[QUE/QFT-06002] QFT-PROBLEM

 $\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}\newcommand{\dd}[2][]{\frac{d#1}{d#2}}${}$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}${}$\newcommand{\ket}[1]{|#1\rangle}$ {} $\newcommand{\bra}[1]{\langle #1|}$

For a real free Dirac field, mass \(m\), compute \[ \matrixelement{0}{\psi(x)\psi(y)}{\vec{p},r; \vec{q},s}\] and show that the result is properly anti-symmetrized wave function for two identical fermions with momenta \(\vec{p},r;\vec{q},s\) and spins \(r,s\). Here \(\ket{\vec{k}, \vec{q}}\) is the state with two fermions with momenta \(\vec{p}, \vec{q}\) and spins \(r,s\).