In quantum field theory, particles are described as excitations of their respective quantum fields. The concept of a field is fundamental to quantum field theory, and it represents a physical quantity that pervades all of space. Fields are characterized by their properties, such as their behavior under various transformations and their equations of motion.
Now, in the case of electrons and positrons, both are considered excitations of the electron field. The electron field is a quantum field that permeates all of spacetime, and it represents the underlying entity associated with electrons. According to the principles of quantum field theory, excitations in the electron field can manifest as particles. These particles can be electrons or their antiparticles, which are positrons.
Particles and antiparticles are related by the concept of particle-antiparticle symmetry. Antiparticles have the same mass as their corresponding particles, but they have opposite electric charge and other quantum numbers. In the case of the electron field, excitations with positive electric charge correspond to positrons, while those with negative electric charge correspond to electrons.
The reason why both electrons and positrons are described by the same field is due to the underlying symmetries of the theory. Quantum field theories are constructed to be mathematically consistent and obey certain symmetries, such as gauge symmetries or symmetries related to the conservation of electric charge. These symmetries allow for the existence of particles and antiparticles within the same field.
In summary, in quantum field theory, particles are described as excitations of their respective quantum fields. In the case of electrons and positrons, both are considered excitations of the electron field, which is a fundamental field that permeates all of spacetime. The underlying symmetries of the theory allow for the existence of both particles and antiparticles within the same field, with opposite electric charge and other quantum numbers.