# Theoretical Physics Reference guide based on:

How to become a GOOD Theoretical Physicist by Gerard 't Hooft

## Table of Contents

- 1. Languages
- 2. Primary mathematics
- 2.1. Natural numbers.
- 2.2. Integers.
- 2.3. Rational numbers.
- 2.4. Real numbers.
- 2.5. Complex numbers.
- 2.6. Set theory: open sets, compacts spaces. Topology.
- 2.7. Algebraic equations. Approximations techniques. Series expansions: Taylor series.
- 2.8. Solving equations with complex numbers. Trigonometry.
- 2.9. Infinitesimals. Differentiation. Differentiate basic functions.
- 2.10. Integration. Integrate basic functions. Differential equations. Linear equations.
- 2.11. Fourier transformation. The use of complex numbers. Convergence of series.
- 2.12. The complex plane. Cauchy theorems and contour integration.
- 2.13. The gamma function.
- 2.14. Gaussian integrals. Probability theory.
- 2.15. Partial differential equations. Dirichlet and Neumann boundary conditions.

- 3. Classical Mechanics
- 3.1. Static mechanics (force, tension). Hydrostatics. Newton's Laws.
- 3.2. The elliptical orbits of planets. The many body system.
- 3.3. The action principle. Hamilton’s equations. The Lagrangean.
- 3.4. The harmonic oscillator. The pendulum.
- 3.5. Poisson’s brackets.
- 3.6. Wave equations. Liquids and gases. The Navier-Stokes equations. Viscosity and friction.

- 4. Optics
- 5. Statistical Mechanics & Thermodynamics
- 5.1. The first, second and third laws of thermodynamics.
- 5.2. The Boltzmann distribution.
- 5.3. The Carnot cycle. Entropy. Heat engines.
- 5.4. Phase transitions. Thermodynamical models.
- 5.5. The Ising Model (postpone techniques to solve the 2-dimensional Ising Model to later).
- 5.6. Planck’s radiation law (as a prelude to Quantum Mechanics).

- 6. Electronics
- 7. Electromagnetism
- 7.1. Maxwell’s Theory for electromagnetism: Homogeneous and inhomogeneous .
- 7.2. 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).
- 7.3. The vector potential and gauge invariance (extremely important).
- 7.4. Emission and absorption on EM waves (antenna).
- 7.5. Light scattering against objects

- 8. Computational Physic
- 9. Quantum Mechanics (Non-relativistic).
- 9.1. Bohr’s atom
- 9.2. De Broglie’s relations (Energy-frequency, momentum-wave number).
- 9.3. Schrödinger’s equation (with electric potential and magnetic field).
- 9.4. Ehrenfest’s theorem.
- 9.5. A particle in a box.
- 9.6. The hydrogen atom, solved systematically. The Zeeman effect. Stark effect.
- 9.7. The quantum harmonic oscillator.
- 9.8. Operators: energy, momentum, angular momentum, creation and annihilation operators.
- 9.9. Their commutation rules.
- 9.10. Introduction to quantum mechanical scattering. The S-matrix. Radio-active decay.

- 10. Atoms & Molecules
- 11. Solid State Physics
- 12. Nuclear Physics
- 13. Plasma Physics
- 14. Advanced Mathematics
- 14.1. Group theory, and the linear representations of groups.
- 14.2. Lie group theory.
- 14.3. Vectors and tensors.
- 14.4. More techniques to solve (partial) differential and integral equations.
- 14.5. Extremum principle and approximation techniques based on that.
- 14.6. Difference equations.
- 14.7. Generating functions.
- 14.8. Hilbert space.
- 14.9. Introduction to the functional integral.

- 15. Special Relativity
- 16. Advanced Quantum Mechanics
- 16.1. Hilbert space.
- 16.2. Atomic transitions.
- 16.3. Emission and absorption of light.
- 16.4. Stimulated emission.
- 16.5. Density matrix.
- 16.6. Interpretation of QM.
- 16.7. The Bell inequalities.
- 16.8. Towards relativistic QM: The Dirac equation, finestructure.
- 16.9. Electrons and positrons.
- 16.10. BCS theory for superconductivity.
- 16.11. Quantum Hall effect.
- 16.12. Advanced scattering theory.
- 16.13. Dispersion relations.
- 16.14. Perturbation expansion.
- 16.15. WKB approximation, Extremum principle.
- 16.16. Bose-Einstein condensation
- 16.17. Superliquid helium

- 17. Phenomenology/Astroparticles
- 18. General Relativity
- 19. Cosmology
- 20. Astrophysics
- 21. Quantum Field Theory
- 21.1. Classical fields: Scalar, Dirac-spinor, Yang-Mills vector fields.
- 21.2. Interactions, perturbation expansion. Spontaneous symmetry breaking, Goldstone mode, Higgs mechanism
- 21.3. 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.
- 21.4. 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.

