Quantum 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- derstand some bits of the topology going on, but for me it was clouded by all kinds 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 declare to myself “I’m going to start learning physics this year,” but then each year my time would be sucked away by writing papers in my research program, 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 learn physics—unless I found a way to incorporate that process into the obligations that I already had. So I arranged to teach a course on this subject. In some ways it was a crazy thing to do, and it had some consequences: for example, my research program took a complete nosedive during this time, because all my energy went into 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 heavily revised what was there, and also added a bunch of additional material. I am 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, Liz Henning, Joseph Loubert, Kristy Pelatt, Min Ro, Dylan Rupel, Patrick Schultz, AJ Stewart, Michael Sun, and Jason Winerip. WARNING: The present document is a work in progress. Certain sections have not been revised, further sections are being added, some things are occasionally 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 *j *j O *j *j *j O *j *j *j O *j *j *j O * TQFT O i O iiii O iiii O iiii tiiii 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 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’s lectures get technical pretty quickly and are also short on examples. We will try to remedy that. 1.0.1. Introduction to TQFTs. Let k be a field (feel free to just think about 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 W h: ∂∆ → M 1 q M2, a linear map φW,h : Z(M1) → Z(M2). Here M 1 denotes M1 with the opposite orientation. ∼ ∼ (3) Isomorphisms Z(∅) = k and Z(M1 q M2) = Z(M1) ⊗ Z(M2) for every closed, oriented d-manifolds M1 and M2. 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 W1 is a cobordism from M1 to M2, and W2 is a cobordism from M2 to M3. An example of this is depicted here: M1 W1 W2 M3 M2 M2 W2 Gluing W1 and W2 results in a “composite” cobordism W3 from M1 to M3, and we ask that φW3 = φW2 ◦ φW1 . (ii) φM×I : Z(M) → Z(M) is the identity map (iii) Each Z(M) is finite-dimensional and the map φ Z(M) ⊗ Z(M) =∼ Z(M q M) −→M×I Z(∅) =∼ k is a perfect pairing. Here M × I is being thought of as a cobordism between M q M and ∅. How do we get invariants of manifolds from a TQFT? If W is a closed, oriented (d + 1)-manifold then it can be thought of as a cobordism from ∅ to itself, and therefore gives rise to a linear map φW : Z(∅) → Z(∅). Since Z(∅) =∼ k, this map is just multiplication by an element of k. This element (which we will also denote as φW ) is an invariant of W . Next let us look at some examples. When d = 0, the oriented d-manifolds are just finite sets with the points labelled by plus and minus signs (for the orientation). Let us write pt+ and pt− for a point with the two orientations. If Z(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.