[QUE/QFT-05007] QFT-PROBLEM

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 \(f\) \[ H{'} = \beta c \sqrt{|\vec{p}|^2 + m^2c^2}.\]