, and assume that a 2n-dimensional model

symplectic vector space (V,Ω) with compatible complex structure J has been fixed. We think

of the symplectic frame bundle for (M,ω) as being modeled on Sp(V ), in the following sense.

Definition 2.3.1. The symplectic frame bundle Sp(M,ω)ρ−→M is a right principal Sp(V )

bundle over M given by

Sp(M,ω)m = b : V → TmM : b is a symplectic linear isomorphism , ∀m ∈M.

The group Sp(V ) acts on the fibers by precomposition.

In the previous section, we defined a metaplectic structure for (M,ω) to be a lifting of the

structure group for (M,ω) from Sp(V ) to Mp(V ). Analogously, a metaplectic-c structure for

(M,ω) is a lifting of the structure group to Mpc(V ).

Definition 2.3.2. A metaplectic-c structure for (M,ω) consists of a right principal Mpc(V )

bundle PΠ−→M and a map P

Σ−→ Sp(M,ω) such that

Σ(q · a) = Σ(q) · σ(a), ∀q ∈ P, ∀a ∈ Mpc(V ),


and such that the following diagram commutes.

PΣ //

Π

Sp(M,ω)

ρyy

(M,ω)

To complete the construction of a metaplectic-c prequantization, we equip the metaplectic-c

structure with a u(1)-valued one-form whose properties are similar to that of the connection

one-form on a prequantization circle bundle.

Definition 2.3.3. A metaplectic-c prequantization of (M,ω) is a metaplectic-c structure

(P,Σ) for (M,ω), together with a u(1)-valued one-form γ on P such that:

(1) γ is invariant under the principal Mpc(V ) action;

(2) for any α ∈ mpc(V ), if ∂α is the vector field on P generated by the infinitesimal action of

α, then γ(∂α) = 12η∗α;

(3) dγ = 1i~Π∗ω.

Note that if (P,Σ, γ) is a metaplectic-c prequantization for (M,ω), then (P, γ) is a circle

bundle with connection one-form over Sp(M,ω). The circle that acts on the fibers of P is the

center U(1) ⊂ Mpc(V ). We will view P as a bundle over M or a bundle over Sp(M,ω) as the

circumstance demands.

For any α ∈ mpc(V ), we can write α = κ ⊕ τ under the identification of mpc(V ) with

sp(V )⊕ u(1). Then γ(∂α) = τ , and naturality of the exponential map and equivariance of the

map Σ with respect to σ ensure that Σ∗∂α = ∂κ, where ∂κ is the vector field on Sp(M,ω)

generated by the infinitesimal action of κ ∈ sp(V ). In fact, ∂α is uniquely specified by the two

conditions γ(∂α) = τ and Σ∗∂α = ∂κ, an observation that will be useful in Chapter 3.

There is a cohomology constraint that must be satisfied in order for (M,ω) to admit a

metaplectic-c prequantization, and that condition is very closely related to the ones we saw for

the prequantization circle bundle and metaplectic structure. Specifically, (M,ω) is metaplectic-

c prequantizable if and only if the cohomology class[

12π~ω

]+ 1

2c1(TM) ∈ H2(M,R) is integral.


Thus, if (M,ω) admits a prequantization circle bundle and a metaplectic structure, then it is

metaplectic-c prequantizable, but the converse is not true.

Two metaplectic-c prequantizations for (M,ω) are considered equivalent if there is a dif-

feomorphism between them that preserves the one-forms and commutes with the respective

maps to Sp(M,ω). If (M,ω) is metaplectic-c prequantizable, then the set of equivalence classes

of metaplectic-c prequantizations for (M,ω) is in one-to-one correspondence with the locally

constant cohomology group H1(M,U(1)), which also parametrizes the circle bundles with flat

connection over (M,ω). An observation that will be used repeatedly is that if M is simply

connected, then the metaplectic-c prequantization of (M,ω) is unique up to isomorphism.

Chapter 3

Metaplectic-c Quantomorphisms

3.1 Introduction

A prequantization circle bundle for a symplectic manifold (M,ω) consists of a circle bundle

Y → M and a connection one-form γ on Y such that dγ = 1i~ω. Souriau [22] defined a

quantomorphism between two prequantization circle bundles (Y1, γ1)→ (M1, ω1) and (Y2, γ2)→

(M2, ω2) to be a diffeomorphism K : Y1 → Y2 such that K∗γ2 = γ1. This condition implies

that K is equivariant with respect to the principal circle actions. Souriau then defined the

infinitesimal quantomorphisms of a prequantization circle bundle (Y, γ) to be the vector fields

on Y whose flows are quantomorphisms. Kostant [14] proved that the space of infinitesimal

quantomorphisms, which we denote Q(Y, γ), is isomorphic to the Poisson algebra C∞(M). In

Section 3.2, we present an explicit construction of the isomorphism from Q(Y, γ) to C∞(M).

The objective of the rest of the chapter is to adapt the concept of an infinitesimal quantomor-

phism to the case where (M,ω) admits a metaplectic-c prequantization (P,Σ, γ). In Section 3.3,

we define a metaplectic-c quantomorphism, which is a diffeomorphism of metaplectic-c prequan-

tizations that preserves all of their structures. Our definition is based on Souriau’s, but includes

a condition that is unique to the metaplectic-c context. We then use the metaplectic-c quanto-

morphisms to define Q(P,Σ, γ), the space of infinitesimal metaplectic-c quantomorphisms. We

show that every property that was proved for Q(Y, γ) has a parallel for Q(P,Σ, γ). In particu-

lar, Q(P,Σ, γ) is isomorphic to the Poisson algebra C∞(M). The construction in Section 3.2 is

used as a model for the proofs in Section 3.3. We indicate when the calculations are analogous,

15

Chapter 3. Metaplectic-c Quantomorphisms 16

and when the metaplectic-c case requires additional steps.

3.2 Kostant-Souriau Quantomorphisms

In this section, we construct a Lie algebra isomorphism from C∞(M) to the space of infinitesimal

quantomorphisms. As we have already noted, the fact that these algebras are isomorphic was

originally stated by Kostant [14] in the context of line bundles with connection. His proof can

be reconstructed from several propositions across Sections 2 – 4 of [14]. Kostant’s isomorphism

is also stated by Sniatycki [20], but much of the proof is left as an exercise. We are not aware of

a source in the literature for a self-contained proof that uses the language of principal bundles,

and this is one of our reasons for performing an explicit construction here.

The other goal of this section is to motivate the analogous constructions for a metaplectic-

c prequantization, which will be the subject of Section 3.3. Each result that we present for

Kostant-Souriau prequantization will have a parallel in the metaplectic-c case. When the proofs

are identical, we will simply refer back to the work shown here, thereby allowing Section 3.3 to

focus on those features that are unique to metaplectic-c structures.

3.2.1 Infinitesimal quantomorphisms of a prequantization circle bundle

Definition 3.2.1. Let (Y1, γ1)p1−→ (M1, ω1) and (Y2, γ2)

p2−→ (M2, ω2) be prequantization circle

bundles for two symplectic manifolds. A diffeomorphism K : Y1 → Y2 is called a quantomor-

phism if K∗γ2 = γ1.

Let K : Y1 → Y2 be a quantomorphism. Notice that for any θ ∈ u(1), the vector field ∂θ on

Y1 is completely specified by the conditions γ1(∂θ) = θ and dγ1(∂θ) = 0, and the same is true

on Y2. Since K∗γ2 = γ1, we see that K∗∂θ = ∂θ for all θ, and so K is equivariant with respect

to the principal circle actions.

Definition 3.2.2. Let (Y, γ)p−→ (M,ω) be a prequantization circle bundle. An infinitesimal

quantomorphism of (Y, γ) is a vector field ζ ∈ X (Y ) whose flow φt on Y is a quantomorphism

from its domain to its range for each t. The space of infinitesimal quantomorphisms of (Y, γ)

is denoted by Q(Y, γ).


Let ζ ∈ X (Y ) have flow φt. The connection form γ is preserved by φt if and only if Lζγ = 0.

Therefore the space of infinitesimal quantomorphisms of (Y, γ) is

Q(Y, γ) = ζ ∈ X (Y ) |Lζγ = 0.

If K : Y1 → Y2 is a quantomorphism, then it induces a diffeomorphism (in fact, a symplec-

tomorphism) K ′ : M1 →M2 such that the following diagram commutes.

Y1K //

p1

Y2

p2

M1K′ //M2

This implies that for any ζ ∈ Q(Y, γ) with flow φt, there is a flow φ′t on M that satisfies

p φt = φ′t p. If ζ ′ is the vector field on M with flow φ′t, then p∗ζ = ζ ′. In other words,

elements of Q(Y, γ) descend via p∗ to well-defined vector fields on M .

3.2.2 The Lie algebra isomorphism

Let (Y, γ)p−→ (M,ω) be a prequantization circle bundle. We will now present an explicit

construction of a Lie algebra isomorphism from C∞(M) to Q(Y, γ). Recall that the vector field

∂2πi on Y satisfies γ(∂2πi) = 2πi ∈ u(1) and p∗∂2πi = 0.

Lemma 3.2.3. For all f, g ∈ C∞(M),

[ξf , ξg] = ξf,g −1

2π~p∗f, g∂2πi.

Proof. It suffices to show that

p∗[ξf , ξg] = p∗

(ξf,g −

1

2π~p∗f, g∂2πi

)and

γ([ξf , ξg]) = γ

(ξf,g −

1

2π~p∗f, g∂2πi

)

First, we have

p∗

(ξf,g −

1

2π~p∗f, g∂2πi

)= ξf,g = [ξf , ξg].


Since p∗ξf = ξf and p∗ξg = ξg, it follows that p∗[ξf , ξg] = [ξf , ξg]. Thus the first equation is

verified.

Next, note that

γ

(ξf,g −

1

2π~p∗f, g∂2πi

)=

1

i~p∗f, g,

and

γ([ξf , ξg]) = − 1

i~(p∗ω)(ξf , ξg) =

1

i~p∗f, g.

Therefore the second equation is also verified.

Lemma 3.2.4. The map E : C∞(M)→ X (Y ) given by

E(f) = ξf +1

2π~p∗f∂2πi, ∀f ∈ C∞(M)

is a Lie algebra homomorphism.

Proof. Let f, g ∈ C∞(M) be arbitrary. We need to show that

ξf,g +1

2π~p∗f, g∂2πi =

[ξf +

1

2π~p∗f∂2πi, ξg +

1

2π~p∗g∂2πi

].

Using Lemma 3.2.3, the left-hand side becomes

[ξf , ξg] + 21


Expanding the right-hand side yields

[ξf , ξg] +

[ξf ,

1

2π~p∗g∂2πi

]+

[1

2π~p∗f∂2πi, ξg

]+

[1

2π~p∗f∂2πi,

1

2π~p∗g∂2πi

].

The fourth term vanishes because ∂θ(p∗f) = ∂θ(p

∗g) = 0 for any θ ∈ u(1). To evaluate the

third term, recall that that [∂θ, ξ] = 0 for any θ ∈ u(1) and ξ ∈ X (M). Therefore [∂2πi, ξg] = 0,

so this term reduces to

− 1

2π~(ξgp

∗f)∂2πi =1



By the same argument, the second term also reduces to

1


Combining these results, we find that the right-hand side of the desired equation is

[ξf , ξg] + 21

i~p∗f, g∂2πi,

which equals the left-hand side.

Lemma 3.2.5. For all f ∈ C∞(M), E(f) ∈ Q(Y, γ).

Proof. We need to show that LE(f)γ = 0. We calculate

LE(f)γ = E(f)y dγ + d(E(f)y γ) =1

i~p∗(ξfyω)− 1

i~p∗df = 0.

So far, we have shown that E : C∞(M)→ Q(Y, γ) is a Lie algebra homomorphism. We will

now construct a map F : Q(Y, γ) → C∞(M), and show that E and F are inverses. This will

complete the proof that C∞(M) and Q(Y, γ) are isomorphic.

Let ζ ∈ Q(Y, γ) be arbitrary. Then Lζγ = ζy dγ+d(γ(ζ)) = 0. This implies that ∂θy (ζy dγ+

d(γ(ζ))) = 0 for any θ ∈ u(1). Since dγ(ζ, ∂θ) = 1i~(p∗ω)(ζ, ∂θ) = 0, it follows that ∂θy d(γ(ζ)) =

L∂θγ(ζ) = 0. We can therefore define the map F : Q(Y, γ)→ C∞(M) so that

− 1

i~p∗F (ζ) = γ(ζ), ∀ζ ∈ Q(Y, γ).

Theorem 3.2.6. The map E : C∞(M) → Q(Y, γ) is a Lie algebra isomorphism with inverse

F .

Proof. Let f ∈ C∞(M) and ζ ∈ Q(Y, γ) be arbitrary. We will show that F (E(f)) = f and

E(F (ζ)) = ζ. Using the definitions of E and F , we have

− 1

i~p∗F (E(f)) = γ(E(f)) = γ

(ξf +

1

2π~p∗f∂2πi

)= − 1

i~p∗f.


This implies that F (E(f)) = f .

To show that E(F (ζ)) = ζ, it suffices to show that γ(E(F (ζ))) = γ(ζ) and p∗E(F (ζ)) = p∗ζ.

By definition,

E(F (ζ)) = ξF (ζ) +1

2π~p∗F (ζ)∂2πi = ξF (ζ) +

1

2πiγ(ζ)∂2πi.

It is immediate that γ(E(F (ζ))) = γ(ζ), and that p∗E(F (ζ)) = ξF (ζ). Observe that

ζy p∗ω = i~ζy dγ = −i~d(γ(ζ)) = p∗(dF (ζ)),

having used Lζγ = 0. Therefore (p∗ζ)yω = dF (ζ), which implies that p∗ζ = ξF (ζ). Thus

p∗E(F (ζ)) = p∗ζ. This concludes the proof that E(F (ζ)) = ζ.

Since E and F are inverses, and we know from Lemma 3.2.5 that E : C∞(M)→ Q(Y, γ) is

a Lie algebra homomorphism, it follows that E and F are the desired Lie algebra isomorphisms.

The primary goal of Section 3.3 is to duplicate the above construction for the infinitesi-

mal quantomorphisms of a metaplectic-c prequantization. However, before moving on to the

metaplectic-c case, we will show how the map E can be used to represent the elements of

C∞(M) as operators on the space of sections of the prequantization line bundle for (M,ω).

This result will also have an analogue in the metaplectic-c case, which we will discuss in Section

3.3.3.

3.2.3 An operator representation of C∞(M)

Let (L,∇) be the complex line bundle with connection associated to (Y, γ). Recall the associ-

ation between a section s of L and an equivariant function s : Y → C from Section 2.1.2. We

note the following properties.

• For any f ∈ C∞(M) and s ∈ Γ(L), the equivariant function corresponding to the section

fs is f s = p∗fs.

• The vector field ∂2πi is generated by the infinitesimal action of 2πi ∈ u(1) on Y . Thus,


for all y ∈ Y ,

(∂2πis)(y) =d

dt

∣∣∣∣t=0

s(y · e2πit) = −2πis(y).

Further recall that the Kostant-Souriau representation map r : C∞(M) → End Γ(L) is

defined by

r(f)s =(i~∇ξf + f

)s, ∀f ∈ C∞(M), s ∈ Γ(L).

Using the preceding observations, we see that

r(f)s =(i~ξf + p∗f

)s =

(i~ξf −

1

2πip∗f∂2πi

)s = i~E(f)s.

Since we proved in Lemma 3.2.4 that E(f, g) = [E(f), E(g)] for all f, g ∈ C∞(M), the

following is immediate.

Theorem 3.2.7. The map r : C∞(M)→ End Γ(L) satisfies

[r(f), r(g)] = i~r(f, g), ∀f, g ∈ C∞(M).

Thus the same map that provides the isomorphism from C∞(M) to Q(Y, γ) also yields the

usual Kostant-Souriau representation of C∞(M) as a space of operators on Γ(L). We will see

a similar result in the case of metaplectic-c prequantization.

3.3 Metaplectic-c Quantomorphisms

Having reviewed the properties of infinitesimal quantomorphisms in Kostant-Souriau prequan-

tization, we will now explore their parallels in metaplectic-c prequantization. In Section 3.3.1,

we develop our definition for a metaplectic-c quantomorphism, and use it to define an infinites-

imal metaplectic-c quantomorphism. The remainder of the chapter is dedicated to proving the

metaplectic-c analogues of the results presented in Section 3.2.

Suppose (M,ω) admits a metaplectic-c prequantization (P,Σ, γ). The space of infinitesimal

quantomorphisms of (P,Σ, γ) consists of those vector fields on P whose flows preserve all of the

structures on (P,Σ, γ). Note that one of these structures is the map PΣ−→ Sp(M,ω), which


does not have a direct analogue in the Kostant-Souriau case. We will show how to incorporate

this additional piece of information in the next section.

3.3.1 Infinitesimal metaplectic-c quantomorphisms

As in Section 3.2.1, we begin by developing the idea of a quantomorphism between metaplectic-c

prequantizations. Let (P1,Σ1, γ1)Σ1−→ Sp(M1, ω1)

ρ1−→ (M1, ω1) and (P2,Σ2, γ2)Σ2−→ Sp(M2, ω2)

ρ2−→ (M2, ω2) be metaplectic-c prequantizations for two symplectic manifolds, and let Πj = ρj

Σj for j = 1, 2. Let K : P1 → P2 be a diffeomorphism. We will determine the conditions that K

must satisfy in order for it to preserve all of the structures of the metaplectic-c prequantizations.

First, by analogy with the Kostant-Souriau definition, assume that K satisfies K∗γ2 = γ1.

Fix m ∈M1, and consider the fiber P1m. For any q ∈ P1m, notice that

TqP1m = ξ ∈ TqP1 |Π1∗ξ = 0 = ker dγ1q.

The same property holds for a fiber of P2 over a point in M2. By assumption, K∗ is an

isomorphism from ker dγ1q to ker dγ2K(q) for all q ∈ P1. Therefore Π2 is constant on K(P1m).

Moreover, since K is a diffeomorphism, we can make the analogous argument with K−1, and

conclude that K(P1m) is in fact a fiber of P2 over M2, and every fiber of P2 is the image of a

fiber of P1. Thus K induces a diffeomorphism K ′′ : M1 →M2 such that the following diagram

commutes.

P1K //

Π1

P2

Π2

M1

K′′ //M2

Lemma 3.3.1. The map K ′′ : M1 →M2 is a symplectomorphism.

Proof. It suffices to show that K ′′∗ω2 = ω1. Using the properties of K, γ1 and γ2, we calculate

Π∗1(K ′′∗ω2) = (K ′′ Π1)∗ω2 = (Π2 K)∗ω2 = K∗(i~dγ2) = i~dγ1 = Π∗1ω1.

Therefore K ′′∗ω2 = ω1, as required.


Assume that a model symplectic vector space (V,Ω) has been fixed. Recall from Definition

2.3.1 that an element b ∈ Sp(M1, ω1)m is a map b : V → TmM1 such that b∗ω1m = Ω. Since K ′′

is a symplectomorphism, the composition K ′′∗ b : V → TK′′(m)M2 satisfies (K ′′∗ b)∗ω2K′′(m) = Ω,

which implies that K ′′∗ b ∈ Sp(M2, ω2)K′′(m). Let K ′′ : Sp(M1, ω1)→ Sp(M2, ω2) be the lift of

K ′′ given by

K ′′(b) = K ′′∗ b, ∀b ∈ Sp(M1, ω1).

Then K ′′ is a diffeomorphism, and it is equivariant with respect to the principal Sp(V ) actions.

Thus, if we assume that K∗γ2 = γ1, we obtain the diffeomorphisms K ′′ : M1 → M2 and

K ′′ : Sp(M1, ω1) → Sp(M2, ω2), where both K and K ′′ are lifts of K ′′. However, K is not

necessarily a lift of K ′′. Indeed, there might not be any map K ′ : Sp(M1, ω1)→ Sp(M2, ω2) of

which K is a lift. A map K for which there is no corresponding K ′ is constructed in Section

3.4, Example 3.4.1. In Section 3.2.1, we showed that a diffeomorphism of prequantization circle

bundles that preserves the connection forms must be equivariant with respect to the principal

circle actions. By contrast, Example 3.4.1 demonstrates that it is possible for K to preserve

the prequantization one-forms without being equivariant with respect to the principal Mpc(V )

actions.

Suppose we make the additional assumption that K(q · a) = K(q) · a for all q ∈ P1 and

a ∈ Mpc(V ). Then K induces a diffeomorphism K ′ : Sp(M1, ω1) → Sp(M2, ω2) that satisfies

K ′Σ1 = Σ2K. Combining this with the map K ′′ : M1 →M2 yields the following commutative

diagram.

P1K //

Σ1

P2

Σ2

Sp(M1, ω1)

K′ //

ρ1

Sp(M2, ω2)

ρ2

M1

K′′ //M2

We now have two maps, K ′ and K ′′, which are diffeomorphisms from Sp(M1, ω1) to Sp(M2, ω2).

By construction, ρ2 K ′ = ρ2 K ′′, and both K ′ and K ′′ are equivariant with respect to the

principal Sp(V ) actions. However, it is still possible for K ′ and K ′′ to be different. A map K

for which K ′ 6= K ′′ is given in Example 3.4.2.


As will be shown in Section 3.3.2, this potential discrepancy between K ′ and K ′′ must be

prevented in order to construct the desired isomorphism between C∞(M) and the infinitesimal

quantomorphisms. We therefore propose the following definition.

Definition 3.3.2. The diffeomorphism K : P1 → P2 is a metaplectic-c quantomorphism

if

(1) K∗γ2 = γ1,

(2) the induced diffeomorphism K ′′ : M1 →M2 satisfies K ′′ Σ1 = Σ2 K.

Let K : P1 → P2 be a metaplectic-c quantomorphism. Given our concept of a quantomor-

phism as a diffeomorphism that preserves all of the structures of a metaplectic-c prequantization,

we would expect that K is equivariant with respect to the Mpc(V ) actions. Equivariance is a

consequence of the definition, as we now show.

