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 ↓


Quantum Electrodynamics


Quantum electrodynamics, often abbreviated as QED, is a fundamental theory in physics that describes how light and matter interact. It is the quantum field theory of the electromagnetic force and plays an essential role in understanding phenomena on the subatomic scale. Developed between the 1920s and 1940s, QED combines quantum mechanics with special relativity and has been remarkably successful in making accurate predictions about a variety of processes involving electromagnetic interactions.

Fundamentals of quantum electrodynamics

At its core, QED deals with the interactions between charged particles and electromagnetic fields. The basic premise is that particles such as electrons and positrons interact with photons, the force carriers of the electromagnetic field. In QED, these interactions are described using the language of quantum fields.

The fields are quantized, meaning that they are divided into discrete quantum states. For example, an electromagnetic field is quantized into photons. Similarly, the electron field is quantized into electrons. These particles are excitations or quanta of their respective fields.

The concept of virtual particles

An important concept in QED is that of virtual particles. In interactions between charged particles, such as the deflection of one electron by another, QED describes these processes as the exchange of virtual photons, which are momentary disturbances in the electromagnetic field that cannot be observed directly but have an effect on the results of experiments.

Virtual photon Electron

Feynman diagrams

To visually represent interactions in QED and simplify calculations, physicists use Feynman diagrams. These diagrams provide a graphical depiction of the behavior of scattering, decaying, or interacting subatomic particles. Each line and vertex in a Feynman diagram corresponds to specific mathematical terms within perturbation theory, a technique for approximating solutions to complex interactions.

In a simple Feynman diagram for electron-electron scattering, two lines (representing electrons) approach each other, exchange a virtual photon, and then move apart once again. These lines are directed, indicating the flow of time, and the photon is typically represented as a wavy line.

Electron Electron Photon

Calculations in QED

Calculations in quantum electrodynamics involve measuring the probabilities of different interaction outcomes. QED is notable because its predictions have been confirmed by experimental observations with astonishing accuracy, sometimes matching measurements to within ten parts per billion.

Calculations are often performed using perturbation theory, which expresses the amplitude of an interaction as an infinite series of terms. The simplest terms correspond to the simplest Feynman diagram, and more complex diagrams add successively smaller contributions. The process of calculating these amplitudes involves evaluating integrals in a space known as momentum space.

A basic interaction can be described mathematically as follows: 

        begin{equation}
        mathcal{M}(e^- e^- to e^- e^-) = ie^2 bar{u}(p_1') gamma^mu u(p_1) frac{g_{munu}}{(p_1'-p_1)^2} bar{u}(p_2') gamma^nu u(p_2)
        end{equation}
Here, (mathcal{M}) is the amplitude, (u) and (bar{u}) are spinor wave functions for electrons, and (gamma^mu) are gamma matrices that describe space-time properties.

Renormalization

One of the significant challenges faced in QED calculations is the appearance of infinities when evaluating certain integrals. This is resolved through a process called renormalization. In essence, renormalization adjusts parameters in the theory, such as charge and mass, to absorb these infinities and produce meaningful finite results.

The success of renormalization in QED not only saved the theory but laid the groundwork for renormalization techniques in other quantum field theories. Through renormalization, QED can provide clear predictions that match closely with experimental findings.

For example, when calculating the magnetic moment of the electron, its experimental value can be reconciled with the theoretical prediction by including higher-order corrections obtained from these diagrams, which contribute to the so-called low mass effect.

Applications and implications

QED has many applications. It is essential in the development of technologies based on semiconductors such as transistors. The concepts of QED are also fundamental to laser technology, providing an explanation of how photons in laser beams interact coherently.

Additionally, QED influences various fields of physics and holds a central place in particle physics and cosmology. When describing the interactions of light with matter, QED is indispensable, explaining not only scattering and absorption but also emission and reflection processes.

At a more theoretical level, QED has been a step toward the development of the Standard Model of particle physics, which describes three of the four known fundamental forces in the universe (all except gravity) and classifies all known subatomic particles.

Experimental verification

One of the remarkable experimental confirmations of the predictions of QED is the precise measurement of the unusual magnetic moment of the electron. The predicted and measured values are in excellent agreement, which underlines the predictive power of QED.

Additionally, the Lamb shift in the energy levels of hydrogen was accurately predicted by QED. The energy levels of the hydrogen atom shift due to the interactions between the electron and vacuum fluctuations of the electromagnetic field. This was one of the early successes of QED and provided a strong confirmation of the theory.

Conclusion

Quantum electrodynamics is a cornerstone of modern physics, providing profound insights into the nature of electromagnetic interactions. Its accuracy and precision in predictions validate its mathematical formulation and conceptual framework, demonstrating the power and beauty of quantum field theories. As an integral part of the Standard Model, QED remains crucial in our quest to understand the fundamental workings of the universe.


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Quantum Electrodynamics

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