Gerard ’t Hooft, Theoretical Physics as a Challenge

This is a web site for young students - and anyone else - who are (like me) thrilled by the challenges posed by real science, and who are - like me - determined to use their brains to discover new things about the physical world that we are living in. In short, it is for all those who decided to study theoretical physics, in their own time.

It so often happens that I receive mail - well-intended but totally useless - by amateur physicists who believe to have solved the world. They believe this, only because they understand totally nothing about the real way problems are solved in Modern Physics. If you really want to contribute to our theoretical understanding of physical laws - and it is an exciting experience if you succeed! - there are many things you need to know. First of all, be serious about it. All necessary science courses are taught at Universities, so, naturally, the first thing you should do is have yourself admitted at a University and absorb everything you can. But what if you are still young, at School, and before being admitted at a University, you have to endure the childish anecdotes that they call science there? What if you are older, and you are not at all looking forward to join those noisy crowds of young students ?

It should be possible, these days, to collect all knowledge you need from the internet. Problem then is, there is so much junk on the internet. Is it possible to weed out those very rare pages that may really be of use? I know exactly what should be taught to the beginning student. The names and topics of the absolutely necessary lecture courses are easy to list, and this is what I have done below. It is my intention to search on the web where the really useful papers and books are, preferably downloadable as well. This way, the costs of becoming a theoretical physicist should not exceed much the price of a computer with internet connection, a printer, and lots of paper and pens. Unfortunately, I still have to recommend to buy text books as well, but it is harder to advise you here; perhaps in a future site. Let's first limit ourselves to the absolute minimum. The subjects listed below must be studied. Any omission will be punished: failure. Do get me right: you don't have to believe anything you read on faith - check it. Try alternative approaches, as many as you can. You will discover, time and again, that really what those guys did indeed was the smartest thing possible. Amazing. the best of the texts come with exercises. Do them. find out that you can understand everything. Try to reach the stage that you discover the numerous misprints, tiny mistakes as well as more important errors, and imagine how you would write those texts in a smarter way.

I can tell you of my own experiences. I had the extreme luck of having excellent teachers around me. That helps one from running astray. It helped me all the way to earn a Nobel Prize. But I didn't have internet. I am going to try to be your teacher. It is a formidable task. I am asking students, colleagues, teachers to help me improve this site. It is presently set up only for those who wish to become theoretical physicists, not just ordinary ones, but the very best, those who are fully determined to earn their own Nobel Prize. If you are more modest than that, well, finish those lousy schools first and follow the regular routes provided by educators and specialized -gogues who are so damn carefully chewing all those tiny portions before feeding them to you. This is a site for ambitious people. I am sure that anyone can do this, if one is gifted with a certain amount of intelligence, interest and determination.

Theoretical Physics is like a sky scraper. It has solid foundations in elementary mathematics and notions of classical (pre-20th century) physics. Don't think that pre-20th century physics is "irrelevant" since now we have so much more. In those days, the solid foundations were laid of the knowledge that we enjoy now. Don't try to construct your sky scraper without first reconstructing these foundations yourself. The first few floors of our skyscraper consist of advanced mathematical formalisms that turn the Classical Physics theories into beauties of their own. They are needed if you want to go higher than that. So, next come many of the other subjects listed below. Finally, if you are mad enough that you want to solve those tremendously perplexing problems of reconciling gravitational physics with the quantum world, you end up studying general relativity, superstring theory, M-theory, Calabi-Yau compactification and so on. That's presently the top of the sky scraper. There are other peaks such as Bose-Einstein condensation, fractional Hall effect, and more. Also good for Nobel Prizes, as the past years have shown. A warning is called for: even if you are extremely smart, you are still likely to get stuck somewhere. Surf the net yourself. Find more. Tell me about what you found. If this site has been of any help to someone while preparing for a University study, if this has motivated someone, helped someone along the way, and smoothened his or her path towards science, then I call this site successful. Please let me know. Here is the list.

Note that this site NOT meant to be very pedagogical. I avoid texts with lots of colorful but distracting pictures from authors who try hard to be funny. Also, the subjects included are somewhat focused towards my own interests.

LIST OF SUBJECTS, IN LOGICAL ORDER (not everything has to be done in this order, but this approximately indicates the logical coherence of the various subjects. Some notes are at a higher level than others).

More resources

The .ps files are PostScript files §.

(In this initial phase this page is still incomplete!)

Languages:
English is a prerequisite. If you haven't mastered it yet, learn it. You must be able to read, write, speak and understand English, but you don't have to be perfect here. The lousy English used in this text is mine. That's enough. All publications are in English. Note the importance of being able to write in English. Sooner or later you will wish to publish your results. People must be able to read and understand your stuff.

Free English courses and resources.

BBC world service learning English.