Let α ∈ mpc(V ) be arbitrary, and write α = κ⊕ τ under the identification of mpc(V ) with

sp(V ) ⊕ u(1). The vector field ∂α on P1 is completely specified by the conditions γ1(∂α) = τ

and Σ1∗∂α = ∂κ, and the same is true on P2. Notice that

γ2(K∗∂α) = γ1(∂α) = τ,

and

Σ2∗K∗∂α = K ′′∗Σ1∗∂α = K ′′∗∂κ = ∂κ,

where the final equality follows from the fact that K ′′ is equivariant with respect to Sp(V ).

Thus K∗∂α = ∂α for all α ∈ mpc(V ), which implies that K is equivariant with respect to

Mpc(V ), as desired.

Now consider a single metaplectic-c prequantized space (P,Σ, γ)Σ−→ Sp(M,ω)

ρ−→ (M,ω)

with Π = ρ Σ.

Definition 3.3.3. A vector field ζ ∈ X (P ) is an infinitesimal metaplectic-c quantomor-

phism if its flow φt is a metaplectic-c quantomorphism from its domain to its range for each

t.


Let ζ ∈ X (P ) have flow φt. Property (1) of a quantomorphism holds for φt if and only if

Lζγ = 0. If we assume that φt satisfies property (1), then there is a flow φ′′t on M such that

Π φt = φ′′t Π. The vector field that it generates on M is Π∗ζ.

Lemma 3.3.1 shows that φ′′t is a family of symplectomorphisms. Therefore we can lift φ′′t

to a flow on Sp(M,ω), denoted by φ′′t, where φ′′t(b) = (φ′′t)∗ b for all b ∈ Sp(M,ω). Let the

vector field on Sp(M,ω) that has flow φ′′t be Π∗ζ. Then property (2) of a quantomorphism

holds for φt if and only if Σ∗ζ is a well-defined vector field on Sp(M,ω) and Σ∗ζ = Π∗ζ.

We conclude that the space of infinitesimal metaplectic-c quantomorphisms of (P,Σ, γ) is

Q(P,Σ, γ) = ζ ∈ X (P )∣∣∣ Lζγ = 0 and Σ∗ζ = Π∗ζ,

where it is understood that the condition Σ∗ζ = Π∗ζ can only be satisfied if Σ∗ζ is well defined.

In the next section, we will construct a Lie algebra isomorphism from C∞(M) to Q(P,Σ, γ).

3.3.2 The Lie algebra isomorphism

We begin with a procedure, given by Robinson and Rawnsley in Section 7 of [17], for lifting

a Hamiltonian vector field on M to Sp(M,ω) and then to P . These steps will be used in

constructing the isomorphism E : C∞(M)→ Q(P,Σ, γ).

Fix f ∈ C∞(M), and let its Hamiltonian vector field ξf have flow ϕt on M . We know that

ϕt∗ preserves ω because Lξfω = 0. Let ϕt be the lift of ϕt to Sp(M,ω) given by

ϕt(b) = ϕt∗ b, ∀b ∈ Sp(M,ω),

and let the vector field on Sp(M,ω) with flow ϕt be ξf . We have ρ∗ξf = ξf by construction.

Also, ϕt commutes with the right principal Sp(V ) action on Sp(M,ω), so [ξf , ∂κ] = 0 for all

κ ∈ sp(V ). Now let ξf be the lift of ξf to P that is horizontal with respect to γ. Then Σ∗ξf = ξf

and γ(ξf ) = 0. A summary of the key properties of ξf , ξf and ξf is below.


(P, γ)

Σ

ξf γ(ξf ) = 0, Σ∗ξf = ξf , Π∗ξf = ξf

Sp(M,ω)

ρ

ξf [ξf , ∂κ] = 0 ∀κ ∈ sp(V ), ρ∗ξf = ξf

(M,ω) ξf

In Section 3.2.2, we made use of the vector field ∂2πi on Y . The corresponding object in

this context is the vector field ∂2πi on P , where 2πi ∈ u(1) ⊂ mpc(V ). This vector field satisfies

γ(∂2πi) = 2πi and Σ∗(∂2πi) = 0.

Lemma 3.3.4. For all f, g ∈ C∞(M),

[ξf , ξg] = ξf,g −1

2π~Π∗f, g∂2πi.

Proof. It suffices to show that

Σ∗[ξf , ξg] = Σ∗

(ξf,g −

1

2π~Π∗f, g∂2πi

)and

γ([ξf , ξg]) = γ

(ξf,g −

1

2π~Π∗f, g∂2πi

).

A calculation establishes that ξf,g = [ξf , ξg]. The rest of the proof proceeds identically to that

of Lemma 3.2.3.

Lemma 3.3.5. The map E : C∞(M)→ X (P ) given by

E(f) = ξf +1

2π~Π∗f∂2πi, ∀f ∈ C∞(M)

is a Lie algebra homomorphism.

Proof. Precisely analogous to Lemma 3.2.4.

Lemma 3.3.6. For all f ∈ C∞(M), E(f) ∈ Q(P,Σ, γ).

Proof. We need to show that LE(f)γ = 0 and Σ∗E(f) = Π∗E(f). The verification that LE(f)γ =

0 is the same as that in Lemma 3.2.5. Note that Σ∗E(f) = ξf and Π∗E(f) = ξf , so Π∗E(f) =

ξf = Σ∗E(f). Thus the necessary conditions are satisfied, and E(f) ∈ Q(P,Σ, γ).


As before, we will construct an inverse for E, and conclude that E is a Lie algebra iso-

morphism. Let ζ ∈ Q(P,Σ, γ) and α ∈ mpc(V ) be arbitrary. Using an identical argument to

the one that precedes Theorem 3.2.6, the fact that ∂αy (Lζγ) = 0 implies that L∂αγ(ζ) = 0.

Therefore we can define F : Q(P,Σ, γ)→ C∞(M) so that

− 1

i~Π∗F (ζ) = γ(ζ), ∀ζ ∈ Q(P,Σ, γ).

Theorem 3.3.7. The map E : C∞(M)→ Q(P,Σ, γ) is a Lie algebra isomorphism with inverse

F .

Proof. Let f ∈ C∞(M) and ζ ∈ Q(P,Σ, γ) be arbitrary. From the definitions of E and F ,

F (E(f)) satisfies

− 1

i~Π∗F (E(f)) = γ(E(f)) = γ

(ξf +

1

2π~Π∗f∂2πi

)= − 1

i~Π∗f.

Thus F (E(f)) = f .

Next, we claim that γ(E(F (ζ))) = γ(ζ) and Σ∗E(F (ζ)) = Σ∗ζ. Observe that

E(F (ζ)) = ξF (ζ) +1

2π~Π∗F (ζ)∂2πi = ξF (ζ) +

1

2πiγ(ζ)∂2πi.

It is immediate that γ(E(F (ζ))) = γ(ζ) and Σ∗E(F (ζ)) = ξF (ζ). From the definition of

Q(P,Σ, γ), we know that Σ∗ζ = Π∗ζ. It remains to show that Π∗ζ = ξF (ζ). We calculate

ζyΠ∗ω = ζy i~dγ = −i~d(γ(ζ)) = Π∗dF (ζ).

This demonstrates that (Π∗ζ)yω = dF (ζ), which implies that Π∗ζ = ξF (ζ) as needed. Thus we

have shown that E(F (ζ)) = ζ, and this completes the proof that E and F are inverses.

If the definition of Q(P,Σ, γ) did not include the condition that Σ∗ζ = Π∗ζ, this proof

would fail in the final step. We would be able to show that Σ∗E(F (ζ)) = ξF (ζ) = Π∗ζ, but

this vector field would not necessarily equal Σ∗ζ, and so F would not be the inverse of E. This

explains why property (2) of a metaplectic-c quantomorphism is necessary in order to obtain a

subalgebra of X (P ) that is isomorphic to C∞(M).


3.3.3 An operator representation of C∞(M)

In [17], Robinson and Rawnsley construct an infinite-dimensional Hilbert space E ′(V ) of holo-

morphic functions on V ∼= Cn, on which the group Mpc(V ) acts via the metaplectic repre-

sentation. They then define the bundle of symplectic spinors for the prequantized system

(P,Σ, γ)Π−→ (M,ω) to be

E ′(P ) = P ×Mpc(V ) E ′(V ).

We omit the details of the metaplectic representation here; the only fact we need is that the

subgroup U(1) ⊂ Mpc(V ) acts on elements of E ′(V ) by scalar multiplication. We write an

element of E ′(P ) as an equivalence class [q, ψ] with q ∈ P and ψ ∈ E ′(V ).

Section 7 of [17] contains the following construction. Let s ∈ Γ(E ′(P )) be given, and define

the map s : P → E ′(V ) so that [q, s(q)] = s(Π(q)) for all q ∈ P . This map s satisfies the

equivariance condition

s(q · a) = a−1s(q), ∀q ∈ P, a ∈ Mpc(V ),

where the action on the right-hand side is that of the metaplectic representation. Conversely,

if s : P → E ′(V ) is any map with the equivariance property above, it can be used to define

a section s ∈ Γ(E ′(P )) by setting s(m) = [q, s(q)] for each m ∈ M and any q ∈ P such that

Π(q) = m.

Let f ∈ C∞(M) be arbitrary, and recall the lifting ξf → ξf → ξf of ξf to P . A standard

calculation establishes that [ξf , ∂α] = 0 for all α ∈ mpc(V ). Thus, if s : P → E ′(V ) is an

equivariant map, then so is ξf s. Define the map D : C∞(M)→ End Γ(E ′(P )) such that for all

f ∈ C∞(M) and s ∈ Γ(E ′(P )), Dfs is the section of E ′(P ) that satisfies

Dfs = ξf s.

Further, define δ : C∞(M)→ End Γ(E ′(P )) by

δfs = Dfs+1

i~fs, ∀f ∈ C∞(M), s ∈ Γ(E ′(P )).


Theorem 7.8 of [17] states that δ is a Lie algebra homomorphism.

We see that the construction of D precisely parallels the construction of the connection ∇

on the prequantization line bundle L associated to a prequantization circle bundle (Y, γ). As

in Section 3.2.3, we make two observations.

• For any s ∈ Γ(E ′(P )) and f ∈ C∞(M), f s = Π∗fs.

• For any equivariant map s : P → E ′(V ), ∂2πis = −2πis.

Therefore

δfs =

(ξf +

1

2π~Π∗f∂2πi

)s = E(f)s.

The fact that δ is a Lie algebra homomorphism then follows immediately from Lemma 3.3.5.

This construction would apply equally well to any associated bundle where the subgroup U(1) ⊂

Mpc(V ) acts on the fiber by scalar multiplication.

3.4 Two Diffeomorphisms That Are Not Quantomorphisms

A metaplectic-c quantomorphism K between metaplectic-c prequantizations (P1,Σ1, γ1)Σ1−→

Sp(M1, ω1)ρ1−→ (M1, ω1) and (P2,Σ2, γ2)

Σ2−→ Sp(M2, ω2)ρ2−→ (M2, ω2) is a diffeomorphism

K : P1 → P2 such that

(1) K∗γ2 = γ1,

(2) the induced diffeomorphism K ′′ : M1 →M2 satisfies K ′′ Σ1 = Σ2 K,

where K ′′ : Sp(M1, ω1)→ Sp(M2, ω2) is the lift of K ′′ given by

K ′′(b) = K ′′∗ b, ∀b ∈ Sp(M1, ω1).

We claimed that condition (1) is insufficient to guarantee that K is the lift of some map

K ′ : Sp(M1, ω1)→ Sp(M2, ω2). In particular, a diffeomorphism K that only satisfies condition

(1) might not be equivariant with respect to the principal Mpc(V ) actions. We further claimed

that K might be equivariant and satisfy condition (1), yet fail to satisfy condition (2). We will

now construct examples to support these claims.


Let M = R2 \ (0, 0) with Cartesian coordinates (p, q) and polar coordinates (r, θ). Equip

M with the symplectic form ω = dp∧ dq = rdr ∧ dθ, and observe that the one-form β = 12r

2dθ

satisfies dβ = ω. Let V = R2 with basis x, y and symplectic form Ω = x∗ ∧ y∗, and consider

the global trivialization of the tangent bundle TM such that for all m ∈M , TmM is identified

with V by mapping x → ∂∂p

∣∣∣m

and y → ∂∂q

∣∣∣m

. Identify Sp(M,ω) with M × Sp(V ) using this

trivialization.

Let P = M ×Mpc(V ), and define the map Σ : P → Sp(M,ω) by Σ(m, a) = (m,σ(a)) for all

m ∈M and a ∈ Mpc(V ). Let ϑ0 be the trivial connection on the product bundle M ×Mpc(V ),

and let γ = 1i~β + 1

2η∗ϑ0. Then (P,Σ, γ) is a metaplectic-c prequantization of (M,ω). In both

of the examples below, we will give a diffeomorphism K : P → P .

To facilitate the construction in Example 3.4.1, we introduce a more explicit representation

for elements of the metaplectic-c group. By definition, Mpc(V ) = Mp(V ) ×Z2 U(1). The

restriction of the projection map Mpc(V )σ−→ Sp(V ) to Mp(V ) yields the double covering

Mp(V )σ−→ Sp(V ). Write an element of Mpc(V ) as an equivalence class [h, e2πit] with h ∈

Mp(V ) and t ∈ R. In terms of this parametrization, the projection map is given by

σ[h, e2πit] = σ(h),

and the determinant map Mpc(V )η−→ U(1) is given by

η[h, e2πit] = (e2πit)2.

Example 3.4.1. We will define a diffeomorphism K : P → P that preserves γ, but that does

not descend through Σ to a well-defined map on Sp(M,ω).

Let µ : R → Mp(V ) be any smooth nonconstant path such that µ(t + 1) = µ(t) for all

t ∈ R. Note that the composition σ µ : R → Sp(V ) is also nonconstant. Now define

F : Mp(V )× U(1)→ Mp(V )× U(1) by

F (h, e2πit) = (hµ(2t), e2πit), ∀h ∈ Mp(V ), t ∈ R.

This map is a diffeomorphism of Mp(V )×U(1), and it descends to a diffeomorphism of Mpc(V ),


which we also denote F . For any [h, e2πit] ∈ Mpc(V ), observe that

η(F [h, e2πit]) = η[hµ(2t), e2πit] = (e2πit)2 = η[h, e2πit].

This implies that for any α ∈ mpc(V ), 12η∗F∗α = 1

2η∗α.

Define the diffeomorphism K : P → P by K(m, a) = (m,F (a)) for all m ∈ M and a ∈

Mpc(V ). Since K is the identity on M , it preserves β. From the property of F shown above,

K also preserves 12η∗ϑ0, and thus it preserves γ. Fix (m, g) ∈ Sp(M,ω), and let h ∈ Mp(V ) be

such that σ(h) = g. Then the fiber of P over (m, g) is P(m,g) = (m, [h, e2πit]) | t ∈ R. Notice

that

Σ K(m, [h, e2πit])) = Σ(m, [hµ(2t), e2πit]) = (m, gσ(µ(2t))),

which is not constant with respect to t. Thus K(P(m,g)) is not contained within a single fiber

of P over Sp(M,ω), which shows that there is no map K ′ : Sp(M,ω) → Sp(M,ω) such that

K ′ Σ = Σ K.

If K : P → P is equivariant with respect to Mpc(V ), then it induces a diffeomorphism

K ′ : Sp(M,ω) → Sp(M,ω) that satisfies K ′ Σ = Σ K. This map and K ′′ are both lifts of

K ′′ : M →M , but they might not be the same map.

Example 3.4.2. We will define a diffeomorphismK : P → P that preserves γ and is equivariant

with respect to Mpc(V ), but where K ′ 6= K ′′.

Let Tλ : M → M be the map that rotates M about the origin by the angle λ, where λ is

not an integer multiple of 2π. Define K : P → P by

K(m, a) = (Tλ(m), a), ∀m ∈M, a ∈ Mpc(V ).

Then K∗γ = γ, and K(q ·a) = K(q)·a for all q ∈ P and a ∈ Mpc(V ). The map K ′ : Sp(M,ω)→

Sp(M,ω) is given by

K ′(m, g) = (Tλ(m), g), ∀m ∈M, g ∈ Sp(V ),


and the map K ′′ : M →M is simply Tλ. If we let Tλ also denote the automorphism of V given

by rotation about the origin by λ, then under our chosen identification of TM with M ×V , we

have

K ′′∗ (m, v) = (Tλ(m), Tλ(v)), ∀m ∈M, v ∈ V.

Therefore K ′′ : Sp(M,ω)→ Sp(M,ω) is given by

K ′′(m, g) = (Tλ(m), Tλ g), ∀m ∈M, g ∈ Sp(V ).

Hence K ′ 6= K ′′.

Chapter 4

Quantized Energy Levels and

Dynamical Invariance

4.1 Introduction

It is well known from physics that the quantum mechanical versions of systems such as the har-

monic oscillator and the hydrogen atom are restricted to discrete energy levels. More generally,

suppose (M,ω) is a quantizable symplectic manifold and H : M → R is a function on M . If

we think of H as a Hamiltonian energy function, then we can ask what it means for a regular

value E of H to be a quantized energy level for the system (M,ω,H).

One possible answer to this question involves the construction of the symplectic reduction.

The orbits of the Hamiltonian vector field ξH partition the level set H−1(E). If ME , the space

of orbits, is a manifold, then ω induces a symplectic form ωE on ME , and (ME , ωE) is the

symplectic reduction of (M,ω) at E. The values of E for which this new symplectic manifold

is quantizable can be taken to be the quantized energy levels of the system. This definition has

been applied to the hydrogen atom in the context of Kostant-Souriau quantization [18], and

to the n-dimensional harmonic oscillator (n ≥ 2) in the context of metaplectic-c quantization

[17]. In both cases, the physically predicted energy spectrum is obtained. However, technical

complications arise in cases where the space of orbits is not a manifold. Also, if the level set is

one-dimensional, as it is for the one-dimensional harmonic oscillator, then the space of orbits is

33

Chapter 4. Quantized Energy Levels and Dynamical Invariance 34

zero-dimensional and the quantization condition is always satisfied trivially. It would be more

convenient to have a quantized energy condition that is evaluated over the original manifold,

rather than the quotient.

In this chapter, we propose a definition for a quantized energy level of (M,ω,H) in the

case where (M,ω) admits a metaplectic-c prequantization. If E is a quantized energy level of

(M,ω,H), and if one additional condition is satisfied, which we describe in Section 4.2.2, then

the symplectic reduction (ME , ωE) is metaplectic-c quantizable, provided that it is a manifold.

However, the quantized energy condition is evaluated on a bundle over the level set H−1(E),

and does not require the existence of the symplectic reduction.

This scenario was first studied by Robinson [16], and our work is strongly motivated by his

results. However, we choose a more robust condition than the one Robinson considered. As we

will show, if H1 and H2 are two functions on (M,ω) with regular values E1 and E2, respectively,

such that H−11 (E1) = H−1

2 (E2), then under our definition, E1 is a quantized energy level of

(M,ω,H1) if and only if E2 is a quantized energy level of (M,ω,H2). In other words, the

quantization condition only depends on the geometry of the level set, and not on the dynamics

of a particular Hamiltonian. We refer to this theorem as the dynamical invariance property. It

is not true of the definition considered in [16].

Section 4.2 summarizes Robinson’s results concerning symplectic reduction, and concludes

with our proposed definition for a quantized energy level of a metaplectic-c quantized system.

The proof of the dynamical invariance of our definition is given in Section 4.3. In Section 4.4,

we consider the example of the harmonic oscillator. This example serves two purposes. First,

we demonstrate a computational technique in which a local change of coordinates on M is lifted

to the metaplectic-c prequantization. Second, we give an example of a function on M that has

exactly one level set in common with the harmonic oscillator, and show that our quantization

condition on that level set is identical for both functions, while Robinson’s is not.

Finally, in Section 4.5, we adapt the quantized energy definition to apply to a Kostant-

Souriau quantizable symplectic manifold. We show that the Kostant-Souriau and metaplectic-c

definitions have almost identical properties, but the metaplectic-c version is the only one that

yields the correct quantized energy levels for the harmonic oscillator.


4.2 Quantized Energy Levels

Given a symplectic manifold (M,ω) and a smooth function H : M → R, the symplectic

reduction of (M,ω) at the regular value E of H is constructed by taking the quotient of the

level set H−1(E) by the flow of the Hamiltonian vector field ξH . Suppose (M,ω) admits a

metaplectic-c prequantization (P,Σ, γ). As discussed in Section 3.3.2, there is a natural lift of

ξH to a vector field ξH on Sp(M,ω); ξH can then be lifted to a vector field ξH on P that is

horizontal with respect to γ. Using some of the results that were given in [17], Robinson [16]

constructed a certain associated bundle PS → H−1(E), on which ξH induces a vector field, and

he examined the conditions under which the quotient of PS by the induced flow of ξH yields a

metaplectic-c prequantization for the symplectic reduction of (M,ω) at E.

In this section, we review the key results from [16, 17], then state our alternative definition

for a quantized energy level of the system (M,ω,H). Our definition is evaluated on the bundle

PS , and ensures that the symplectic reduction acquires a metaplectic-c prequantization under

the same conditions as Robinson’s definition. Further, as we will show in Section 4.3, our

definition depends only on the geometry of the level set H−1(E), and not on the specific choice

of H. This is an improvement over the definition from [16].

4.2.1 Vector subspaces and quotients

Assume that the model vector space V has been chosen as in Section 2.3.1. Let W ⊂ V be a

real subspace of codimension 1. Define the symplectic orthogonal of W to be

W⊥ = v ∈ V : Ω(v,W ) = 0 .

