# Derivation of Fermion field operator

I want a detailed explicit calculation of obtaining fermion field operator from the Dirac equation.In all the books the field operator is written directly just writing like "using Fourier mode expansion".The dirac equation is a matrix equation ..i am unable to obtain the fermion field operator by calculating on my own.Please let me know if I misinterpreted any thing here..I am new to these topic.

• You are assuming that the solution (a spinor indeed) of the free Dirac equation can be computed using a Fourier series. Then, you promote the coefficients of the series to operators.
– Jon
Commented Feb 23, 2021 at 14:42
• Are you asking about the relationship between the matrix-ness and the operator-ness? The Dirac matrices $\gamma^\mu$ are still ordinary matrices in the quantum-field version of the model. The Dirac spinor $\psi$ is a column matrix whose components $\psi_a$ are operators that act on a Hilbert space. Observables like $\psi^\dagger_a(x)\psi_b(y)$ are expressed in terms of these operators. These operators aren't derived. Their properties are postulated as part of the definition of the quantum-field model. Commented Feb 24, 2021 at 0:41