- 22. Supersymmetry/Supergravity

## 1 Languages

- Spell checkers: `http://aspell.net/`
- Grammar checkers: LanguageTool — Open source language checker, written in Java. `https://www.languagetool.org/`
- Vocabulary: `https://www.vocabulary.com/`
- Bilingual Dictionary. Linguee `https://www.linguee.com/`

- The
**acquisition-learning hypothesis**is a**hypothesis**that forms part of*Stephen Krashen*'s

theory of second language

**acquisition**. It states that there are two independent ways in which we develop our linguistic skills:**acquisition**and learning. According to Krashen acquisition is more important than learning. `https://en.wikipedia.org/wiki/Input_hypothesis`- Learn through audiobook:

- Librivox: free public domain audiobooks `https://librivox.org/`

- Learn reading:

- Open library: `https://openlibrary.org/`

- Learn through audiobook:
- Learning:

- Duoling: `https://www.duolingo.com`
- Coffe Break: `https://radiolingua.com/`
- Anki. Anki is a program which makes remembering things easy: `https://apps.ankiweb.net/`

### 1.1 **English**.

English is a prerequisite. All sciences nowadays are in English.

- Lexical database of English: Wordnet `https://wordnet.princeton.edu/`

- Learning:

- BBC learning English: `https://www.bbc.co.uk/learningenglish/`
- engVid free english lessons: `https://www.engvid.com/`

### 1.2 Some French, German, Spanish and Italian may be useful.

- Learning:

- DW Learning German: `https://learngerman.dw.com/en/overview`, `https://www.youtube.com/user/dwlearngerman`
- BBC learning French: `http://www.bbc.co.uk/languages/french/`
- Coffe Break Italian: `https://radiolingua.com/coffeebreakitalian/`

## 2 Primary mathematics

### 2.1 Natural numbers.

### 2.2 Integers.

### 2.3 Rational numbers.

### 2.4 Real numbers.

### 2.5 Complex numbers.

### 2.6 Set theory: open sets, compacts spaces. Topology.

### 2.7 Algebraic equations. Approximations techniques. Series expansions: Taylor series.

### 2.8 Solving equations with complex numbers. Trigonometry.

### 2.9 Infinitesimals. Differentiation. Differentiate basic functions.

### 2.10 Integration. Integrate basic functions. Differential equations. Linear equations.

### 2.11 Fourier transformation. The use of complex numbers. Convergence of series.

### 2.12 The complex plane. Cauchy theorems and contour integration.

### 2.13 The gamma function.

### 2.14 Gaussian integrals. Probability theory.

### 2.15 Partial differential equations. Dirichlet and Neumann boundary conditions.

## 3 Classical Mechanics

### 3.1 Static mechanics (force, tension). Hydrostatics. Newton's Laws.

### 3.2 The elliptical orbits of planets. The many body system.

### 3.3 The action principle. Hamilton’s equations. The Lagrangean.

### 3.4 The harmonic oscillator. The pendulum.

### 3.5 Poisson’s brackets.

A function \(f(p, q, t)\) of the phase space coordinates of the system and time as total time derivative

\[ \frac{d f}{d t}=\frac{\partial f}{\partial t}+\sum_{i}\left(\frac{\partial f}{\partial q_{i}} \dot{q}_{i}+\frac{\partial f}{\partial p_{i}} \dot{p}_{i}\right), \]

often written as

\[ \frac{d f}{d t}=\frac{\partial f}{\partial t} + \{H, f\}, \]

where \(\{H, f\}\) is the *Poisson bracket*. In canonical coordinates \((q_i, p_i)\) given (*any*) two functions \(f(p_i, q_i, t)\) and \(g(p_i, q_i, t)\) the
Poisson bracket takes the form,

\[ \{f, g\}=\sum_{i=1}^{N}\left(\frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}}-\frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial q_{i}}\right). \]

The canonical coordinates satisfy the fundamental Poisson bracket relations

- \(\left\{q^{i}, q^{j}\right\}=0\)
- \(\left\{p_{i}, p_{j}\right\}=0\)
- \(\left\{q_{i}, p_{j}\right\}=\delta_{i j}\)

The Poisson bracket has the following properties:

- \(\{f, g\} = -\{g, f\}\),
- \(\{f, c\} = 0\),
- \(\{f, q_i\} = -\frac{\partial f}{\partial p_i}\),
- \(\{f, p_i\} = \frac{\partial f}{\partial q_i}\),
- \(\{f_1+f_2, g\} = \{f_1, g\} + \{f_2, g\}\),
- \(\{f_1f_2, g\} = f_1\{f_2, g\} + \{f_1, g\}f_2\),
- \(\frac{\partial}{\partial t}\{f, g\} = \left\{\frac{\partial f}{\partial t}, g\right\} + \left\{f, \frac{\partial g}{\partial t}\right\}\),

In case two, \(c\) is a constant of the motion, also called an *integral of the motion*.

