Final answer:
In canonical quantization, the Klein-Gordon equation is expressed in terms of the field operator φ (phi), not the state. To express the evolution of quantum states, one uses the Hamiltonian, which includes these field operators and acts on the states in the Hilbert space.
Step-by-step explanation:
In the context of canonical quantization in quantum field theory (QFT), the Klein-Gordon equation is expressed in terms of a field operator denoted by φ (phi). This field operator is an operator that acts on quantum states in the Hilbert space and is not the state itself. When we quantize a classical field theory, we replace classical fields with quantum field operators that create and annihilate particles at each point in space. In the quantum Klein-Gordon equation, φ is such a field operator, representing the quantum version of the classical field.
If we wanted to express the quantum Klein-Gordon equation in terms of the state, we would use the state vector representation, which is abstract and typically denoted by a ket, |psi>, in the Dirac notation. However, one cannot simply rewrite the quantum Klein-Gordon equation directly in terms of states since the equation describes the evolution of field operators over spacetime and the dynamics they impose on the states. Instead, the states evolve according to the Hamiltonian of the system which includes the field operators.