Primary calculation involving the Dirac gama matrices

[YSM]

Homework Statement

How to work out a calculation involving properties of gama matrices and the dirac equation.

Relevant Equations

showed below

When working on the exercise 3.2 of Peskin's QFT, I find one of the calculating steps confused for me. I read the solution, which is showed in the picture. I just don't understand the boxed part.

I know it involved the Dirac equation, and the solution seems to treat the momentum as a operator, because only in this way can I relate the momentum in the equation with the partial derivative in the Dirac Equation. But I don't think the momentum in the solution of Dirac field serve as an operator.

 

[Antarres]

Momentum in Dirac equation indeed is an operator, in fact: $p_\mu = i\partial_\mu$. So if that's the only problem, there's your answer.

Edit: Momentum in solutions of Dirac equation is eigenvalue of momentum operator, though they're usually denoted with the same letter.

 

[YSM]

Thank you for your answer, but why no minus sign in front of p?

 

[Antarres]

It's the sign convention where metric is given by $diag(1, -1, -1, -1)$. So in that convention the energy operator is given by $p_0 = i\partial_t$ as it should be, and 3-momentum operator is given by $\textbf{p} = -i\nabla$ because $p^i = -p_i$ for spatial indices.