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 → Quantum Electrodynamics ↓


Renormalization in quantum electrodynamics


In the world of physics, scientists attempt to understand the fundamental forces that govern the universe. One of these forces is the electromagnetic force, which is described by quantum electrodynamics (QED), a quantum field theory. However, to understand the calculations in QED, physicists use a technique called "reparameterization." This concept is not just a mathematical trick, but a profound tool that helps us understand the nature of the universe at a much deeper level.

Understanding quantum electrodynamics

Before diving into renormalization, let's briefly understand what quantum electrodynamics (QED) is. QED is a quantum field theory that describes electromagnetic interactions between charged particles and the electromagnetic field. It is one of the cornerstones of the Standard Model of particle physics, providing accurate predictions for processes involving electrons, positrons, and photons.

To predict outcomes in QED, physicists use a mathematical series known as perturbation theory. This involves calculating probabilities as a power series in terms of the fine-structure constant, approximately 1/137. Each term in this series can be interpreted as a different possible set of interactions between particles, represented mathematically by Feynman diagrams.

The challenge of infinity

One of the important challenges faced in QED is the emergence of infinities when calculating certain probabilities. These infinities arise because, at very small distances or high energies, interactions between particles produce infinite results. For example, if we try to calculate the bare charge or bare mass of a particle directly from the fundamental equations of QED, we end up with infinite values.

Let's consider an example involving the self-energy of an electron. In QED, an electron surrounded by a cloud of virtual particles can be represented graphically as follows:

   ,  
  , 
 ,  
   ,

In this case, the electron, represented as a line, interacts with virtual particles that appear momentarily due to quantum fluctuations represented by the loop. Traditional calculations provide infinite self-energy for the electron, which is physically unacceptable because the electron's energy must be finite.

Introduction to renormalization

Renormalization is the process by which these infinities are removed or reinterpreted to obtain finite, meaningful quantities. The central idea is to redefine physical quantities, such as charge and mass, in terms of measurable values, rather than attempting to calculate them directly.

In technical terms, renormalization involves introducing a "cutoff" scale. This limit prevents the theory from dealing with extremely high energies that contribute to infinity. The resulting calculations at these energy scales produce finite results, which correspond to physical measurements. This process involves redefining several key parameters in the theory.

Basic quantities and physical quantities

In renormalization, it is necessary to distinguish between "bare" quantities and "physical" quantities. Bare quantities are the initial parameters in our equations, which are inherently infinite. Physical quantities, on the other hand, are finite and they are the ones we can measure in experiments.

Let us illustrate this with the concept of charge renormalization. Consider the bare charge ( e_0 ) and the physical, measurable charge ( e ). Through the renormalization process, the relation between them can be expressed as:

e^2 = Z_3 e_0^2
    e^2 = Z_3 e_0^2

Here, ( Z_3 ) is a factor known as the renormalization constant. The role of ( Z_3 ) is to absorb the infinity, so that ( e^2 ), the observed charge squared, remains finite and consistent with the experimentally observed values.

Renormalization group

A remarkable insight provided by renormalization consists in a concept called the "renormalization group." This idea provides a powerful way to understand how physical quantities change with different energy scales or distances. The result is a set of equations, known as the renormalization group equations, that describe how coupling constants, such as electromagnetic coupling, change with energy scale.

Imagine you are looking at an electron at different distances. A particular energy scale will define the effective charge that the electron appears to have. As you get closer to the electron (higher energy scale), the charge observed differs from the charge you observe at lower energies (farther away) due to the effects of virtual particles around the electron. The renormalization group provides a formal way to track these changes.

Implications and successes

Renormalization has played a key role in making QED one of the most successful and accurate theories of physics. Predictions made by QED, when combined with renormalization techniques, have been confirmed with incredible accuracy. This includes calculations of the anomalous magnetic moment of the electron and the Lamb shift in hydrogen.

Diagrammatic representation: walking coupling constant

Another important aspect of renormalization involves the visualization of the "walk" of the coupling constants, which describes how these constants evolve with changing energy levels. Consider the following diagram, where ( alpha(mu) ) represents the electromagnetic coupling constant at a given energy scale ( mu ):

                   ,
                  . _________> energy scale (mu)
    C .
    O......
    You ...
    P ...
    I ...
    I ..
    N.
      ,

As energy increases (moving to the right), the value of the coupling constant changes due to interactions with virtual particles. This visualization helps to understand why the coupling constant appears to "move" with energy.

Challenges and expansion

Despite its achievements, renormalization is not without challenges. Within some theories, it proves difficult or impossible to remove all infinities, a problem that points to the need for new physics or more thorough theoretical frameworks. Furthermore, extensions of renormalization consider its applications to other interactions, going beyond electromagnetic interactions into areas such as quantum chromodynamics (QCD) and weak forces.

Conclusion

In short, the concept of renormalization in quantum electrodynamics is more than just a means of fixing infinity - it is a way of understanding the profound connection between physical laws and observational scales. Whether by calculating the effect of the fine structure constant or studying the nature of virtual particles, renormalization remains a key principle for understanding the quantum dynamics of the universe, which ultimately shapes our understanding of particle physics and fundamental forces.


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