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


The Standard Model of particle physics is one of the most important achievements in the world of physics. It provides a framework for understanding the fundamental forces and particles that make up the universe. As an essential component of quantum field theory, the Standard Model describes how particles known as fermions and bosons interact, and these interactions provide insights into the nature of matter and energy.

Introduction to the Standard Model

At its core, the Standard Model posits that all matter is made up of elementary particles, which can be divided into two main categories: fermions and bosons. Fermions include quarks and leptons, while bosons are force-carrying particles.

The Standard Model also describes three of the four fundamental forces that govern the universe: the electromagnetic, weak, and strong forces. The fourth force, gravity, is not included in the Standard Model.

Fermions: the building blocks of matter

Fermions are the particles that make up matter. They obey the Pauli exclusion principle, which means that no two fermions can be in the same quantum state at the same time.

Fermions are divided into two groups:

  • Quarks: These particles are the basic components of protons and neutrons. There are six types of quarks: up, down, charm, strange, top, and bottom.
  • Leptons: These include electrons and neutrinos. Like quarks, there are six types of leptons: electron, muon, tau, electron neutrino, muon neutrino, and tau neutrino.

Imagine six elements of quarks and leptons:

Up quark (u) Down quark (d) Charm quark (C) Strange quarks Top quark (t) Bottom quark (b)

Similarly, leptons can be visualized as:

electron (e) Electron neutrino Muon (μ) Muon neutrino Tau (τ) tau neutrino

Bosons: force carriers

Bosons are particles that mediate the interactions between fermions. Unlike fermions, they do not obey the Pauli exclusion principle. The standard model includes three principal bosons:

  • Photon (γ): Mediates electromagnetic interactions. It is responsible for light and all electromagnetic radiation.
  • W and Z bosons (W +, W−, Z): Responsible for the weak interactions. These interactions are important in nuclear decay processes.
  • Gluon (g): Carries the strong force. This force holds quarks together within protons and neutrons.

Additionally, the Higgs boson (H) is associated with the Higgs field, which imparts mass to particles via the Higgs mechanism.

Photon (γ) W bosons Z boson Gluon (g) Higgs (H)

Formulations and interactions

The Standard Model describes the interactions of particles using mathematical formulas. The interactions are represented by Lagrangians, which are constructed using symmetry principles. Lagrangians describe the dynamics of the field and the interactions of particles:

L = L kinetic + L mass + L Yukawa + L gauge + L Higgs

These terms represent kinetic energy, mass, interaction (Yukawa coupling), gauge field (carrier force) and the Higgs field, respectively.

Electromagnetic contact

The electromagnetic force affects particles with electrical charge. It is described using quantum electrodynamics (QED). The force-carrier particle for electromagnetic interactions is the photon, which is a massless boson that allows it to mediate long-range forces.

Maxwell's equations describe classical electromagnetic fields, and their quantum counterpart requires a complicated formulation involving photons:

∇⋅E=ρ/ε₀
∇ ⋅ b = 0
∇ × e = -∂b / ∂t
∇ × B = μ₀ J + μ₀ε₀ ∂E / ∂t

Weak interaction

The weak force is responsible for processes such as beta decay. It affects all fermions and is mediated by the W and Z bosons. These bosons are massive, which limits the range of weak interactions.

Weak interactions are described by the electroweak theory, which unifies weak and electromagnetic interactions. This was a significant achievement, a major step forward in understanding the fundamental forces. The theory includes the Weinberg angle, which describes how weak forces mix with electromagnetic interactions.

Strong interaction

The strong force is the most powerful of all the fundamental forces. It acts on quarks through the exchange of gluons. Quantum chromodynamics (QCD) is the theory that describes strong interactions, which are characterized by color charge.

Unlike electric charge, which has only positive and negative states, color charge has three types: red, green and blue, as well as their anticolors. Gluons help quarks change color, maintaining the strong nuclear force that holds atomic nuclei together.

Red Green Blue

The binding of quarks within protons and neutrons, as well as the phenomenon of asymptotic freedom, emerge from these color charge properties. Asymptotic freedom describes how quarks behave like free particles at short distances.

Higgs mechanism and mass

The Higgs field plays a key role in the Standard Model by providing mass to elementary particles via the Higgs mechanism. As particles interact with the Higgs field, they gain mass, a phenomenon confirmed by the discovery of the Higgs boson at CERN in 2012.

The mass production process involves the following interaction Lagrangian:

L Higgs = (d μ φ) † (d μ φ) − V(φ)

The potential V(φ) results in spontaneous symmetry breaking, causing particles to have mass. This discovery was important in validating the Standard Model's explanation of mass.

Limitations and unexplained phenomena

Despite its many successes, the Standard Model does not cover all physical phenomena. Notably, it does not take into account the following:

  • Gravity: While the other three forces have been explained, a quantum theory of gravity is needed to include gravity, which is still unclear.
  • Dark matter and dark energy: Evidence suggests that these components exist in much greater numbers than normal matter, but they are not described by the standard model.
  • Neutrino mass: The discovery of neutrino oscillations implies that they have mass, which is not accounted for in the original Standard Model.

Conclusion

The Standard Model represents a landmark scientific theory that describes the known fundamental particles and their interactions through three of the four fundamental forces. The inclusion of quantum field theories created a coherent framework for understanding nature at its most elementary level.

While the Standard Model comprehensively explains many aspects of particle physics, physicists continue to search for new theories and experimental evidence to resolve its limitations. The search for a more comprehensive theory, potentially unifying gravity with quantum mechanics or taking into account dark matter and energy, drives research in modern theoretical physics.


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Standard model of particle physics

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