Reference: References: Robert D. Klauber, Student Friendly Quantum Field Theory, (Sandtrove Press, 2013) – Chapter 4, Problem 4.11.
The Dirac equation in condensed form is
where the gamma matrices have been defined earlier. The Hermitian conjugate of the gamma matrix is given by the Hermiticity condition
To get the adjoint form of the Dirac equation, we use the adjoint solutions
The Hermitian conjugate of 1 is
Remember that when we take the Hermitian conjugate of a matrix equation we must take the Hermitian conjugate of each matrix and also reverse the order of matrix multiplication, which is why the term appears at the end on the LHS. The is a differential operator, not a matrix, so it retains its position in front of the .
Using 2, we can write this as
Then, by post-multiplying by and using , the identity matrix, we get, using the definition 3
Note that although the terms in this adjoint equation are adjoint solutions, the gamma matrices are the original (that is, not the Hermitian conjugate) gamma matrices.