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9) ∂fk hω ¯ For any given field, here is a succession of operations we have to trace. First, the component T44 of the energy-momentum tensor is determined. Then, by integrating over the whole three-dimensional space, we obtain the Hamiltonian density H. Next, we proceed to the series expansion of the state vector |ψ >, associated to the field quanta, as cˆk |φk >, |ψ > = k where we take into account the fact that, if cˆk is an operator, then |ψ > is also an operator that acts on the number of particles.

4) yields ˆ i ˆ ∂H = [H, qk ]. 5) Suppose that qk is a state function of the form qk = ck e−iωt . 7) Alternative methods of field quantization where ˆ ∂H . 8) ˆ ˆ ∂H i ˆ ∂H = − iω e−iωt = [H, ck ] e−iωt , ∂pk ∂fk ¯ h or ˆ 1 ˆ ∂H =− [H, cˆk ]. 9) ∂fk hω ¯ For any given field, here is a succession of operations we have to trace. First, the component T44 of the energy-momentum tensor is determined. Then, by integrating over the whole three-dimensional space, we obtain the Hamiltonian density H. Next, we proceed to the series expansion of the state vector |ψ >, associated to the field quanta, as cˆk |φk >, |ψ > = k where we take into account the fact that, if cˆk is an operator, then |ψ > is also an operator that acts on the number of particles.

0. 1) is satisfied by the eigenvalue α = 0. The eigenvalues taken by α are then α = 0, 1, 2, 3, ..... showing that α can be any natural number, including zero. This fact straightforwardly suggests that α is a number of particles, α = n. The value α = 0 corresponds to the vacuum state. 14) c+ |n >= n + 1 |n + 1 > . 14). Indeed, √ √ (c c+ − c+ c)|n >= n + 1 c |n + 1 > − nc+ |n − 1 > √ √ = ( n + 1)2 |n > −( n)2 |n >= |n > . 1), among other things, leads to the definition of the vector ket |n >, which describes the state of a system of n particles.

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