Table Of ContentQuantum Theory for Topologists
Daniel Dugger
Contents
Preface 5
Chapter 1. Introduction 7
Part 1. A first look into modern physics 11
Chapter 2. Classical mechanics 13
2.1. Lagrangian mechanics 13
2.2. Harmonic oscillators 18
2.3. Variational approach to the Euler-Lagrange equations 28
2.4. Hamiltonian mechanics 34
Chapter 3. Quantum mechanics 39
3.1. Preamble to quantum mechanics 39
3.2. Quantum mechanics 46
3.3. Examples of one-dimensional quantum systems 52
3.4. Introduction to path integrals 67
Chapter 4. Maxwell’s equations 73
4.1. Maxwell’s equations 73
4.2. Differential forms 80
4.3. A second look at Maxwell’s equations 87
4.4. The Aharanov-Bohm effect 92
4.5. Magnetic monopoles and topology 97
Part 2. Delving deeper into quantum physics 101
Chapter 5. Spin and Dirac’s theory of the electron 103
5.1. The Spin groups 104
5.2. Projective representations and other fundamentals 108
5.3. The story of electron spin 114
5.4. Spin one and higher 121
5.5. Lie algebra methods 122
5.6. The Schr¨odinger-Pauli equation 125
5.7. Spin and relativity: mathematical foundations 131
5.8. The Dirac equation 133
Chapter 6. Gauge theory 135
6.1. Principal bundles 135
6.2. Connections and curvature 141
6.3. Yang-Mills theory 147
3
4 CONTENTS
6.4. Digression on characteristic classes and Chern-Simons forms 149
Part 3. Quantum field theory and topology 157
Chapter 7. Quantum field theory and the Jones polynomial 159
7.1. A first look at Chern-Simons theory 159
7.2. Linking numbers and the writhe 161
7.3. Polyakov’s paper 167
7.4. The Jones Polynomial 169
Chapter 8. Quantum field theory 171
8.1. Introduction to functional integrals 171
Part 4. Appendices 175
Appendix A. Background on differential geometry 177
A.1. Introduction to connections 177
A.2. Curvature 184
A.3. DeRham theory with coefficients in a bundle 195
A.4. Principal connections 201
Appendix B. Background on compact Lie groups and their representations 207
B.1. Root systems 207
B.2. Classification of simply-connected, compact Lie groups 210
B.3. Representation theory 213
B.4. The groups SO(n) and Spin(n). 224
Appendix. Bibliography 231
PREFACE 5
Preface
For a long time I have wanted to learn about the interactions between math-
ematical physics and topology. I remember looking at Atiyah’s little book The
geometry and physics of knots [At] as a graduate student, and understanding very
little. Over the years, as my knowledge of mathematics grew, I periodically came
back to that little book—as well as other introductory texts on the subject. Each
time I found myself almost as lost as when I was a graduate student. I could un-
derstandsomebitsofthetopologygoingon, butformeitwascloudedbyallkinds
of strange formulas from physics and differential geometry (often with a morass of
indices) for which I had no intuition. It was clear that learning this material was
going to take a serious time commitment, and as a young professor—eventually
one with a family—time is something that is hard to come by. Each year I would
declaretomyself“I’mgoingtostartlearningphysicsthisyear,”buttheneachyear
mytimewouldbesuckedawaybywritingpapersinmyresearchprogram,teaching
courses, writing referee reports, and all the other things that go with academic life
and are too depressing to make into a list.
Finally I realized that nothing was ever going to change, and I would never
learnphysics—unlessIfoundawaytoincorporatethatprocessintotheobligations
that I already had. So I arranged to teach a course on this subject. In some ways
itwasacrazythingtodo,andithadsomeconsequences: forexample,myresearch
programtookacompletenosediveduringthistime,becauseallmyenergywentinto
learning the material I was putting into my lectures. But it was fun, and exciting,
and somehow this process got me over a hump.
The notes from this course have evolved into the present text. During the
course, my lectures were put into LaTeX by the attending students. Afterwards I
heavilyrevisedwhatwasthere,andalsoaddedabunchofadditionalmaterial. Iam
grateful to the students for doing the intial typesetting, for asking good questions
during the course, and for their patience with my limited understanding: these
students were Matthew Arbo, Thomas Bell, Kevin Donahue, John Foster, Jaree
Hudson,LizHenning,JosephLoubert,KristyPelatt,MinRo,DylanRupel,Patrick
Schultz, AJ Stewart, Michael Sun, and Jason Winerip.
