[QUE/QFT-06010] QFT-PROBLEM

Verify that free Dirac Lagrangian is invariant under phase transformation \begin{equation*} \psi(x) \longrightarrow \psi{'}(x) = e^{i\alpha}\psi(x). \end{equation*} Find the corresponding conserved quantity \(Q\). For \(\tt classical Dirac field \), express it in terms of \(a^{(r)}(p),a^{(r)\dagger}(p), b^{(r)}(p)\) and \(b^{(r)\dagger}(p)\), and show that \begin{equation} Q = \sum_{r=1}^2 \int \Big(\frac{M}{E_p}\Big) d^3p \big\{a^{(r)}(p)a^{(r)\dagger}(p) + b^{(r)\dagger}(b^{(r)}(p) \big\} \end{equation} Notation is same as Gasiorowicz, {\it Elementary Particle Physics}, John Wiley and Sons, New York (1966).