ABSTRACT
In this chapter we move from the treatment of the electromagnetic field to that of the Dirac fields which describe spin 1 2 $ \frac{1}{2} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315369723/48d43afe-85b0-4e2b-a932-df2bd1af07be/content/inline-math6_536.tif"/> particles. The problem posed is how to treat fields which must obey anticommutation rules and the Pauli exclusion principle. In the elementary treatments of field theory we saw that a free field which describes non-interacting bosons is equivalent to a set of harmonic oscillators, one for each state in which a particle can be found. Focusing on a single oscillator we can define the creation and destruction operators, a † , a $ a^{\dagger },\,a $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315369723/48d43afe-85b0-4e2b-a932-df2bd1af07be/content/inline-math6_537.tif"/> which obey the commutation rules [ a , a † ] = 1 . $$ \begin{aligned}[a,\,a^{\dagger }] = 1 \ . \end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315369723/48d43afe-85b0-4e2b-a932-df2bd1af07be/content/math6_238.tif"/>