An important identity satisfied by the Poisson brackets is the **Jacob identity**

\[ \{f,\{g, h\}\} + \{g,\{h, f\}\} + \{h,\{f, g\}\} = 0. \]

### 3.6 Wave equations. Liquids and gases. The Navier-Stokes equations. Viscosity and friction.

## 4 Optics

### 4.1 Fraction and reflection.

### 4.2 Lenses and mirrors.

### 4.3 The telescope and the microscope.

### 4.4 Introduction to wave propagation.

### 4.5 Doppler effect.

### 4.6 Huijgens’ principle of wave superposition.

### 4.7 Wave fronts.

### 4.8 Caustics

## 5 Statistical Mechanics & Thermodynamics

### 5.1 The first, second and third laws of thermodynamics.

### 5.2 The Boltzmann distribution.

### 5.3 The Carnot cycle. Entropy. Heat engines.

### 5.4 Phase transitions. Thermodynamical models.

### 5.5 The Ising Model (postpone techniques to solve the 2-dimensional Ising Model to later).

### 5.6 Planck’s radiation law (as a prelude to Quantum Mechanics).

## 6 Electronics

### 6.1 Ohm’s law, capacitors, inductors, using complex numbers to calculate their effects.

### 6.2 Transistors, diodes.

## 7 Electromagnetism

### 7.1 Maxwell’s Theory for electromagnetism: Homogeneous and inhomogeneous .

### 7.2 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).

### 7.3 The vector potential and gauge invariance (extremely important).

### 7.4 Emission and absorption on EM waves (antenna).

### 7.5 Light scattering against objects

## 8 Computational Physic

### 8.1 Numerical Integration and Differentiation

#### 8.1.1 Derivative by:

- Finite difference `https://en.wikipedia.org/wiki/Finite_difference_method`

The derivative of a function \(f(x)\) can be approximated by the

*forward difference*based on the size*\(h\)*\[ f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \approx \frac{f(x+h)-f(x)}{h} - \frac{h}{2}f^{\prime\prime}(\xi). \]

First order of a Taylor expansion of \(f(x)\) with truncation error \(O(h) = hf(\xi)^{\prime\prime}/2\) linearly related to \(h\). The approximation is poor unless \(h\) is sufficient small. A final expression of this example and its order

\[ \frac{f(x+h)-f(x)}{h} = f^{\prime}(x) + O(h). \]

The derivative can be also approximated by the

*backward difference*, replacing \(h\) by \(-h\) in the Taylor expansion,\[ f^{\prime}(x) \approx \frac{f(x)-f(x-h)}{h}. \]

Subtracting the backward difference from the forward difference and dividing everything by \(2h\), we find the

*central difference*\[ \begin{aligned} f^{\prime}(x) &=\frac{f(x+h)-f(x-h)}{2 h}-\left(\frac{f^{\prime \prime \prime}(x)}{3 !} h^{2}+\frac{f^{(5)}(x)}{5 !} h^{4}+\cdots\right) \\ & \approx \frac{f(x+h)-f(x-h)}{2 h}, \end{aligned} \] with the second order truncation error \(O(h)\). Adding the forward difference to the backward difference and solving for \(f(x)^{\prime\prime}\) we can get the approximation for the second order derivative

\[ \boxed{f(x)^{\prime\prime} \approx \frac{f(x-h)-2 f(x)+f(x+h)}{h^{2}}} \]

- Interpolation

`https://en.wikipedia.org/wiki/Interpolation` `https://en.wikipedia.org/wiki/Linear_interpolation`

The derivative of a given function \(f(t)\) can be approximated by that of its polynomial interpolation based on a set of \(n+1\) points \(y_i = f(x_i), (i=0,\cdots, n)\), such as the Lagrange polynomial interpolation (`https://en.wikipedia.org/wiki/Polynomial_interpolation`)

\[ f(x) \approx L_{n}(x)=\sum_{i=0}^{n} y_{i} l_{i}(x). \]