French, German, Spanish and Italian may be useful too, but they are not at all necessary. They are nowhere near the foundations of our sky-scraper, so don't worry. You do need the Greek alphabet. Greek letters are used a lot. Learn their names, otherwise you make a fool of yourself when giving an oral presentation. Now, here begins the serious stuff. Don't complain that it looks like being a lot. You won't get your Nobel Prize for free, and remember, all of this together takes our students at least 5 years of intense study (at least one reader was surprised at this statement, saying that (s)he would never master this in 5 years; indeed, I am addressing people who plan to spend most of their time to this study). More than rudimentary intelligence is assumed to be present, because ordinary students can master this material only when assisted by patient teachers. It is necessary to do exercises. Some of the texts come with exercises. Do them, or better, invent your own exercises. Try to outsmart the authors, but please refrain from mailing to me your alternative theories until you have studied the entire lot; if you do this well you will discover that many of these authors were not so stupid after all.

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Primary Mathematics:
Are you comfortable with numbers, adding, subtracting, square roots, etc.?

*Beginning Algebra* (West Texas A&M)
*Intermediate Algebra* (West Texas A&M)

Natural numbers: 1, 2, 3, ...
Integers: ..., -3, -2, -1, 0, 1, 2, ...
Rational numbers (fractions): ¼, ½, ¾, ²³⁷⁹¹ / 773, ...
Real numbers: Sqrt(2) = 1.4142135... , π = 3.14159265... , e = 2.7182818..., ...
Complex numbers: 2+3i, e^ia= cos(a) + i sin( a), ... they are very important!

Dave E. Joyce's trigonometry course

This is a must:
Prof. James Binney's course on complex numbers(PDF)

(nearly) complete overview of primary mathematics(K.Kubota, Kentucky)

See also Chris Pope's lecture notes:
Methods1-ch1
Methods1-ch2

The complex plane. Cauchy theorems and contour integration (G. Cain, Atlanta)

Set theory: open sets, compact spaces. Topology.You may be surprised to learn that they do play a role indeed in physics!

Algebraic equations. Approximation techniques. Series expansions: the Taylor series. Solving equations with complex numbers. Trigonometry: sin(2x)=2sin x cos x, etc.

Infinitesimals. Differentiation. Differentiate basic functions (sin, cos, exp). Integration. Integrate basic functions, when possible. Differential equations. Linear equations.
The Fourier transformation. The use of complex numbers. Convergence of series.

The complex plane. Cauchy theorems and contour integration (now this is fun).
The Gamma function (enjoy studying its properties).
Gaussian integrals. Probability theory.

Partial differential equations. Dirichlet and Neumann boundary conditions.

This is for starters. Some of these topics actually come as entire lecture courses. Much of those are essential ingredients of theories in Physics. You don't have to finish it all before beginning with what follows next, but remember to return to those subjects skipped during the first round.

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Classical Mechanics:

An intermediate level course on Analytical Classical Dynamics by R. Fitzpatrick, Univ. of Texas, Austin.

A good set of Lecture notes from Harvard.

A short course on Classical Mechanics by Prof. J. J. Binney

Static mechanics (forces, tension); hydrostatics. Newton's Laws.
The elliptical orbits of planets. The many-body system.
The action principle. Hamilton's equations. The Lagrangean. (Don't skip - extremely important!)
The harmonic oscillator. The pendulum.
Poisson's brackets.
Wave equations. Liquids and gases. The Navier-Stokes equations. Viscosity and friction.

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Statistical Mechanics and Thermodynamics:

The course "Statistical Mechanics" by Alfred Huan

Prof. Kelly's lecture notes on Statistical Physics

Gould/Tobochnik lecture notes

Intermediate level course on Statistical Mechanics by R. Fitzpatrick

The first, second and third laws of thermodynamics.
The Boltzmann distribution.
The Carnot cycle. Entropy. Heat engines.
Phase transitions. Thermodynamical models.
The Ising Model (postpone techniques to solve the 2-dimensional Ising Model to later).
Planck's radiation law (as a prelude to Quantum Mechanics)

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Electromagnetism:

W.J.Spence, Electromagnetism

Notes on Classical Electromagnetism by R. Fritzpatrick.

Bo Thide's EM Field theory text(advanced)

Worked out exercises from Jackson's book,
set 1 / set 2

Maxwell's Theory for electromagnetism

homogeneous and inhomogeneous

Maxwell's laws in a medium. Boundaries. Solving the equations in:

vacumm and homogeneous medium (electromagnetic waves)
in a box (wave guides)
at boundaries (fraction and reflection)

The vector potential and gauge invariance (extremely important)
emission and absorption on EM waves (antenna)
light scattering against objects.

