Derive the quantised Hamiltonian corresponding to the Dirac equation


 * 1) Show that \(\mathcal{L}=\bar{\phi}\left(\gamma^{\mu}p_{\mu} - m\right)\phi\) is the Lagrangian representation of the Dirac equation, where \(\bar{\phi} = \gamma^0\phi^{\dagger}\) and \(\left(\gamma^0\right) = 1\).
 * 2) Separate into time and space parts: \(\mathcal{L}=\bar{\phi}\left(\gamma^{0}i\hbar\partial_t + \gamma\cdot i\hbar\nabla - m\right)\phi\)
 * 3) Input the Lagrangian into the Euler-Lagrange equation, \(\frac{\partial \mathcal{L}}{\partial\phi} - d_t\frac{\partial \mathcal{L}}{\partial\dot{\phi}} - \nabla\frac{\partial \mathcal{L}}{\partial\nabla\phi} = 0\): \(\bar{\phi}\frac{\partial}{\partial\phi}m\phi - \gamma^{0}i\hbar d_t \bar{\phi} \frac{\partial}{\partial\dot{\phi}}\phi - \nabla\bar{\phi} \frac{\partial}{\partial\nabla\phi}\gamma\cdot i\hbar\nabla\phi = 0\)
 * 4) Solve: \(m\bar{\phi} - \gamma^{0}i\hbar \dot{\bar{\phi}} - \bar{\phi}\cdot \nabla\cdot\gamma i\hbar = 0\)
 * 5) Find the equivalent Hamiltonian.
 * 6) Find the canonical momentum: \(\Pi_{\phi} = \frac{\partial \mathcal{L}}{\partial \dot{\phi}} = i\hbar\phi^{\dagger}\) \(\Pi_{\bar{\phi}} = \frac{\partial \mathcal{L}}{\partial \bar{\dot{\phi}}} = 0\)
 * 7) Solve for the Hamiltonian density, \(\mathcal{H}=\Pi_{\phi}\dot{\phi} + \Pi_{\bar{\phi}}\bar{\dot{\phi}} - \mathcal{L}\): \(\mathcal{H}=i\hbar\phi^{\dagger}\phi - \bar{\phi}\left(\gamma^{0}i\hbar\partial_t + \gamma\cdot i\hbar\nabla - m\right)\phi\)
 * 8) Simplify: \(\mathcal{H}=\bar{\phi}\left(-i\hbar\underline{\gamma}\cdot\nabla + m\right)\phi\)
 * 9) Substitute in the Dirac equation, \(\left(-i\hbar\underline{\gamma}\cdot\nabla + m\right)\phi = i\hbar\gamma^0\partial_t\phi\): \(\mathcal{H}=\bar{\phi}i\hbar\gamma^0\partial_t\phi\)
 * 10) Simplify: \(\mathcal{H}=i\hbar\phi^{\dagger}\partial_t\phi\)
 * 11) Integrate over \(x\) to find the Hamiltonian: \(\mathbf{\hat{H}}=i\int dx \phi^{\dagger}\partial_t\phi\)
 * 12) Input the general solution, \(\phi(x,t)=\int \frac{dk}{(2\pi)^3 2\omega_k} \sum_{s=1,2}\left[c_{k,s}(t)u_{k,s}e^{i(k\cdot x - \omega_k t)} + d_{k,s}^{\dagger}(t)v_{k,s}e^{-i(k\cdot x - \omega_k t)}\right]\) into the Hamiltonian: \(\mathbf{\hat{H}}=i\int \int \int \frac{dk\,dq}{(2\pi)^6 4 \omega_k \omega_q} dx \sum_{s,s\prime} \left(c_{k,s}^{\dagger}(t)u_{k,s}^{\dagger}e^{-i(k\cdot x - \omega_k t)} + d_{k,s}(t)v_{k,s}^{\dagger}e^{i(k\cdot x - \omega_k t)}\right) \left(-i\omega_q c_{q,s\prime}(t)u_{q,s\prime}e^{-i(k\cdot x - \omega_q t)} + i\omega_q d_{q,s\prime}^{\dagger}(t)v_{q,s\prime}e^{i(k\cdot x - \omega_q t)}\right)\)
