To fully understand the construction of the electric field in Quantum Field Theory (QFT), there are a few additional concepts and mathematical tools that you may need to familiarize yourself with. Here are some areas of study that would be beneficial:
Special Relativity: QFT is based on the principles of special relativity, which describes the behavior of physical phenomena in the absence of gravity. It is important to have a solid understanding of Lorentz transformations, spacetime intervals, and the relativistic formulation of classical electromagnetism.
Lagrangian and Hamiltonian Formalism: In QFT, the Lagrangian and Hamiltonian formalisms are employed to describe the dynamics of fields. Familiarize yourself with the principles of classical field theory, including Lagrangian densities, Euler-Lagrange equations, and canonical quantization.
Group Theory: Group theory plays a crucial role in QFT, particularly in understanding the symmetries of the underlying theory. Study concepts like Lie groups, representations, and the theory of continuous symmetry groups, as they are fundamental to the construction of QFT.
Second Quantization: Quantum Field Theory extends quantum mechanics to systems with an infinite number of degrees of freedom. Second quantization is the formalism used to describe the quantization of fields, where creation and annihilation operators are introduced to represent particles.
Scattering Theory: QFT provides a framework to calculate scattering amplitudes, which describe the probabilities of particle interactions. Familiarity with scattering theory and techniques for calculating scattering amplitudes, such as Feynman diagrams and perturbation theory, would be beneficial.
Renormalization: In QFT, certain calculations may yield infinities or divergent results. Renormalization techniques are employed to remove these infinities and obtain physically meaningful predictions. Understanding the basics of renormalization is crucial in dealing with the subtleties of QFT calculations.
Gauge Theory: Gauge theories, such as Quantum Electrodynamics (QED), describe the interactions between charged particles and electromagnetic fields. Developing a solid understanding of gauge theories, gauge transformations, and covariant derivatives will be essential to comprehend the construction of the electric field in QFT.
It's worth noting that Quantum Field Theory is a highly advanced and mathematically sophisticated subject. The above list provides a starting point, but it's advisable to consult textbooks and resources specifically focused on QFT to deepen your understanding further.