1. Introduction

1.1. Preface

I have a very bad memory. I am able to memorize quite a lot of things short term, but I am not able to remember most formulas from quantum mechanics over the long term (e.g. like over the summer). I don’t remember formulas for perturbation theory (neither time dependent or time independent), I don’t remember Feynman rules in quantum field theory, I don’t even remember the Dirac equation exactly (where the i should be, if there is m or m^2, …). The thing about quantum field theory is not that some particular steps are difficult, but that there are so many of them and one has to master all of them at once, in order to really “get it”.

I never got QFT, because once I mastered one part sufficiently, I forgot some other part and it took so much time to master that other part that I forgot the first part again. However, I was determined that I would get it. In order to do so, I realized I need to keep notes of things I understood, written in my own way. Then, when I relearn some parts that I forgot, it just takes me a few minutes to go over my reference notes to get into it quickly. My own style of understanding is that the notes should be complete (no need to consult external books), yet very short and getting directly to the point, and also with every single calculation carried out explicitly.

See also the preface to the QFT part.

If you want to study physics, learn math the physics way (as opposed to the usual mathematics way of a definition, theorem, proof, …). When I was beginning my undergrad physics studies (and even on a high school), I also had this common misconception, that I need to study math and understand every proof and then I’ll be somehow prepared for physics. I was very wrong. I used to study calculus by myself and then trying to learn the proofs, and Lebesgue integral and I was learning that from the mathematics books. At the university, I always did all my math exams first (as far as I remember, I always got A from those), hoping that would be a good start for the physics exams, but I always found out that it was mostly useless.

Now I know that the only way to study physics is to go and do physics directly and learn the math on the way as needed. The math section of this book reviews all the math, that is necessary for studying theoretical physics (graduate level).

There are actually quite a lot of good math books written by physicists as well as many excellent physics books, covering everything that I cover here. But I really like to have all the theoretical physics and the corresponding math explained in one book, and to keep it as short as possible. Also everyone has a bit different style and amount of rigor and I have not found a book that would perfectly suite my own style, thus I wrote one.

1.2. Introduction

The Theoretical Physics Reference is an attempt to derive all theoretical physics equations (that are ever needed for applications) from the general and special relativity and the standard model of particle physics.

The goals are:

  • All calculations are very explicit, with no intermediate steps left out.

  • Start from the most general (and correct) physical theories (general relativity or standard model) and derive the specialized equations from them (e.g. the Schrödinger equation).

  • Math is developed in the math section (not in the physics section).

  • Theory should be presented as short and as explicitly as possible. Then there should be arbitrary number of examples, to show how the theory is used.

  • There should be just one notation used throughout the book.

  • It should serve as a reference to any physics equation (exact derivation where it comes from) and the reader should be able to understand how things work from this book, and be ready to understand specialized literature.

This is a work in progress and some chapters don’t conform to the above goals yet. Usually first some derivation is written, as we understood it, then the mathematical tools are extracted and put into the math section, and the rest is fit where it belongs. Sometimes we don’t understand some parts yet, then those are currently left there as they are.

There are many excellent books about theoretical physics, that one can consult about particular details. The goal of this book (when completed) is to show where things come from and serve as a reference to any particular field, so that one doesn’t get lost when reading specialized literature.

Here is an incomplete list of some of the best books in theoretical physics (we only picked those that we actually read):

  1. Landau, L. D.; Lifshitz, E. M: Course of Theoretical Physics

  2. Richard Feynman: The Feynman Lectures on Physics

  3. Walter Greiner: “Classical Theoretical Physics” series of texts

  4. Herbert Goldstein: Classical Mechanics

  5. J.D. Jackson: Classical Electrodynamics

  6. Charles W. Misner, Kip S. Thorne, John Wheeler: Gravitation

  7. Bernard Schutz: A First Course in General Relativity

  8. Carrol S.: The Lecture Notes on General Relativity

  9. J.J. Sakurai: Advanced Quantum Mechanics

  10. Brown L. S.: Quantum Field Theory

  11. Mark Srednicki: Quantum Field Theory

  12. Claude Itzykson, Jean-Bernard Zuber: Quantum Field Theory

  13. Zee A.: Quantum Field Theory in a Nutshell

  14. Steven Weinberg: The Quantum Theory of Fields

  15. L.H. Ryder: Quantum Field Theory

  16. Jiří Hořejší: Fundamentals of Electroweak Theory

  17. Michele Maggiore: A Modern Introduction to Quantum Field Theory

  18. M.E. Peskin & D.V. Schroeder: An Introduction to Quantum Field Theory

  19. J.W. Negele, H. Orland: Quantum Many-Particle Systems

  20. X-G. Wen: Quantum Field Theory of Many-Body Systems

  21. Dirac, P.A.M.: General Theory of Relativity