Taking the derivative of \(f(x)\),

\[ f^{\prime}(x) \approx L_{n}^{\prime}(x)=\sum_{i=0}^{n} y_{i} l_{i}^{\prime}(x)=\sum_{i=0}^{n} y_{i} d_{i}(x), \] where

\[ d_{i}(x)=\frac{d}{d x} l_{i}(x)=\frac{d}{d x}\left(\prod_{j=0, j \neq i}^{n} \frac{x-x_{j}}{x_{i}-x_{j}}\right). \]

The error is

\[ \begin{aligned} E_{n}(x)=R_{n}^{\prime}(x) &=\frac{1}{(n+1) !} \frac{d}{d x}\left(\left(f^{(n+1)}(\xi(x)) l(x)\right)\right.\\ &=\frac{1}{(n+1) !}\left[l^{\prime}(x) f^{(n+1)}(\xi(x))+l(x) \frac{d}{d x} f^{(n+1)}(\xi(x))\right]. \end{aligned} \]

The error is unknown in general since \(\xi\) is not known explicit. But we can have for each node \(x_i\) where \(w(x_i)\),

\[ E_{n}(x)=R_{n}^{\prime}(x)=l^{\prime}(x) \frac{f^{(n+1)}(\xi(x))}{(n+1) !}. \]

- Special cases:

- \(n=1\), linear interpolation based on \(n+1 = 2\) points:

\[ \begin{array}{l} d_{0}=l_{0}^{\prime}(x)=\frac{d}{d x}\left(\frac{x-x_{1}}{x_{0}-x_{1}}\right)=\frac{1}{x_{0}-x_{1}}, \\ d_{1}=l_{1}^{\prime}(x)=\frac{d}{d x}\left(\frac{x-x_{0}}{x_{1}-x_{0}}\right)=\frac{1}{x_{1}-x_{0}}, \end{array} \]

and

\[ f^{\prime}(x) \approx L_{1}^{\prime}(x)=y_{0} d_{0}(x)+y_{1} d_{1}(x)=\frac{y_{1}-y_{0}}{x_{1}-x_{0}}, \quad f^{\prime \prime}(x)=0. \]

- \(n=2\), quadratic interpolation based on \(n+1 = 3\) points:

\[ \begin{aligned} d_{0}(x) &=l_{0}^{\prime}(x)=\frac{d}{d x}\left(\frac{\left(x-x_{1}\right)\left(x-x_{2}\right)}{\left(x_{0}-x_{1}\right)\left(x_{0}-x_{2}\right)}\right)=\frac{2 x-x_{1}-x_{2}}{\left(x_{0}-x_{1}\right)\left(x_{0}-x_{2}\right)}, \\ d_{1}(x) &=l_{1}^{\prime}(x)=\frac{d}{d x}\left(\frac{(x-x_{0})(x-x_{2})}{\left(x_1-x_{0}\right)\left(x_1-x_{2}\right)}\right)=\frac{2 x-x_{0}-x_{2}}{\left(x_{1}-x_{0}\right)\left(x_{1}-x_{2}\right)}, \\ d_{2}(x) &=l_{2}^{\prime}(x)=\frac{d}{d x}\left(\frac{\left(x-x_{0}\right)\left(x-x_{1}\right)}{\left(x_{2}-x_{0}\right)\left(x_{2}-x_{1}\right)}\right)=\frac{2 x-x_{0}-x_{1}}{\left(x_{2}-x_{0}\right)\left(x_{2}-x_{1}\right)}, \end{aligned} \]

and

\[ \begin{aligned} & f^{\prime}(x) \approx L_{2}^{\prime}(x)=\sum_{i=1}^{2} d_{i} y_{i} \\ =& \frac{2 x-x_{1}-x_{2}}{\left(x_{0}-x_{1}\right)\left(x_{0}-x_{2}\right)} y_{0}+\frac{2 x-x_{0}-x_{2}}{\left(x_{1}-x_{0}\right)\left(x_{1}-x_{2}\right)} y_{1}+\frac{2 x-x_{0}-x_{1}}{\left(x_{2}-x_{0}\right)\left(x_{2}-x_{1}\right)} y_{2}. \end{aligned} \]

If the 3 points are equally spaced with \(x_1=x, x_0 = x - h, x_2 = x + h\) one can have

\[ f^{\prime}(x) \approx \frac{x-(x+h)}{2 h^{2}} y_{0}+\frac{2 x-(x-h)-(x+h)}{-h^{2}} y_{1}+\frac{x-(x-h)}{2 h^{2}}=\frac{1}{2 h}\left(y_{2}-y_{0}\right). \]

with error \(O(h^2)\).

