a Hilbert space and to construct a positive operator as a quantum operator on this Hilbert space in such a way that it gives this evolution for the operators as given by the preceding equation, and to show that contains a vacuum state on which has zero eigenvalue. In practice, this construction is a difficult problem for interacting field theories, and has been solved completely only in a few simple cases via the methods of constructive quantum field theory. Many of these issues can be sidestepped using the Feynman integral as described for a particular in the article on scalar field theory. In the case of a free field, with = 0, the quantization procedure is relatively straightforward. It is convenient to Fourier transform the fields, so that The reality of the fields implies that . The classical Hamiltonian may be expanded in Fourier modes as where . This Hamiltonian is thus recognizable as an infinite sum of classical normal mode oscillator excitations , each one of which is quantized in the standard manner, so the free quantum Hamiltonian looks identical. It is the s that have become operators obeying the standard commutation relations, [, †] = [†, ] = iħ, with all others vanishing. The collective Hilbert space of all these oscillators is thus constructed using creation and annihilation operators constructed from these modes, for which [, †] = 1 for all , with all other commutators vanishing. The vacuum is taken to be annihilated by all of the , and is the Hilbert space constructed by applying any combination of the infinite collection of creation operators † to . This Hilbert space is called Fock space. For each , this construction is identical to a quantum harmonic oscillator. The quantum field is an infinite array of quantum oscillators. The quantum Hamiltonian then amounts to , where may be interpreted as the number operator giving the number of particles in a state with momentum . This Hamiltonian differs from the previous expression by the subtraction of the zero-point energy of each harmonic oscillator. This satisfies the condition that must annihilate the vacuum, without affecting the time-evolution of operators via the above exponentiation operation. This subtraction of the zero-point energy may be considered to be a resolution of the quantum operator ordering ambiguity, since it is equivalent to requiring that all creation operators appear to the left of annihilation operators