by Gerard 't Hooft
Note: This web site will soon be removed from its
present address. An updated and renewed version is available at:
http://www.staff.science.uu.nl/~Gadda001/goodtheorist/index.html
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.
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.
Return to List
Now, first things first :
Primary Mathematics:
Are you comfortable with numbers, adding, subtracting, square roots, etc.?
- 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!
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.
Return to List
Classical Mechanics:
- 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.
Return to List
Optics:
- fraction and reflection.
- lenses and mirrors.
- The telescope and the microscope.
- Introduction to wave propagation.
- Doppler effect.
- Huijgens' principle of wave superposition.
- Wave fronts.
- Caustics.
Return to List
Statistical Mechanics and Thermodynamics:
- 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)
Return to List
Electronics:
(Only some very basic things about electronic circuits)
- Ohm's law, capacitors, inductors, using complex numbers to calculate
their effects.
- Transistors, diodes (how these actually work comes later).
Return to List
Electromagnetism:
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.
Return to List
Computational Physics :
Even the pure sang theorist may be interested in some aspects of
Computational physics.
Return to List
Quantum Mechanics (Non-relativistic):
- 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.
Return to List
Atoms and Molecules:
- Chemical binding
- Orbitals
- Atomic and molecular spectra
- Emission and absorption of light.
- Quantum selection rules
- Magnetic moments.
Return to List
Solid State Physics:
- Crystal groups
- Bragg reflection
- Dielectric and diamagnetic constants
- Bloch spectra
- Fermi level
- Conductors, semiconductors and insulators
- Specific heat
- Electrons and holes
- The transistor
- Supraconductivity
- Hall effect.
Return to List
Nuclear Physics
- Isotopes
- Radio-activity
- Fission and fusion
- Droplet model
- Nuclear quantum numbers
- Magic nuclei
- Isospin
- Yukawa theory
Return to List
Advanced Mathematics:
- 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
Return to List
Special Relativity
- The Lorentz transformation
- Lorentz contraction, time dilatation
- E =
mc^{2}
- 4-vectors and 4-tensors
- Transformation rules for the Maxwell field
- Relativistic Doppler effect
Return to List
Advanced Quantum Mechanics:
- 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
Return to List
Phenomenology:
subatomic particles (mesons, baryons, photons, leptons, quarks) and
cosmic rays; property of materials and chemistry; nuclear isotopes; phase
transitions; astrophysics (planetary system, stars, galaxies, red shifts,
supernovae); cosmology (cosmological models, inflationary universe theories,
microwave background radiation); detection techniques.
Return to List
General Relativity:
- The metric tensor
- Space-time curvature
- Einstein's gravity equation
- The Schwarzschild black hole
- Reissner-Nordström black hole
- Periastron shift
- Gravitational lensing
- Cosmological models
- Gravitational radiation
Return to List
Quantum Field Theory:
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.
Return to List
There are many more
lecture notes to be found on the web.
There are numerous good books on all sorts of topics in Theoretical Physics.
Just to name a few:
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: Joăo 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.
5. I found also extremely helpful
MIT's OpenCourseWare: Lewin's lectures in basic Physics are terrific
(in video).
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)
Back to homepage.
Your visitor number is: (since 21/8/03)
Last revised: 2009
Design by CJ