Quantizing a field in quantum field theory (QFT) involves expressing the field in terms of creation and annihilation operators, which allows for the quantization of its properties. In QFT, fields are treated as operators that act on a quantum state, and the quantization process assigns discrete and quantized values to the field's properties.
In QFT, each type of particle is associated with a field. For example, the electron field is associated with electrons, the photon field with photons, and so on. The field is described as a mathematical object that spans all of space and time, and it can be decomposed into a sum of Fourier modes or wavefunctions, as you mentioned.
To quantize the field, we introduce creation and annihilation operators for each Fourier mode or wavefunction. The creation operator creates a particle in a particular state of the field, while the annihilation operator removes a particle from that state. These operators satisfy specific commutation or anticommutation relations depending on the statistics of the particles (bosons or fermions).
By quantizing the field, we obtain a quantized representation of the relevant particle's properties. The creation and annihilation operators allow us to describe the number of particles present, their momentum, energy, and other observable quantities. The quantization process imposes quantized energy levels on the particles and introduces the concept of particle "excitations" or "quanta" associated with the field.
Quantization also leads to the idea of particle-antiparticle pairs. For example, in the electron field, the creation operator can create an electron, while the annihilation operator can create a positron (the antiparticle of the electron). The presence or absence of these particles is represented by the occupation of the corresponding modes in the field.
It's important to note that quantizing a field does not imply that all the properties of the particles are quantized. Some properties, such as the spin of a particle, are intrinsic and do not change during the quantization process. However, the quantization of the field provides a framework for understanding and calculating the probabilistic behavior and interactions of particles in quantum field theory.