Then W⊥ is a one-dimensional subspace of W . The form Ω induces a symplectic form on the

quotient space W/W⊥. This new symplectic vector space has symplectic group Sp(W/W⊥) and

metaplectic-c group Mpc(W/W⊥). The complex structure J on V induces a complex structure

on W/W⊥ in such a way that

W/W⊥ ∼= (W⊥ ⊕ JW⊥)⊥, (4.2.1)


where the isomorphism respects both the symplectic structures and the complex structures [17].

We write an element of W/W⊥ as an equivalence class [w] for some w ∈W .

Let Sp(V ;W ) be the subgroup of Sp(V ) consisting of those symplectic automorphisms that

preserve W :

Sp(V ;W ) = g ∈ Sp(V ) : gW = W .

Then elements of Sp(V ;W ) also preserve W⊥, and there is a group homomorphism

Sp(V ;W )ν−→ Sp(W/W⊥),

defined by

(νg)[w] = [gw], ∀g ∈ Sp(V ;W ), ∀w ∈W.

Let Mpc(V ;W ) be the preimage of Sp(V ;W ) under the projection map Mpc(V )σ−→ Sp(V ).

Robinson and Rawnsley [17] constructed a group homomorphism Mpc(V ;W )ν−→ Mpc(W/W⊥)

such that the diagram below commutes.

Mpc(V ) ⊃ Mpc(V ;W )ν //

σ

Mpc(W/W⊥)

σ

Sp(V ) ⊃ Sp(V ;W )ν // Sp(W/W⊥)

The map ν has the following property. Given a ∈ Mpc(V ;W ), let χ(a) = DetR(σ(a)|W⊥).

Then χ is a real-valued character on Mpc(V ;W ), and it satisfies

η ν(a) = η(a)signχ(a), ∀a ∈ Mpc(V ;W ).

Since signχ(a) is identically 1 on a neighborhood of I, we see that η∗ ν∗ = η∗ as maps between

Lie algebras.

4.2.2 Quotients and reduction over a manifold

Let (M,ω) be a symplectic manifold that admits a metaplectic-c prequantization (P,Σ, γ). The

definitions from Section 2.3.2 yield the three-level structure

(P, γ)Σ−→ Sp(M,ω)

ρ−→ (M,ω), ρ Σ = Π.


Let H : M → R be a smooth function on M . We define the Hamiltonian vector field ξH

using the convention that

ξHyω = dH.

Let the flow of ξH be φt, and note that φt is a symplectomorphism from its domain to its range

for each t. The lift of φt to a flow φt on Sp(M,ω) is given by

φt(b) = φt∗ b, ∀b ∈ Sp(M,ω),

where we interpret elements of Sp(M,ω) as symplectic isomorphisms as discussed in Section

2.3.2. Let ξH be the vector field on Sp(M,ω) whose flow is φt. Define the vector field ξH on P

to be the lift of ξH that is horizontal with respect to the one-form γ, and let the flow of ξH be

φt.

We now restrict our attention to a particular level set. Let E ∈ R be a regular value of H,

and let S = H−1(E). The null foliation TS⊥ is defined fiberwise over S by

TsS⊥ = ζ ∈ TsM : ω(ζ, TsS) = 0 , ∀s ∈ S.

Then TsS⊥ is a one-dimensional subspace of the tangent space TsS, and TsS

⊥ = span ξH(s).

The constructions that follow are taken from [16]. As in Section 4.2.1, fix a subspace W ⊂ V

of codimension 1. By definition, an element b ∈ Sp(M,ω)m is a symplectic linear isomorphism

b : V → TmM . Let Sp(M,ω;S) be the subset of Sp(M,ω)|S given by

Sp(M,ω;S)s = b ∈ Sp(M,ω)s : bW = TsS , ∀s ∈ S.

Then Sp(M,ω;S) is a principal Sp(V ;W ) bundle over S. Note that any b ∈ Sp(M,ω;S)s maps

W⊥ to TsS⊥.

Viewing P as a circle bundle over Sp(M,ω), let PS be the result of restricting P to

Sp(M,ω;S). Then PS is a principal Mpc(V ;W ) bundle over S. Let γS be the pullback of

the one-form γ to PS . After all of these restrictions, we have constructed the new three-level


structure

(PS , γS)→ Sp(M,ω;S)→ S.

For each s ∈ S, TsS/TsS⊥ is a symplectic vector space with symplectic structure induced

by ωs. Let Sp(TS/TS⊥) be the symplectic frame bundle for TS/TS⊥, modeled on W/W⊥:

Sp(TS/TS⊥)s =b′ : W/W⊥ → TsS/TsS

⊥ : b′ is a symplectic linear isomorphism, ∀s ∈ S.

Recall the group homomorphism ν : Sp(V ;W ) → Sp(W/W⊥) from Section 4.2.1. The bundle

associated to Sp(M,ω;S) by the map ν can be naturally identified with Sp(TS/TS⊥).

Next, let PS → S be the bundle associated to PS → S by the group homomorphism

ν : Mpc(V ;W ) → Mpc(W/W⊥). Then PS is a principal Mpc(W/W⊥) bundle over S, and a

circle bundle over Sp(TS/TS⊥). The one-form γS induces a connection one-form γS on PS .

This completes the construction of another three-level structure,

(PS , γS)→ Sp(TS/TS⊥)→ S.

Now let us consider the actions of φt, φt and φt on all of these bundles. First, it is clear that

φt preserves S, since ξH(s) ∈ TsS at each s ∈ S. Therefore φt restricts to a flow on Sp(M,ω;S),

and so φt restricts to a flow on PS . One can verify that φt and φt are equivariant with respect

to the principal Sp(V ) and Mpc(V ) actions, respectively, which implies that φt induces a flow

on the associated bundle Sp(TS/TS⊥), and φt induces a flow on PS . We let φt and φt also

denote the respective flows induced on Sp(TS/TS⊥) and PS .

Suppose the null foliation is fibrating, so that the quotient of S by the flow of ξH is a

manifold. Let the quotient manifold be ME , and let ωE be the symplectic form on ME induced

by ω. Further suppose that the quotient of Sp(TS/TS⊥) by φt is well defined, in which case

it is naturally isomorphic to the symplectic frame bundle Sp(ME , ωE). This is the additional

condition that was referenced in the introduction. As discussed in [16], to ensure that the

quotient of Sp(TS/TS⊥) by φt is well defined, it is sufficient to require that if s ∈ S is fixed

by φt for some t, then φt∗ is the identity on TsS. The culmination of all of these steps is shown

below.


(P, γ)

Σ

(PS , γS)incl.oo

ν // (PS , γS)

Sp(M,ω)

ρ

Sp(M,ω;S)incl.oo

ν // Sp(TS/TS⊥)

/φt // Sp(ME , ωE)

(M,ω) S

incl.oo = // S/φt // (ME , ωE)

The obvious way to complete the picture is to factor (PS , γS) by φt. Robinson [16] addressed

the question of when this yields a metaplectic-c prequantization for (ME , ωE) with the following

theorem.

Theorem 4.2.1. (Robinson, 1990) Assume that the symplectic reduction (ME , ωE) is a man-

ifold, and that its symplectic frame bundle Sp(ME , ωE) can be identified with the quotient of

Sp(TS/TS⊥) by the induced flow φt. If γS has trivial holonomy over all of the closed orbits of

φt on Sp(TS/TS⊥), then the quotient of (PS , γS) by φt is a metaplectic-c prequantization for

(ME , ωE).

Robinson then observed that the holonomy condition in Theorem 4.2.1 is satisfied if γS has

trivial holonomy over all closed orbits of φt on Sp(M,ω;S). The quantization condition that

was explored in the remainder of [16] is this latter holonomy condition: the regular value E

is quantized if γS has trivial holonomy over the closed orbits of φt on Sp(M,ω;S). However,

we argue that a more robust condition arises when we evaluate the holonomy over orbits in

Sp(TS/TS⊥). As such, we propose the following definition.

Definition 4.2.2. When γS has trivial holonomy over all closed orbits of φt on Sp(TS/TS⊥),

we say that E is a quantized energy level of the system (M,ω,H).

In those cases where E is a quantized energy level and (ME , ωE) exists, Theorem 4.2.1

gives a sufficient condition for (ME , ωE) to admit a metaplectic-c prequantization. However,

Definition 4.2.2 can also be applied to systems where the symplectic reduction does not exist.

For the remainder of the chapter, we do not assume that the symplectic reduction exists.

The next section contains the proof of the dynamical invariance property. We will show that

if S is a regular level set of two functions H1 and H2 on M , then S corresponds to a quantized

energy level for (M,ω,H1) if and only if it does so for (M,ω,H2).


4.3 Dynamical Invariance

We continue to use all of the definitions established in Section 4.2. To prove that the quantized

energy condition is dynamically invariant, we proceed in two stages. In Section 4.3.1, we prove

Theorem 4.3.2, which states that if E is a quantized energy level for the system (M,ω,H),

then for any diffeomorphism f : R → R, the value f(E) is a quantized energy level for the

system (M,ω, f H). While this is a less general statement, it allows us to establish some

useful preliminary observations. Then, in Section 4.3.2, we prove Theorem 4.3.5, which states

that if the functions H1, H2 : M → R have regular values E1 and E2, respectively, such that

H−11 (E1) = H−1

2 (E2), then E1 is a quantized energy level for (M,ω,H1) if and only if E2 is a

quantized energy level for (M,ω,H2).

4.3.1 Invariance under a diffeomorphism

Let H : M → R be given, and fix a regular value E of H. Let f : R→ R be a diffeomorphism,

and let S = H−1(E) = (f H)−1(f(E)).

Denote the Hamiltonian vector field for H by ξH , and let it have flow φt on M . Then ξH

lifts to ξH on Sp(M,ω) and to ξH on P , with flows φt and φt, respectively, as in Section 4.2.2.

Similarly, denote the Hamiltonian vector field for f H by ξfH , and let it have flow ρt on M .

By the same process, we obtain the lifts ξfH on Sp(M,ω) and ξfH on P , with flows ρt and

ρt, respectively.

Observe that for any m ∈M ,

d(f H)|m =df(H)

dH

∣∣∣∣H(m)

dH|m,

which implies that

ξfH(m) =df(H)

dH

∣∣∣∣H(m)

ξH(m).

Since S is a level set of H,df(H)

dH

∣∣∣∣H(m)

is constant over S. Let c ∈ R be that constant value

on S. Then ξfH = cξH everywhere on S, which implies that ρt = φct everywhere on S.

The symplectic vector bundle Sp(TS/TS⊥) is naturally identified with the bundle associated

to Sp(M,ω;S) by the group homomorphism ν : Sp(V ;W )→ Sp(W/W⊥). We write an element


of W/W⊥ as an equivalence class [w] for some w ∈ W , and we write an element of TsS/TsS⊥

as an equivalence class [ζ] for some ζ ∈ TsS. As discussed in Section 4.2.2, the lifted flow φt on

Sp(M,ω) maps Sp(M,ω;S) to itself and induces a flow φt on Sp(TS/TS⊥). More explicitly,

for any b′ ∈ Sp(TS/TS⊥), we define φt(b′) by choosing b ∈ Sp(M,ω;S) such that [bw] = b′[w]

for all w ∈W , and setting

φt(b′)[w] = [φt(b)w], ∀w ∈W.

These remarks apply equally well to the flows ρt on Sp(M,ω) and Sp(TS/TS⊥).

Lemma 4.3.1. ρt = φct on Sp(TS/TS⊥).

Proof. Fix s ∈ S and b′ ∈ Sp(TS/TS⊥)s. Let b ∈ Sp(M,ω;S)s be such that for all w ∈ W ,

[bw] = b′[w]. Since ρt = φct on S, we have ρt∗|TS = φct∗ |TS. Therefore, for any w ∈W ,

ρt(b)w = ρt∗|s(bw) = φct∗ |s(bw) = φct(b)w,

where we used the fact that bw ∈ TsS. Using the definitions of φt and ρt on Sp(TS/TS⊥), it

follows that

ρt(b′)[w] = [ρt(b)w] = [φct(b)w] = φct(b′)[w].

Thus ρt = φct on Sp(TS/TS⊥).

Lemma 4.3.1 implies that ξfH = cξH on Sp(TS/TS⊥). Since ξfH is just a constant

multiple of ξH , it is clear that the flows of both vector fields have the same orbits. Thus, if

γS has trivial holonomy over all closed orbits of φt on Sp(TS/TS⊥), then it also has trivial

holonomy over all closed orbits of ρt. The following is now immediate.

Theorem 4.3.2. If E is a quantized energy level for (M,ω,H) and f : R→ R is a diffeomor-

phism, then f(E) is a quantized energy level for (M,ω, f H).

4.3.2 Regular level set of multiple functions

Let H1, H2 : M → R be two smooth functions, and assume that there are E1, E2 ∈ R such that

Ej is a regular value of Hj for j = 1, 2, and H−11 (E1) = H−1

2 (E2). Let this shared level set be


S. Let the Hamiltonian vector fields ξH1 and ξH2 on M have flows φt and ρt, respectively. The

induced flows φt and ρt on Sp(TS/TS⊥) are defined as in Section 4.3.1.

The key observation in this scenario is that TS and TS⊥ do not depend on H1 or H2. For

all s ∈ S, we have

TsS⊥ = span ξH1(s) = span ξH2(s) ,

which implies that there is a map c : S → R such that ξH2(s) = c(s)ξH1(s) for all s ∈ S. The

map c need not be constant on S, and ξH1 and ξH2 need not be parallel away from S.

In Lemma 4.3.1, we were able to establish a relationship between ρt and φt. In the more

general case, we will not be able to relate ρt and φt directly, but must instead work in terms of

their time derivatives.

For any s ∈ S, ρt∗|s and φt∗|s preserve TS. Therefore, for all ζ ∈ TsS, the tangent vectors

ddt

∣∣t=0

(ρt∗|sζ

)and d

dt

∣∣t=0

(φt∗|sζ

)are elements of TζTS. Note that the vector space TsS can

be naturally identified with its tangent space TζTsS, which is a subspace of TζTS. Thus

ξH1(s) ∈ TsS can also be viewed as an element of TζTS. This identification is used in the

statement of the following result.

Lemma 4.3.3. For all s ∈ S and all ζ ∈ TsS,

d

dt

∣∣∣∣t=0

(ρt∗|sζ

)= c(s)

d

dt

∣∣∣∣t=0

(φt∗|sζ

)+ (ζc)ξH1(s).

Proof. We ignore M , and treat S as a (2n−1)-dimensional manifold. The vector fields ξH1 and

ξH2 on S have flows φt and ρt, respectively, and satisfy ξH2(s) = c(s)ξH1(s).

Fix s ∈ S, and let U be a coordinate neighborhood for s with coordinates X = (X1, . . . ,

X2n−1). We write φt = (φt1, . . . , φt2n−1) and ρt = (ρt1, . . . , ρ

t2n−1) with respect to the local

coordinates. Then φt∗|s becomes a (2n− 1)× (2n− 1) matrix with the (j, k)th entry given by

(φt∗|s

)jk

=∂φtj(X)

∂Xk

∣∣∣∣∣X=s

,

and similarly for ρt∗|s. Also, since ξH2 = cξH1 , we have

d

dt

∣∣∣∣t=0

ρtj = cd

dt

∣∣∣∣t=0

φtj , j = 1, . . . , 2n− 1,


everywhere on U .

Let ζ ∈ TsS be arbitrary. Then ζ =∑2n−1

k=1 ak∂

∂Xk

∣∣∣X=s

for some coefficients a1, . . . , a2n−1 ∈

R, and (φt∗|sζ

)j

=2n−1∑k=1

ak∂φtj(X)

∂Xk

∣∣∣∣∣X=s

= ζφtj .

The same process establishes that (ρt∗|sζ)j = ζρtj . When we take the time derivative of the

latter equation, we find

d

dt

∣∣∣∣t=0

(ρt∗|sζ

)j

= ζ

(d

dt

∣∣∣∣t=0

ρtj

)= ζ

(cd

dt

∣∣∣∣t=0

φtj

).

Now we apply the Leibniz rule and recall that ddt

∣∣t=0

φtj(s) = ξH1(s)j . The result is

d

dt

∣∣∣∣t=0

(ρt∗|sζ

)j

= c(s)d

dt

∣∣∣∣t=0

(φt∗|sζ)j + (ζc)ξH1(s)j . (4.3.1)

The coordinates on U determine coordinates for TS|U , which in turn provide a basis for

TζTS. In terms of that basis,

d

dt

∣∣∣∣t=0

(φt∗|sζ

)=

(ξH1(s)1, . . . , ξH1(s)2n−1,

d

dt

∣∣∣∣t=0

(φt∗|sζ

)1, . . . ,

d

dt

∣∣∣∣t=0

(φt∗|sζ

)2n−1

),

d

dt

∣∣∣∣t=0

(ρt∗|sζ

)=

(ξH2(s)1, . . . , ξH2(s)2n−1,

d

dt

∣∣∣∣t=0

(ρt∗|sζ

)1, . . . ,

d

dt

∣∣∣∣t=0

(ρt∗|sζ

)2n−1

),

and

ξH1(s) = (0, . . . , 0, ξH1(s)1, . . . , ξH1(s)2n−1)

as an element of TζTS. Substituting Equation (4.3.1) into the expression for ddt

∣∣t=0

(ρt∗|sζ

)and

comparing the result with the expression for ddt

∣∣t=0

(φt∗|sζ

)shows that

d

dt

∣∣∣∣t=0

(ρt∗|sζ

)= c(s)

d

dt

∣∣∣∣t=0

(φt∗|sζ

)+ (ζc)ξH1(s),

as desired.


Lemma 4.3.4. For all s ∈ S and all b′ ∈ Sp(TS/TS⊥)s,

ξH2(b′) = c(s)ξH1(b′).

Proof. Within the model vector space V , fix a symplectic basis (x1, . . . , xn, y1, . . . , yn). Let

W = span x1, . . . , xn, y2, . . . , yn ,

so that

W⊥ = span x1 and W/W⊥ = span [x2], . . . , [xn], [y2], . . . , [yn] .

For convenience, let (z1, . . . , z2n) = (x1, . . . , xn, y2, . . . , yn, y1). Then

W = span z1, . . . , z2n−1 , W⊥ = span z1 , W/W⊥ = span [z2], . . . , [z2n−1] .

For this proof, we will not view an element of the symplectic frame bundle as a map, but as

a basis for a tangent space. Explicitly, over m ∈M , we identify b ∈ Sp(M,ω)m with the ordered

2n-tuple (ζ1, . . . , ζ2n) ∈ (TmM)2n, where bzj = ζj for j = 1, . . . , 2n. Similarly, over s ∈ S, we

identify b′ ∈ Sp(TS/TS⊥)s with the ordered (2n− 2)-tuple ([ζ2], . . . , [ζ2n]) ∈(TsS/TsS

⊥)2n−2,

where b′[zj ] = [ζj ] for j = 2, . . . , 2n− 1.

Let s ∈ S and b ∈ Sp(M,ω;S)s be given. Let bzj = ζj ∈ TsM for j = 1, . . . , 2n. Then

ζj ∈ TsS for j = 1, . . . , 2n − 1 and ζ1 ∈ TsS⊥. The flow ρt on Sp(M,ω;S), evaluated at b, is

given by

ρt(b) = ρt∗|m b =(ρt∗|mζ1, . . . , ρ

t∗|mζ2n

),

and the analogous expression holds for φt.

When we descend to Sp(TS/TS⊥), the image of b is the element b′ = ([ζ2], . . . , [ζ2n−1]) ∈

(TsS/TsS⊥)2n−2. The induced flow ρt on Sp(TS/TS⊥), evaluated at b′, is

ρt(b′) =([ρt∗|sζ2], . . . , [ρt∗|sζ2n−1]

).

As t varies, this expression describes a curve in (TS/TS⊥)2n−2, through b′. The tangent vector


to this curve at b′ is

ξH2(b′) =d

dt

∣∣∣∣t=0

ρt(b′) =

(d

dt

∣∣∣∣t=0

[ρt∗|sζ2], . . . ,d

dt

∣∣∣∣t=0

[ρt∗|sζ2n−1]

)∈ T[ζ2](TS/TS

⊥)× . . .× T[ζ2n−1](TS/TS⊥).

The pushforward of the quotient map TS → TS/TS⊥, based at ζj , is a linear surjection

TζjTS → T[ζj ](TS/TS⊥) whose kernel is TζjTsS

⊥. If we identify TζjTsS⊥ with TsS

⊥, then we

get a natural isomorphism between T[ζj ](TS/TS⊥) and TζjTS/TsS

⊥. Applying these isomor-

phisms to the components of ξH2(b′) yields

ξH2(b′) =

([d

dt

∣∣∣∣t=0

(ρt∗|sζ2)

], . . . ,

[d

dt

∣∣∣∣t=0

(ρt∗|sζ2n−1)

])∈ Tζ2TS/TsS⊥ × . . .× Tζ2n−1TS/TsS

⊥.

Since ζj ∈ TsS for j = 2, . . . , 2n− 1, we can use Lemma 4.3.3:

d

dt

∣∣∣∣t=0

(ρt∗|sζj

)= c(s)

d

dt

∣∣∣∣t=0

(φt∗|sζj

)+ (ζjc)ξH1(s),

where each term in the above equation is viewed as an element of TζjTS. Upon descending to

TζjTS/TsS⊥, the multiple of ξH1(s) vanishes and we find that

[d

dt

∣∣∣∣t=0

(ρt∗|sζj

)]= c(s)

[d

dt

∣∣∣∣t=0

(φt∗|sζj

)].

Therefore

ξH2(b′) =

(c(s)

[d

dt

∣∣∣∣t=0

(φt∗|sζ2)

], . . . , c(s)

[d

dt

∣∣∣∣t=0

(φt∗|sζ2n−1)

])= c(s)ξH1(b′).

This completes the proof.

We can now prove the result that was the objective of this section.

