Quantization of the field is accomplished by demanding the coefficients ck's to satisfy the following commutation rules:

If a number operator Nk = ck*ck is defined such that it operates on the state vector |nk> to generate:

Nk|nk> = nk|nk>

where nk is the number of particles in the k state; it can be shown that

ck*|nk> = (nk+1)1/2|nk+1>

ck|nk> = nk1/2|nk-1>

Thus ck* increases the number of particles in the k state by 1, while ck reduces the number of particles in the k state by 1. They are called creation and annihilation operator respectively. The complete set of eigenvectors is given by:

for all values of kl and nl. They form an abstract space called the Hilbert space with all the eigenvectors orthogonal (perpendicular) to each others and the norm (length) equal to 1. .

In particular, the vacuum state is:

which corresponds to no particle in any state - the vacuum.