To turn a classical field theory into a quantum field theory, you follow a process known as quantization. Here are the general steps involved in the quantization of a classical field theory:
Identify the classical field: Start with a classical field theory that describes the dynamics of a particular field, such as the electromagnetic field or a scalar field.
Promote the field to an operator: In quantum field theory, the classical field is promoted to an operator. These operators act on a quantum state to create or annihilate particles associated with the field.
Impose canonical commutation relations: The field operators are subjected to canonical commutation relations or canonical anticommutation relations, depending on the nature of the field (bosonic or fermionic). These relations describe the fundamental commutation/anticommutation properties between the field operators and their conjugate momenta.
Expand the field in terms of creation and annihilation operators: The field operator is expanded in terms of creation and annihilation operators. The creation operators create particles with positive energies, while the annihilation operators remove particles.
Define the vacuum state: Identify the vacuum state, which is the state with no particles present. The vacuum state is defined as the state that is annihilated by all the annihilation operators.
Apply the Hamiltonian operator: Replace the classical Hamiltonian with a quantum Hamiltonian operator, which is constructed from the field operators and their conjugate momenta.
Solve the quantum field equations: Solve the quantum field equations of motion, which are obtained from the quantum Hamiltonian. These equations determine the time evolution of the quantum field and its associated particles.
Compute observables and correlation functions: Use the field operators to compute expectation values of observables and correlation functions, which provide information about the behavior and interactions of the quantum field and particles.
It's important to note that the quantization procedure can be more involved and intricate depending on the specific field theory and its symmetries. Different quantization techniques, such as canonical quantization or path integral quantization, may be employed based on the nature of the theory and the desired mathematical framework.
Overall, the goal of quantization is to describe the classical field theory in terms of quantum operators and states, enabling the study of quantum phenomena and the interactions of particles associated with the field.