In quantum field theory, creation and annihilation operators are mathematical operators that describe the creation and annihilation of particles within a quantum field. They do not create the field itself, but rather they represent the creation and destruction of particles associated with that field.
Quantum field theory treats particles as excitations of underlying fields that permeate space. These fields are described by operators that create and annihilate particles. The creation operator creates a particle in the field, while the annihilation operator removes a particle from the field.
The creation and annihilation operators operate on the state of the field, and when applied, they modify the state by adding or subtracting particles. For example, applying a creation operator to the vacuum state of a field creates a one-particle state, while applying an annihilation operator to a one-particle state removes the particle, returning to the vacuum state.
The relationship between creation and annihilation operators and the field is given by the field operator. The field operator is a superposition of creation and annihilation operators at different positions in space. It describes the field itself and how it interacts with particles. By applying the field operator to a state, you can obtain the value of the field at a specific point.
So, in summary, creation and annihilation operators do not create the field itself, but rather they represent the creation and annihilation of particles associated with the field. The field operator describes the field, and by applying creation and annihilation operators to the field, you can add or remove particles.