The correspondence between quantum fields and particle types is a fundamental aspect of quantum field theory. While quantum fields are indeed operator-valued, the connection to particle types arises from the way these operators act on the vacuum state and create or annihilate particles.
In quantum field theory, particles are understood as excitations or quanta of their corresponding quantum fields. Each type of particle is associated with a specific field, such as the electron field or the photon field. The field operators create and annihilate particles by acting on the vacuum state of the system.
The creation operator of a quantum field creates a particle of its associated type, while the annihilation operator removes a particle from the system. These operators have specific properties and transformation rules, which determine the behavior of the corresponding particles.
The particle types are defined by the properties of the fields themselves, such as their symmetries, transformation properties under various operations, and the equations of motion they satisfy. For example, the electron field satisfies the Dirac equation and describes fermionic particles with spin-1/2, while the photon field corresponds to massless bosonic particles with spin-1.
While the quantum field operators are mathematical objects, it is their action on the vacuum state and the resulting creation and annihilation processes that give rise to the particle interpretation. The specific form of the operators and their commutation or anti-commutation relations are determined by the underlying field equations and the desired properties of the corresponding particles.
In summary, the correspondence between quantum fields and particle types arises from the way the field operators act on the vacuum state and create or annihilate particles, as dictated by the properties and equations of motion of the fields themselves. The operator-valued nature of the fields is an essential mathematical framework to describe and quantify the behavior of particles in quantum field theory.