**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**

**1. Linear and abstract algebra**

**Elementary**

Hoffman and Kunze, Linear Algebra

**Introductory**

Mac Lane and Birkhoff, Algebra

Halmos, Finite-dimensional linear algebra

Dummit and Foote, Abstract algebra

**2. Real analysis and elements of complex analysis**

**Elementary**

Spivak, Calculus

Piskunov, Differential and integral calculus, vols 1 & 2

Pugh, Real mathematical analysis

Palka, An introduction to complex function theory

**Introductory**

2.1 Apostol, Calculus, vols 1 and 2

Hardy, A course of pure mathematics

2.2 Courant, Differential and integral calculus, vols 1 & 2

Kolmogorov and Fomin, Introductory real analysis

Priestley, Introduction to complex analysis

2.3 Rudin, Principles of mathematical analysis

Conway, Functions of one complex variable

Stein and Shakarchi, Complex analysis

**3. Ordinary differential equations (ODEs)**

**Elementary**

Zill et al, Differential equations with boundary value problems

Brenner, Problems in differential equations

**Introductory**

Arnold, Ordinary differential equations

Perko, Differential equations and dynamical systems

Hirsch and Smale, Differential equations, dynamical systems and an introduction to chaos

**4. Partial differential equations (PDEs)**

**Elementary**

Olver, Introduction to partial differential equations

Berg and Mc. Gregor, Elementary partial differential equations

**Introductory**

Strauss, Partial differential equations: an introduction

Courant and Hilbert, Methods of mathematical physics, vols 1 & 2

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

**Elementary**

Steinberg, Representation theory of finite groups: an introductory approach

Hall, Lie groups, Lie algebras and their representations

**Introductory**

Serre, Linear representations of finite groups

Brocker, Representations of compact Lie groups

**6. Functional analysis and operator theory **

**Elementary**

Bachman and Narici, Functional analysis

Retherford, Hilbert space:compact operators and the trace theorem

**Introductory**

Kresyzig, Introductory functional analysis with applications

Young, An introduction to Hilbert space

Helmberg, Introduction to spectral theory in Hilbert space

Halmos, A Hilbert space problem book

**7. Differential geometry**

**Elementary**

O'Neill, Elementary differential geometry

Topogonov, Differential geometry of curves and surfaces

Lee, Introduction to smooth manifolds

**Introductory**

Munkres, Topology

Spivak, A comprehensive introduction to differential geometry, vol 1

Lee, Riemannian manifolds: an introduction to curvature