Theorem 4.3.5. If H1, H2 : M → R are smooth functions such that H−11 (E1) = H−1

2 (E2) for


regular values Ej of Hj , j = 1, 2, then E1 is a quantized energy level for (M,ω,H1) if and only

if E2 is a quantized energy level for (M,ω,H2).

Proof. Using Lemma 4.3.4, we see that ξH1 and ξH2 are parallel on Sp(TS/TS⊥), although the

multiplicative factor that relates them is not necessarily constant. This implies that φt and ρt

have identical orbits in Sp(TS/TS⊥), and if γS has trivial holonomy over the closed orbits of φt,

then the same must also be true for ρt. Thus, if E1 is a quantized energy level for (M,ω,H1),

then E2 is a quantized energy level for (M,ω,H2), and vice versa.

4.4 The Harmonic Oscillator

In this section, we apply our quantized energy condition to the n-dimensional harmonic oscilla-

tor. The metaplectic-c quantization of this system has already been examined in [16] and [17],

and it comes as no surprise that we obtain the same quantized energy levels that are derived

in those treatments using prequantization data:

EN = ~(N +

n

2

),

where N is an integer such that EN is positive1. Our first objective in studying this example is to

present a useful computational technique: we will locally change coordinates from Cartesian to

symplectic polar on the base manifold, then show how to lift this change to the symplectic frame

bundle and prequantization bundle. Once symplectic polar coordinates have been established,

we will turn to our second objective, which is to construct an explicit example that illustrates

the principal of dynamical invariance.

4.4.1 Initial choices

Let (x1, . . . , xn, y1, . . . , yn) be a basis for the model vector space V such that Ω =∑n

j=1 x∗j ∧ y∗j .

All elements of V will be written as ordered 2n-tuples with respect to this basis. Assume

that V is identified with Cn by mapping the point (a1, . . . , an, b1, . . . , bn) ∈ V to the point

1Both [16] and [17] derive the more familiar condition N ≥ 0 by proceeding from prequantization to quan-tization and introducing a polarization. Since our quantized energy condition uses only the metaplectic-c pre-quantization, we do not obtain this constraint on the starting point for N .


(b1 + ia1, . . . , bn + ian) ∈ Cn. Then the complex structure J on V is given in matrix form by

J =

0 I

−I 0

, where I is the n× n identity matrix.

A note concerning notation: given expressions aj and bj , j = 1, . . . , n, we let (aj)1≤j≤n

represent the n × n diagonal matrix diag(a1, . . . , an), and we let

aj

bj

1≤j≤n

represent the

2n× 1 column vector (a1, . . . , an, b1, . . . , bn)T .

Let M = R2n, with Cartesian coordinates (p1, . . . , pn, q1, . . . , qn) and symplectic form ω =∑nj=1 dpj ∧ dqj . The energy function for the harmonic oscillator is H = 1

2

∑nj=1

(p2j + q2

j

),

which has Hamiltonian vector field

ξH =

n∑j=1

(qj

∂

∂pj− pj

∂

∂qj

).

Let this vector field have flow φt on M .

There is a global trivialization of TM given by identifying xj 7→ ∂∂pj

∣∣∣m

, yj 7→ ∂∂qj

∣∣∣m

at every

point m ∈ M , which induces a global trivialization of Sp(M,ω). Let P be the trivial bundle

M ×Mpc(V ), with bundle projection map PΠ−→M . Define the map P

Σ−→ Sp(M,ω) by

Σ(m, a) = (m,σ(a)), ∀m ∈M, ∀a ∈ Mpc(V ),

where the ordered pair on the right-hand side is written with respect to the global trivialization

stated above. Then (P,Σ) is a metaplectic-c structure for (M,ω).

On M , define the one-form β by

β =1

2

n∑j=1

(pjdqj − qjdpj),

so that dβ = ω. Let ϑ0 be the trivial connection on the product bundle M × Mpc(V ), and

define the one-form γ on P by

γ =1

i~Π∗β +

1

2η∗ϑ0.

Then (P,Σ, γ) is a metaplectic-c prequantization for (M,ω). In fact, it is the unique metaplectic-


c prequantization up to isomorphism, since M is contractible.

Fix E > 0, a regular value of H, and let S = H−1(E). Let m0 ∈ S be given. Observe

that the entire system (P,Σ, γ) → (M,ω,H) is invariant under following transformations: (1)

rotation of the pjqj-plane about the origin by any angle, for any j, and (2) simultaneous rotations

of the pjpk- and qjqk-planes about the origin by the same angle, for any j 6= k. Thus, after

suitable rotations, we can assume without loss of generality that m0 = (p01, . . . , p0n, 0, . . . , 0)

in Cartesian coordinates, where p0j 6= 0 for all j. It is easily established that

φt(m0) =

(cos t)1≤j≤n (sin t)1≤j≤n

(− sin t)1≤j≤n (cos t)1≤j≤n

p0j

0

1≤j≤n

.

This expression describes a periodic orbit with period 2π. Let C be the orbit of ξH through m0:

C =φt(m0) : t ∈ R

.

4.4.2 From Cartesian to symplectic polar coordinates

On the Manifold

By symplectic polar coordinates, we mean the local coordinates (s1, . . . , sn, θ1, . . . , θn) given by

sj =1

2

(p2j + q2

j

), θj = tan−1

(qjpj

), j = 1, . . . , n, (4.4.1)

whenever these expressions are defined. The polar angles θj are all defined modulo 2π. For

later reference, the inverse coordinate transformations are

pj =√

2sj cos θj , qj =√

2sj sin θj , j = 1, . . . , n. (4.4.2)

Let

U =

(p1, . . . , pn, q1, . . . , qn) ∈M : p2j + q2

j > 0, j = 1, . . . , n,

so that symplectic polar coordinates and the corresponding vector fields

∂∂s1, . . . , ∂

∂sn, ∂∂θ1

, . . . ,

∂∂θn

are defined everywhere on U . Observe that the orbit C is contained in U . We will construct

a local trivialization for Sp(M,ω) over U , and a local trivialization for P over C.


When we convert to symplectic polar coordinates on U , we find

ω =n∑j=1

dsj ∧ dθj ,

which implies that for all m ∈ U ,

∂∂s1

∣∣∣m, . . . , ∂

∂sn

∣∣∣m, ∂∂θ1

∣∣∣m, . . . , ∂

∂θn

∣∣∣m

is a symplectic basis

for TmM . Further,

β =n∑j=1

sjdθj , H =n∑j=1

sj , ξH = −n∑j=1

∂

∂θj, (4.4.3)

and m0 = (s01, . . . , s0n, 0, . . . , 0), where s0j = 12p

20j , j = 1, . . . , n. The orbit C has the form

C = (s01, . . . , s0n, τ, . . . , τ) : τ ∈ R/2πZ .

The goal is to lift this change of coordinates to the symplectic frame bundle, then to the

metaplectic-c prequantization. To facilitate this process, we introduce the following notation.

Let Φc : U → R2n be the Cartesian coordinate map, and let Uc = Φc(U). Similarly, let

Φp : U → Rn × (R/2πZ)n be the symplectic polar coordinate map, and let Up = Φp(U). Let F

denote the transition map Φp Φ−1c . Then we have the following commutative diagram.

UΦc

zz

Φp

''Uc ⊂ R2n F // Up ⊂ Rn × (R/2πZ)n

This will serve as a model for the changes of coordinates on Sp(M,ω) and P .

On the Symplectic Frame Bundle

An element of the fiber Sp(M,ω)m is a symplectic isomorphism from V to TmM . Over U , we

define the sections bc and bp of Sp(M,ω) as follows. At each m ∈ U , they are given by

bc(m) : V → TmM such that

xj

yj

1≤j≤n

7→

∂∂pj

∣∣∣m

∂∂qj

∣∣∣m

1≤j≤n

,

bp(m) : V → TmM such that

xj

yj

1≤j≤n

7→

∂∂sj

∣∣∣m

∂∂θj

∣∣∣m

1≤j≤n

.


Using these sections, we define two maps Φc, Φp on Sp(M,ω) over U :

Φc : Sp(M,ω)|U → Uc × Sp(V )

is defined by

Φc(bc(m) · g) = (Φc(m), g), ∀m ∈ U, ∀g ∈ Sp(V ),

and

Φp : Sp(M,ω)|U → Up × Sp(V )

is defined by

Φp(bp(m) · g) = (Φp(m), g), ∀m ∈ U, ∀g ∈ Sp(V ).

In other words, the section bc determines the Cartesian trivialization of Sp(M,ω) over U , and

the section bp determines the symplectic polar trivialization. Lifting the change of coordinates

to Sp(M,ω)|U is equivalent to finding a map F : Uc × Sp(V ) → Up × Sp(V ) that is a lift of

F : Uc → Up and such that the following commutes.

Sp(M,ω)|UΦc

ww

Φp

''Uc × Sp(V )

F // Up × Sp(V )

If such a map F exists, then it must satisfy

F Φc(bc) = Φp(bc),

and indeed, once this condition is satisfied, then the value of F everywhere else will be de-

termined by the Sp(V ) group action and the fact that F is a lift of F . Let us then evaluate

Φp(bc).

For all m ∈ U , let

G(m) =

(∂pj∂sj

∣∣∣m

)1≤j≤n

(∂qj∂sj

∣∣∣m

)1≤j≤n(

∂pj∂θj

∣∣∣m

)1≤j≤n

(∂qj∂θj

∣∣∣m

)1≤j≤n

,


so that

G(m)

∂∂pj

∣∣∣m

∂∂qj

∣∣∣m

1≤j≤n

=

∂∂sj

∣∣∣m

∂∂θj

∣∣∣m

1≤j≤n

.

Then G(m) is a symplectic matrix, and can be viewed as an element of Sp(V ). From the

definitions of bc(m) and the group action, we see that

bc(m) : G(m)

xj

yj

1≤j≤n

7→ G(m)

∂∂pj

∣∣∣m

∂∂qj

∣∣∣m

1≤j≤n

=

∂∂sj

∣∣∣m

∂∂θj

∣∣∣m

1≤j≤n

.

Thus

Φp(bc(m)) = (Φp(m), G(m)),

and more generally,

Φp(bc(m) · g) = (Φp(m), G(m)g), ∀g ∈ Sp(V ).

Hence we define the map F by

F (Φc(m), g) = (Φp(m), G(m)g), ∀m ∈ U, ∀g ∈ Sp(V ).

Observe, using Equation (4.4.2), that the entries of G(m) are singly defined with respect to

the angles θj , so G(m) is singly defined as m traverses the curve C, or any other closed path

through U . This is to be expected, since the symplectic polar coordinates are singly defined

everywhere on U . However, we will encounter different behavior when we lift the change of

variables to the metaplectic-c prequantization.

On the Metaplectic-c Prequantization

When we lift the change of variables to P , we restrict our attention further from U to C. Let

Cc = Φc(C) and let Cp = Φp(C). To emphasize that all of our calculations take place over the

closed curve C, we let m(τ) = (s01, . . . , s0n, τ, . . . , τ) ∈ C for all τ ∈ R/2πZ, and we abbreviate

G(m(τ)) by G(τ). Then G(τ) is a closed loop through Sp(V ) with period 2π.


Recall that P = M ×Mpc(V ), and that Σ : P → Sp(M,ω) is given by Σ(m, a) = (m,σ(a))

for all (m, a) ∈ P , where the right-hand side is written with respect to the Cartesian trivial-

ization. In other words, P was constructed to be consistent with Cartesian coordinates. To

formalize this property, we let the map

Φc : P |C → Cc ×Mpc(V )

be defined by

Φc(m(τ), a) = (Φc(m(τ)), a), ∀(m(τ), a) ∈ P |C .

Then Φc is compatible with Φc in the sense that the following diagram commutes.

P |C Σ //

Φc

Sp(M,ω)|C

Φc

Cc ×Mpc(V )σ // Cc × Sp(V )

In the bottom line, σ maps Cc ×Mpc(V ) to Cc × Sp(V ) by acting on the second component.

Our goal is to construct a map Φp : P |C → Cp ×Mpc(V ) that is compatible with Φp in the

same sense. To do so, we will find a map F such that the diagram below commutes, then set

Φp = F Φc.

P |C

Φc

Σ

**Cc ×Mpc(V )

F //

σ

**

Cp ×Mpc(V )σ

**

Sp(M,ω)|C

Φc

Φp

Cc × Sp(V )

F // Cp × Sp(V )

Since F must be a lift of F and therefore of F , we assume that F takes the form

F (Φc(m(τ)), a) = (Φp(m(τ)), G(τ)a), ∀m(τ) ∈ C, ∀a ∈ Mpc(V ),


where G(τ) ∈ Mpc(V ). From the condition that

σ F Φc = F σ Φc,

it follows that we must have σ(G(τ)) = G(τ) for all τ ∈ R/2πZ. That is, G(τ) must be a lift of

G(τ) to Mpc(V ). More specifically, in order for F to be singly defined, G(τ) must be a closed

loop in Mpc(V ) with period 2π.

There is another consideration: the effect of this change of coordinates on the one-form γ.

Recall that γ is defined on P by

γ =1

i~Π∗β +

1

2η∗ϑ0,

where ϑ0 is the trivial connection on the product bundle M ×Mpc(V ). When we change to

symplectic polar coordinates, we would like γ to retain the same form, where β can be written

in polar coordinates as in Equation (4.4.3), and where ϑ0 now represents the trivial connection

on the product bundle Cp×Mpc(V ). Since ker(η) = Mp(V ), we can accomplish this by requiring

that the path G(τ) lie within Mp(V ).

Recall the parametrization of Mpc(V ) that was described in Section 2.3.1. If G(τ) is

a lift of G(τ) to Mp(V ), then the parameters of G(τ) have the form (G(τ), µ(τ)) where

µ(τ)2DetCCG(τ) = 1, and where

CG(τ) =1

2(G(τ)− JG(τ)J).

To determine µ(τ), we must examine the matrix G(τ) ∈ Sp(V ) explicitly. Using Equation

(4.4.2), we find

G(τ) =

(

1√2s0j

cos τ

)1≤j≤n

(1√2s0j

sin τ

)1≤j≤n(

−√

2s0j sin τ)

1≤j≤n(√

2s0j cos τ)

1≤j≤n

.

Now we calculate CG(τ) using the complex structure noted in Section 4.4.1, then convert to an


n× n complex matrix. The result is the diagonal matrix

CG(τ) =1

2

((√2s0j +

1√2s0j

)eiτ

)1≤j≤n

,

which has complex determinant

DetCCG(τ) =

n∏j=1

1

2

(√2s0j +

1√2s0j

)eiτ = Keinτ ,

where K is a positive real value that is constant over C. Thus, a lift of G(τ) to Mp(V ) is

parametrized by

G(τ) 7→(G(τ),

1√Ke−inτ/2

). (4.4.4)

Notice that (G(τ + 2π), µ(τ + 2π)) = (G(τ), µ(τ)e−inπ). When n is even, we get a closed

loop through Mp(V ) with period 2π, which is the optimal outcome. When n is odd, however,

traversing C once multiplies µ(τ) by −1. This shows that we cannot always lift G(τ) to a closed

path through Mp(V ). Nevertheless, we choose the path G(τ) as defined in Equation (4.4.4)

because it preserves the form of γ. If we let εn ∈ Mpc(V ) be the element whose parameters are

(I, e−iπn), then G(τ + 2π) = εnG(τ).

To prevent F from being multi-valued, let C = m(τ) ∈ C : τ ∈ (0, 2π), and let Cc and

Cp be the images of C in Cartesian and symplectic polar coordinates, respectively. Then let

F : Cc ×Mpc(V )→ Cp ×Mpc(V ) be given by

F (Φc(m(τ)), a) = (Φp(m(τ)), G(τ)a), ∀(m(τ), a) ∈ P |C ,

and identify P |C with Cp×Mpc(V ) under the map Φp = F Φc. This gives us symplectic polar

coordinates for P over C. On Cp ×Mpc(V ), γ takes the form

γ =1

i~

n∑j=1

s0jdθj +1

2η∗ϑ0,

where ϑ0 is the trivial connection on the product bundle, and where the values s0j are constant

over Cp. We will manually adjust for the fact that closing the loop multiplies G(τ) by εn: if


there is a lift of C to P |C whose image in Cp×Mpc(V ) has the form (m(τ), a(τ)) for some path

a(τ) in Mpc(V ), then this lifted path is closed over C if and only if a(τ + 2π) = εna(τ).

4.4.3 Quantized energy levels of the harmonic oscillator

Having established the change of coordinates, we will now drop the explicit use of Φp, F and

their lifts. Elements of U will be written with respect to the symplectic polar coordinates

(s1, . . . , sn, θ1, . . . , θn), which for convenience we abbreviate by Xk, k = 1, . . . , 2n. Using the

identifications xj 7→ ∂∂sj

∣∣∣m, yj 7→ ∂

∂θj

∣∣∣m

for all m ∈ U , j = 1, . . . , n, we write

Sp(M,ω)|U = U × Sp(V ) and P |C = C ×Mpc(V ).

At the level of tangent spaces, we have

T(m,I) Sp(M,ω) = TmM × sp(V ), ∀m ∈ U,

and

T(m,I)P = TmM ×mpc(V ) = TmM × (sp(V )⊕ u(1)) , ∀m ∈ C.

On U , ξH = −∑n

j=1∂∂θj

. Therefore TmS⊥ = span

∑nj=1

∂∂θj

∣∣∣m

at each m ∈ U . Within

the model vector space V , let

W⊥ = span y1 + . . .+ yn .

Then

W = span x1 − x2, . . . , xn−1 − xn, y1, . . . , yn ,

and

W/W⊥ = span [x1 − x2], . . . , [xn−1 − xn], [y1 − y2], . . . , [yn−1 − yn] .

The local trivialization of Sp(M,ω)|U induces the local trivializations Sp(M,ω;S)|U∩S = U ∩

S × Sp(V ;W ) and Sp(TS/TS⊥)|U∩S = U ∩ S × Sp(W/W⊥).

By definition, PS is the bundle associated to PS by the group homomorphism ν : Mpc(V )→


Mpc(W/W⊥). Over C, we have the local trivializations P |C = C × Mpc(V ), PS |C = C ×

Mpc(V ;W ) and PS |C = C ×Mpc(W/W⊥), where the associated bundle map ν : PS |C → PS |Cacts by

ν(m, a) = (m, ν(a)), ∀(m, a) ∈ PS |C .

Using properties of ν noted in Section 4.2.1, we have ν(εn) = εn ∈ Mpc(W/W⊥) for all n.

Therefore, within PS , the same factor of εn is required to close the loop from C to C. Further,

since η∗ ν∗ = η∗, the one-form γS induced on PS does not change form under the map ν:

γS |C =1

i~

n∑j=1

s0jdθj +1

2η∗ϑ0, (4.4.5)

where ϑ0 is now the trivial connection on the product bundle C ×Mpc(W/W⊥). Lastly, the

relevant tangent spaces are

T(m,I) Sp(TS/TS⊥) = TmM × sp(W/W⊥), ∀m ∈ U,

and

T(m,I)PS = TmM × (sp(W/W⊥)⊕ u(1)), ∀m ∈ C.

We chose the initial point m0 = (s01, . . . , s0n, 0, . . . , 0), and that C is the orbit of ξH through

m0. We will show that the orbit of ξH through (m0, I) is closed in Sp(TS/TS⊥). Note that

we already know that the orbits of ξH are closed because H generates a circle action on M .

However, the calculation serves as a useful model for the example that appears in Section

4.4.4. We first need to lift φt to the flow φt on Sp(M,ω). By definition, for any m ∈ U ,

φt(m, I) = (φt(m), φt∗|m). This implies that

ξH(m, I) =d

dt

∣∣∣∣t=0

φt(m, I) =

(ξH(m),

d

dt

∣∣∣∣t=0

φt∗|m).

We write φt = (φt1, . . . , φt2n) with respect to the symplectic polar coordinates. Then φt∗|m is

a 2n× 2n matrix, which we interpret as an element of Sp(V ), and its components are given by


(φt∗)jk =∂φtj∂Xk

∣∣∣m

. Noting that ddt

∣∣t=0

φtj = (ξH)j , we compute

(d

dt

∣∣∣∣t=0

φt∗|m)jk

=∂

∂Xk

∣∣∣∣m

(ξH)j . (4.4.6)

But ξH = −∑n

j=1∂∂θj

on U , which has constant components, so ddt

∣∣t=0

φt∗ is identically 0. Thus

ξH(m, I) = (ξH(m), 0), ∀m ∈ U.

In particular, since the sp(V ) component of ξH is constant over the orbit C, we can find the

flow for ξH on Sp(M,ω) through (m0, I) by exponentiating: it is simply

φt(m0, I) = (φt(m0), I).

This is clearly a closed orbit with period 2π. The induced flow on the bundle Sp(TS/TS⊥) also

takes the form

φt(m0, I) = (φt(m0), I).

Now we restrict our attention to the curve C, and lift ξH to PS , horizontally with respect

to γS , where γS is given in Equation (4.4.5). Using the fact that∑n

j=1 s0j = E, we find that

for any m ∈ C, the horizontal lift of ξH to T(m,I)PS is

ξH(m, I) =

(ξH(m), 0⊕ E

i~

).

The mpc(V ) component is constant over C, so the flow is again calculated by exponentiating:

φt(m0, I) =

(φt(m0), exp

(0⊕ Et

i~

))= (φt(m0), eEt/i~),

where eEt/i~ ∈ U(1) ⊂ Mpc(W/W⊥).

The quantized energy levels are those values of E for which this orbit closes. However, we

must remember that one circuit about C introduces an extra factor of εn. Referring to the

properties of the parametrization stated in Section 2.3.1, we see that φ0(m0, I) = φ2π(m0, I)

if and only if e2πE/i~ = e−inπ, or in other words, when 2πEi~ + inπ = −2πiN for some N ∈ Z.


Upon rearranging and recalling that E is positive, we find that the quantized energy levels take

the form

E = ~(N +

n

2

), N ∈ Z, N > −n

2.