 * 13) Expand: \(\mathbf{\hat{H}}=i\int \int \int \frac{dk\,dq}{(2\pi)^6 4 \omega_k \omega_q} dx \sum_{s,s\prime} \left(-i\omega_q c_{k,s}^{\dagger}(t)u_{k,s}^{\dagger}e^{-i(k\cdot x - \omega_k t)}c_{q,s\prime}(t)u_{q,s\prime}e^{-i(k\cdot x - \omega_q t)}\right.\) \( -i\omega_q d_{k,s}(t)v_{k,s}^{\dagger}e^{i(k\cdot x - \omega_k t)}c_{q,s\prime}(t)u_{q,s\prime}e^{-i(k\cdot x - \omega_q t)}\) \( + i\omega_q c_{k,s}^{\dagger}(t)u_{k,s}^{\dagger}e^{-i(k\cdot x - \omega_k t)}d_{q,s\prime}^{\dagger}(t)v_{q,s\prime}e^{i(k\cdot x - \omega_q t)}\) \(\left. + i\omega_q d_{k,s}(t)v_{k,s}^{\dagger}e^{i(k\cdot x - \omega_k t)}d_{q,s\prime}^{\dagger}(t)v_{q,s\prime}e^{i(k\cdot x - \omega_q t)}\right)\)
 * 14) Integrate over \(x\): \(\mathbf{\hat{H}}=i \int \int \frac{dk\,dq}{(2\pi)^6 4 \omega_k \omega_q} \sum_{s,s\prime} \left(-i\omega_q c_{k,s}^{\dagger}(t)u_{k,s}^{\dagger}c_{q,s\prime}(t)u_{q,s\prime}\delta(q-x)e^{i(\omega_q-\omega_x) t)}\right.\) \( -i\omega_q d_{k,s}(t)v_{k,s}^{\dagger}c_{q,s\prime}(t)u_{q,s\prime}\delta(q+x)e^{i(\omega_q+\omega_x) t)}\) \( + i\omega_q c_{k,s}^{\dagger}(t)u_{k,s}^{\dagger}d_{q,s\prime}^{\dagger}(t)v_{q,s\prime}\delta(q+x)e^{-i(\omega_q+\omega_x) t)}\) \(\left. + i\omega_q d_{k,s}(t)v_{k,s}^{\dagger}d_{q,s\prime}^{\dagger}(t)v_{q,s\prime}\delta(q-x)e^{-i(\omega_q-\omega_x) t)}\right)\)
 * 15) Cancel out \(\omega_q\) and integrate over \(q\): \(\mathbf{\hat{H}}=\int \frac{dk}{(2\pi)^6 4 \omega_k} \sum_{s,s\prime} \left(c_{k,s}^{\dagger}(t)u_{k,s}^{\dagger}c_{k,s\prime}(t)u_{k,s\prime} + d_{k,s}(t)v_{k,s}^{\dagger}c_{-k,s\prime}(t)u_{-k,s\prime}e^{i(\omega_k+\omega_{-k}) t)} - c_{k,s}^{\dagger}(t)u_{k,s}^{\dagger}d_{-k,s\prime}^{\dagger}(t)v_{-k,s\prime}e^{-i(\omega_k+\omega_{-k}) t)} - d_{k,s}(t)v_{k,s}^{\dagger}d_{k,s\prime}^{\dagger}(t)v_{k,s\prime}\right)\)
 * 16) Simplify using the fact that \(u_{k, s}v_{k,s\prime}^{\dagger} = 0\) and \(u_{k, s}u_{k,s\prime}^{\dagger} = \delta_{ss\prime}2\omega_k\): \(\mathbf{\hat{H}}=\int \frac{dk}{(2\pi)^6 2 \omega_k} \omega_k \sum_{s=1,2}\left[c_{k,s}^{\dagger}c_{k,s}-d_{k,s}^{\dagger}d_{k,s}\right]\)