Tru Physics

Quantum Electrodynamics (QED)

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Introduction

Quantum Electrodynamics (QED) is a quantum field theory that describes how light and matter interact. It is the first theory where full agreement between quantum mechanics and special relativity is achieved. QED mathematically describes all phenomena involving electrically charged particles interacting by means of exchange of photons and represents the quantum counterpart of classical electromagnetism.

Basic Principles

QED is built upon the principles of quantum mechanics and special relativity. It treats particles not as point-like objects, but as excitations of the underlying quantum fields. In QED, particles interact by exchanging virtual photons, the quanta of the electromagnetic field.

The fundamental equation of QED is the Dirac equation:

i\hbar\frac{\partial}{\partial t}\psi(x,t) = \left[\dfrac{\hbar c}{i}\vec{\alpha}\cdot\vec{\nabla} + mc^2\beta\right]\psi(x,t)

where \psi(x,t) is the wave function of the electron, \hbar is the reduced Planck’s constant, c is the speed of light, \vec{\alpha} and \beta are matrices introduced by Dirac, and m is the mass of the electron.

Feynman Diagrams

Feynman diagrams are a key tool in QED and other quantum field theories. Named after physicist Richard Feynman, these diagrams provide a pictorial representation of the interactions described by QED.

A typical Feynman diagram involves lines representing particles meeting at vertices, with the lines representing the propagation of particles and the vertices representing the interactions between them.

QED Lagrangian

In QED, the dynamics of the electromagnetic field and its interaction with matter (charged particles) are described by the QED Lagrangian:

\mathcal{L}{QED} = \overline{\psi}(i\gamma^\mu D\mu - m)\psi - \dfrac{1}{4}F_{\mu\nu}F^{\mu\nu}

where \psi is the Dirac spinor, \gamma^\mu are the gamma matrices, D_\mu is the covariant derivative, m is the mass of the electron, F_{\mu\nu} is the electromagnetic field tensor.

Quantization of the Electromagnetic Field

Just like matter, the electromagnetic field can also be quantized in QED. The quantum of the electromagnetic field is the photon, which is a massless particle with spin-1.

The quantization of the electromagnetic field is achieved by promoting the classical electromagnetic potentials to quantum operators and imposing commutation relations.

The Concept of Virtual Particles

QED, like other quantum field theories, involves the concept of “virtual particles”. These are not particles in the usual sense but are disturbances in the fields that mediate interactions between “real” particles. In the case of QED, virtual photons mediate the electromagnetic interactions between charged particles.

QED and the Fine-Structure Constant

One of the triumphs of QED is the accurate prediction of the fine-structure constant, a dimensionless constant that characterizes the strength of the electromagnetic interaction between elementary charged particles. It is given by:

\alpha = \dfrac{e^2}{4\pi\epsilon_0\hbar c} \approx 1/137

where e is the elementary charge, \epsilon_0 is the vacuum permittivity, \hbar is the reduced Planck’s constant, and c is the speed of light.

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