Thus the quantization condition correctly reproduces the expected n2 -shift in the energy levels

of the harmonic oscillator.

4.4.4 Example of dynamical invariance

Fix n = 2. It is convenient to use the definitions sj = 12(p2

j + q2j ) and ∂

∂θj= pj

∂∂qj− qj ∂

∂pjevery-

where on M , for j = 1, 2. We will point out when a statement only holds on the neighborhood

U .

Let k ∈ R be a positive constant, and consider the two functions H1, H2 : M → R given by

H1 = s1 + s2 − k, H2 = (s1 + s2 − k)(s1 + 2s2 + 1).

The function H1 is just the energy function for the harmonic oscillator, shifted by k: its

quantized energy levels are

EN = ~N − k, N ∈ Z, N >k

~.

Notice that H−11 (0) = H−1

2 (0). Let this shared level set be S. The energy E = 0 is a quantized

energy for the system (M,ω,H1) if and only if k~ ∈ N.

The two Hamiltonian vector fields are

ξH1 = − ∂

∂θ1− ∂

∂θ2,

ξH2 = (s1 + 2s2 + 1)ξH1 −H1

(∂

∂θ1+ 2

∂

∂θ2

)= −(2s1 + 3s2 − k + 1)

∂

∂θ1− (3s1 + 4s2 − 2k + 1)

∂

∂θ2.

Since H1 = 0 on S, we see that ξH2 = (s1 + 2s2 + 1)ξH1 everywhere on S. Thus the vector fields

are parallel on S, as expected, and so they share the same orbits in S. However, ξH1 and ξH2

are not parallel away from S. Let φt be the flow of ξH1 , and let ρt be the flow of ξH2 .


Consider the initial point m0 ∈ S ∩ U , where m0 = (s01, s02, 0, 0) with s01 + s02 = k and

s01, s02 6= 0. The orbit of both φt and ρt through m0 is

C = (s01, s02, τ, τ) : τ ∈ R/2πZ .

From Section 4.4.3, we know that

ξH1(m, I) = (ξH1(m), 0), ∀m ∈ C,

and therefore

φt(m0, I) = (φt(m0), I).

By the same calculation, ξH2(m, I) = (ξH2(m), ddt

∣∣t=0

ρt∗|m) for m ∈ C. Applying Equation

(4.4.6) to the components of ξH2 yields

d

dt

∣∣∣∣t=0

ρt∗|m =

0 0 0 0

0 0 0 0

−2 −3 0 0

−3 −4 0 0

,

which we interpret as an element of sp(V ). Let this matrix be denoted by κ. Then

ξH2(m, I) = (ξH2(m), κ),

and since the Lie algebra component is constant over C, we obtain the flow on Sp(M,ω) through

(m0, I) by exponentiating:

ρt(m0, I) = (ρt(m0), exp(tκ)).


A calculation establishes that

exp(tκ) =

1 0 0 0

0 1 0 0

−2t −3t 1 0

−3t −4t 0 1

,

which is clearly not periodic. Thus ρt has no closed orbits on Sp(M,ω) over S ∩ U .

Now let us transfer to Sp(TS/TS⊥). If we apply the definitions and identifications laid out at

the beginning of Section 4.4.3, now setting n = 2, then we find W⊥ = span y1 + y2, W/W⊥ =

span [x1 − x2] , [y1 − y2], and the identification of Sp(M,ω)|C with C × Sp(V ) induces an

identification of Sp(TS/TS⊥)|C with C × Sp(W/W⊥). Notice that

exp(tκ)(x1 − x2) = x1 − x2 + t(y1 + y2),

exp(tκ)(y1 − y2) = y1 − y2.

Therefore the path through Sp(W/W⊥) induced by exp(tκ) is

ν(exp(tκ)) =

1 0

0 1

.

Thus, on Sp(TS/TS⊥)|C ,

ρt(m0, I) =(ρt(m0), I

),

which coincides with φt(m0, I).

The above calculation omits certain cases: namely, if the starting point m0 has s01 = 0

or s02 = 0. We can no longer eliminate such cases by performing a rotation, because H2 is

not symmetric with respect to s1 and s2. If m0 /∈ U , then we have to modify our approach.

For example, if s01 = 0, then we retain Cartesian coordinates for the p1q1-plane and convert

to symplectic polar on the p2q2-plane. The calculation is more complicated, but the result is

similar: over C, we find that ξH2 = (ξH2 , κ) for some constant value of κ ∈ sp(V ). The path

ρt(m0, I) = (ρt(m0), exp(tκ)) does not close in Sp(M,ω), but the induced path in Sp(TS/TS⊥)


coincides with φt(m0, I). The same pattern holds if we take s02 = 0.

Thus, if the quantized energy condition were stated in terms of the holonomy of γS over

closed orbits in Sp(M,ω;S), as it was in [16], then the value E = 0 would satisfy the condition

vacuously for the system (M,ω,H2), regardless of the value of k. It is only by descending

to Sp(TS/TS⊥) that we recover the quantization condition k~ ∈ N. Hence our definition of a

quantized energy level is dynamically invariant, while that in [16] is not.

4.5 Comparison with Kostant-Souriau Quantization

4.5.1 The quantized energy condition and its properties

The quantized energy condition in Definition 4.2.2 can be easily adapted to Kostant-Souriau

prequantization. Indeed, the Kostant-Souriau case is simpler, since it does not involve the

symplectic frame bundle.

Assume that (M,ω) admits a prequantization circle bundle (Y, γ), and let H : M → R

be a smooth function. We now mimic the constructions in Section 4.2.2, using Y in place of

P . Denote the Hamiltonian vector field corresponding to H by ξH as before, and let ξH be

the lift of ξH to Y that is horizontal with respect to γ. Fix E, a regular value of H, and let

S = H−1(E) ⊂M . Let Y S be the restriction of Y to S, and let γS be the pullback of γ to Y S .

(Y, γ)

(Y S , γS)incl.oo

(M,ω) S

incl.oo

The construction stops here, and we give the definition of a quantized energy level using

(Y S , γS).

Definition 4.5.1. If the connection one-form γS has trivial holonomy over all closed orbits of

the Hamiltonian vector field ξH on S, then E is a Kostant-Souriau (KS) quantized energy

level for the system (M,ω,H).

This definition is dynamically invariant, and the proof is much simpler than that in Section

4.3.2.


Theorem 4.5.2. If H1, H2 : M → R are smooth functions such that H−11 (E1) = H−1

2 (E2) for

regular values Ej of Hj , j = 1, 2, then E1 is a KS quantized energy level for (M,ω,H1) if and

only if E2 is a KS quantized energy level for (M,ω,H2).

Proof. We argued in Section 4.3.2 that ξH1 and ξH2 are parallel on S, which implies that they

have the same orbits. Therefore γS has trivial holonomy over the orbits of one if and only if it

has trivial holonomy over the orbits of the other. The KS version of the dynamical invariance

theorem is immediate.

In Section 4.2.2, we examined the case in which the symplectic reduction of (M,ω) at

E is a manifold. Theorem 4.2.1, due to Robinson [16], gives the conditions under which the

quantization condition is sufficient to imply that the symplectic reduction admits a metaplectic-

c prequantization. A similar theorem can be given in the context of prequantization circle

bundles.

First, we state a general result, which was used in [16] to prove Theorem 4.2.1. Let S

be an arbitrary manifold, and suppose that (Z, δ) is a principal circle bundle with connection

one-form over S. Let the curvature of δ be $. Suppose that F is a foliation of S whose leaf

space SF is a smooth manifold. Denote the leaf projection map by Sπ−→ SF . If δ has trivial

holonomy over all of the leaves of F , then Z can be factored to produce a well-defined circle

bundle ZF → SF . Further, δ descends to a connection one-form δF on ZF , and the curvature

$F of δF satisfies π∗$F = $.

Now apply this result to the circle bundle (Y S , γS)→ S, where the foliation is given by the

orbits of the vector field ξH on the level set S, and the symplectic reduction (ME , ωE) is its

leaf space.

Theorem 4.5.3. Suppose that the symplectic reduction (ME , ωE) for (M,ω) at E is a manifold.

If γS has trivial holonomy over all closed orbits of ξH on S, then the quotient of (Y S , γS) by

the orbits of ξH is a prequantization circle bundle for (ME , ωE).

Thus the Kostant-Souriau version of the quantized energy condition is sufficient to ensure

that the symplectic reduction admits a prequantization circle bundle, whenever the symplectic

reduction is a manifold. This is a slight improvement over the metaplectic-c result, since it does

not depend on a quotient of the symplectic frame bundle being well defined.


4.5.2 Lack of half-shift in the harmonic oscillator

In this section, we will determine the KS quantized energy levels of the n-dimensional harmonic

oscillator. The calculation will be significantly simpler than that in Section 4.4, but it will yield

the wrong answer.

Let M = R2n, with Cartesian coordinates (p1, . . . , pn, q1, . . . , qn) and symplectic form ω =∑nj=1 dpj ∧ dqj . The energy function and corresponding Hamiltonian vector field for the har-

monic oscillator are

H =1

2

n∑j=1

(p2j + q2

j ), ξH =

n∑j=1

(qj

∂

∂pj− pj

∂

∂qj

).

Let the flow of ξH on M be φt. We know from Section 4.4.1 that all of the orbits of ξH are

circles, and that φt+2π(m) = φt(m) for all m ∈M .

Let Y = M × U(1), with projection map YΠ−→M . We define

β =1

2

n∑j=1

(pjdqj − qjdpj)

on M , and let γ = 1i~Π∗β + ϑ0 on Y , where ϑ0 is the trivial connection on the product bundle

M × U(1). Then (Y, γ) is a prequantization circle bundle for (M,ω), and it is unique up to

isomorphism.

Let E > 0 be arbitrary, and let S = H−1(E). At any point s ∈ S, we calculate that

ξHyβ = −E. Therefore the lifted vector field ξH at the point (s, I) ∈ Y is

ξH(s, I) =

(ξH(s),

E

i~

),

where we identify the tangent space T(s,I)Y with TsM × u(1). Note that the u(1) component

is constant over all of S, so in particular it is constant over an orbit. Let the flow of ξH be φt.

By exponentiation, we find that

φt(s, I) =(φt(s), eEt/i~

).


From this, it follows that the holonomy of γS over the orbits in S is trivial if and only if

E = N~ for some N ∈ N. This is not consistent with the quantum mechanical prediction of

E = ~(N + n

2

)when the dimension n is odd.

This shortcoming in the Kostant-Souriau prequantization of the harmonic oscillator is well

known, and the standard solution is to proceed from prequantization to quantization while in-

troducing the half-form correction, as described in Section 2.2.2. In this context, the quantized

energy levels for the system (M,ω,H) can be taken to be the eigenvalues of the operator cor-

responding to the energy function H. When the quantization recipe is applied to the harmonic

oscillator, the presence of the half-form bundle adds the n2 shift to the energy eigenvalues. How-

ever, this correction comes at the cost of introducing a choice of polarization, and the quantized

energy definition can no longer be evaluated over a single level set of H.

By comparison, when we use the metaplectic-c formulation of a quantized energy level,

we find that the correct harmonic oscillator energies are encoded in the geometry of the level

sets of the energy function. The result is dynamically invariant, independent of polarization,

and consistent with physical prediction. This example illustrates the benefits of metaplectic-c

quantization and our quantized energy definition.

Chapter 5

The Hydrogen Atom

5.1 Introduction

The quantized energy levels of the hydrogen atom have been calculated for various physical

models, using various flavors of geometric quantization. Notable examples include:

• Simms [18], who used the observation that the space of orbits corresponding to a fixed

negative energy is isomorphic to S2 × S2, and determined those energies for which the

reduced manifold admits a prequantization circle bundle;

• Sniatycki [21], who looked at the 2-dimensional relativistic Kepler problem and computed

a Bohr-Sommerfeld condition for the completely integrable system given by considering

the energy and angular momentum functions simultaneously;

• Duval, Elhadad and Tuynman [7], who took the phase space to include the spins of the

electron and proton, then chose a polarization and determined the Kostant-Souriau quan-

tized operator corresponding to the energy function with fine and hyperfine interaction

terms.

These examples exist at one of two possible extremes. On one hand, the quantized energy

condition can be evaluated over the symplectic reduction of a particular level set of the energy

function, as in [18]. This definition only looks at one energy level at a time, but it requires

constructing the symplectic reduction and establishing that the result is a smooth manifold.

65

Chapter 5. The Hydrogen Atom 66

On the other hand, the quantized energy levels can be determined from properties of the

quantized system as a whole, as in [21] or [7]. These approaches are characterized by requiring

a polarization, or the equivalent information – the Hamiltonian vector fields corresponding to

the Poisson-commuting functions in a completely integrable system generate a real polarization

– and the calculation is not restricted to a single level set of the energy function in question.

Our proposed definition for a quantized energy level acts as a middle ground between these

two extremes. The objective of this chapter is to apply the metaplectic-c quantized energy

condition to the hydrogen atom, using the physical model that is equivalent to the Kepler

problem. We will show that the quantized energy levels are in agreement with the quantum

mechanical prediction.

In Section 5.2, we set up our model of the hydrogen atom and construct a metaplectic-c

prequantization for its phase space. Section 5.3 presents the Ligon-Schaaf regularization map,

which is a symplectomorphism from the negative-energy domain of the hydrogen atom to an

open submanifold of TS3. We show how to relate the energy function for the hydrogen atom

to that of a free particle on S3, a process that makes use of the dynamical invariance property

of our definition. Finally, in Section 5.4, we determine the quantized energy levels of a free

particle on S3, and use these to determine the quantized energy levels for the hydrogen atom.

We will apply the definitions and constructions that were given in Section 4.2. In addition,

we will make repeated use of the notation and conventions that are described below.

5.1.1 Choices for subsequent calculations

In the sections that follow, we will need model symplectic vector spaces of several different

dimensions. Let us fix some standardized choices.

For any n ∈ N, let (Vn,Ωn) be a 2n-dimensional symplectic vector space. Let (v1, . . . , vn,

w1, . . . , wn) be a symplectic basis for Vn, and write all elements of Vn as ordered 2n-tuples with

respect to this basis. The symplectic form can be written in terms of the dual basis as

Ωn =n∑j=1

v∗j ∧ w∗j .


Assume that each real vector (a1, . . . , an, b1, . . . , bn) ∈ Vn is identified with the complex vector1

(b1 + ia1 . . . , bn+ ian) ∈ Cn. The resulting complex structure J on Vn is written in matrix form

as J =

0 I

−I 0

, where I is the n× n identity matrix.

When we require a subspace of Vn of codimension 1, we choose

Wn = span v1, . . . , vn, w1, . . . , wn−1 .

Then

W⊥n = span vn and Wn/W⊥n = span [v1], . . . , [vn−1], [w1], . . . , [wn−1] .

Using Equation (4.2.1), it is immediate that Wn/W⊥n is isomorphic to Vn−1 as a symplectic vec-

tor space and a complex vector space. The commutative diagram from Section 4.2.1 containing

the group homomorphisms ν and ν can be rewritten as follows.

Mpc(Vn) ⊃ Mpc(Vn;Wn)ν //

σ

Mpc(Vn−1)

σ

Sp(Vn) ⊃ Sp(Vn;Wn)

ν //// Sp(Vn−1)

5.2 Quantizing the Hydrogen Atom

5.2.1 Setup

Let R3 represent R3 with the origin removed, and let M = T R3 = R3 × R3. We use Cartesian

coordinates q = (q1, q2, q3) on R3 and p = (p1, p2, p3) on R3. Equip M with the symplectic form

ω =∑3

j=1 dqj ∧ dpj .

We consider the model of the hydrogen atom that is equivalent to the Kepler problem.

Assume that a proton is fixed at the origin in R3, and that an electron of mass me interacts with

it via the electrostatic force, which obeys an inverse-square law with constant of proportionality

k. Then M is the phase space for the motion of the electron, where q and p represent its position

1We use this sign convention for consistency with [17].


and momentum, respectively. The Hamiltonian energy function is

H =1

2me|p|2 − k

|q|,

and the corresponding Hamiltonian vector field is

ξH =3∑j=1

(1

mepj

∂

∂qj− k

|q|3qj

∂

∂pj

).

Let ξH have flow φt on M .

The solutions to the Kepler problem are well known [1, 4]. The angular momentum vector

L = q×p is a constant of the motion, as is the eccentricity vector e = 1mek

p×L− q|q| . In position

space, the orbits corresponding to a given energy E ∈ R are conic sections with eccentricity

|e| =

√1 +

2E|L|2mek2

, (5.2.1)

having the origin as a focus.

In particular, suppose E < 0. Then the value of |L| lies in the interval

[0,√−mek2

2E

]. If

|L| > 0, then the orbit is an ellipse, with |L| =√−mek2

2E being the special case of a circle. All

elliptical orbits with energy E have period 2πΛ , where Λ =

√− 8E3

mek2. If |L| = 0, however, then

e = 1 and the orbit is a line segment. Physically, this represents the case where the electron

begins from rest and collapses on a straight-line trajectory into the proton. Such motion is

not periodic, which implies that the level set H−1(E) contains orbits of ξH that are not closed.

Further, the collapse occurs in finite time, meaning that the vector field ξH is not complete.

The objective of this chapter is to determine the quantized energy levels for (M,ω,H). In

the next section, we construct a metaplectic-c prequantization for (M,ω), and formulate our

approach for performing the quantized energy calculation.


5.2.2 Metaplectic-c prequantization of (M,ω)

We choose the model symplectic vector space V3, as described in Section 5.1.1. The tangent

bundle TM can be identified with M × V3 with respect to the global trivialization

vj 7→∂

∂qj

∣∣∣∣m

, wj 7→∂

∂pj

∣∣∣∣m

, ∀m ∈M, j = 1, 2, 3.

This yields an identification of the symplectic frame bundle Sp(M,ω) with M × Sp(V3).

Let P = M ×Mpc(V3), with bundle projection map PΠ−→ M . Define the map Σ : P →

Sp(M,ω) by

Σ(m, a) = (m,σ(a)), ∀m ∈M, ∀a ∈ Mpc(V3),

where the right-hand side is written with respect to the global trivialization above. Let

β =

3∑j=1

(qjdpj + d(qjpj)) (5.2.2)

on M , so that dβ = ω. The reason for this choice of β will be made clear in Section 5.3.1. Let

γ be the u(1)-valued one-form on P given by

γ =1

i~Π∗β +

1

2η∗ϑ0,

where ϑ0 is the trivial connection on the product bundle. Then (P,Σ, γ) is the metaplectic-c

prequantization for (M,ω) (unique up to isomorphism).

By definition, the quantized energy levels of (M,ω,H) are those regular values E of H

such that the holonomy of γS is trivial over all closed orbits of ξH on Sp(TS/TS⊥), where

S = H−1(E). If E ≥ 0, then the quantization condition can be evaluated immediately. From

Equation (5.2.1) for the eccentricity of the orbit, we see that if E = 0, then the orbits are

parabolas in position space, and if E > 0, then they are hyperbolas. In these cases, ξH has

no closed orbits in S, which implies that ξH cannot have any closed orbits in Sp(TS/TS⊥).

Therefore the holonomy condition is satisfied vacuously, and all nonnegative energy levels are

quantized energy levels. This is consistent with the physical prediction from quantum mechan-

ics: a particle that is not spatially confined has a continuous energy spectrum.


It remains to consider the orbits corresponding to negative energy. Let

N = m ∈M : H(m) < 0 .

Then N is an open, simply connected submanifold of M . The symplectic form on M restricts

to one on N , and the metaplectic-c prequantization for M restricts to one for N . We use the

same symbols to denote the restricted objects: (N,ω) is a symplectic manifold, and (P,Σ, γ) is

its metaplectic-c prequantization. Since N is simply connected, every metaplectic-c prequanti-

zation for (N,ω) is isomorphic to (P,Σ, γ).

Let E < 0 be fixed, and let S = H−1(E) ⊂ N . Through the process described in Section

4.2.2, we obtain the three-level structures

(PS , γS)→ Sp(N,ω;S)→ S

and

(PS , γS)→ Sp(TS/TS⊥)→ S.

Lift ξH to ξH on Sp(N,ω;S). It induces a vector field on Sp(TS/TS⊥), which we also denote

by ξH .

To evaluate the quantization condition using these bundles, we would have to determine

the closed orbits of ξH on Sp(TS/TS⊥), then lift ξH to ξH on PS , horizontally with respect

to γS , and ensure that every lift of a closed orbit in Sp(TS/TS⊥) is closed in PS . However,

this procedure is computationally prohibitive in all but the special case of the circular orbit.

Instead, we will use Ligon-Schaaf regularization to transform the quantized energy calculation

on (N,ω) into one on an open submanifold of TS3. This is the subject of Section 5.3.

5.3 Ligon-Schaaf Regularization

Let T+S3 represent the result of removing the zero section from TS3. In Section 5.2.1, we noted

that the vector field ξH is not complete. The Ligon-Schaaf map is a symplectomorphism from

(N,ω) to an open submanifold of T+S3, having the property that ξH is mapped to a vector


field that is complete on T+S3. This map was presented in [15], and an in-depth discussion of

it can be found in [4].

In this section, we will state the Ligon-Schaaf map and list its relevant properties. Then we

will construct a metaplectic-c prequantization for TS3, and show how to lift the Ligon-Schaaf

map to the level of symplectic frame bundles, and of metaplectic-c prequantizations. Using

the lifted maps and the dynamical invariance property, we will be able to relate the quantized

energy levels of the hydrogen atom to those of a free particle on S3.

Our conventions and notation largely follow those in [4]. We state results without proof;

much more detail can be found in [4].