The 2nd order derivative:

\[ \begin{aligned} & f^{\prime \prime}(x) \approx L_{2}^{\prime \prime}(x)=\frac{d}{d x} L_{2}^{\prime}(x) \\ =& \frac{d}{d x}\left[\frac{2 x-x_{1}-x_{2}}{\left(x_{0}-x_{1}\right)\left(x_{0}-x_{2}\right)} y_{0}+\frac{2 x-x_{0}-x_{2}}{\left(x_{1}-x_{0}\right)\left(x_{1}-x_{2}\right)} y_{1}+\frac{2 x-x_{0}-x_{1}}{\left(x_{2}-x_{0}\right)\left(x_{2}-x_{1}\right)} y_{2}\right] \\ =& 2\left[\frac{y_{0}}{\left(x_{0}-x_{1}\right)\left(x_{0}-x_{2}\right)}+\frac{y_{1}}{\left(x_{1}-x_{0}\right)\left(x_{1}-x_{2}\right)}+\frac{y_{2}}{\left(x_{2}-x_{0}\right)\left(x_{2}-x_{1}\right)}\right] \end{aligned} \]

If the 3 points are equally spaced we have

\[ f^{\prime}(x) \approx \frac{2 x-x_{1}-x_{2}}{2 h^{2}} y_{0}-\frac{2 x-x_{0}-x_{2}}{h^{2}} y_{1}+\frac{2 x-x_{0}-x_{1}}{2 h^{2}} y_{2}, \]

and

\[ \begin{aligned} f^{\prime}\left(x_{0}\right)&=\frac{1}{2 h}\left(-3 y_{0}+4 y_{1}-y_{2}\right), \\ f^{\prime}\left(x_{1}\right)&=\frac{1}{2 h}\left(-y_{0}+y_{2}\right), \\ f^{\prime}\left(x_{2}\right)&=\frac{1}{2 h}\left(y_{0}-4 y_{1}+3 y_{2}\right). \end{aligned} \]

The second derivative is

\[ f^{\prime \prime}(x) \approx \frac{d}{d x}\left[\frac{2 x-x_{1}-x_{2}}{2 h^{2}} y_{0}-\frac{2 x-x_{0}-x_{2}}{h^{2}} y_{1}+\frac{2 x-x_{0}-x_{1}}{2 h^{2}} y_{2}\right] \] \[ \boxed{f^{\prime \prime}(x) = \frac{1}{h^{2}}\left(y_{0}-2 y_{1}+y_{2}\right)}. \]

- Special cases:

### 8.2 Programming languages

#### 8.2.1 Julia

- Packages

- Latexify.jl

This is a package for generating LaTeX maths from julia objects.

- SugarBLAS

This package provides macros for BLAS functions representing polynomials

- Flux

Flux is an elegant approach to machine learning.

- DifferentialEquations.jl

This is a suite for numerically solving differential equations written in Julia and available for use in Julia, Python, and R.

- Knet

Knet (pronounced "kay-net") is the Koç University deep learning framework implemented in Julia by Deniz Yuret and collaborators. It supports GPU operation and automatic differentiation using dynamic computational graphs for models defined in plain Julia.

- Latexify.jl

## 9 Quantum Mechanics (Non-relativistic).

### 9.1 Bohr’s atom

### 9.2 De Broglie’s relations (Energy-frequency, momentum-wave number).

### 9.3 Schrödinger’s equation (with electric potential and magnetic field).

### 9.4 Ehrenfest’s theorem.

### 9.5 A particle in a box.

### 9.6 The hydrogen atom, solved systematically. The Zeeman effect. Stark effect.

### 9.7 The quantum harmonic oscillator.

### 9.8 Operators: energy, momentum, angular momentum, creation and annihilation operators.

### 9.9 Their commutation rules.

### 9.10 Introduction to quantum mechanical scattering. The S-matrix. Radio-active decay.

## 10 Atoms & Molecules

### 10.1 Chemical binding.

### 10.2 Orbitals.

### 10.3 Atomic and molecular spectra.

### 10.4 Emission and absorption of light.

### 10.5 Quantum selection rules.

### 10.6 Magnetic moments.

## 11 Solid State Physics

### 11.1 Crystal groups.

### 11.2 Bragg reflection.

### 11.3 Dielectric and diamagnetic constants.

### 11.4 Bloch spectra.

### 11.5 Fermi level.

### 11.6 Conductors, semiconductors and insulators.

### 11.7 Specific heat.

### 11.8 Electrons and holes.

### 11.9 The transistor.

### 11.10 Superconductivity.

### 11.11 Hall effect.

## 12 Nuclear Physics

### 12.1 Isotopes. Radio-activity. Fission and Fusion.

#### 12.1.1 Geiger-Nuttall law

The Geinger-Nuttall relates the decay constant of a radioactive isotope with the energy of the alpha particles.

\[ \log _{10} \lambda=-a_{1} \frac{Z}{\sqrt{E}}+a_{2} \]

where \(\lambda\) is the decay constant, \(Z\) the atomic number, \(E\) the total kinetic energy and \(a_1\) and \(a_2\) are constants.

