# Math person wants to go into Mathematical Physics

1. Feb 21, 2009

### Unknot

Hi.

I don't know much about physics, and I wanted to get into some mathematical physics. I want to learn, from a mathematician's perspective, quantum mechanics and general relativity (and if I am right these are two most popular "flavours" of mathematical physics?).

Someone (not a physicist) recommended Gustafson/Sigal "Mathematical Methods in Quantum Mechanics" and Wald's "General Relativity." Now I do know that there are books that grad students in physics usually look at. Goldstein, Jackson, Sakurai, etc. Do I need to look at those first? Or can I just jump into those two books? Also if you have some other suggestions that would be great.

2. Feb 21, 2009

### Unknot

Oops I think I posted in the wrong section. I suppose it should really be in "science book discussion"

3. Feb 21, 2009

### ice109

how much differential geometry do you know? i would start with a classical mechanics book like arnold v.i's if you're not that good at differential geometry or walter thirring's book if you are that good.

4. Feb 21, 2009

### Unknot

I would say I know enough to pass a qual in differential geometry.

5. Feb 21, 2009

### cristo

Staff Emeritus
I'm pretty much like you: I learnt quantum mechanics and relativity from a mathematician's perspective (mainly because the courses were taught by the maths department!) How much classical mechanics have you studied? If you haven't studied any, then it will be best to start off there, but I imagine you've got at least a little under your belt. For quantum mechanics, the set text for my course was Gasiorowicz: quantum physics, which isn't a bad book, from what I remember, and was chosen by the lecturer as being a good book for mathematicians. As for relativity, you should probably learn special relativity first. I can't remember what I used, but see here (http://math.ucr.edu/home/baez/physics/Administrivia/rel_booklist.html) for a good review of textbooks. For GR I would recommend d'Inverno or Schutz as both being good books, but again, see the list for more info.

6. Feb 21, 2009

### George Jones

Staff Emeritus
Last edited by a moderator: May 4, 2017
7. Oct 14, 2009

### n!kofeyn

Last edited by a moderator: Apr 24, 2017
8. Nov 15, 2009

### Prologue

Thanks for the suggestions on books, they all look good.

9. Dec 21, 2009

### Astronuc

Staff Emeritus
I also found this book and agree with George's assessment.

Contents
Preface vii

Chapter 1. Prologue 1
1.1. Linguistic prologue: notation and terminology 1
1.2. Physical prologue: dimensions, units, constants, and particles 5
1.3. Mathematical prologue: some Lie groups and Lie algebras 8

Chapter 2. Review of Pre-quantum Physics 13
2.1. Mechanics according to Newton and Hamilton 13
2.2. Mechanics according to Lagrange 18
2.3. Special relativity 22
2.4. Electromagnetism 25

Chapter 3. Basic Quantum Mechanics 33
3.1. The mathematical framework 33
3.2. Quantization 42
3.3. Uncertainty inequalities 51
3.4. The harmonic oscillator 53
3.5. Angular momentum and spin 56
3.6. The Coulomb potential 60

Chapter 4. Relativistic Quantum Mechanics 65
4.1. The Klein-Gordon and Dirac equations 66
4.2. Invariance and covariance properties of the Dirac equation 70
4.3. Consequences of the Dirac equation 74
4.4. Single-particle state spaces 83
4.5. Multiparticle state spaces 89

Chapter 5. Free Quantum Fields 97
5.1. Scalar fields 97
5.2. The rigorous construction 105
5.3. Lagrangians and Hamiltonians 107
5.4. Spinor and vector fields 112
5.5. The Wightman axioms 119

Chapter 6. Quantum Fields with Interactions 123
6.1. Perturbation theory 123
6.2. A toy model for electrons in an atom 128
6.3. The scattering matrix 136
6.4. Evaluation of the S-matrix: first steps 143
6.5. Propagators 147
6.6. Feynman diagrams 154
6.7. Feynman diagrams in momentum space 162
6.8. Cross sections and decay rates 167
6.9. QED, the Coulomb potential, and the Yukawa potential 172
6.10. Compton scattering 177
6.11. The Gell-Mann–Low and LSZ formulas 180

Chapter 7. Renormalization 191
7.1. Introduction 192
7.2. Power counting 196
7.3. Evaluation and regularization of Feynman diagrams 200
7.4. A one-loop calculation in scalar field theory 206
7.5. Renormalized perturbation theory 211
7.6. Dressing the propagator 214
7.7. The Ward identities 219
7.8. Renormalization in QED: general structure 224
7.9. One-loop QED: the electron propagator 234
7.10. One-loop QED: the photon propagator and vacuum polarization 237
7.11. One-loop QED: the vertex function and magnetic moments 244
7.12. Higher-order renormalization 251

Chapter 8. Functional Integrals 257
8.1. Functional integrals and quantum mechanics 257
8.2. Expectations, functional derivatives, and generating functionals 265
8.3. Functional integrals and Boson fields 271
8.4. Functional integrals and Fermion fields 278
8.5. Afterword: Gaussian processes 287

Chapter 9. Gauge Field Theories 291
9.1. Local symmetries and gauge fields 291
9.2. A glimpse at quantum chromodynamics 296
9.3. Broken symmetries 299
9.4. The electroweak theory 303

Bibliography 317
Index 323

Amazon allows one to browse some of Chapter 1.

Publishes pages allows review of Chapter 2.
http://www.ams.org/bookstore?fn=20&arg1=mathphys&ikey=SURV-149

Last edited by a moderator: Apr 24, 2017
10. Dec 22, 2009