WARNING: The present document is a work in progress. Certain sections
havenotbeenrevised,furthersectionsarebeingadded,somethingsareoccasionally
moved around, etc. There are plenty of false or misleading statements that are
gradually being identified and removed. Use at your own risk!
CHAPTER 1
Introduction
For this first lecture all you need to know about quantum field theory is that
it is a mysterious area of modern physics that is built around the “Feynman path
integral”. I say it is mysterious because these path integrals are not rigorously
defined and often seem to be divergent. Quantum field theory seems to consist
of a collection of techniques for getting useful information out of these ill-defined
integrals.
Work dating back to the late 1970s shows that QFT techniques can be used to
obtain topological invariants of manifolds. Schwartz did this in 1978 for an invari-
ant called the Reidemeister torsion, and Witten did this for Donaldson invariants
and the Jones polynomial in several papers dating from the mid 80s. Shortly after
Witten’s work Atiyah wrote down a definition of “topological quantum field the-
ory”, and observed that Witten’s constructions factored through this. One has the
following picture:
QFT techniques
(cid:15)O(cid:15)O(cid:15)O(cid:15)O *j *j *j *j *j *j *j *j *j *j *j **
(cid:15)O(cid:15)(cid:15)(cid:15)O(cid:15)O(cid:15)O ttiiiiiiiiiiiiiiiiiTQFT
topological invariants of manifolds
Here the squiggly lines indicate non-rigorous material!
Much effort has been spent over the past twenty-years in developing rigorous
approaches to constructing TQFTs, avoiding the path integral techniques. The
names Reshetikin and Turaev come to mind here, but there are many others as
well. However, this course will not focus on this material. Our goal will instead be
to understand what we can about Feynman integrals and the squiggly lines above.
We will try to fill in the necessary background from physics so that we have some
basic idea of what QFT techniques are all about. There’s quite a bit of material to
cover:
//
Classical mechanics Classical field theory
(cid:15)(cid:15) (cid:15)(cid:15)
//
Quantum mechanics Quantum field theory.
The plan is to spend about a week on each of these areas, giving some kind of
survey of the basics.
7
8 1. INTRODUCTION
A good reference for this foundational material is the set of lecture notes by
Rabin[R]. However,Rabin’slecturesgettechnicalprettyquicklyandarealsoshort
on examples. We will try to remedy that.
1.0.1. IntroductiontoTQFTs. Letkbeafield(feelfreetojustthinkabout
k =C). A “(d+1)-dimensional TQFT” consists of
(1) For every closed, oriented d-manifold M a vector space Z(M) over k;
(2) For every oriented (d + 1)-manifold W together with a homeomorphism
h: ∂∆W →M qM , a linear map
1 2
φ : Z(M )→Z(M ).
W,h 1 2
Here M denotes M with the opposite orientation.
1 1
(3) Isomorphisms Z(∅) ∼= k and Z(M qM ) ∼= Z(M )⊗Z(M ) for every closed,
1 2 1 2
oriented d-manifolds M and M .
1 2
This data must satisfy a long list of properties which are somewhat complicated
to state in this form. For this introductory lecture let us just give the basic ideas
behind a few of the properties:
(i) (Composition) Suppose W is a cobordism from M to M , and W is a
1 1 2 2
cobordism from M to M . An example of this is depicted here:
2 3
M W W M
1 1 2 3
M M
2 2
W
2
Gluing W and W results in a “composite” cobordism W from M to M ,
1 2 3 1 3
and we ask that φ =φ ◦φ .
W3 W2 W1
(ii) φ : Z(M)→Z(M) is the identity map
M×I
(iii) Each Z(M) is finite-dimensional and the map
Z(M)⊗Z(M)∼=Z(M qM)φ−M→×I Z(∅)∼=k
is a perfect pairing. Here M ×I is being thought of as a cobordism between
M qM and ∅.