*H. Geiger and J.M. Nuttall (1911) "The ranges of the α particles from various radioactive substances and a relation between range and period of transformation," Philosophical Magazine, Series 6, vol. 22, no. 130, pages 613-621. See also: H. Geiger and J.M. Nuttall (1912) "The ranges of α particles from uranium," Philosophical Magazine, Series 6, vol. 23, no. 135, pages 439-445.*

*G. Gamow (1928) "Zur Quantentheorie des Atomkernes" (On the quantum theory of the atomic nucleus), Zeitschrift für Physik, vol. 51, pages 204-212.*

### 12.2 Droplet model.

### 12.3 Nuclear quantum numbers.

### 12.4 Magic nuclei.

### 12.5 Isospin.

### 12.6 Yukawa theory.

## 13 Plasma Physics

### 13.1 Magnetohydrodynamics.

### 13.2 Alfvén waves.

## 14 Advanced Mathematics

### 14.1 Group theory, and the linear representations of groups.

### 14.2 Lie group theory.

### 14.3 Vectors and tensors.

### 14.4 More techniques to solve (partial) differential and integral equations.

### 14.5 Extremum principle and approximation techniques based on that.

### 14.6 Difference equations.

### 14.7 Generating functions.

### 14.8 Hilbert space.

### 14.9 Introduction to the functional integral.

## 15 Special Relativity

These annotations are based on `http://jacobi.luc.edu/index.html` and `http://jacobi.luc.edu/Useful.html` i.e., the website of Prof. Robert A. McNees

Spacetime coordinates, 4-vectors, metric tensor \(g_{\mu\nu}\), etc:

Spacetime is any mathematical model which fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum, within of special relativity it relies on the Minkowski geometry.

### 15.1 4-vectors.

In relativistic expressions four-vectors are written as

\[ A^{\mu} = (A_0, \vec{A}) = (A_0, A^1, A^2, A^3). \]

Four-vectors with superscript are called *contravariant vectors* (regarded as a column
vector). *Covariant vectors*, denoted by subscript are defined as

\[ A_{\mu} = g_{\mu\nu}A^{\nu}, \]

with the *metric tensor* \(g_{\mu\nu}\) that reads in Cartesian coordinates in Minkowski space.

\[ \eta_{\mu \nu}=\eta^{\mu \nu}=\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{array}\right); \qquad \eta^{\mu}{ }_{\nu}=\delta^{\mu}{ }_{\nu}=\left(\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right). \]

Raising and lowering indices:

\[ A^{\mu}=\eta^{\mu \nu} A_{\nu}; \qquad A_{\mu}=\eta_{\mu \nu} A^{\nu}; \]

Einstein summation convention:

\[ A^{\mu}B_{\mu} = A^0B_0 + A^{1}B_{1} + A^2B_2 + A^3B_3 = A^0B^0 - \vec{A}\cdot\vec{B}. \]

The inverse of the metric tensor is given by

\[ g^{\mu\nu} = g_{\mu\nu}. \]

A scalar product of two four-vectors is then defined by

\[ A \cdot B = A_{\mu}B^{\mu} = A^{\mu}B_{\mu} = A^0B^0 - \vec{A}\cdot\vec{B}. \]

The invariant square of the four-vector is thus

\[ A^2 = A_{\mu}A^{\mu} = A_0^2 - \vec{A}^2. \]

#### 15.1.1 Spacetime coordinates:

\[x^{\mu} = (x^0, \vec{x}) = (ct, \vec{x}); \qquad x_{\mu} = (ct, -\vec{x}); \qquad \vec{x} = (x^1, x^2, x^3) = (x, y, z); \]

#### 15.1.2 Four-momentum:

\[ p^{\mu} = (E/c, \vec{p}); \qquad p_{\mu} = (E/c, -\vec{p}); \qquad p^{\mu} = (p^1, p^2, p^3) = (p_x, p_y, p_z); \]

#### 15.1.3 Derivatives:

\[ \partial_{\mu} \equiv \frac{\partial}{\partial x^{\mu}} = \left(\frac{1}{c} \frac{\partial}{\partial t}, \vec\nabla\right); \qquad \partial^{\mu} \equiv \frac{\partial}{\partial x_{\mu}} = \left(\frac{1}{c} \frac{\partial}{\partial t}, -\vec\nabla\right); \]

