Minimal curriculum

Minimal undergraduate curriculum

Modern physics consists of five domains, which are distinguished by their focus, methods and intent:

I. Theoretical physics

II. Phenomenology

III. Numerical physics (eg. lattice QCD, numerical gravity)

IV. Experimental and observational physics (eg. high energy experiment, observational astrophysics)

V. Applied and industrial physics

Of these, phenomenology and numerical physics are closer to theoretical than to experimental and observational physics, though they are less concerned with conceptual questions or mathematical rigor. Each of the five domains is further subdivided according to subject.

Competence in any of these five domains requires a good grasp of fundamental physics, which is the main focus of standard undergraduate programs. It is impossible to be a good phenomenologist, numerical physicist, experimental or observational physicist or even applied or industrial physicist without having a good grasp of the theoretical and mathematical underpinnings of fundamental physics.

A specialization in theoretical physics requires not only an excellent grasp of fundamental physics but also competence in advanced subjects (including areas of advanced mathematics) which are not covered at undergraduate level. Mathematical physics requires an excellent understanding of mathematics, which goes beyond what is expected of theoretical physicists.

A student interested in theoretical or mathematical physics will benefit much more from learning mathematics during undergraduate years than from focusing on lab work, since advanced theoretical physics subjects require graduate level mathematics. Theoretical and mathematical physicists do not deal with experiments or experimental data directly.

A standard undergraduate course in fundamental physics covers five core physics subjects:

1. Classical mechanics

2. Thermodynamics and statistical physics

3. Electromagnetism and special relativity

4. Quantum mechanics

5. General relativity and elements of cosmology

as well as the following seven core mathematics subjects:

1. Linear algebra and elements of abstract algebra

2. Real analysis and elements of complex analysis (the elementary part of analysis includes what in the US system is called "calculus")

3. ODEs

4. PDEs

5. Finite groups, Lie groups, Lie algebras and their representations

6. Functional analysis and operator theory

7. Differential geometry.

Each of these subjects can be covered at four levels:

Level 0: Elementary

Level 1: Introductory

Level 2: Intermediate

Level 3: Advanced.

Levels 2 and 3 correspond to the advanced undergraduate/beginning graduate and advanced graduate levels respectively. "Level -1" indicated below is preparatory, i.e. material that should normally be covered in high school. References at intermediate and advanced levels can be found here and here. Students who find the references below too easy are encouraged to try those instead. Students who find the elementary level too easy should jump directly to the introductory, intermediate or advanced levels.

Note. The minimal undergraduate curriculum detailed below is far from sufficient for doing quality research in any area of modern theoretical or mathematical physics. Students who wish to work in those subjects will have to learn much more at the graduate level. They are encouraged to try the intermediate and advanced levels during their undergraduate years on any topic which is of particular interest to them. Students who are serious about mathematical physics are encouraged to take at least some classes in the Mathematics department of their university, ideally as a replacement for elective labs or for elective courses concerned with subjects that fall outside the core areas of theoretical physics.

Some elementary (Level 0) and introductory (Level 1) references

A. Physics

Some standard elementary and introductory series to undergraduate physics are listed here and will not be repeated below. They cover all core physics subjects except for general relativity.

Elementary

Halliday, Resnick and Walker, Fundamentals of Physics

Alonso and Finn, Physics

Introductory

The Berkeley Physics Course, vols 1-4

Other recommended books by subject and level (out of four possible levels, 0-3), listed in approximate order of increasing difficulty.

Note. In some cases, introductory references are grouped by sublevels of difficulty which are indicated in increasing order.

1. Classical mechanics

Elementary

Kleppner and Kolenkov, An introduction to Mechanics

Introductory

1.1 Taylor, Classical mechanics

1.2 Goldstein et al, Classical Mechanics

Arnold, Mathematical Methods of Classical Mechanics

2. Thermodynamics and statistical physics

Elementary

Callen, Thermodynamics and an introduction to thermostatistics

Schroeder, An introduction to thermal physics

Kittel and Kroemer, Thermal physics

Introductory

2.1 Rief, Fundamentals of statistical and thermal physics

2.2 Kardar, Statistical physics of particles

Huang, Statistical mechanics

3. Electromagnetism and special relativity

Elementary

Griffiths, Introduction to electrodynamics

Introductory

Greiner, Classical electrodynamics

Zangwill, Modern electrodynamics

Brau, Modern problems in classical electrodynamics

Scheck, Classical field theory

French, Special relativity

Taylor and Wheeler, Spacetime physics

Resnick, Introduction to special relativity

Tsamparlis, Special relativity

4. Quantum mechanics

Elementary

Eisberg and Resnick, Quantum physics

Zettili, Quantum mechanics

Introductory

4.1 Townsend, A modern approach to quantum mechanics

Griffiths, Introduction to quantum mechanics

4.2 Shankar, Principles of quantum mechanics

Messiah, Quantum mechanics, vols 1 & 2

5. General relativity and elements of cosmology

Elementary

Hartle, Gravity: an introduction to Einstein's general relativity

Zee, Einstein gravity in a nutshell

Introductory

5.1 Carroll, Spacetime and geometry: an introduction to general relativity

Schutz, A first course in general relativity

5.2 Wald, General relativity

5.3. Hughston and Tod, An introduction to general relativity

Natario, An introduction to mathematical relativity

B. Mathematics

Preparatory and foundational subjects

Note. One cannot understand mathematics without a good grasp on its logical methods and basic proof theory. All aspiring theoretical physicists should be fluent in reading, understanding and composing mathematical proofs. They should also be able to read pure mathematics literature with ease. This is doubly so for mathematical physicists.

-1. Structure and basic methods of mathematical proofs

Hammack, Book of Proof

Lakatos, Proofs and refutations

Stewart and Tall, The foundations of mathematics

0. Set-theoretic foundations

Elementary

Halmos, Naive set theory

Introductory

Fraenkel and Bar-Hillel, Foundations of set theory

Kleene, Mathematical logic

Fraenkel, Abstract set theory

Core undergraduate subjects