## Quantum Field Theory

### Second Quantization

**See the 2021 version.**

Using the neutral scalar meson field in the last section as an example, quantization of the field is accomplished by demanding the coefficients c_{k}'s to satisfy the following commutation rules:

| ---------- (4) |

If a number operator N_{k} = c_{k}c_{k} is defined such that it operates on the state vector |n_{k} to generate:

N_{k}|n_{k} = n_{k}|n_{k}

where n_{k} is the number of particles in the k state; it can be shown that

c_{k}|n_{k} = (n_{k}+1)^{1/2}|n_{k}+1

c_{k}|n_{k} = n_{k}^{1/2}|n_{k}-1

Thus c_{k} increases the number of particles in the k state by 1, while c_{k} 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:

| ---------- (5) |

for all values of k^{l} and n_{l}.
They form an abstract space called the Fock space with all the eigenvectors orthogonal (perpendicular) to each others and the norm (length) equal to 1.

In particular, the vacuum state is:

| ---------- (6) |

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

See more about second quantization in "Quantization and Field Equations" and "The Different Perspectives of the 1st and 2nd Quantizations".
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