In quantum field theory (QFT), particles and fields are described by field operators, which are promoted from classical solutions of the corresponding classical field equations. Let's discuss the interpretation of classical solutions and their promotion to field operators for the Klein-Gordon, Dirac, and Proca equations individually:
Klein-Gordon Equation: The classical Klein-Gordon equation describes scalar particles. Its classical solutions represent classical scalar fields. These solutions are interpreted as classical wave-like excitations in the field, corresponding to particles with definite energy and momentum. However, in QFT, these classical solutions are promoted to quantum field operators, known as Klein-Gordon fields. The quantum interpretation of the Klein-Gordon field is that it describes a collection of particles called scalar bosons (like the Higgs boson) and their quantum excitations, which can be created or annihilated.
Dirac Equation: The classical Dirac equation describes spinor particles, such as electrons. Its classical solutions represent classical spinor fields. These solutions describe classical wave-like excitations of the spinor field, representing particles with definite energy, momentum, and spin. In QFT, the classical solutions of the Dirac equation are promoted to quantum field operators, called Dirac fields. The Dirac field operator describes the behavior of fermionic particles (like electrons) and their quantum excitations. It allows for the creation and annihilation of particles, and the quantum interpretation is in terms of fermionic particles obeying the rules of quantum statistics.
Proca Equation: The classical Proca equation describes vector particles, such as gauge bosons. Its classical solutions represent classical vector fields. These solutions correspond to classical wave-like excitations in the field, representing particles with definite energy, momentum, and spin. In QFT, the classical solutions of the Proca equation are promoted to quantum field operators, known as Proca fields. The Proca field operator describes the behavior of vector bosons (e.g., photons, W and Z bosons) and their quantum excitations. It allows for the creation and annihilation of vector particles, and the quantum interpretation is in terms of vector bosons with quantized properties.
In summary, the classical solutions of the Klein-Gordon, Dirac, and Proca equations represent classical wave-like excitations in the corresponding fields, which can be interpreted as particles with definite energy, momentum, and other properties. In quantum field theory, these classical solutions are promoted to quantum field operators, describing the quantum behavior of the corresponding particles and allowing for particle creation and annihilation.