#### 15.1.4 Four-momentum operator

\[ \hat{P}^{\mu} = i\partial^{\mu}. \]

#### 15.1.5 Lorentz-invariant *d'Alembert operator*

\[ \partial^{\mu}\partial_{\mu} = \frac{\partial^2}{\partial t^2} - \vec\nabla^2 = \Box. \]

#### 15.1.6 Four-current

\[ j^{\mu} = (\rho, \vec{j}). \]

#### 15.1.7 Electromagnetic potential

\[ A^{\mu} = (\phi, \vec{A}). \]

### 15.2 The Lorentz transformation.

\[ A'_{\mu} = \Lambda_{\mu}{}^{\nu}A_{\nu}; \qquad A'^{\mu} = \Lambda^{\mu}{}_{\nu}A^{\nu}; \\ \eta^{\mu\nu} = \Lambda^{\mu}{}_{\alpha}\Lambda^{\nu}{}_{\beta}\,\eta^{\alpha\beta}; \qquad \eta_{\mu\nu} = \Lambda_{\mu}{}^{\alpha}\Lambda_{\nu}{}^{\beta}\,\eta_{\alpha\beta}. \]

- Properties:
- \[\eta^{\mu \nu}=\Lambda_{\alpha}^{\mu} \Lambda_{\beta}^{\nu} \eta^{\alpha \beta}\]
- \[ \left|\operatorname{det} \Lambda_{\nu}^{\mu}\right|=1 \]
- \[ \left(\Lambda_{1}\right)^{\mu} \alpha\left(\Lambda_{2}\right)_{\nu}^{\alpha}=\left(\Lambda_{12}\right)_{\nu}^{\mu} \]
- \[ \left|\Lambda_{0}^{0}\right| \geq 1 \]

### 15.3 Lorentz contraction, time dilatation.

### 15.4 \(E=mc^2\).

### 15.5 Transformation rules for the Maxwell field.

### 15.6 Relativistic Doppler effect.

## 16 Advanced Quantum Mechanics

### 16.1 Hilbert space.

### 16.2 Atomic transitions.

### 16.3 Emission and absorption of light.

### 16.4 Stimulated emission.

### 16.5 Density matrix.

### 16.6 Interpretation of QM.

### 16.7 The Bell inequalities.

### 16.8 Towards relativistic QM: The Dirac equation, finestructure.

### 16.9 Electrons and positrons.

### 16.10 BCS theory for superconductivity.

### 16.11 Quantum Hall effect.

### 16.12 Advanced scattering theory.

### 16.13 Dispersion relations.

### 16.14 Perturbation expansion.

### 16.15 WKB approximation, Extremum principle.

### 16.16 Bose-Einstein condensation

### 16.17 Superliquid helium

## 17 Phenomenology/Astroparticles

### 17.1 Subatomic particles (mesons, baryons, photons, leptons, quarks) and cosmic rays

## 18 General Relativity

### 18.1 4-tensors

#### 18.1.1 The metric tensor.

In local coordinates \(x^{\mu}\) the metric can be written in the form

\[ g=g_{\mu \nu} d x^{\mu} \otimes d x^{\nu}. \]

The factors \(dx^{\mu}\) are **one-form gradient** of the scalar fields \(x^{\mu}\). The coefficients
\(g_{\mu\nu}\) are a set of 16 real-valued functions. The metric is symmetric

\[
g_{\mu \nu} = g_{\nu \mu},
\]
giving 10 independents coefficients. Being \(dx^{\mu}\) the components of an infinitesimal displacement
four-vector, the metric determines the invariant square of an infinitesimal line element,
*interval*, denoted as

\[ d s^{2}=g_{\mu \nu} d x^{\mu} d x^{\nu}. \]

The interval \(ds^2\) imparts information about the causal structure of the spacetime.

