Graduate
  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

Graduate → Quantum mechanics ↓


Quantum field theory


Quantum field theory (QFT) is a fundamental theory in physics that describes the behavior and interactions of quantum fields, used to model particles such as electrons, photons, and other elementary entities. QFT combines classical field theory, special relativity, and quantum mechanics, making it a powerful framework for understanding the quantum nature of the universe.

Basic concepts of quantum field theory

Fields and particles

In QFT, the basic objects are fields, which are defined throughout space and time. A field can be viewed as a continuous value that exists at every point in space, similar to the temperature distribution in a field.

Particle

Particles are interpreted as excited states of the underlying fields. This concept is important because it allows particles to be created and destroyed, which are phenomena observed in nature.

Quantization of fields

Quantization involves taking a classical field such as the electromagnetic field and applying quantum rules to it. In simple terms, this means turning waves into particles. For example, photons are quantized excitations of the electromagnetic field.

Mathematical framework

Lagrangian and Hamiltonian formulations

Lagrangian is a function that summarizes the dynamics of the field. It is used to derive the equations of motion for the field via the principle of least action. The Hamiltonian, another key concept, often represents the total energy of the system.

L = int d^3x left( frac{1}{2} partial_mu phi partial^mu phi - frac{1}{2} m^2 phi^2 right)

This is an example of a Lagrangian for a scalar field (phi), where (m) is the mass of the particle associated with the field.

Path integral formulation

Developed by Richard Feynman, the path integral formulation is an alternative approach to quantum mechanics. It calculates probabilities by considering all the possible paths a particle can take from point A to point B and adding up the contributions from each path.

Z = int mathcal{D}phi  e^{i S[phi]/hbar}

In this expression, Z represents the partition function, (mathcal{D}phi) indicates integration over all field configurations, and S[phi] is the action of the field.

Interactions and Feynman diagrams

A key aspect of QFT is understanding the interactions between particles. Feynman diagrams are a powerful visual tool used to calculate these interactions. They serve as intuitive representations of particle processes where lines represent particle paths and vertices represent interactions.

interaction

This simple diagram can represent the electron–positron interaction, with the lines representing paths and the vertex representing the point where they meet or interact.

Quantum electrodynamics (QED)

Quantum electrodynamics is a field theory that describes how light and matter interact. It is one of the most accurate theories of physics, providing highly accurate predictions of quantities such as the magnetic moment of the electron. QED uses photons as mediating particles of the electromagnetic force.

Symmetries and conservation laws

Symmetries in QFT lead to important conservation rules, explained by Noether's theorem. Symmetry refers to a property of a system that remains unchanged under a transformation, such as a change in time or space.

Gauge symmetry

Gauge symmetry is a type of symmetry that applies to fields in QFT. It is the basis of many particle physics theories, including the Standard Model. Gauge symmetry is important to ensure the mathematical consistency of the theory.

Renormalization

Renormalization is a technique used to deal with infinite quantities that may appear in calculations of particle interactions. It systematically removes infinities by redefining physical parameters (such as mass and charge) to ensure that predictions match observations.

For example, calculating the charge of an electron can lead to infinite results when interactions are considered at the fundamental level. Renormalization allows us to adjust the calculations so that the predictions remain finite and align with empirical measurements.

Renormalization group

The renormalization group is a conceptual framework that describes how physical processes change when one looks at them at different scales. It helps physicists understand phenomena ranging from the behavior of elementary particles to phase transitions in substances.

Standard model of particle physics

The Standard Model is a theory composed of quantum field theories that describe the electromagnetic, weak, and strong nuclear forces. It includes particles such as quarks, leptons, and gauge bosons.

For example, the strong force is explained by quantum chromodynamics (QCD), in which quarks interact through the exchange of force-carrying particles, gluons.

Quarks

This diagram symbolizes the strong force interactions between quarks, and shows their complex and dynamic interactions through exchanges created by gluons.

Beyond the Standard Model

While the standard model has been extremely successful, it does not explain gravity or dark matter and has other limitations. Quantum field theorists are exploring different approaches, such as supersymmetry, string theory, and quantum gravity, to enhance our understanding of the universe.

Conclusion

Quantum field theory is the cornerstone of modern physics, providing deep insight into the workings of the universe at the smallest scales. From explaining fundamental particles to creating models that help understand forces and interactions, QFT is essential for both theoretical studies and experimental breakthroughs. Learning and appreciating the beauty of QFT can illuminate the complex tapestry of the universe.


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Quantum field theory

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