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 ↓


Gauge invariance in quantum electrodynamics


Gauge invariance is a fundamental concept in quantum electrodynamics (QED) and more broadly quantum field theory (QFT). It represents a symmetry of a field theory, and it plays a key role in the formulation and understanding of the fundamental interactions described by the Standard Model of particle physics. In this lengthy explanation, we will take a deep look at what gauge invariance is, why it is necessary, and how it leads to a successful description of electromagnetism within the framework of QED.

Basic concept of gauge invariance

Gauge invariance essentially refers to the idea that certain physical systems are more elegantly described by variables that can undergo specific transformations without affecting observable quantities. Such symmetries are called gauge symmetries.

What is gauge conversion?

Gauge transformations in electromagnetism involve transforming the scalar potential Φ and the vector potential A in such a way that the electric field E and the magnetic field B remain unchanged. In mathematical terms, if we have a scalar function λ (x,t), the transformation can be written as:

Φ' = Φ + ∂λ/∂t
a' = a − ∇λ

The electric and magnetic fields are given as follows:

E = -∇Φ - ∂A/∂t
B = ∇×A

These fields remain unchanged under the gauge transformations outlined above.

Why is gauge invariance important?

Gauge invariance is important because it is the basis of fundamental interactions in nature. In the case of quantum electrodynamics, gauge invariance ensures that the theory respects the local symmetries governed by electromagnetism. This means that no matter how we describe the potentials, the physical predictions (such as forces and particle momentum) will not change.

Mathematical description of gauge invariance

Field equations

The electromagnetic field tensor F μν is a principal component that captures the electric and magnetic fields in a covariant form. It is defined in terms of the four-potential A μ as follows:

F μν = ∂ μ A ν − ∂ ν A μ

Maxwell's equations, which are the fundamental equations of electromagnetism, can be derived from the Lagrangian of the electromagnetic field, which is gauge invariant:

L = -1/4 * F μν F μν

Principle of minimum action

The action principle states that the path taken by a system is the one that leads the action to a peak. For electromagnetism, the action S is given by:

S = ∫ L d⁴x

The differentiability of the action with respect to the potential should vanish, leading to the Euler–Lagrange equations. These transformations should not change the physics, due to gauge invariance.

Charge protection

Gauge invariance has profound implications in terms of charge conservation. The continuity equation for charge conservation is given as:

∂μjμ = 0

where J μ is the four-current. Gauge invariance in the Lagrangian ensures that this equation is true, tying the symmetry and conservation law together via Noether's theorem.

Electromagnetic interaction in Q.E.D.

Quantum field theory setup

In quantum field theory, particles are excitations of underlying fields. The electromagnetic interaction in QED involves the interaction between electromagnetic fields and matter fields. The fundamental particles (electrons and positrons) are described by the Dirac field, Ψ, while the electromagnetic interaction is mediated by the photon, which is described by the electromagnetic potential Aμ.

Inclusion of interaction term

The Dirac equation for free particles is given as:

(iγ μ ∂ μ − m)ψ = 0

To include electromagnetism, we replace the partial derivative with the covariant derivative:

∂Dμ = ∂μ + ∂Aμ

Thus, the interaction is introduced by modifying the derivative in the Dirac equation:

(iγ μ D μ − m)ψ = 0

Feynman diagrams

Quantum electrodynamics is represented using Feynman diagrams, which depict the interactions between particles. A simple process in QED is the scattering of an electron and a positron, which can be viewed as an exchange of photons.

e − + e + → γ → e − + e +

In this diagram, the black lines represent the electron and positron, and the blue line represents the exchanged photon. Conservation of charge and energy is assured at each vertex, and this is an effect of gauge symmetry manifesting at the level of interactions.

Beyond Q.E.D.: Gauge theories in the standard model

While we have focused on QED as an example, the concept of gauge invariance extends beyond the other forces described in the Standard Model, including the weak and strong nuclear forces. Each interaction is described by its own gauge theory with its symmetry properties.

The symmetry group for the electroweak interaction is SU(2) L × U(1) Y, which leads to a unified theory of the electroweak force. The theory of the strong interaction is governed by the SU(3) C symmetry of quantum chromodynamics (QCD).

Conclusion

Gauge invariance is a cornerstone of modern physics, integral to the formulation and understanding of fundamental forces. Through the framework of QED, it not only explains the success of electromagnetic theory but also guides the larger construction of the Standard Model. In essence, gauge invariance highlights the beauty and consistency of the laws of nature, unifying diverse physical phenomena under a single, symmetrical umbrella.


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Gauge invariance in quantum electrodynamics

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