Mathematically, there are several key differences between the classical electromagnetic (EM) field in classical electromagnetism and the EM field in Quantum Electrodynamics (QED). Here are some of the fundamental distinctions:
Classical EM Field (Maxwell's Equations):
- Description: Classical electromagnetism, based on Maxwell's equations, treats the EM field as a continuous, deterministic field.
- Equations: Maxwell's equations involve differential equations, such as the Gauss's law, Gauss's law for magnetism, Faraday's law of electromagnetic induction, and Ampere's law with Maxwell's addition.
- Continuous Fields: In classical EM, the EM field is described by continuous functions of space and time, represented by vector fields (electric field E and magnetic field B).
- Determinism: Classical EM is deterministic, meaning that the evolution of the EM field can be calculated precisely given the initial conditions and the governing equations.
Quantum Electrodynamics (QED):
- Description: Quantum Electrodynamics is a quantum field theory that combines quantum mechanics with special relativity to describe the interactions of photons (EM radiation) with charged particles.
- Equations: QED employs the mathematical framework of quantum field theory and treats the EM field as a quantized field composed of particles called photons. It uses a Lagrangian formalism and Feynman diagrams to calculate probabilities of various particle interactions.
- Quantized Fields: In QED, the EM field is quantized, meaning it is described in terms of discrete, quantized particles (photons) and their corresponding wave functions.
- Probabilistic Nature: QED incorporates the probabilistic nature of quantum mechanics, where calculations involve probability amplitudes and probabilities rather than deterministic quantities.
- Virtual Particles: QED introduces the concept of virtual particles, which are temporary particle-antiparticle pairs that can emerge from the vacuum and mediate interactions between charged particles.
- Renormalization: QED requires renormalization techniques to account for infinities that arise in certain calculations. These techniques involve adjusting parameters in the theory to match experimental results.
In summary, the classical EM field in classical electromagnetism is described by continuous vector fields and deterministic equations, while the EM field in QED is described by quantized fields of photons and incorporates probabilistic calculations, virtual particles, and the need for renormalization techniques. QED provides a more comprehensive framework that includes the quantum behavior of the EM field and its interactions with charged particles.