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Quantum Mechanics (Non-relativistic):

Introduction to QM and special relativity: Michael Fowler

Niels Walet lecture course on QM (Manchester)

Lecture Notes on QM from MIT

lecture notes

James Branson, Quantum Physics (UCSD)

Bohr's atom.
DeBroglie's relations (Energy-frequency, momentum-wave number)
Schrödinger's equation (with electric potential and magnetic field).
Ehrenfest's theorem.
A particle in a box.
The hydrogen atom, solved systematically. The Zeeman effect.Stark effect.
The quantum harmonic oscillator.
Operators: energy, momentum, angular momentum, creation and annihilation operators.
Their commutation rules.
Introduction to quantum mechanical scattering. The S-matrix. Radio-active decay.

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Advanced Mathematics:

See John Heinbockel, Virginia.
See Chr. Pope: Methods2§

Mathematics textbooks list

G.'t Hooft: Lie groups in Physics, (now also in English) + exercises

For Lie Groups, see also the last section of Chr. Pope's lectures (under "General Relativity")

The special functions and polynomials(PDF) (just understand the principles)

Group theory, and the linear representations of groups
Lie group theory
Vectors and tensors
More techniques to solve (partial) differential and integral equations
Extremum principle and approximation techniques based on that
Difference equations
Generating functions
Hilbert space
Introduction to the functional integral

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Advanced Quantum Mechanics:

Prof. Stringari's course on Ultracold Fluids

Introduction to the Quantum Hall effect by A.H. MacDonald

Introduction to Coherent States and Quantum Information Theory by K. Fujii

Tutorial on Quantum information by Peter Zoller

Intoduction to Quantum Computation by A. Chatterjee

Advanced QM by Freeman J. Dyson

K. Schulten's notes on advanced QM

Hans Paar, Advanced Quantum Theory

Hilbert space
Atomic transitions
Emission and absorption of light
Stimulated emission
Density matrix
Interpretation of QM
The Bell inequalities
Towards relativistic QM: The Dirac equation, finestructure
Electrons and positrons
BCS theory for supraconductivity
Quantum Hall effect
Advanced scattering theory
Dispersion relations
Perturbation expansion
WKB approximation, Extremum principle
Bose-Einstein condensation
Superliquid helium

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Quantum Field Theory:

Pierre van Baal's notes on QFT

The Conceptual Basis of Quantum Field Theory by G. 't Hooft
a chapter in Handbook of the Philosophy of Science

Magnetic monopoles and instantons

Classical fields: Scalar, Dirac-spinor, Yang-Mills vector fields.

Interactions, perturbation expansion. Spontaneous symmetry breaking, Goldstone mode, Higgs mechanism.

Particles and fields: Fock space. Antiparticles. Feynman rules. The Gell-Mann-Lévy sigma model for pions and nuclei. Loop diagrams. Unitarity, Causality and dispersion relations. Renormalization (Pauli-Villars; dimensional ren.) Quantum gauge theory: Gauge fixing, Faddeev-Popov determinant, Slavnov identities, BRST symmetry. The renormalization group. Asymptotic freedom.

Solitons, Skyrmions. Magnetic monopoles and instantons. Permanent quark confinement mechanism. The 1/N expansion. Operator product expansion. Bethe-Salpeter equation. Construction of the Standard Model. P and CP violation. The CPT theorem. Spin and statistics connection. Supersymmetry.

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Classical Mechanics:

Classical Mechanics - 3rd ed. - Goldstein, Poole & Safko
Classical dynamics: a contemporary approach - Jorge V. José, Eugene J. Saletan
Classical Mechanics - Systems of Particles and Hamiltonian Dynamics - W. Greiner
Mathematical Methods of Classical Mechanics, 2nd ed. - V.I. Arnold
Mechanics 3rd ed. - L. Landau, E. Lifshitz
Statistical Mechanics:
L. E. Reichl: A Modern Course in Statistical Physics, 2nd ed.
R. K. Pathria: Statistical Mechanics
M. Plischke & B. Bergesen: Equilibrium Statistical Physics
L. D. Landau & E. M. Lifshitz: Statistical Physics, Part 1
S.-K. Ma, Statistical Mechanics, World Scientific
Quantum Mechanics:
Quantum Mechanics - an Introduction, 4th ed. - W. Greiner
R. Shankar, Principles of Quantum Mechanics, Plenum
Quantum Mechanics - Symmetries 2nd ed. - W. Greiner, B. Muller
Quantum Mechanics - Vol 1&2 - Cohen-Tannoudji
J.J. Sakurai, Advanced Quantum Mechanics, Addison-Wesley
Electrodynamics:
J.D. Jackson, Classical Electrodynamics, 3rd ed., Wiley & Sons.
Electromagnetic Fields And Waves - lorrain and corson
Classical Electrodynamics - W. Greiner
Introduction to Electrodynamics - D. Griffiths
Quantum Electrodynamics - 3rd ed., - W. Greiner, J. Reinhardt
Optics:
Principles of Optics - M.Born, E. Wolf
Principles Of Nonlinear Optics - Y. R. Shen
Thermodynamics:
Thermodynamics and an Introduction to Thermostatistics 2ed - H. Callen
Thermodynamics and statistical mechanics - Greiner, Neise, Stoecker
Solid State Physics:
Solid State Physics - Ashcroft, Neil W, Mermin, David N
Introduction to Solid State Physics 7th edition- Kittel, Charles
Special Relativity:
Classical Mechanics - Point Particles And Relativity - W. Greiner
Introduction to the theory of relativity and the principles of modern physics - H. Yilmaz
General Relativity:
J.B. Hartle, Gravity, An Introduction to Einstein's General Relativity, Addison Wesley, 2003.
T.-P. Cheng, Relativity, Gravitation and Cosmology, A Basic Introduction, Oxford Univ. Press, 2005.
Particle Physics:
Introduction to Elementary Particles - D. Griffiths
Fundamentals in Nuclear Physics - From Nuclear Structure to Cosmology - Basdevant, Rich, Spiro
Field Theory:
B. de Wit & J. Smith, Field Theory in Particle Physics, North-Holland
C. Itzykson & J.-B. Zuber, Quantum Field Theory, McGraw-Hill.
String Theory:
Barton Zwiebach, A First Course in String Theory, Cambridge Univ. Press, 2004
M.B. Green, J.H. Schwarz & E. Witten, Superstring theory, Vols. I & II, Cambridge Univ. Press
Cosmology:
An Introduction to cosmology, 3rd Ed – Roos
Relativity, thermodynamics, and cosmology - Tolman R.C.
General:
J.B. Marion & W.F. Hornyak, Principles of Physics, Saunders College Publishing, 1984, ISBN 0-03-049481-8
H. Margenau and G.M. Murphy, The Mathematics of Physics and Chemistry, D. v.Nostrand Comp.
R. Baker, Linear Algebra, Rinton Press

Find lists of other useful textbooks here: Mathematics, Physics (most of these are rather for amusement than being essential for understanding the World),
or a little bit more seriously: Physics.

There already is some response. I thank: Rob van Linden, Robert Tough, Thuy Nguyen, Tina Witham, Jerry Blair, Jonathan Martin, David Cuthbertson, Trent Strong, and many others.

Mr. Hisham Kotry came with an important question:

"...You sketch the path for potential students through the forest of college level physics... Two years ago I decided to self-study theoretical physics by following the syllabus of a renown university and the advice from your page and now I'm half-way through the journey but I was wondering about what happens next? Quoting you from the former page "In short, it is for all those who decided to study theoretical physics, in their own time.", Do you know of anyone who got tenure at a physics department or any research institute based on studies he did in his own time without holding a university degree?"

This is not so easy to answer, unfortunately. What I can say, is:
Eventually, whether you like it or not, you will have to obtain some University degree, if you wish a self-supporting career in theoretical Physics. One possibility is to follow a Master course such as the one offered by our University. I don't know about your qualifications, but I suspect that, with enough determination, you may be able to comply.

This is not a burocratic argument but a very practical one. It is also advisable not to wait until you think your self-study is completed. You must allow your abilities to be tested, so that you get the recognition that you may well deserve. Also, I frequently meet people who get stuck at some point. Only by intense interactions with teachers and peers one can help oneself across such barriers. I have not yet met anyone who could do the entire study all by him/herself without any guidance. If you really think you have reached a professional level in your studies, you can try to get admitted to schools, conferences and workshops in topics of your interest.

3/04/06: Message received from John Glasscock, Bloomington, IN:
The only one I know of currently is John Moffatt at U Toronto, who was a student of Abdus Salam at Imperial College, London. He started life as a painter in Paris, had no undergraduate degree, taught himself, corresponded with Einstein, and was admitted, based on his demonstrated original work, at IC. (Source: João Magueijo, _Faster than the Speed of Light_. Perseus Publishing, Cambridge, MA. 2003.)

Suggestions for further lecture notes from Alvaro Véliz:

1. The archimedeans webpage: It has a lot of lecture notes in Physics and Mathematics from Part I and II from Cambridge.

2. David Tong's (DAMTP) lectures in classical mechanics I found these lectures fascinating.

3. Bo Thide's (Uppsala) lectures in electromagnetic theory.

4. Angel Uranga's lectures in String Theory._

5. I found also extremely helpful MIT's OpenCourseWare: Lewin's lectures in basic Physics are terrific (in video).

6. Michael Fowler's (Virginia) lectures in Quantum Mechanics
(not checked, G. 't H.)

I thank Aldemar Torres Valderrama for his assistance in updating and renewing numerous links on this page.

§ Note that .ps files are PostScript files, so defend yourself against Microsoft PaintShop, that often wants to appropriate .ps files. PostScript files are read using GhostView (gsview)