5.3.1 TS3 and the Ligon-Schaaf map

Consider TR4 = R4 × R4 with Cartesian coordinates (x, y), where x = (x1, x2, x3, x4) and

y = (y1, y2, y3, y4). Let TR4 have symplectic form

ω4 =

4∑j=1

dxj ∧ dyj ,

and let

β4 = −4∑j=1

yjdxj (5.3.1)

on TR4, so that dβ4 = ω4. The submanifold TS3 is given by

TS3 =

(x, y) ∈ TR4 : |x|2 = 1, x · y = 0⊂ TR4,

where we take the usual Euclidean inner product on R4.

A calculation shows that the restriction of ω4 to TS3 yields a symplectic form on TS3. Let

(TS3, ω3) be the resulting symplectic manifold. Let β3 be the restriction of β4 to TS3, so that

dβ3 = ω3.

Let T+S3 represent TS3 with the zero section removed:

T+S3 =

(x, y) ∈ TS3 : |y| > 0.


Define the map D : T+S3 → R by

D(x, y) = −mek2

2|y|2, ∀(x, y) ∈ T+S3.

As shown in [4], the Hamiltonian vector field for D on T+S3 ⊂ TR4 is

ξD =

4∑j=1

(mek

2

|y|4yj

∂

∂xj− mek

2

|y|2xj

∂

∂yj

),

and its flow is

ψtD(x, y) =

cos mek2

|y|3 t1|y| sin

mek2

|y|3 t

−|y| sin mek2

|y|3 t cos mek2

|y|3 t

x

y

.

The map D is called the Delaunay Hamiltonian, and ξD is the Delaunay vector field. Since the

orbits of ξD must preserve |y|, it is clear from the form of ψtD that ξD is complete on T+S3,

and every orbit is closed.

Now let S3n represent S3 with the north pole (0, 0, 0, 1) removed. The Ligon-Schaaf map

LS : N → T+S3n is given by

LS(q, p) = (A sinϕ+B cosϕ,−νA cosϕ+ νB sinϕ),

where

ν =

√− mek2

2H(q, p), ϕ =

1

ν(q · p),

A =

(q

|q|− 1

mek(q · p)p, 1

ν(q · p)

), B =

(1

ν|q|p, 1

mek|p|2|q| − 1

).

This map has the following properties:

(1) LS is a diffeomorphism between N and T+S3n;

(2) LS∗β3 =∑3

j=1 (qjdpj + d(qjpj)) = β (recall Equation (5.2.2));

(3) H = D LS.

Assertions (1) and (2) imply that LS is a symplectomorphism; from that and (3), it follows

that LS∗ξH = ξD.


5.3.2 Metaplectic-c prequantization for TS3

We begin by constructing a metaplectic-c prequantization for (TR4, ω4), and let it induce a

metaplectic-c prequantization for (TS3, ω3). For (TR4, ω4), we proceed in precisely the same

way as we constructed (P,Σ, γ) for (M,ω) in Section 5.2.2. Choose the model vector space

V4, and identify the symplectic frame bundle Sp(TR4, ω4) with TR4 × Sp(V4) using the global

trivialization for the tangent bundle given by

vj 7→∂

∂xj

∣∣∣∣(x,y)

, wj 7→∂

∂yj

∣∣∣∣(x,y)

, ∀(x, y) ∈ TR4, j = 1, . . . , 4.

Let Q4 = TR4 ×Mpc(V4) with bundle projection map Q4Π4−→ TR4. Define the map Γ4 : Q4 →

Sp(TR4, ω4) by

Γ4(x, y, a) = (x, y, σ(a)), ∀(x, y) ∈ TR4, ∀a ∈ Mpc(V4).

Lastly, define the u(1)-valued one-form δ4 on Q4 by

δ4 =1

i~Π∗4β4 +

1

2η∗ϑ0,

where ϑ0 is the trivial connection on Q4, and where β4 was defined in Equation (5.3.1). Then

(Q4,Γ4, δ4) is the unique metaplectic-c prequantization for (TR4, ω4) up to isomorphism.

In order to construct the metaplectic-c prequantization of (TS3, ω3) induced by (Q4,Γ4, δ4),

we proceed by the process of symplectic reduction. Let R : TR4 → R be given by

R(x, y) =1

2|x|2, ∀(x, y) ∈ TR4.

The Hamiltonian vector field for R on TR4 is

ξR = −4∑j=1

xj∂

∂yj,

and its flow on TR4 is

ψtR(x, y) = (x, y − tx), ∀(x, y) ∈ TR4.


It is straightforward to show that TS3 can be identified with the space of orbits of ξR on the

level set A = R−1(12). The orbit projection map p : A→ TS3 is

p(x, y) = (x, y − (x · y)x), ∀(x, y) ∈ A.

Let i : A→ TR4 be the inclusion map. Recall that ω3 is the symplectic form on TS3 obtained

by restricting ω4. A calculation shows that p∗ω3 = i∗ω4. Therefore (TS3, ω3) is identified with

the symplectic reduction of (TR4, ω4) at the level set corresponding to R = 12 .

Choose the model subspace W4 as described in Section 5.1.1, and model TzA ⊂ TzTR4 on

W4 ⊂ V4 for all z ∈ A. Following the procedure described in Section 4.2.2, we construct the

three-level structure

(QA4 , δA4 )

ΓA4−→ Sp(TR4, ω4;A)→ A

by restriction, then construct the three-level structure

(Q4A, δ4A)Γ4A−→ Sp(TA/TA⊥)→ A

by taking associated bundles. Then Sp(TA/TA⊥) is a principal Sp(V3) bundle over A, and Q4A

is a principal Mpc(V3) bundle over A.

Let the symplectic frame bundle Sp(TS3, ω3) be modeled on V3, so that Sp(TS3, ω3) is

a principal Sp(V3) bundle over TS3. Then Sp(TA/TA⊥) and Sp(TS3, ω3) have isomorphic

fibers. If we were to complete the symplectic reduction process, we would identify Sp(TS3, ω3)

with a quotient of Sp(TA/TA⊥). However, since TS3 ⊂ A, it is more convenient to identify

Sp(TS3, ω3) with a subbundle of Sp(TA/TA⊥).

For each z ∈ A, p∗|z appears in the short exact sequence

0→ TzA⊥ → TzA

p∗|z−→ Tp(z)TS3 → 0.

Note that the projection map p is the identity on TS3 ⊂ A. It follows that for all z ∈

TS3, p∗|z : TzA/TzA⊥ → TzTS

3 is a symplectic isomorphism. Therefore Sp(TS3, ω3) and

Sp(TA/TA⊥)|TS3 are isomorphic as principal Sp(V3) bundles over TS3. Using that isomor-


phism, we view Sp(TS3, ω3) as a subset of Sp(TA/TA⊥).

Let (Q,Γ, δ) be the result of restricting (Q4A,Γ4A, δ4A) to Sp(TS3, ω3) ⊂ Sp(TA/TA⊥).

Then (Q,Γ, δ) is the metaplectic-c prequantization for (TS3, ω3) (unique up to isomorphism).

We will also use the notation (Q,Γ, δ) to denote the metaplectic-c prequantizations for T+S3

and T+S3n obtained by restriction.

5.3.3 Lifting the Ligon-Schaaf map

Recall that the Ligon-Schaaf map LS : N → T+S3n is a symplectomorphism. Define the map

LS : Sp(N,ω)→ Sp(T+S3n, ω3) by

LS(b) = LS∗ b, ∀b ∈ Sp(N,ω),

and observe that this is an isomorphism of principal Sp(V3) bundles.

In Section 5.2.2, we used the global trivialization

vj 7→∂

∂qj

∣∣∣∣m

, wj 7→∂

∂pj

∣∣∣∣m

, ∀m ∈ N, j = 1, 2, 3,

to identify Sp(N,ω) with N × Sp(V3). From the map LS, we see that Sp(T+S3n, ω3) can be

identified with T+S3n × Sp(V3) with respect to the global trivialization

vj 7→ LS∗∂

∂qj

∣∣∣∣LS(m)

, wj 7→ LS∗∂

∂pj

∣∣∣∣LS(m)

, ∀LS(m) ∈ T+S3n, j = 1, 2, 3.

In terms of these trivializations for Sp(N,ω) and Sp(T+S3n, ω3), LS is given simply by

LS(m, g) = (LS(m), g), ∀m ∈ N, ∀g ∈ Sp(V3). (5.3.2)

As we have seen before, once we have a global trivialization for the symplectic frame bundle,

it is straightforward to construct a metaplectic-c prequantization. Let Q′ = T+S3n ×Mpc(V3)

with bundle projection map Q′Π′−→ T+S3

n. Define the map Γ′ : Q′ → Sp(T+S3n, ω3) by

Γ′(z, a) = (z, σ(a)), ∀z ∈ T+S3n, ∀a ∈ Mpc(V3).


Let

δ′ =1

i~Π′∗β3 +

1

2η∗ϑ0,

where β3 is the restriction to T+S3n of the one-form defined in Equation (5.3.1), and where ϑ0 is

the trivial connection on Q′. Then (Q′,Γ′, δ′) is a metaplectic-c prequantization for (T+S3n, ω3).

Recall that P = N ×Mpc(V3). Define LS : P → Q′ by

LS(m, a) = (LS(m), a).

This is clearly an isomorphism of principal Mpc(V3) bundles. Since LS∗β3 = β, we have

LS∗δ′ = γ. Lastly, it follows from Equation (5.3.2) that Γ′ LS = LS Σ. Therefore LS is an

isomorphism of metaplectic-c prequantizations.

All of these observations combine to yield the following commutative diagram.

(P, γ)

Σ

LS // (Q′, δ′)

Γ′

Sp(N,ω)

LS // Sp(T+S3n, ω3)

(N,ω)

LS // (T+S3n, ω3)

Each of the maps LS, LS, and LS is an isomorphism. From these isomorphisms and the fact

that D LS = H, it follows that E < 0 is a quantized energy level of (N,ω,H) if and only if

it is a quantized energy level of (T+S3n, ω3, D).

Recall that we constructed the metaplectic-c prequantization (Q,Γ, δ) for (T+S3n, ω3) in the

previous section. Since T+S3n is simply connected, (Q′,Γ′, δ′) must be isomorphic to (Q,Γ, δ).

Therefore we can calculate the quantized energy levels for the system (T+S3n, ω3, D) using

(Q,Γ, δ).

Moreover, we claim that E < 0 is a quantized energy level of (N,ω,H) if and only if it

is a quantized energy level of (T+S3, ω3, D). When we place the north pole back into S3,

we acquire more closed orbits: namely, those with x-components that pass through (0, 0, 0, 1).

However, due to the rotational symmetry of the system (Q,Γ, δ)→ (T+S3, ω3, D), an orbit that


passes through (0, 0, 0, 1) can always be transformed into one that does not, without altering

the holonomy condition. Thus the quantized energy levels of (T+S3n, ω3, D) and (T+S3, ω3, D)

are identical.

5.3.4 Rescaling the Delaunay Hamiltonian

So far, we have shown that the negative quantized energy levels of the hydrogen atom are the

same as the quantized energy levels of the Delaunay Hamiltonian on T+S3. In this section, we

will make one final transformation that relates these energies to the quantized energy levels of

a free particle on S3.

Let K : TS3 → R be given by

K(x, y) =1

2|y|2, ∀(x, y) ∈ TS3.

When this function is interpreted as a Hamiltonian energy function, it has a kinetic energy

term but no potential energy term: that is, it describes a free particle on S3. It is shown in [4]

that the corresponding Hamiltonian vector field on TS3 is

ξK =4∑j=1

(yj

∂

∂xj− |y|2xj

∂

∂yj

),

and the flow of this vector field is

ψtK(x, y) =

cos |y|t 1|y| sin |y|t

−|y| sin |y|t cos |y|t

x

y

, ∀(x, y) ∈ TS3 such that |y| > 0. (5.3.3)

If |y| = 0, then the particle is stationary, and the flow is simply ψtK(x, 0) = (x, 0). Clearly

K = 0 on the zero section of TS3, and K > 0 on T+S3.

The range of the Delaunay Hamiltonian D is R<0. Let f : R>0 → R<0 be given by

f(z) = −mek2

4z, ∀z ∈ R>0. (5.3.4)

Note that f is a diffeomorphism, and D = f K on T+S3. Using the dynamical invariance


property of quantized energy levels, we see that E is a quantized energy level of (T+S3, ω3, D)

if and only if f−1(E) is a positive quantized energy level of (TS3, ω3,K). It remains to find the

positive quantized energy levels for a free particle on S3.

5.4 Quantization of a Free Particle on S3

5.4.1 Orbits of ξK and local coordinates for TS3

Let E > 0 be arbitrary, and let S = K−1(E) ⊂ TS3. From the flow of ξK in Equation (5.3.3),

it is apparent that all orbits of ξK in S are closed, with period 2π√2E . Let z0 = (x0, y0) ∈ S be

an arbitrary initial point, and let C ⊂ S be the the orbit of ξK through z0:

C =ψtK(z0) : t ∈ R

⊂ TS3,

We can use the rotational symmetry of the system (Q,Γ, δ) → (TS3, ω3,K) to make some

simplifying assumptions about C. If we view x0 and y0 as two perpendicular vectors in R4, then

there is some rotation about the origin in R4 that carries them both to the x3x4-plane. By

performing this rotation in x-space and y-space, we can assume without loss of generality that

x0 and y0 take the form x0 = (0, 0, x30, x40) and y0 = (0, 0, y30, y40), where |x|2 = x230 +x2

40 = 1,

x · y = x30y30 + x40y40 = 0, and |y|2 = y230 + y2

40 = 2E . The orbit C then lies in x3x4y3y4-space.

Thus far, we have treated TS3 as a submanifold of TR4, using the coordinates (x, y) on TR4

to describe points in TS3. Now we make a local change of coordinates on TR4 that will yield

symplectic coordinates for TS3 on a neighborhood that contains C. Specifically, we introduce

4-dimensional spherical coordinates and their conjugate momenta. Let U ⊂ TR4 be the open

set

U =

(x, y) ∈ TR4 : x23 + x2

4 > 0.


On U , let the new spatial coordinates be (a, b, c, r), where

a = arctan

√x2

2 + x23 + x2

4

x1,

b = arctan

√x2

3 + x24

x2,

c = arctanx4

x3,

r =√x2

1 + x22 + x2

3 + x24.

The angles a and b are defined modulo π, and the angle c is defined modulo 2π. Let ρ1 =√x2

2 + x23 + x2

4 and ρ2 =√x2

3 + x24. The conjugate momenta corresponding to the spherical

coordinates are

pa =x1

ρ1(x2y2 + x3y3 + x4y4)− ρ1y1,

pb =x2

ρ2(x3y3 + x4y4)− ρ2y2,

pc =x3y4 − x4y3,

pr =1

r(x1y1 + x2y2 + x3y3 + x4y4).

Later, we will need the inverse transformations, which are

x1 = r cos a,

x2 = r sin a cos b,

x3 = r sin a sin b cos c,

x4 = r sin a sin b sin c,

(5.4.1)

y1 = pr cos a− par

sin a,

y2 = pr sin a cos b+par

cos a cos b− pbr sin a

sin b

y3 = pr sin a sin b cos c+par

cos a sin b cos c+pb

r sin acos b cos c− pc

r sin a sin bsin c,

y4 = pr sin a sin b sin c+par

cos a sin b sin c+pb

r sin acos b sin c+

pcr sin a sin b

cos c.

(5.4.2)

For convenience, we let aj , j = 1, . . . , 4, range over a, b, c, r.


On U , one can verify that

β4 = −4∑j=1

yjdxj = −4∑j=1

pajdaj , (5.4.3)

and so

ω4 =4∑j=1

dxj ∧ dyj =4∑j=1

daj ∧ dpj .

The submanifold TS3 is characterized by the constant values r = 1 and pr = 0, which implies

that the restrictions to TS3 ∩ U of β4 and ω4 are

β3 = −3∑j=1

pajdaj , ω3 =3∑j=1

daj ∧ dpaj .

Thus (a, b, c, pa, pb, pc) are symplectic coordinates for TS3 ∩U . On this neighborhood, the map

K takes the form

K =1

2

(p2a +

p2b

sin2 a+

p2c

sin2 a sin2 b

).

Several times now, once we had a set of symplectic coordinates such as (a, b, c, r, pa, pb, pc, pr),

we used the trivialization of the symplectic frame bundle given by the coordinate vector fields to

construct a metaplectic-c prequantization. We could apply this same procedure to U ; however,

since U is not simply connected, it is not necessarily the case that the metaplectic-c prequanti-

zation so constructed would be isomorphic to the result of restricting (Q4,Γ4, δ4) to U . Instead,

we must show how the local change of variables from Cartesian to spherical coordinates can be

lifted from TR4 to Q4. This is the subject of the next section.

5.4.2 Change of variables over C

Recall that the integral curve of ξK through the initial point z0 = (x0, y0) is

ψtK(x0, y0) =

cos√

2Et 1√2E sin

√2Et

−√

2E sin√

2Et cos√

2Et

x0

y0

,

and its image C is a closed curve lying in x3x4y3y4-space. Since C ⊂ U , the points in C can

be rewritten in spherical coordinates. Upon converting, we find that any point in C satisfies


pa = pb = 0 and a = b = π2 . Further, pc is a constant value over C satisfying p2

c = 2E . Since

C ⊂ TS3, we also have r = 1 and pr = 0. Therefore, in spherical coordinates, the orbit takes

the form

C =(π

2,π

2, c, 1, 0, 0, pc, 0

): c ∈ R/2πZ

.

Let z(c) represent the point(π2 ,

π2 , c, 1, 0, 0, pc

)∈ C.

The change of coordinates from Cartesian to spherical must now be lifted to the symplectic

frame bundle and the metaplectic-c prequantization for (TR4, ω4). The change of coordinates

on Sp(TR4, ω4) will take place over U , and that on Q4 will take place over C. We follow the

same steps that we applied to the harmonic oscillator in Section 4.4.2.

On the neighborhood U , we have two different coordinate maps: the Cartesian map Φc :

U → R8, and the spherical map Φs : U → R4× (R/πZ)2×R/2πZ×R. The change of variables

on U is simply the transition map F = Φs Φ−1c . Let Φc(U) = Uc and Φs(U) = Us. Then each

of the following maps is a diffeomorphism.

UΦc

~~

Φs

Uc

F // Us

Let bc be the section of Sp(TR4, ω4) over U given by

bc(z) : V4 → TzTR4 such that vj 7→∂

∂xj

∣∣∣∣z

, wj 7→∂

∂yj

∣∣∣∣z

, ∀z ∈ U, j = 1, . . . , 4.

That is, bc is the section that defines the trivialization of Sp(TR4, ω4)|U with respect to Cartesian

coordinates. Let the map Φc : Sp(TR4, ω)|U → Uc × Sp(V4) be given by

Φc(bc(z) · g) = (Φc(z), g), ∀z ∈ U, ∀g ∈ Sp(V4).

Similarly, let bs be the section of Sp(TR4, ω4)|U given by

bs(z) : V4 → TzTR4 such that vj 7→∂

∂aj

∣∣∣∣z

, wj 7→∂

∂paj

∣∣∣∣z

, ∀z ∈ U, j = 1, . . . , 4,


and define the map Φs : Sp(TR4, ω)|U → Us × Sp(V4) by

Φs(bs(z) · g) = (Φs(z), g), ∀z ∈ U, ∀g ∈ Sp(V4).

To perform the change of coordinates on the level of the symplectic frame bundle, we must

lift F to a map F : Uc × Sp(V4) → Us × Sp(V4) in such a way that the following diagram

commutes.

Sp(TR4, ω)|UΦc

vv

Φs

((Uc × Sp(V4)

F // Us × Sp(V4)

In Equations (5.4.1) and (5.4.2), we gave explicit formulas for xj and yj in terms of ak and

pak . At each z ∈ U , let G(z) be the 8 × 8 matrix consisting of the partial derivatives of the

Cartesian coordinates with respect to the spherical ones:

G(z) =

∂xk∂aj

∣∣∣z

∂yk∂aj

∣∣∣z

∂xk∂paj

∣∣∣z

∂yk∂paj

∣∣∣z

1≤j,k≤4

.

Then

G(z)

∂∂xk

∣∣∣z

∂∂yk

∣∣∣z

1≤k≤4

=

∂∂aj

∣∣∣z

∂∂paj

∣∣∣z

1≤j≤4

, ∀z ∈ U.

Since the Cartesian and spherical coordinate vectors are both symplectic bases for TzTR4, G(z)

is a symplectic matrix and can be treated as an element of Sp(V4). By an identical argument

to that in Section 4.4.2, the desired map F is given by

F (Φc(z), g) = (Φs(z), G(z)g), ∀z ∈ U, ∀g ∈ Sp(V4).

The final lift to the metaplectic-c prequantization will take place over C. Let Cc and Cs be

the images of C under Φc and Φs, respectively. Recall that points in C are denoted by z(c) with

c ∈ R/2πZ. We write G(c) for G(z(c)). The components of G(c) are single-valued with respect

to c, so G(c) is a closed path through Sp(V4).

It is clear from the construction of Q4 how to define a local trivialization with respect to the


Cartesian coordinates. Let Φc : Q4|C → Cc ×Mpc(V4) be given by Φc(z(c), a) = (Φc(z(c)), a)

for all z(c) ∈ C and a ∈ Mpc(V4). Then Φc and Φc make the following diagram commute.

Q4|CΓ4 //

Φc

Sp(TR4, ω4)|C

Φc

Cc ×Mpc(V4)σ // Cc × Sp(V4)

We require a local trivialization of Q4 with respect to the spherical coordinates that has the

analogous relationship to Φs. To find the appropriate map Φs : Q4|C → Cs ×Mpc(V4), we will

construct a map F that satisfies the diagram shown below, then define Φs = F Φc.

Q4|C

Φc

Γ4

**Cc ×Mpc(V4)

F //

σ

**

Cs ×Mpc(V4)σ

**

Sp(TR4, ω4)|C

Φc

Φs

Cc × Sp(V4)

F // Cs × Sp(V4)

As before, if F exists, then it has the form

F (Φc(z(c)), a) = (Φs(z(c)), G(c)a), ∀z(c) ∈ C, ∀a ∈ Mpc(V4),

where G(c) ∈ Mpc(V4) and σ(G(c)) = G(c) for all c. To preserve the form of δ4, we lift the

path G(c) to a path in Mp(V4), then check whether G(c) is single-valued with respect to c.

