Deriving special relativistic dynamics, including relativistic mechanics and electrodynamics, from quantum electrodynamics (QED) is a complex and challenging task. While QED is a relativistic quantum field theory that incorporates both quantum mechanics and special relativity, directly deriving the classical equations of motion from its mathematical structure is not straightforward.
QED describes the interactions between charged particles and electromagnetic fields in terms of quantum fields, specifically the electron field and the electromagnetic field. These fields satisfy certain mathematical equations, such as the Dirac equation for the electron field and Maxwell's equations for the electromagnetic field.
To connect QED to classical dynamics, one needs to consider the limit where quantum effects become negligible and recover the classical behavior. This limit is known as the classical or semiclassical limit. In this limit, the quantum fields reduce to classical fields, and the quantum operators are replaced by classical observables.
In principle, it is possible to derive classical dynamics from QED using a technique called the semiclassical approximation. In this approach, one expands the quantum fields and operators in powers of the Planck constant (h) and keeps only the leading-order terms. The resulting equations can then be manipulated to obtain the classical equations of motion.
However, deriving classical dynamics directly from QED is highly nontrivial due to the inherent complexity of the theory. QED involves intricate calculations, renormalization procedures, and advanced mathematical techniques like Feynman diagrams. Furthermore, the transition from the quantum to classical regime is not well understood in general and often requires additional assumptions or approximations.
While it may be possible to obtain some classical equations of motion from QED in specific cases or under simplifying assumptions, it is challenging to rigorously derive the complete framework of relativistic mechanics and electrodynamics solely from the mathematical structure of QED. The connection between the two domains is a rich and active area of research, and further developments in theoretical physics may shed more light on this intriguing relationship.