Classical electromagnetism and Quantum Electrodynamics (QED) describe the behavior of electromagnetic waves at different levels of abstraction and mathematical formalism. While classical electromagnetism is based on classical physics, QED is a quantum field theory that incorporates quantum mechanics. The mathematical relationship between the two can be understood by considering the formalism of QED as a quantum mechanical extension of classical electromagnetism.
In classical electromagnetism, electromagnetic waves are described by Maxwell's equations, which are a set of partial differential equations that relate the electric and magnetic fields to their sources (charges and currents). These equations are deterministic and provide a complete description of the classical electromagnetic phenomena.
On the other hand, in QED, the behavior of electromagnetic waves is described by a quantum field theory. The fundamental entities in QED are the electromagnetic field and charged particles, such as electrons and positrons. The electromagnetic field is quantized, meaning it is treated as a collection of particles called photons. These photons are the quanta of the electromagnetic field and carry energy and momentum.
The mathematical relationship between classical electromagnetism and QED can be seen through a process known as quantization, where classical fields are promoted to quantum fields. In QED, the electromagnetic field is quantized using the principles of quantum mechanics. This leads to the introduction of creation and annihilation operators that create and destroy photons. These operators act on a quantum state, describing the number and properties of photons present.
By using perturbation theory and Feynman diagrams, QED provides a framework to calculate probabilities and amplitudes for various electromagnetic processes involving charged particles and photons. These calculations involve the interaction of particles through the exchange of photons, and the resulting amplitudes are related to the probabilities of different outcomes.
Overall, QED reduces to classical electromagnetism in the appropriate limit, known as the classical limit, where quantum effects become negligible. This means that classical electromagnetism can be seen as an approximation or an average behavior of the underlying quantum mechanical processes described by QED. However, it's important to note that the mathematical formalism and concepts in QED go beyond what is described by classical electromagnetism, providing a more complete and accurate description of electromagnetic phenomena at the quantum level.