PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD → Quantum field theory → Standard model of particle physics ↓


Grand Unified Theories


The quest to find a singular theoretical framework that explains the fundamental forces of nature has fascinated scientists for decades. Ambitions to achieve this include the concept of Grand Unified Theories (GUTs) in the context of quantum field theory and the standard model of particle physics.

Understanding the basics

First, the Standard Model is a theory that describes three of the four known fundamental forces in the universe: electromagnetic, weak nuclear, and strong nuclear forces, while omitting gravity. It also classifies all known fundamental particles.

Forces in Standard Model: - Electromagnetic Force - Weak Nuclear Force - Strong Nuclear Force

Elementary particles

The elementary particles in the Standard Model include fermions (which make up matter) and bosons (which carry forces). Fermions are divided into quarks and leptons. Bosons are force carriers, including the photon for electromagnetism, the W and Z bosons for the weak force, and the gluon for the strong force. The discovery of the Higgs boson was a key part of the Standard Model.

Quarks Leptons Gluon Bosons

Origins of Grand Unified Theories

The Grand Unified Theory aims to extend the Standard Model by proposing that at extremely high energies, the electromagnetic, weak, and strong forces are unified into a single force governed by a large symmetry group. This hypothesis states that these three forces are manifestations of a singular fundamental force that separates into separate forces at low energies.

To understand this, consider how electricity and magnetism were once considered separate phenomena, but were eventually unified into electromagnetism through Maxwell's equations.

Symmetry in the GUT

Symmetry plays a key role in these theories. In GUTs, one starts with a single symmetric state or structure, which is beautiful and simple. As the universe cools and conditions change, this symmetry breaks down into the less symmetric forms we observe, such as different forces.

Example of symmetry breaking: Unified Symmetry Group → Electroweak Symmetry + Strong Symmetry → Electromagnetism + Weak + Strong

Mathematical framework

In GUTs, one of the primary challenges is to identify a suitable symmetry group (often denoted as SU(n), where n is the dimension of the symmetry) that can accommodate all the small symmetries found in the Standard Model.

SU(5) Electricity Strong Gravity

The simplest GUT, SU(5), was developed in the 1970s and generated significant interest because it neatly reconciles the forces, although it faced experimental challenges.

Experimental evidence and challenges

While the Standard Model is incredibly successful, GUTs must agree with the myriad experimental results observed in particle physics. One challenge is the prediction of proton decay, which many GUTs predict as a result of the unification of forces. However, extremely rare decay rates have been observed, and current detectors have not confirmed the predictions, placing constraints on the theory parameters.

Proton decay

In many GUTs, the unification of forces allows quarks to transform into leptons, meaning that protons made of quarks can decay. This hypothetical phenomenon presents an important test for GUTs. If a proton can spontaneously decay over a very long timescale, it would provide crucial evidence in favour of these theories.

Hypothetical Proton Decay: Proton → Positron + Neutral Meson

Moving beyond the standard model

While GUTs offer exciting possibilities, they are not the only avenues being explored for a deeper understanding of physics beyond the Standard Model. The search continues with Theories of Everything (ToEs), which aim to incorporate gravity into a unified framework by introducing super-symmetry and string theories.

For example, string theory proposes that the basic building blocks of the universe are one-dimensional "strings" rather than point particles. These strings can vibrate at different frequencies, creating the observed particles.

String theory visualization

Although difficult to visualize beyond three spatial dimensions, imagine that strings are tiny loops that vibrate in unique patterns, and contain the various forces and particles that appear in our universe.

String Particle

GUTs remain an essential topic of research as physicists attempt to understand the complex origins and fundamental laws of the universe. This journey is far from complete, and ongoing experimental and theoretical work continues to test and refine these great ideas.

Grand Unified Theories hold the promise of inspiring discoveries in the future that may one day transform our understanding of the universe, and shed light on hitherto invisible relationships.


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Grand Unified Theories

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