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 mechanics → Relativistic quantum mechanics ↓


Quantum field theory


Quantum field theory (QFT) is a fundamental framework in physics that combines classical field theory, special relativity, and quantum mechanics. It provides a comprehensive description of how particles interact with each other through fields. QFT is essential for understanding the behavior of particles at high energy levels, such as those encountered in particle accelerators and astrophysical phenomena.

Basics of quantum mechanics

To understand QFT, we must first review some basic concepts in quantum mechanics. Quantum mechanics describes physical systems in terms of wave functions, which are mathematical functions that encode the probabilities of finding a particle in various positions and states.

ψ(x) = A * exp(i(px - Et)/ħ)

Here, ψ(x) denotes the wave function, x is the position, p is the momentum, E is the energy, ħ is the reduced Planck constant, and A is a standardization factor.

Special relativity

Special relativity, proposed by Albert Einstein, states that the laws of physics are the same for all non-accelerating observers. A key insight of special relativity is that space and time are interconnected, forming a four-dimensional spacetime.

The relation between energy and momentum in special relativity is given as:

E² = p²c² + m²c⁴

where E is energy, p is momentum, c is the speed of light, and m is the mass of the particle.

The need for quantum field theory

The combination of quantum mechanics and special relativity creates the need for a new type of theory: quantum field theory. An important issue in quantum mechanics is the concept of particle antiparticles and the creation and destruction of particles, which cannot be adequately described by standard quantum theory alone.

Fields and particles

In QFT, fields are fundamental, and particles are excitations (or quanta) of these fields. For example, the electromagnetic field gives rise to particles called photons.

The visualization of the concept is as follows: let's say the field is like a calm lake. A particle is like a wave traveling on this lake. The lake itself (the field) has energy and potential for waves at any point, but you 'see' a wave only when the energy is concentrated in one region (the particle).

Particle

Quantization of fields

In QFT, we apply the principle of quantization to fields. This involves promoting classical field variables into operators that act on quantum states. This process is similar to the quantization of mechanical systems where position and momentum become operators.

The fundamental exchange relation for fields is given by:

[φ(x), π(y)] = iħδ(xy)

where φ(x) is the area operator and π(y) is the conjugate momentum operator, and δ(xy) is the Dirac delta function.

Feynman diagrams

Feynman diagrams are pictorial representations of the mathematical expressions that govern the behavior of particles in quantum field theory. These diagrams provide an intuitive way to understand the interactions of particles.

Here's a simple Feynman diagram example for electron-positron annihilation:

E⁻e⁺γE⁻e⁺

Interaction and perturbation theory

In QFT, the interactions between fields are handled using perturbation theory. This is a way of approximating complex quantum systems by starting with a simple system and adding interactions as small perturbations.

H = H_0 + H_I

where H is the total Hamiltonian, H_0 is the Hamiltonian of the free field, and H_I is the interaction Hamiltonian. The effects of H_I are calculated as a series of corrections on top of the solutions of H_0.

Renormalization

Renormalization is a process for dealing with infinite quantities arising from interactions in QFT. It involves adjusting the parameters of the theory (such as mass and charge) so that observable quantities can be accurately predicted. The adjusted parameters are called renormalized quantities.

Imagine having to properly tune the radio to avoid static noise in order to hear the music clearly.

Standard model of particle physics

QFT forms the basis of the Standard Model of particle physics, which describes the electromagnetic, weak and strong nuclear forces. These forces are mediated by gauge bosons that emerge from quantized fields.

QuarksGluonLeptons

Applications of quantum field theory

QFT is used in various areas of physics, including:

  • Condensed Matter Physics: Understanding phenomena such as superconductivity and quantum phase transitions.
  • Particle Physics: Investigating the fundamental particles and forces in nature.
  • Cosmology: Exploring events in the early universe, such as the Big Bang.

The way forward

Quantum field theory is a constantly evolving field. Researchers are still discovering new insights into the specifics of QFT, including attempts to reconcile it with general relativity in theories of quantum gravity such as string theory and loop quantum gravity.

In conclusion, quantum field theory serves as an essential pillar of modern physics, enabling us to understand the complex interactions of the universe at a fundamental level.


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Quantum field theory

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