- \(ds^2<0 \rightarrow\) timelike
- \(ds^2=0 \rightarrow\) lightlike
- \(ds^2>0 \rightarrow\) spacelike

Under a change of coordinates \(x^{\mu} \rightarrow x^{\overline{\mu}}\), the components transform as

\[ g_{\bar{\mu} \bar\nu}=\frac{\partial x^{\rho}}{\partial x^{\bar{\mu}}} \frac{\partial x^{\sigma}}{\partial x^{\bar{\nu}}} g_{\rho \sigma}=\Lambda_{\bar{\mu}}^{\rho} \Lambda_{\bar{\nu}}^{\sigma} g_{\rho \sigma}. \]

#### 18.1.2 Space-time curvature.

The metric \(g\) completely determines the curvature of spacetime. On any semi-Riemannian manifold there is a unique connection \(\nabla\) that is compatible with the metric and torsion-free. The Levi-Civita connection. The Christoffel symbols of this connection are given in terms of partial derivatives of the metric in local coordinates

\[ \Gamma^{\lambda}_{}{\mu \nu}=\frac{1}{2} g^{\lambda \rho}\left(\frac{\partial g_{\rho \mu}}{\partial x^{\nu}}+\frac{\partial g_{\rho \nu}}{\partial x^{\mu}}-\frac{\partial g_{\mu \nu}}{\partial x^{\rho}}\right)=\frac{1}{2} g^{\lambda \rho}\left(g_{\rho \mu, \nu}+g_{\rho \nu, \mu}-g_{\mu \nu, \rho}\right), \]

the curvature of spacetime is then given by the Riemann curvature tensor which is defined in terms of the Levi-Civita connection \(\nabla\),

\[ R^{\rho}{}_{\sigma \mu \nu} = \partial_{\mu} \Gamma^{\rho}{}_{\nu \sigma}-\partial_{\nu} \Gamma^{\rho}{}_{\mu \sigma} + \Gamma^{\rho}{}_{\mu \lambda} \Gamma^{\lambda}{}_{\nu \sigma} - \Gamma^{\rho}{}_{\nu \lambda} \Gamma^{\lambda}{}_{\mu \sigma}. \]

- Ricci tensor

\[ R_{\mu \nu}=\delta^{\sigma}{ }_{\rho} R^{\rho}{}_{\mu \sigma \nu}. \]

Scalar curvature

\[ R = g^{\mu\nu}R_{\mu\nu}. \]

- Schouten tensor

\[ S_{\mu \nu}=\frac{1}{d-1}\left(R_{\mu \nu}-\frac{1}{2 d} g_{\mu \nu} R\right); \qquad \nabla^{\nu} S_{\mu \nu}=\nabla_{\mu} S^{\nu}{}_{\nu}. \]

- Weyl tensor

\[ C^{\lambda}{}_{\mu \sigma \nu} = R^{\lambda}{}_{\mu \sigma \nu} + g^{\lambda}{}_{\nu} S_{\mu \sigma} - g^{\lambda}{}_{\sigma} S_{\mu \nu} + g_{\mu \sigma} S^{\lambda}{}_{\nu} - g_{\mu \nu} S^{\lambda}{}_{\sigma}. \]

- Commutators of Covariant Derivatives

\[ {\left[\nabla_{\mu}, \nabla_{\nu}\right] A_{\lambda} = R_{\lambda \sigma \mu \nu} A^{\sigma}}; \qquad {\left[\nabla_{\mu}, \nabla_{\nu}\right] A^{\lambda} = R^{\lambda}{}_{\sigma \mu \nu} A^{\sigma}}. \]

- Bianchi Identity

\[ \nabla_{\kappa} R_{\lambda \mu \sigma \nu}-\nabla_{\lambda} R_{\kappa \mu \sigma \nu}+\nabla_{\mu} R_{\kappa \lambda \sigma \nu}=0; \qquad \nabla^{\nu} R_{\lambda \mu \sigma \nu}=\nabla_{\mu} R_{\lambda \sigma}-\nabla_{\lambda} R_{\mu\sigma}; \qquad \nabla^{\nu} R_{\mu \nu}=\frac{1}{2} \nabla_{\mu} R. \]

- Bianchi identity for Weyl

\[ \nabla^{\nu} C_{\lambda \mu \sigma \nu}=(d-2) \left(\nabla_{\mu} S_{\lambda\sigma}-\nabla_{\lambda} S_{\mu \sigma}\right); \qquad \nabla^{\lambda} \nabla^{\sigma} C_{\lambda\mu \sigma \nu} = \frac{d-2}{d-1}\left[\nabla^{2} R_{\mu\nu}-\frac{1}{2 d} g_{\mu \nu} \nabla^{2} R-\frac{d-1}{2 d} \nabla_{\mu} \nabla_{\nu}R - \left(\frac{d+1}{d-1}\right) R_{\mu}{}^{\lambda} R_{\nu \lambda} + C_{\lambda \mu \sigma \nu} R^{\lambda \sigma}+\frac{(d+1)}{d(d-1)} R R_{\mu \nu}+\frac{1}{d-1} g_{\mu \nu}\left(R^{\lambda \sigma} R_{\lambda \sigma}-\frac{1}{d} R^{2}\right)\right] \]