If G(c) ∈ Mp(V4), then its parameters (recall Section 2.3.1) take the form (G(c), µ(c)), where

µ(c)2DetCCG(c) = 1 and CG(c) = 12(G(c)−JG(c)J). To determine µ(c), we must calculate G(c).

Using the expressions in Equations (5.4.1) and (5.4.2), we compute the partial derivatives that

form the matrix G, and evaluate at the point z(c) =(π2 ,

π2 , c, 1, 0, 0, pc, 0

)∈ Cs. Using the


abbreviations S(c) = sin c and C(c) = cos c, the result is

G(c) =

−1 0 0 0 0 0 0 0

0 −1 0 0 0 0 0 0

0 0 −S(c) C(c) 0 0 −pcC(c) −pcS(c)

0 0 C(c) S(c) 0 0 pcS(c) −pcC(c)

0 0 0 0 −1 0 0 0

0 0 0 0 0 −1 0 0

0 0 0 0 0 0 −S(c) C(c)

0 0 0 0 0 0 C(c) S(c)

.

Next, we evaluate CG(c), using the matrix form for J noted in Section 5.1.1, and convert it

to a 4× 4 complex matrix. We find that

CG(c) =

−1 0 0 0

0 −1 0 0

0 0 −S(c)− i2pcC(c) C(c)− i

2pcS(c)

0 0 C(c) + i2pcS(c) S(c)− i

2pcC(c)

,

which has complex determinant

DetCCG(c) = −1− 1

4p2c .

This value is real and constant over C. Thus we can define G(c) to be the element of Mp(V4) ⊂

Mpc(V4) with parameters(G(c), i

(1 + 1

4p2c

)−1/2)

, for all c ∈ R/2πZ, whereupon G(c) is the

desired closed path through Mp(V4).

Having determined G(c), we now define the map F : Cc ×Mpc(V4)→ Cs ×Mpc(V4) by

F (Φc(z(c)), a) = (Φs(z(c)), G(c)a) ∀z(c) ∈ C, ∀a ∈ Mpc(V4).

The local trivialization of Q4|C that is compatible with spherical coordinates comes about by

setting Φs = F Φc. The one-form δ4 on Q4 induces a one-form δ4s on Cs×Mpc(V4) that takes


the form

δ4s =1

i~Π∗β4 +

1

2η∗ϑ0,

where β4 is written in spherical coordinates as in Equation (5.4.3), and where ϑ0 is now the

trivial connection on Cs ×Mpc(V4).

5.4.3 Restrictions to C ⊂ TS3

From now on, we no longer write the maps Φs, Φs and Φs explicitly, but write elements of

TR4, Sp(TR4, ω4)|U and Q4|C with respect to spherical coordinates and the local spherical

trivializations. Further, we treat TS3 as a six-dimensional manifold with local coordinates

(a, b, c, pa, pb, pc) on TS3 ∩ U , and with symplectic form ω3 =∑3

j=1 daj ∧ dpaj .

Recall from Section 5.3.2 that we defined the map R(x, y) = 12 |x|

2 on TR4, and set

A = R−1(12). We argued that the symplectic frame bundle Sp(TS3, ω3) is isomorphic to

Sp(TA/TA⊥)|TS3 , and that the metaplectic-c prequantization (Q,Γ, δ) for (TS3, ω3) is ob-

tained by restricting (Q4A,Γ4A, δ4A) to Sp(TA/TA⊥)|TS3 .

In spherical coordinates, we have

R(z) =1

2r2, ∀z ∈ TR4 ∩ U,

which has Hamiltonian vector field

ξR = r∂

∂pr.

It is immediate that at each point z ∈ A ∩ U ,

TzA = span

∂

∂a,∂

∂b,∂

∂c,∂

∂pa,∂

∂pb,∂

∂pc,∂

∂pr

∣∣∣∣z

and TzA⊥ = span

∂

∂pr

∣∣∣∣z

,

which implies that

TzA/TzA⊥ = span

[∂

∂a

],

[∂

∂b

],

[∂

∂c

],

[∂

∂pa

],

[∂

∂pb

],

[∂

∂pc

]∣∣∣∣z

.

We see that over A ∩ U , the local trivialization of Sp(TR4, ω4)|U with respect to spherical


coordinates induces a local trivialization

Sp(TR4, ω4;A)|A∩U = A ∩ U × Sp(V4;W4).

The group homomorphism ν : Sp(V4;W4)→ Sp(V3) then induces a local trivialization

Sp(TA/TA⊥)|A∩U = A ∩ U × Sp(V3).

Restricting further to TS3 ∩ U ⊂ A ∩ U , we find that

Sp(TS3, ω3)|TS3∩U = TS3 ∩ U × Sp(V3),

with the local trivialization given by

vj 7→∂

∂aj

∣∣∣∣z

, wj 7→∂

∂paj

∣∣∣∣z

, ∀z ∈ TS3 ∩ U, j = 1, 2, 3.

On the level of metaplectic-c bundles, we can write

QA4 |C = C ×Mpc(V4;W4)

by restricting Q4|C to Sp(TR4, ω;A)|C . Then the group homomorphism ν : Mpc(V4;W4) →

Mpc(V3) induces the local trivialization

Q4A|C = Q|C = C ×Mpc(V3).

The prequantization one-form on Q is

δ =1

i~Π∗β3 +

1

2η∗ϑ0,

where ϑ0 is the trivial connection on C ×Mpc(V3), and where

β3 = −3∑j=1

pjdaj = −pcdc


over C.

5.4.4 Quantized energy levels for (TS3, ω3, K)

In the local coordinates (a, b, c, pa, pb, pc), the map K : TS3 → R is given by

K =1

2

(p2a +

p2b

sin2 a+

p2c

sin2 a sin2 b

),

and its Hamiltonian vector field is

ξK = pa∂

∂a+

pb

sin2 a

∂

∂b+

pc

sin2 a sin2 b

∂

∂c+

(p2b

sin3 acos a+

p2c cos a

sin3 a sin2 b

)∂

∂pa+

p2c cos b

sin2 a sin3 b

∂

∂pb.

Recall that we fixed E > 0 and defined S = K−1(E), and that z0 is a point in S. The

closed curve C is the orbit of ξK through z0. We will show that the orbit of ξK through

(z0, I) ∈ Sp(TS/TS⊥) is closed, and then we will determine the values of E for which the orbit

of ξK through (z0, I) ∈ QS is also closed.

The lift of the flow ψtK to ψtK on Sp(TS3, ω3) is given by

ψtK(z, I) = (z, ψtK∗|z), ∀z ∈ TS3 ∩ U,

which implies that the lifted vector field ξK is

ξK(z, I) =

(ξK(z),

d

dt

∣∣∣∣t=0

ψtK∗|z), ∀z ∈ TS3 ∩ U.

As a 6× 6 matrix, ddt

∣∣t=0

ψtK∗|z can be interpreted as an element of the Lie algebra sp(V3), and

its components are (d

dt

∣∣∣∣t=0

ψt∗|z)jk

=∂(ξK)j∂Xk

∣∣∣∣X=z

,

where Xk ranges over (a, b, c, pa, pb, pc). In particular, when we evaluate this matrix of partial


derivatives at z(c) ∈ C, we obtain

d

dt

∣∣∣∣t=0

ψtK∗|z(c) =

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

−p2c 0 0 0 0 0

0 −p2c 0 0 0 0

0 0 0 0 0 0

.

This value is constant over C; we denote it by κ. Thus

ξK(z(c), I) = (ξK(z(c)), κ), ∀z(c) ∈ C,

which implies that the flow of ξK through (z0, I) ∈ Sp(TS3, ω3) is

ψtK(z0, I) = (ψtK(z0), exp(tκ)).

Let λ =√p2c =√

2E . A calculation shows that

exp(tκ) =

cosλt 0 0 1λ sinλt 0 0

0 cosλt 0 0 1λ sinλt 0

0 0 1 0 0 0

−λ sinλt 0 0 cosλt 0 0

0 −λ sinλt 0 0 cosλt 0

0 0 0 0 0 1

.

Thus the orbit through (z0, I) is closed, with period 2πλ .

Over C, ξK reduces to

ξK = pc∂

∂c.


Therefore, for all z(c) ∈ C, we have

Tz(c)S⊥ = span

∂

∂c

∣∣∣∣z(c)

, Tz(c)S = span

∂

∂a,∂

∂b,∂

∂c,∂

∂pa,∂

∂pb

∣∣∣∣z(c)

,

and

Tz(c)S/Tz(c)S⊥ = span

[∂

∂a

],

[∂

∂b

],

[∂

∂pa

],

[∂

∂pb

]∣∣∣∣z(c)

.

We identify Sp(TS/TS⊥)|C with C × Sp(V2) in the obvious way.

Upon descending to Sp(TS/TS⊥)|C , the induced flow takes the form

ψtK(z0, I) = (ψtK(z0), ν(exp(tκ))),

where ν is the group homomorphism Sp(V3;W3)ν−→ Sp(V2). We calculate that

ν(exp(tκ)) =

cosλt 0 1λ sinλt 0

0 cosλt 0 1λ sinλt

−λ sinλt 0 cosλt 0

0 −λ sinλt 0 cosλt

.

The corresponding vector field on Sp(TS/TS⊥)|C is

ξK(z(c), I) = (ξK(z(c)), κ), ∀z(c) ∈ C,

where

κ =d

dt

∣∣∣∣t=0

ν(exp(tκ)) =

0 0 1 0

0 0 0 1

−λ2 0 0 0

0 −λ2 0 0

∈ sp(V2).

The local trivializations lift to the level of metaplectic-c bundles, so thatQS |C = C×Mpc(V2).

The induced one-form δS on QS takes the form

δS = − 1

i~pcdc+

1

2η∗ϑ0


over C. Therefore the horizontal lift of ξK to QS |C is

ξK(z(c), I) =

(ξK(z(c)), κ⊕ λ2

i~

), ∀z(c) ∈ C.

Since the mpc(V2) component is constant, the orbit of ξK through (z0, I) ∈ QS is

ψtK(z0, I) =

(ψtK(z0), exp

(tκ⊕ λ2t

i~

)).

We can write the Mpc(V2) component as exp(tκ ⊕ 0)eλ2t/i~, where exp(tκ ⊕ 0) ∈ Mp(V2) ⊂

Mpc(V2) and eλ2t/i~ ∈ U(1) ⊂ Mpc(V2).

The parameters of exp(tκ⊕0) have the form (exp tκ, µ(t)) where µ(t)2DetCCexp(tκ) = 1. As

a 2× 2 complex matrix,

Cexp(tκ) =

cosλt+ i2

(λ+ 1

λ

)sinλt 0

0 cosλt+ i2

(λ+ 1

λ

)sinλt

,

which has determinant

DetCCexp(tκ) =

(cosλt+

i

2

(λ+

1

λ

)sinλt

)2

.

This value circles the origin twice as t ranges from 0 to 2π/λ. Therefore the parameters of

exp(tκ⊕ 0) are (exp(tκ),

(cosλt+

i

2

(λ+

1

λ

)sinλt

)−1),

which describes a closed orbit in Mp(V2) with period 2π/λ.

The orbit in QS is closed if and only if the U(1) term is also a closed orbit with period

2π/λ. We require eλ2t/i~ = 1 when t = 2π/λ, or equivalently, λ2

i~2πλ = −2πin for some n ∈ Z.

Rearranging and recalling that λ2 = p2c = 2E results in E = 1

2n2~2 for some n ∈ Z.

We now apply the argument made in Section 5.3.4. We are only concerned with the strictly

positive quantized energies ofK, because those correspond to quantized energies of the Delaunay

Hamiltonian D on T+S3. Thus we can ignore the energy level corresponding to n = 0, and we


can also dismiss n < 0 as redundant. This leaves us with

En =1

2n2~2, n ∈ N.

The positive value En is a quantized energy level of K if and only if f(En) is a quantized energy

level of D, where the diffeomorphism f is given in Equation (5.3.4), and the quantized energy

levels of D are precisely the negative quantized energy levels of the hydrogen atom. At last, we

find that the negative quantized energy levels of the hydrogen atom are

En = f(En) = −mek2

2n2~2, n ∈ N,

which is exactly the quantum mechanical prediction for our model of the hydrogen atom.

Chapter 6

Generalization of the Quantized

Energy Condition

The definition of a quantized energy level that was presented in Chapter 4 has a natural gen-

eralization from a regular value of a single function to that of a family of k Poisson-commuting

functions. In this chapter, we present the generalized definition, and prove the corresponding

generalized version of the dynamical invariance theorem. When the family of functions is of

maximal size, we show that the quantization condition reduces to a Bohr-Sommerfeld condition.

A comment concerning notation: in this chapter, superscripts j, l range over elements of the

Poisson-commuting family, while subscripts a, c range over components with respect to local

coordinates.

6.1 The Quantization Condition

Let (M,ω) be a 2n-dimensional symplectic manifold, and let H1, . . . ,Hk be a family of Poisson-

commuting functions, where 1 ≤ k ≤ n. Then H = (H1, . . . ,Hk) is a map from M to Rk. Let

E = (E1, . . . , Ek) be a regular value of H, and let S = H−1(E).

Let the Hamiltonian vector field for the function Hj be denoted by ξj , 1 ≤ j ≤ k. Note

that because E is a regular value of H, the Hamiltonian vector fields ξ1, . . . , ξk are linearly

independent everywhere on S. Further, for any s ∈ S, TsS⊥ = span

ξ1(s), . . . , ξk(s)

⊂ TsS,

which shows that TsS is a coisotropic subspace of TsM .

92

Chapter 6. Generalization of the Quantized Energy Condition 93

Let (V,Ω) be a 2n-dimensional model symplectic vector space, and let W ⊂ V be a

coisotropic subspace of codimension k. Then W/W⊥ acquires a symplectic structure from

Ω. We now perform an identical construction to that given in Section 4.2.1. Let Sp(V ;W ) be

the subgroup of Sp(V ) that preserves W , and let Mpc(V ;W ) be the preimage of Sp(V ;W ) in

Mpc(V ). Robinson and Rawnsley [17] showed that there is a lift of the natural group homo-

morphism Sp(V ;W )ν−→ Sp(W/W⊥) to the level of metaplectic-c groups.

Mpc(V ) ⊃ Mpc(V ;W )ν //

σ

Mpc(W/W⊥)

σ

Sp(V ) ⊃ Sp(V ;W )ν // Sp(W/W⊥)

Let (P,Σ, γ) be a metaplectic-c prequantization for (M,ω). By exactly the same process as

in Section 4.2.2, we construct the following three-level structures.

(P, γ)

Σ

(PS , γS)incl.oo

ν // (PS , γS)

Sp(M,ω)

ρ

Sp(M,ω;S)incl.oo

ν // Sp(TS/TS⊥)

(M,ω) S

incl.oo = // S

The second column is obtained by restriction, and the third column is obtained by taking

associated bundles. The Hamiltonian vector fields ξ1, . . . , ξk lift to ξ1, . . . , ξk on the level of

symplectic frame bundles, and to ξ1, . . . , ξk on the level of metaplectic-c bundles.

With these constructions established, we can now state the generalized version of our quan-

tized energy condition.

Definition 6.1.1. Let E = (E1, . . . , Ek) ∈ Rk be a regular value for the family H = (H1, . . . ,

Hk) of Poisson-commuting functions on M . If the one-form γS has trivial holonomy over any

closed path u(t) in Sp(TS/TS⊥) that satisfies u(t) ∈ spanξ1(u(t)), . . . , ξk(u(t))

for all t,

then the value E is a quantized energy level for the system (M,ω,H).

In Section 6.2, we state and prove a generalized version of the dynamical invariance theorem.

The proof follows essentially the same steps as in Section 4.3.2, and as such, we omit some of the

details of the calculations. Then, in Section 6.3, we examine the special case of a completely


integrable system. When k = n, we will see that the quantization condition reduces to a

Bohr-Sommerfeld condition.

6.2 Generalized Dynamical Invariance

Let H = (H1, . . . ,Hk) and L = (L1, . . . , Lk) be two families of Poisson commuting functions

on M . Suppose E = (E1, . . . , Ek) and F = (F 1, . . . , F k) are regular values of H and L,

respectively, such that H−1(E) = L−1(F ). We will show that the quantization condition is

identical for H at E and for L at F .

Let ξ1, . . . , ξk be the Hamiltonian vector fields corresponding to H1, . . . ,Hk, and let these

vector fields have flows φ1t, . . . , φkt. Similarly, let η1, . . . , ηk be the Hamiltonian vector fields

corresponding to L1, . . . , Lk, and let these vector fields have flows ρ1t, . . . , ρkt.

Let S = H−1(E) = L−1(F ). Then S is a (2n − k)-dimensional submanifold of M . For all

s ∈ S, TsS⊥ is a k-dimensional subspace of TsS, and

TsS⊥ = span

ξ1(s), . . . , ξk(s)

= span

η1(s), . . . , ηk(s)

.

The above relationship implies that there is a matrix-valued function C : S → Rk×k such that

ηj(s) =k∑l=1

Cjl(s)ξl(s), ∀s ∈ S.

For any 1 ≤ j ≤ k, ξj(s) and ηj(s) are elements of TsS, which is naturally identified with

TζTsS for any ζ ∈ TsS. Since TζTsS ⊂ TζTS, we can think of ξj(s) or ηj(s) as an element of

TζTS. The following lemma is stated in terms of that identification.

Lemma 6.2.1. For all s ∈ S and all ζ ∈ TsS,

d

dt

∣∣∣∣t=0

(ρjt∗ |sζ

)=

k∑l=1

(Cjl(s)

d

dt

∣∣∣∣t=0

(φlt∗ |sζ

)+(ζCjl

)ξl(s)

),

for any 1 ≤ j ≤ k.

Proof. We work on the (2n− k)-dimensional manifold S. Let U ⊂ S be a coordinate neighbor-

hood for s ∈ S, and let X = (X1, . . . , X2n−k) be local coordinates on U . With respect to these


coordinates, we write φjt = (φjt1 , . . . , φjt2n−k) near s, and likewise for ρjt.

Since ηj(s) =∑k

l=1Cjl(s)ξl(s) on S, we have ηja(s) =

∑kl=1C

jl(s)ξla(s) on U for each

1 ≤ a ≤ 2n− k, and so

d

dt

∣∣∣∣t=0

ρjta (s) =

k∑l=1

Cjl(s)d

dt

∣∣∣∣t=0

φlta (s)

on U .

The pushforward φjt∗ |s is a (2n − k) × (2n − k) matrix with entries given by(φjt∗ |s

)ac

=

∂φjta∂Xc

∣∣∣X=s

. For any ζ ∈ TsS,(φjt∗ |sζ

)a

= ζφjta . The same relationship holds for ρjt∗ |sζ. An

analogous calculation to that in the proof of Lemma 4.3.3 yields

d

dt

∣∣∣∣t=0

(ρjt∗ |sζ

)a

=

k∑l=1

(Cjl(s)

d

dt

∣∣∣∣t=0

(φlt∗ |sζ

)a

+(ζCjl

)ξla(s)

), (6.2.1)

for 1 ≤ j ≤ k and 1 ≤ a ≤ 2n− k.

Given the coordinates X on U , we get natural coordinates for TS|U , and thence coordinates

for TζTS. In terms of these,

d

dt

∣∣∣∣t=0

(ρjt∗ |sζ

)=

(ηj1(s), . . . , ηj2n−k(s),

d

dt

∣∣∣∣t=0

(ρjt∗ |sζ

)1, . . . ,

d

dt

∣∣∣∣t=0

(ρjt∗ |sζ

)2n−k

),

d

dt

∣∣∣∣t=0

(φjt∗ |sζ

)=

(ξj1(s), . . . , ξj2n−k(s),

d

dt

∣∣∣∣t=0

(φjt∗ |sζ

)1, . . . ,

d

dt

∣∣∣∣t=0

(φjt∗ |sζ

)2n−k

),

ξl(s) =(

0, . . . , 0, ξl(s)1, . . . , ξl(s)2n−k

).

If we apply Equation (6.2.1) in the first line and compare with the second two lines, recalling

that ηj(s) =∑k

l=1Cjl(s)ξl(s), we conclude that

d

dt

∣∣∣∣t=0

(ρjt∗ |sζ

)=

k∑l=1

(Cjl(s)

d

dt

∣∣∣∣t=0

(φlt∗ |sζ

)+(ζCjl

)ξl(s)

).

Now we use Lemma 6.2.1 to establish a relationship between the vector fields η1, . . . ηk and

ξ1, . . . , ξk on Sp(TS/TS⊥).


Lemma 6.2.2. For all s ∈ S and all b′ ∈ Sp(TS/TS⊥)s,

ηj(b′) =k∑l=1

Cjl(s)ξl(b′), 1 ≤ j ≤ k.

Proof. For V , choose a symplectic basis x1, . . . , xn, y1, . . . , yn. Let

W = span x1, . . . , xn, yk+1, . . . , yn .

Consequently,

W⊥ = span x1, . . . , xk , W/W⊥ = span [xk+1], . . . , [xn], [yk+1], . . . , [yn] .

For convenience, let

(z1, . . . , z2n) = (x1, . . . , xn, yk+1, . . . , yn, y1, . . . , yk)

so that

W = span z1, . . . , z2n−k , W⊥ = span z1, . . . , zk , W/W⊥ = span [zk+1], . . . , [z2n−k] .

