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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 \(r,s\).

Let \(\Gamma\) denote a product of Dirac matrices. Define \(\widetilde{\Gamma}\) by \begin{equation*} \bar{u}(s,q) \widetilde{\Gamma}u(r,p) = (\bar{u}(r,p)^\dagger\Gamma u(s,q))^* . \end{equation*} Use the above definition of \(\widetilde{\Gamma}\) and show that

1. \(\widetilde{\Gamma} = \gamma_0 \Gamma^\dagger \gamma_0. \)
2.      \(\widetilde{\gamma^\mu} = \gamma^\mu \)
3.     \(\widetilde{\gamma_5} = -\gamma_5\)
4.     \(\widetilde{\gamma^\mu \gamma_5}= - \gamma^\mu \gamma_5\)

Compute \(\widetilde{\sigma}_{0k}\) and \(\widetilde{\sigma}_{ij}\) and hence show      that \(\widetilde{\sigma}_{\mu\nu}=\sigma_{\mu\nu}\)

Let \(p^\mu\) be a time like momentum  vector. Let \(n^\mu\)  be a four vector
such that
\begin{equation}
 n^\mu p_\mu=0,  \text{ and } n^\mu n_\mu=-1.
\end{equation}
How many such independent four vectors \(n^\mu\) exist? How that the operators
\begin{equation}
 \Pi^\pm_n =\frac{1}{2}\big(1\pm \gamma_5 \slashed{n}\big)
\end{equation}
are projection operators satisfying.
\begin{equation}
 \Pi^{(+)2}_n = \Pi^{(-)2}_n = I, \qquad \Pi^{(+)}_n \Pi^{(-)}_n =0.
\end{equation}
The notation here is same as in Bjorken and Drell, Gasiorowicz. 

Let \(\displaystyle N=-\Big(\frac{i\beta\vec{\alpha}\cdot\vec{p} }{2mc}\Big)\, f\Big(\frac{|\vec{p}|}{mc}\Big)\)

• Prove that \begin{equation} \exp(iN) = \cos\big(\frac{|\vec{p}|f}{2mc} \big) + \frac{\beta\vec{\alpha}\cdot\vec{p}}{|\vec{p}|} \sin \big(\frac{|\vec{p}|f}{2mc} \big), \end{equation} where \(H\) is Dirac Hamiltonian \(H=c\vec{\alpha}.\cdot\vec{p} + \beta mc^2\)
• Find the real function \(f\) such that \begin{equation} H{'} = e^{iN} H e^{-iN} \end{equation} is free of operators odd operators. For this choice of \(f\) \[ H{'} = \beta c \sqrt{|\vec{p}|^2 + m^2c^2}.\]

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).

Consider an electron in a uniform and constant magentic field \(\vec{B}\) along the \(z-\)axis. Obtain the most general four component positive energy eigennfunctions. Show that the energy eigenvalues are given by \[ E= \sqrt{m^2c^4 + c^2p_3^2 + 2ne\hbar c|\vec{B}|}\] with \(n=0,1,2,...\). List all the constants of motion.

Show that under time reversal \(\bar{u}(p) \to u(—p)B \) and \(v(p) \to \bar{v}(—p) B\). Use this to show that under \(P T\) together \[\bar{u}(p{'}) \gamma_{\alpha_1} \gamma_{\alpha_2}\ldots \gamma_{\alpha_n} u(p) \to \bar{u}(p)\gamma_{\alpha_n}\gamma_{\alpha_{n-1}}\ldots \gamma_{\alpha_1} u(p{'}) \]

Let \(a_{\mu\nu} \) give the Lorentz transformation \[x'^{\mu}=a_{\mu\nu} x^\nu.\] Show that there exists an invertible matrix \(S\) such that \[S\gamma_\mu S^{-1}= a_{\mu\nu}\gamma^\nu . \]

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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 \(r,s\).