HowdowegetinvariantsofmanifoldsfromaTQFT?IfW isaclosed,oriented
(d+1)-manifold then it can be thought of as a cobordism from ∅ to itself, and
therefore gives rise to a linear map
φ : Z(∅)→Z(∅).
W
Since Z(∅) ∼= k, this map is just multiplication by an element of k. This element
(which we will also denote as φ ) is an invariant of W.
W
Next let us look at some examples. When d =0, the oriented d-manifolds are
justfinitesetswiththepointslabelledbyplusandminussigns(fortheorientation).
Letuswritept andpt forapointwiththetwoorientations. IfZ(pt )=V,then
+ − +
1. INTRODUCTION 9
we know Z(pt ) = V∗ (canonically), and in general Z of any oriented 0-manifold
−
will be a tensor product of V’s and V∗’s.
All cobordisms between oriented 0-dimensional manifolds will break up into
identity cobordisms and the two types
+ +
− −
Applying φ to the first gives a map k →V ⊗V∗, and applying φ to the second
gives V ⊗V∗ → k. A little work shows that the second map is just evaluation,
namely it sends a tensor u⊗α to α(u). The first map is “coevaluation”, which is
slightly harder to describe. If e ,...,e is a basis for V, and f ,...,f is the dual
1 n 1 n
basis for V∗, then coevaluation sends 1 to P e ⊗f . One can check that this is
i i i
independent of the choice of basis for V. (For a basis-independent construction,
use the canonical isomorphism V ⊗V∗ ∼= End(V), and then take k → End(V) to
send 1 to the identity map).
What invariant does this TQFT assign to a circle? By breaking up the circle
into its two hemispheres, one sees that φ : k →k is the composite
S1
k −co→ev V ⊗V∗ −e→v k.
P P
This map sends 1 to f (e )= 1=dimV.
i i i i
Exercise 1.0.2. What invariant does the TQFT assign to a disjoint union of
r circles?
Now letusturnto d=1. Closed1-manifoldsare just disjoint unions ofcircles,
so the Z(−) construction is in some sense determined by Z(S1). Let V =Z(S1).
The two corbodisms
give maps (cid:15): k →V and tr: V →k. The “pair of pants” cobordism
givesamapµ: V⊗V →V. WeclaimthatthisproductmakesV intoanassociative,
commutative algebra with unit (cid:15) such that the pairing V ⊗V →k given by a,b7→
tr(ab) is perfect. One sees this by drawing lots of pictures:
????
10 1. INTRODUCTION
Exercise 1.0.3. Determine the invariant that this TQFT assigns to each ori-
ented surface Σ , in terms of the above structure on V. [Hint: The invariant
g
attached to Σ is tr(1).]
0
Finally let us look at d = 2. The point of this whole discussion is to observe
that while things are very easy for d=0 and d=1, for d=2 the situation is much
harder. OnemustgiveeachvectorspaceZ(Σ ),andahomomorphismφ foreach
g W
oriented 3-manifold. This is a lot of information!
The most basic example of a (2+1)-dimensional TQFT is something called
Chern-Simons theory. It was constructed by Witten in his 1989 paper [W] using
QFTtechniques,andmuchworkhassincebeendonetocreatearigorousconstruc-
tion. Here is a pathetic outline, only intended to show how daunting the theory is
to a newcomer. We will only say how to get the invariants of closed 3-manifolds
φ .
M
FixacompactLiegroupG,andfixanintegerk ∈Z (calledthe“level”). There
will be one TQFT for each choice of G and k. Let g be the Lie algebra of G.
Let M be a 3-manifold, and let P →M be the trivial G-bundle over M. If A
is a g-valued connection on P, define the “action”
k Z 2
S(A)= tr(A∧dA+ A∧A∧A).
4π 3
M
Then
Z
φ = eiS(A)dA
M
A
wherethisisaFeynmanintegraloverthe(infinite-dimensional)spaceofallconnec-
tions A on P. This integral is not really defined! Witten shows that if C ,...,C
1 k
are knots in M then he could also use this kind of integral formalism to construct
invariants
φ
M,C1,...,Ck
and in this way he recovered the Jones polynomial for knots in R3.
This now lets me state the second goal of the course: Understand the basic
constructions underlying Chern-Simons theory and Witten’s originalpaper[W] on
this subject.
We will see how far we get with all this!
