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


Introduction to quantum field theory

Quantum field theory (QFT) is a fundamental theoretical framework in physics that blends quantum mechanics (QM) and special relativity into a unified description. It forms the basis of our understanding of particle physics and forms the basis of the Standard Model which describes the fundamental forces of the universe except gravity.

Quantum field theory allows particles to be created and destroyed, materializing the principles of quantum mechanics in the fields that pervade the universe. So when we talk about electrons or photons, we are really talking about excitations of their respective fields.

Historical context and development

The development of QFT began when scientists tried to reconcile the rules of quantum mechanics with the rules of special relativity. QFT pioneers such as Paul Dirac, Richard Feynman, Julian Schwinger, and Tomonaga Shinichiro established the basic aspects during the early and mid-20th century. They built on the foundations of earlier quantum theory established by pioneers such as Niels Bohr and Werner Heisenberg.

Basics of fields and particles

Let's take a deeper look at the fields and particles that are at the heart of quantum field theory.

Fields: Building blocks

The field is an abstract concept in which each point in space is associated with a value. Consider classical examples such as the temperature distribution in a given region or the electromagnetic field around a charged particle; both represent similar concepts.

temperate zones

In QFT, fields can represent forces such as electromagnetism or physical properties such as quantum properties of matter. These fields are quantized, enabling them to exhibit wave-particle duality and have quantum properties.

Particles as excitations

In classical mechanics, particles are viewed as individual entities. In QFT, a particle is viewed as an excitation or disturbance in its associated field. For example, an electron is an excitation in the electron field, and a photon is an excitation in the electromagnetic field.

Particles as excitations

This beautiful conceptual shift allows us to view particles not as isolated objects, but as interconnected manifestations of their internal fields, leading to much more profound insights into the nature of matter and energy.

Mathematics of quantum field theory

The mathematics supporting quantum field theory is quite complex, involving advanced calculus, linear algebra, and abstract algebra. Here is a simple introduction to some of the main mathematical tools used in QFT.

Operators and states

In quantum mechanics, physical quantities such as position and momentum are represented by operators, while the states of physical systems are described by state vectors in a Hilbert space. In QFT, operators correspond to physical observables, and states often represent possible configurations or excitations of fields.

|Psirangle = int phi(mathbf{x}) |mathbf{x}rangle , d^3x

Here, |Psirangle represents the state vector, while phi(mathbf{x}) is the field configuration at the location.

Exchange relations

Operators in quantum mechanics and QFT obey specific exchange relations. For a field operator hat{phi}(mathbf{x},t) and its conjugate momentum hat{pi}(mathbf{x},t), the exchange relation is required:

[hat{phi}(mathbf{x},t), hat{pi}(mathbf{y},t)] = ihbar delta^3(mathbf{x} - mathbf{y})

These relations are important in defining the properties and behaviour of fields in quantum theory.

Lagrangian and path integral formulation

QFT often uses the Lagrangian to express dynamics concisely. The Lagrangian is a mathematical function that describes the overall dynamics of a field or system:

mathcal{L} = frac{1}{2}(partial_t phi)^2 - frac{1}{2}(nabla phi)^2 - V(phi)

In this expression, mathcal{L} is the Lagrangian density for a scalar field phi, where the potential energy V(phi) determines the interactions and self-interactions.

Another powerful method in QFT is the path integral formulation, which allows calculations to be performed over the possible histories of a system:

Z = int mathcal{D}phi , e^{iS[phi]/hbar}

Here, Z is the partition function, and S[phi] represents the action obtained from integrating the Lagrangian over spacetime.

Quantum electrodynamics (QED)

The development of quantum electrodynamics (QED), the relativistic quantum field theory of electrodynamics, was one of the first major successes of QFT. QED describes how light and matter interact and has been remarkably accurate in its predictions and measurements.

Feynman diagrams

To represent particle interactions in QED and later other field theories, physicist Richard Feynman introduced the use of Feynman diagrams. These diagrams visually represent the time evolution of particle interactions and help simplify complex calculations.

Feynman Diagram Example

Each line and vertex in a Feynman diagram has a specific interpretation related to the region and particle it represents. For example, lines can represent particles such as electrons and photons, while vertices show interaction points.

Renormalization

An important challenge in QFT is to manage infinities that appear in simple perturbative calculations. Physicists developed renormalization techniques to systematically understand these infinities and make meaningful predictions.

Simply put, renormalization involves redefining physical parameters (such as mass and charge) so as to absorb infinity, leading to well-defined, finite results. This concept is important and has provided deep insights into understanding the scale-dependence of physical processes.

Applications of quantum field theory

The reach and impact of quantum field theory extend far beyond the tiny particles it describes. Here are some of the major areas where QFT is applied:

Standard model of particle physics

The Standard Model is a QFT framework that describes three of the four fundamental forces: electromagnetism, the weak nuclear force, and the strong nuclear force. It divides elementary particles into fermions, which make up matter, and bosons, which mediate the forces.

Standard Model Quarks Leptons Above Below Attraction Strange Top Bottom Electron Muon Tau Neutrino

Beyond the Standard Model

Theories aiming to explain phenomena beyond the standard model, such as supersymmetry or quantum gravity, also rely heavily on quantum field theoretical concepts.

Condensed matter physics

QFT plays an important role in understanding various phenomena in condensed matter systems, such as superconductivity and quantum phase transitions.

Conclusion

Quantum field theory serves as the cornerstone of modern physics, enabling us to understand the microcosm of the universe and the intricate dance of particles in different fields. Despite being challenging, the beauty of QFT lies in its ability to unravel the deep connections between complex interactions and fundamental forces of nature.

Despite its success, quantum field theory remains an evolving field, continually providing new insights and fueling exploration into the riddles of the quantum realm. With its complex mathematics and profound implications, QFT continues to shape the landscape of physics and influence the search for a deeper understanding of our universe.


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