In quantum field theory, electron annihilation is described using the framework of second quantization and the creation and annihilation operators associated with the electron field.
In quantum field theory, particles are represented as excitations of underlying fields that permeate space. The electron field is one such field that describes the behavior of electrons. The electron field operator, denoted by ψ(x), is a function of both position (x) and time.
The electron annihilation process is mathematically expressed by the annihilation operator, denoted by a, acting on the initial electron state. The annihilation operator removes an electron from the system, and it is associated with the destruction of an electron excitation. The annihilation operator is defined as:
a(p) |e(p)>,
where p represents the momentum of the electron, and |e(p)> is the initial electron state.
The annihilation operator creates negative energy solutions to the Dirac equation, which correspond to the electron's antiparticle, the positron. So, electron annihilation can lead to the creation of a positron. The annihilation process can be represented as:
a(p) |e(p)> -> |0>,
where |0> represents the vacuum state, i.e., the state with no particles.
It's important to note that this description is a simplified version, and in reality, the full mathematical treatment involves spinors, Dirac equations, and more advanced mathematical machinery. However, the concept of annihilation operators acting on quantum states provides the basic framework for understanding electron annihilation in quantum field theory.