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 ↓


Renormalization theory


Renormalization is an important method in quantum field theory (QFT), which helps deal with infinities that arise in calculations. In simple terms, renormalization means refining infinities into meaningful, finite results. Let's take a deeper look at this powerful technique in physics with examples and simple language.

Understanding the problem of infinity

When physicists developed quantum theories to describe forces and particles on the atomic scale, they encountered a peculiar problem: the results of some calculations were infinite. For example, when calculating the probability of a particle interacting, the answers sometimes came out as infinite, which does not make sense in physical reality.

The persistence of infinity

First consider a simple example from classical physics. Imagine you are calculating the area under a curve, and the curve extends to infinity.

∫ (1/x^2) dx from 1 to ∞ = ∞

This integral diverges, just as some calculations in QFT diverge. Such infinities are problematic because they hinder our ability to make accurate predictions.

Simplified example in QFT

In quantum field theories, particles can be represented as fields. For example, an electron can move in an electromagnetic field and cause disturbances. When you try to calculate the interaction force between electrons through an electromagnetic field, you may encounter discrete integrals. Santa these calculations are:

∫ dk × f(k)

where f(k) is some function related to the field configuration, and k is the momentum. If it diverges, we get an infinite result, which is not useful.

The idea of renormalization

The main idea behind renormalization is to incorporate these infinities into the definitions of physical quantities such as mass and charge. This allows us to redefine the parameters in such a way that these infinities are removed from our predictions.

Renormalization of electromagnetic field

Let us consider electrons interacting through an electromagnetic field. A simple calculation might suggest that the charge of the electron is infinite. The process of renormalization involves adjusting the observed charge, denoted as e.

Here's a simple sequence of steps:

  1. Calculate the electron interaction using a bare charge, e_b, which may be infinite.
  2. Subtract the infinities that arise in the calculation to isolate the actual charge, e.
  3. The result is a finite, observable charge.

How renormalization works: a visual example

Imagine that the electrons are in a region where they can interact via virtual particles. Each interaction can lead to higher-order terms in the calculation.

Explanation of the diagram

This histogram beautifully depicts two electrons interacting via virtual photons. The wavy line represents these photons, and every time a photon is exchanged, it contributes to the calculation of the charge.

Reparameterization in this context means rethinking the infinite calculations at each point of the interaction and redefining them with respect to the finite values observable in experiments.

Renormalization in a mathematical framework

In more formal terms, suppose that our theory leads to a perturbation series in some coupling constant, g. Here's the basic idea:

F(g) = F_0 + F_1g + F_2g^2 + F_3g^3 + ...

Suppose some terms in this series are different, then our observed quantity, say F_observed, will be related to the measured values.

F_measured = lim (Λ → ∞) (F_bare(Λ) + antiword(Λ))

where Λ is a high-energy cutoff, and counterterms are adjustments made to balance infinity.

Renormalization groups: beyond simple renormalization

The renormalization group (RG) takes the concept of renormalization further, allowing us to study how theories behave when the energy scale changes. This means that we study how parameters adjust when we zoom in or out of a process.

RG flow

Parameter flows in parameter space under changes in the energy scale. Suppose a parameter, such as the coupling strength, μ changes with scale:

dG/d(log(μ)) = β(G)

Here, β(g) is a beta function, which represents how the coupling changes with energy scale, leading us to the idea of fixed points, where the physics does not change that much with energy changes.

Applications of renormalization

The applications of renormalization also extend beyond high-energy physics. For example, it applies to important phenomena in thermal phase transitions, polymer physics, and condensed matter physics, where many ideas of quantum fields find parallel uses.

Critical incident example

Consider how water transforms between solid, liquid, and gas. Close to critical points, water exhibits scale fluctuations, which give an indication of how renormalization can deal with the scale-dependence of these transformations.

Think of the scaling rules near these points as being similar to those in QFT. Similarly:

criticality_exponent = f(scaling factor)

Conclusion: The effect of renormalization

Renormalization plays a key role in understanding physical phenomena at various scales, from particles to the entire universe. By efficiently redefining infinity, it provides tools for precise calculations and predictions based on empirical observations, bringing consistency to theories that initially seemed abstract and disparate.

With its mathematical rigour and deep physical connections, reparameterisation demonstrates the power of theoretical physics and bridges the gap between observed facts and underlying theories.


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Renormalization theory

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