For any m ∈ M and any b ∈ Sp(M,ω)m, identify b with the 2n-tuple (ζ1, . . . , ζ2n) ∈

(TmM)2n where ζc = bzc for 1 ≤ c ≤ 2n. Similarly, for any s ∈ S and any b′ ∈ Sp(TS/TS⊥)s,

identify b′ with the (2n− 2k)-tuple ([ζk+1], . . . , [ζ2n−k]) ∈(TsS/TsS

⊥)2n−2k, where [ζc] = b′[zc]

for k + 1 ≤ c ≤ 2n− k.

Let s ∈ S and b ∈ Sp(M,ω;S)s be arbitrary, and let ζc = bzc for 1 ≤ c ≤ 2n. Note that

ζc ∈ TsS for all 1 ≤ c ≤ 2n−k and ζc ∈ TsS⊥ for all 1 ≤ c ≤ k. The lifted flows ρjt on Sp(M,ω)

act on b ∈ Sp(M,ω;S) by

ρjt(b) = ρjt∗ |s b =(ρjt∗ |sζ1, . . . ρ

jt∗ |sζ2n

).

The element b descends to the element b′ ∈ Sp(TS/TS⊥) that is identified with ([ζk+1], . . . ,


[ζ2n−k]) ∈(TsS/TsS

⊥)2n−2k. The induced flows ρjt on Sp(TS/TS⊥) act on b′ by

ρjt(b′) =([ρjt∗ |sζk+1], . . . , [ρjt∗ |sζ2n−k]

),

which is a path in (TS/TS⊥)2n−2k. If we take the time derivative, we get

ηj(b′) =d

dt

∣∣∣∣t=0

ρjt(b′) =

(d

dt

∣∣∣∣t=0

[ρjt∗ |sζk+1], . . . ,d

dt

∣∣∣∣t=0

[ρjt∗ |sζ2n−k]

)∈ T[ζk+1](TS/TS

⊥)× . . .× T[ζ2n−k](TS/TS⊥).

The pushforward of the projection map TS → TS/TS⊥, based at ζc, is a linear surjection

TζcTS → T[ζc](TS/TS⊥), with kernel equal to the tangent space TζcTsS

⊥. Identifying this

tangent space with the vector space TsS⊥ yields a natural identification between T[ζc](TS/TS

⊥)

and TζcTS/TsS⊥. Using this, we get

ηj(b′) =

([d

dt

∣∣∣∣t=0

ρjt∗ |sζk+1

], . . . ,

[d

dt

∣∣∣∣t=0

ρjt∗ |sζ2n−k

])∈ Tζk+1

TS/TsS⊥ × . . .× Tζ2n−kTS/TsS

⊥.

Now, since ζc ∈ TsS for all k + 1 ≤ c ≤ 2n− k, Lemma 6.2.1 applies and we have

d

dt

∣∣∣∣t=0

ρjt∗ |sζc =

k∑l=1

(Cjl(s)

d

dt

∣∣∣∣t=0

φlt∗ |sζc + (ζcCjl)ξl(s)

)

for all c. Recall that ξl(s) ∈ TsS⊥ for all 1 ≤ l ≤ k. Therefore, upon taking equivalence classes

with respect to the quotient by TsS⊥, the terms of the form (ζcC

jl)ξl(s) all vanish and we are

left with [d

dt

∣∣∣∣t=0

ρjt∗ |sζc]

=

[k∑l=1

Cjl(s)d

dt

∣∣∣∣t=0

φjt∗ |sζc

], k + 1 ≤ c ≤ 2n− k.

It follows that

ηj(b′) =

k∑l=1

Cjl(s)ξl(b′),

as desired.

We can now prove the generalized version of the dynamical invariance theorem.


Theorem 6.2.3. If H = (H1, . . . ,Hk) and L = (L1, . . . , Lk) are two families of Poisson-

commuting functions on M such that H−1(E) = L−1(F ) for some regular values E of H and

F of L, then E is a quantized energy level for (M,ω,H) if and only if F is a quantized energy

level for (M,ω,L).

Proof. From Lemma 6.2.2, each of the Hamiltonian vector fields η1, . . . , ηk on Sp(TS/TS⊥)

is a linear combination of the vector fields ξ1, . . . , ξk. Therefore, if there is a closed curve in

Sp(TS/TS⊥) whose tangent at every point is in the subspace spanned byη1, . . . , ηk

, then

that tangent is also in the subspace spanned byξ1, . . . , ξk

, and vice versa. The desired result

follows from our definition of a quantized energy level.

6.3 Completely Integrable Systems and Bohr-Sommerfeld Con-

ditions

In this section, we consider the special case in which the Poisson-commuting family is of max-

imal size: namely, k = n. As we will show, the quantization condition simplifies to a Bohr-

Sommerfeld condition in this case. First, we briefly review Bohr-Sommerfeld conditions in the

context of Kostant-Souriau quantization. Our summary is based on the more detailed treat-

ments available in [12, 20, 21, 26].

6.3.1 Kostant-Souriau quantization and Bohr-Sommerfeld conditions

A completely integrable system on the 2n-dimensional manifold (M,ω) is a family of n Poisson-

commuting functions H = (H1, . . . ,Hn) whose differentials are linearly independent almost

everywhere. Given a completely integrable system, outside a closed set of measure zero, the

corresponding Hamiltonian vector fields ξ1, . . . , ξn span a real polarization: that is, an invo-

lutive, Lagrangian subbundle of the tangent bundle. A regular leaf of the polarization is a

Lagrangian submanifold of M .

Assume that (M,ω) is equipped with a Kostant-Souriau prequantization line bundle (L,∇).

A leaf of a real polarization F is called Bohr-Sommerfeld provided there exists a global, non-

vanishing section of L over S that is horizontal in the directions of F , with respect to the


connection ∇. Equivalently, a leaf S is Bohr-Sommerfeld if the connection has trivial holon-

omy over every closed curve in S. Since the connection is flat over a Lagrangian submanifold,

homotopic curves have equal holonomy. Therefore it suffices to check the holonomy of ∇ over

the generators of the fundamental group π1(S).

Now assume that (M,ω) also admits a metaplectic structure, which implies that we can

construct the half-form bundle ∧1/2F . Then we modify the Bohr-Sommerfeld condition so that

the Bohr-Sommerfeld leaves are those over which there is a horizontal section of L⊗∧1/2F . This

corrected Bohr-Sommerfeld condition is necessary in order to obtain the quantum mechanical

energy levels for the system of n independent one-dimensional harmonic oscillators [21]. In

the next sections, we will show that our quantization condition reduces to a Bohr-Sommerfeld

condition when k = n, and specifically to one that replicates the half-form correction for the

system of harmonic oscillators.

6.3.2 Quantized energy condition for k = n

Let H = (H1, . . . ,Hn) be a family of n Poisson-commuting functions on M , with Hamiltonian

vector fields ξ1, . . . , ξn. As usual, let E = (E1, . . . , En) be a regular value of H, and let S =

H−1(E). The crucial observation is that for all s ∈ S, TsS⊥ = TsS = span

ξ1(s), . . . , ξn(s)

,

which implies that Sp(TS/TS⊥) has trivial fiber and can be identified with S. The bundle

(PS , γS) is a principal circle bundle with connection one-form over Sp(TS/TS⊥), and can now

be viewed as a principal circle bundle with connection one-form over S. Since S ⊂ M is

Lagrangian, we have dγS = 0.

Per our definition, E is a quantized energy level of (M,ω,H) if γS has trivial holonomy

over all closed paths in Sp(TS/TS⊥) whose tangents are in the span of the lifted Hamiltonian

vector fields ξ1, . . . , ξn. But now Sp(TS/TS⊥) can be identified with S, which means that ξj

on Sp(TS/TS⊥) can be identified with ξj on S, 1 ≤ j ≤ n. Further, the vector fields ξ1, . . . , ξn

span all of TS. Thus E is a quantized energy level of (M,ω,H) if γS has trivial holonomy

over all closed paths in S. This is a Bohr-Sommerfeld condition, as described in the previous

section.

Our primary examples in previous chapters have been the harmonic oscillator and the hy-

drogen atom. In both cases, the Hamiltonian energy function can be made part of a completely


integrable system. In the remainder of this chapter, we revisit each example and apply the

generalized quantization condition to a regular level set of the completely integrable system.

6.3.3 The n-dimensional harmonic oscillator

Let M = R2n, with Cartesian coordinates (p1, . . . , pn, q1, . . . , qn) and symplectic polar coordi-

nates (s1, . . . , sn, θ1, . . . , θn). We use all of the same definitions that were established in Sections

4.4.1 and 4.4.2. In particular, the metaplectic-c prequantization for (M,ω) is (P,Σ, γ), where

P = M × Mpc(V ), Σ : P → Sp(M,ω) is defined with respect to the global trivialization of

Sp(M,ω) given by the symplectic frame(

∂∂p1

, . . . , ∂∂pn

, ∂∂q1, . . . , ∂

∂qn

), and

γ =1

i~Π∗β +

1

2η∗ϑ0,

where ϑ0 is the trivial connection on P and

β =1

2

n∑j=1

(pjdqj − qjdpj)

on M .

Let H = (H1, . . . ,Hn), where

Hj =1

2(p2j + q2

j ) = sj

for each 1 ≤ j ≤ n. In Cartesian coordinates, the corresponding Hamiltonian vector fields are

ξj = qj∂

∂pj− pj

∂

∂qj.

These vector fields clearly Poisson commute, and they are linearly independent unless qj = pj =

0 for some j. Thus a regular value of the family takes the form E = (E1, . . . , En) ∈ Rk, where

Ej > 0 for all j. Assume such a regular value E has been fixed, and consider the regular level

set S = H−1(E). Then S is an n-torus. In symplectic polar coordinates, which are defined

everywhere on S, a point S takes the form (E1, . . . , En, θ1, . . . , θn).

We know from Section 4.4.2 that we can change coordinates from Cartesian to symplectic


polar over the open subset of U ⊂M where sj > 0 for all j. In terms of these coordinates, we

have the Hamiltonian vector fields ξj = − ∂∂θj

for all j. Let φjt be the flow for the vector field

ξj . It is clear that all orbits in the level set S are closed, with period 2π. If we fix a starting

point in S, then the orbits of the Hamiltonian vector fields ξj through that point generate the

fundamental group π1(S). Therefore we need to find the values of E for which γS has trivial

holonomy over each of these orbits.

Our procedure is exactly the same as that in Sections 4.4.2 and 4.4.3: we locally change vari-

ables from Cartesian to symplectic polar, lift that change of variables to the level of metaplectic-c

structures, and construct local trivializations for all of the relevant bundles over an orbit C. The

only difference is that the curve C is now the orbit for ξj . We omit the details of the calculations

when they are repetitions of those given in Section 4.4.

Without loss of generality, fix the starting point m0 = (E1, . . . , En, 0, . . . , 0) ∈ S. Let C be

the the orbit for ξj through m0. Then a point in C takes the form

m(τ) = (E1, . . . , En, 0, . . . , τ, . . . , 0),

where τ ∈ R/2πZ is the jth angle coordinate.

For all m ∈ U , we construct the matrix of partial derivatives G(m) that corresponds to

the change of variables, just as in Section 4.4.2, and then we restrict it to lie over C. Let

G(τ) = G(m(τ)). In order to lift the change of variables to the level of metaplectic-c structures,

we calculate

DetCCG(τ) =1

2

(√2Ej +

1√2Ej

)eiτ ,

which we denote by Keiτ where K is a positive real constant. Then a lift of G(τ) to Mp(V )

has parameters

G(τ) 7→(G(τ),

1√Ke−iτ/2

).

The matrix G(τ) is single-valued with respect to τ , but e−iτ/2 is not. As τ ranges from

0 to 2π, G(τ) ranges from (I, 1) to (I,−1). We use G(τ) to perform the change of variables

over the subset C = m(τ) : τ ∈ (0, 2π), then manually correct for the change in sign of the µ

parameter when we close the loop from C to C.


Let (x1, . . . , xn, y1, . . . , yn) be the usual symplectic basis for V . Let W = span y1, . . . , yn,

so W⊥ = W and W/W⊥ = 0. Using the local trivializations over U given by xj 7→ ∂∂sj

and

yj 7→ ∂∂θj

, we obtain the identifications Sp(M,ω)|U = U×Sp(V ), Sp(M,ω;S)|U = U×Sp(V ;W ),

and Sp(TS/TS⊥)|U = U × I = U .

Over C, the lifted change of variables gives us the identifications P |C = C ×Mpc(V ), PS |C =

C ×Mpc(V ;W ), and PS |C = C × U(1). The one-form γS takes the form

γS |C =1

i~

n∑j=1

Ejdθj +1

2η∗ϑ0.

Since Sp(TS/TS⊥)|U is identified with U , the lift of ξj(m(τ)) to Sp(TS/TS⊥) is simply

ξj(m(τ), I) = ξj(m(τ)), ∀m(τ) ∈ C.

Then the horizontal lift to PS with respect to γS is

ξj(m(τ), I) =

(ξj(m(τ)),− 1

i~Ej), ∀m(τ) ∈ C.

Since the Lie algebra component is constant over the orbit, we obtain the lifted flow by expo-

nentiating:

φjt(m0, I) =(φjt(m0), e−E

jt/i~).

On S, φjt has period 2π. Because of the change in sign introduced by the definition of G(τ),

the U(1) component is closed with period 2π if e−2πEj/i~ = −1. Using these observations and

the fact that Ej is positive, we find that the quantization condition is

2πEj

i~= −2πi

(Nj +

1

2

)

for some Nj ∈ Z, Nj ≥ 0, which implies that

Ej = ~(Nj +

1

2

), Nj ∈ Z, Nj ≥ 0.

Note that the Hamiltonian energy function for the n-dimensional harmonic oscillator is


just H1 + . . . + Hn. Using the generalized dynamical invariance property, we see immedi-

ately that if we describe S as a level set of the completely integrable system (H1 + . . . +

Hn, H2, . . . ,Hn), the values of H1 + · · ·+Hn on the quantized level sets are E1 + . . .+ En =

~(N1 + 1

2 + . . .+Nn + 12

)= ~

(N + n

2

), for some N ∈ Z, N ≥ 0. This result reproduces the

n2 -shift in the quantized energy levels that was seen in the analysis from Section 4.4. In addition,

the quantized energy levels begin at N = 0, which is consistent with the quantum mechanical

calculation.

6.3.4 The 2-dimensional hydrogen atom

For simplicity, we restrict our attention to the two-dimensional version of the hydrogen atom.

Our initial definitions are special cases of those that appear in Section 5.2.

Let M = R2 × R2 with Cartesian coordinates (q1, q2, p1, p2) and symplectic form ω =∑2j=1 dqj ∧ dpj . As before, the energy function for the hydrogen atom is

H =1

2me|p|2 − k

|q|,

where k,me > 0. Let (P,Σ, γ) be the metaplectic-c prequantization for M in which P =

M ×Mpc(V4), Σ : P → Sp(M,ω) is defined with respect to the global trivialization of Sp(M,ω)

given by the Cartesian coordinate vector fields(

∂∂q1, ∂∂q2, ∂∂p1

, ∂∂p2

), and

γ =1

i~Π∗β +

1

2η∗ϑ0,

where ϑ0 is the trivial connection on P and

β =

2∑j=1

(qjdpj + d(qjpj))

on M .

Let (r, θ) be polar coordinates for R2, with conjugate momenta (pr, pθ) on R2. The change


of coordinates is given by

r =√q2

1 + q22,

θ = tan−1

(q2

q1

),

pr =1

r(q1p1 + q2p2) ,

pθ =q1p2 − q2p1,

and the inverse transformation is

q1 =r cos θ,

q2 =r sin θ,

p1 =pr cos θ − 1

rpθ sin θ,

p2 =pr sin θ +1

rpθ cos θ.

Note that polar coordinates are defined everywhere on M , so we could have defined Σ : P →

Sp(M,ω) in terms of the global trivialization for Sp(M,ω) given by the symplectic frame(∂∂ ,

∂∂θ ,

∂∂pr

, ∂∂pθ

), and let γ be as given above, with β written in polar coordinates. However,

since (M,ω) is not simply connected, it is not obvious that the metaplectic-c prequantization so

constructed is isomorphic to the one that we obtain using Cartesian coordinates. We therefore

begin with the Cartesian trivialization, and lift the change of coordinates to the metaplectic-c

level.

At any point m ∈M , the matrix of partial derivatives representing the transformation from

Cartesian to polar coordinates is

G(m) =

∂q1∂r

∂q2∂r

∂p1∂r

∂p2∂r

∂q1∂θ

∂q2∂θ

∂p1∂θ

∂p2∂θ

∂q1∂pr

∂q2∂pr

∂p1∂pr

∂p2∂pr

∂q1∂pθ

∂q2∂pθ

∂p1∂pθ

∂p2∂pθ


=

cos θ sin θ 1r2pθ sin θ − 1

r2pθ cos θ

−r sin θ r cos θ −pr sin θ − 1rpθ cos θ pr cos θ − 1

rpθ sin θ

0 0 cos θ sin θ

0 0 −1r sin θ 1

r cos θ

.

To determine a lift of G(m) to Mp(V4), we calculate CG(m), convert to a complex matrix, and

take the determinant. The result is

DetCCG(m) = r +1

r+

1

4r3p2θ.

While this is not necessarily constant, it is real and positive everywhere on M . Therefore, over

any closed orbit in M , we can consistently take√

DetCCG(m). Thus a lift of G(m) to Mp(V4)

is given by

G(m) =

(G(m),

1√DetCCG(m)

),

which is single-valued over any closed orbit in M . This demonstrates that the metaplectic-

c prequantization that we obtain using the symplectic polar global trivialization is, in fact,

isomorphic to (P,Σ, γ).

From now on, we use the global trivializations of P and Sp(M,ω) given by the symplectic

frame(∂∂r ,

∂∂θ ,

∂∂pr

, ∂∂pθ

). The one-form γ is given by

γ =1

i~Π∗β +

1

2ϑ0,

where

β = 2rdpr + prdr − pθdθ

on M .

In terms of the polar coordinates,

H =1

2me

(p2r +

1

r2p2θ

)− k

r,


and ω = dpr ∧ dr + dpθ ∧ dθ. A calculation establishes that

ξH =

(k

r2− 1

mer3p2θ

)∂

∂pr− 1

mepr∂

∂r− 1

mer2pθ∂

∂θ.

Now consider pθ. Physically, pθ is the angular momentum of the system. Let E < 0 be

a fixed negative energy value. As discussed in Section 5.2.1, the angular momentum can take

values in

[0,√−mek2

2E

]. The value pθ = 0 represents a degenerate elliptical orbit in which the

electron collapses into the proton in finite time. The value pθ =√−mek2

2E represents a circular

orbit.

Observe that

ξpθ = − ∂

∂θ,

from which we see that H, pθ = 0. Therefore the system (H, pθ) Poisson commutes. Further,

ξH and ξpθ are linearly independent unless pr = 0 and kr2− 1

mer3p2θ = 0. These two conditions

imply that r = − k2E and p2

θ = −mek2

2E : that is, ξH and ξpθ are linearly independent everywhere

except on the circular orbit.

Let (E,L) ∈ R2 be such that E < 0 and 0 < L <√−mek2

2E . Then (E,L) is a regular value

of the family (H, pθ). Let S be the corresponding level set. Then, from Section 5.2.1, all orbits

of ξH in S are closed with period 2πΛ , where Λ =

√− 8E3

mek2. All orbits of ξpθ in S are closed with

period 2π.

Given a starting point m0 ∈ S, we argue that the orbits of ξH and ξpθ through m0 generate

π1(S). By fixing E and L, we determine the lengths of the major and minor axes of the elliptic

orbit of the electron in q-space. Within S, we can either fix a particular elliptical orbit and let

m0 revolve around it, or we can fix the position of m0 on its orbit and rotate the orbit itself

about its focal point at the origin. The latter rotation has the effect of rotating m0 = (q0, p0)

about the origin at the same rate in q-space and p-space, keeping the relative orientation of

the position and momentum vectors constant. The flow of ξH clearly represents the revolution

about a particular elliptic orbit. The flow of ξpθ , on the other hand, represents the rotation of


an orbit about the origin. This is most easily seen in Cartesian coordinates:

ξpθ = q1∂

∂q2− q2

∂

∂q1+ p1

∂

∂p2− p2

∂

∂p1,

which has flow

φtpθ(q, p) =

cos t − sin t 0 0

sin t cos t 0 0

0 0 cos t − sin t

0 0 sin t cos t

q1

q2

p1

p2

as needed.

Since S is a Lagrangian submanifold of M , we have TS⊥ = TS, and so Sp(TS/TS⊥) is

naturally identified with S. We must therefore evaluate the holonomy of γS over the orbits ξH

and ξpθ through m0. Let ξH have flow φtH , and let pθ have flow φtpθ .

First, we consider ξpθ . Let m be any point on the orbit of ξpθ through m0. On Sp(TS/TS⊥),

we simply have

ξpθ(m, I) = ξpθ(m).

Using the expression for β in polar coordinates, we calculate that the horizontal lift to PS is

ξpθ(m, I) =

(ξpθ(m),− L

i~

).

Upon exponentiating, we obtain the integral curve through (m0, I):

φtpθ(m0, I) =(φtpθ(m0), e−Lt/i~

).

Since the orbit in S has period 2π, we see immediately that the quantization condition for the

angular momentum is

L = N~, N ∈ Z.

This is consistent with the Bohr model of the hydrogen atom, in which angular momentum is

assumed to be quantized in units of ~.

Now we apply the same process to ξH . Let m be any point on the orbit of ξH through m0.


Since

ξHyβ =2k

r− 1

me

(p2r +

1

r2p2θ

)= −2E,

we see that the horizontal lift to PS is

ξH(m, I) =

(ξH(m),

2E

i~

).

Then the integral curve through m0 is

φtH(m0, I) =(φtH(m0), e2Et/i~

).

Using the fact that the period of the orbit in S is 2πΛ where Λ =

√− 8E3

mek2, we find that the

quantization condition for the energy is

E = − mek2

2~2N2, N ∈ N.

This agrees with our result from Chapter 5.

Bibliography

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