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  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 ↓


Feynman diagrams and propagators


Feynman diagrams and propagators are fundamental tools in the framework of quantum field theory (QFT), which is important for understanding particle physics, the interaction of particles, and the dynamics of quantum fields. Let us understand these complex concepts in an accessible way.

Understanding Feynman diagrams

Feynman diagrams are pictorial representations of particle processes and interactions in the quantum realm. Proposed by Richard Feynman, these diagrams simplify and clarify the mathematics involved in complex quantum equations.

What do Feynman diagrams represent?

At its core, a Feynman diagram is a graphical shorthand that represents the interactions of particles through lines and vertices, which represent particles and interactions, respectively. These diagrams do not depict the actual paths of particles, but rather they provide a visualization of mathematical terms in perturbation theory.

Components of a Feynman diagram

Feynman diagrams consist of several elements:

  • Outlines: represent incoming and outgoing particles.
  • Internal lines: indicate exchanged particles or virtual particles within an interaction.
  • Vertices: Points where lines meet, representing interactions between particles.

Example of a simple Feynman diagram: electron–muon scattering

Consider the scattering of an electron from a muon (denoted as μ). The Feynman diagram depicting this interaction can be represented as follows:

   e^- ----------->
                 ,
                  ,
                   ,
                    γ
                   ,
                  ,
                 ,
   μ^- ----------->

In this diagram:

  • The straight lines represent the electron (e-) and muon (μ-) paths.
  • The wavy line represents the photons (γ) exchanged during the interaction.
  • The vertex is the point where the photon is emitted or absorbed.

Formalism and computation using Feynman diagrams

Feynman diagrams allow physicists to perform calculations by providing a direct route to obtaining the interaction amplitude. The rules for these calculations emerge from the Feynman rules associated with a given quantum field theory.

Basic Feynman rules

Here are some general steps in using Feynman diagrams for calculations:

  1. Identify the types of particles involved and their associated interactions.
  2. Draw all topologically different diagrams for a given process in the desired order in perturbation theory.
  3. Apply Feynman rules to convert diagrams into mathematical expressions.
  4. Sum the amplitudes in quantum mechanics to get the total probability amplitude.

Example calculation: electron–photon interaction in QED

In this process the electron interacts with the photon via quantum electrodynamics (QED). The vertex of this interaction involves the electron, the positron, and the photon, which is represented in the Feynman diagram as follows:

    e^- 
      , 
       ,
        ,
         γ
        ,
       ,
  e^+

Using Feynman rules, the calculations involve determining the corresponding peak factors, the propagators for the photons, and the wave functions for each particle.

Overview of propagators

Propagators are a crucial component of Feynman diagrams, which describe the probability amplitude of a particle traveling between two points in spacetime. They encapsulate the dynamics of how a field propagates.

Analogies to classical physics

In classical physics, if you throw a ball, the trajectory of its path can be described by knowing its initial conditions and applying Newton's laws. Similarly, in quantum field theory a propagator provides a means of calculating how quantum particles "travel" between interacting lattices.

Exponent in mathematical terms

The mathematical form of the propagator is different for different particles and theories. It can be understood as a Green's function for the field equation.

An example of a scalar particle propagator in a simplified form is as follows:

    D_F(x - y) = ∫ (dk^4 / (2π)^4) (e^(−ik (x−y))) / (k^2 + m^2 − iε)

where k is the wave vector, m is the mass of the particle, and ε is an infinitesimal positive number to ensure proper convergence.

Different types of propagators

Depending on the particle and its properties, propagators can vary considerably:

  • Fermion propagator: describes particles that obey Fermi–Dirac statistics, such as electrons.
  • Gluon propagator: used in quantum chromodynamics (QCD) to describe the force-carrying particles of the strong force.
  • Photon propagator: Theory of electromagnetic forces, used in Q.E.D.

Interactions and dimensions

The core of understanding Feynman diagrams and propagators is understanding how interactions occur and how to calculate amplitudes, which describe the probability of particular outcomes in particle interactions.

Vertex and interaction Hamiltonian

In a Feynman diagram, each vertex represents an interaction, derived from the interaction Hamiltonian of the field theory:

    H_int = g ∫ ϕ(x) ψ^†(x) ψ(x) dx

where g is the coupling constant, φ is the field, and ψ and ψ^† represent the field operators of the particle.

Calculating dimensions

To calculate the amplitude of a process:

  • Identify involved areas and particles.
  • Determine the essential diagrams that contribute to the process.
  • Add up the dimensions from each diagram.

Such calculations make it possible to make predictions about the scattering cross section or decay rate of particles, which is extremely important for experimental particle physics.

Visual examples of Feynman diagrams

Feynman diagrams can depict various interactions beautifully and effectively. Let us consider a visual representation of typical interactions:

Electron–positron annihilation

    e^- --
          ,
           ,  
           --γ 
           , 
          ,
    e^+ --

Here, an electron and positron combine to form a photon, depicted as a wavy line.

Neutron beta decay

    N ----> P
               ,
                 w^-
                 ,
                  e^-

In this process, a neutron decays into a proton, emitting a W-boson, which in turn decays into an electron and an antineutrino (omitted for simplicity).

Conclusion

Feynman diagrams and propagators provide physicists with an essential toolkit for understanding and calculating complex interactions in quantum field theory. They make otherwise complex mathematical expressions more accessible through visualization and provide a simplified method to evaluate interactions. Mastery of these tools is a gateway to furthering our understanding of the fundamental forces and particles of the quantum world.

Further studies and research

For those interested in deepening their knowledge, the following texts are suitable:

  • Quantum Field Theory for the Talented Amateur by Tom Lancaster and Stephen J. Blundell
  • Introduction to Quantum Field Theory by Michael E. Peskin and Daniel V. Schroeder
  • The Quantum Theory of Fields by Steven Weinberg

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Feynman diagrams and propagators

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