Quantizing K-G equation: Basic concepts

\(\ddot{\phi} = (\nabla^2 - m^2)\phi\)

1. Action: (Lagrangian density and Lagrangian) \[S[\phi(\mathbf{x},t)] = \int_{t_1}^{t_2} dt \int d^3 x \mathcal{L}(\phi(\mathbf{x},t), \dot{\phi}(\mathbf{x},t), \nabla\phi(\mathbf{x},t))\]

   Example:

   1. Small transverse fluctuations of a violin strings. \(\mathcal{L} = \frac{1}{2}\rho \dot{\phi}^2 - \frac{1}{2} T (\phi')^2\)

   2. EM in \(A_0 = 0\) gauge. \(\mathcal{L} = -\frac{1}{4}F_{\mu \nu}F^{\mu \nu}\)

2. Eular-Lagrangian Equation. Excise: Derive this from Hamilton’s principle, or variation principle. Cf. Page 7.

   \[\partial_t \frac{\partial \mathcal{L}}{\partial (\partial_t \phi)} + \nabla\cdot \frac{\partial \mathcal{L}}{\partial (\nabla \phi)} = \frac{\partial \mathcal{L}}{\partial \phi}\]

   Relativistic version:

   \[\partial_\mu \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} = \frac{\partial \mathcal{L}}{\partial \phi}\]

   Example on \(\partial_\mu\) and on a given Lagrangian.

   Note: different people treat different terms as K.E. or P.E..

3. Lagrangian – Hamiltonian

   Review: \(H = \sum_i p_i \dot{q}_i - L\). Quantization: \([\hat{p}_i, \hat{q}_j] = -i\hbar \delta_{ij}\)

   Canonical “momentum” field conjugate to \(\phi(\mathbf{x})\): \[\pi(\mathbf{x})\equiv \frac{\partial \mathcal{L}}{\partial \dot{\phi}}\]

   Moral: canonical momentum fields are often not related to what you think of as actual momentum.

   \[H = \left(\int d^3 x \pi(\mathbf{x}, t)\dot{\phi}(\mathbf{x}, t)\right) - L = \int d^3 x \mathcal{H}\] with \[\mathcal{H} = \pi(\mathbf{x}, t)\dot{\phi}(\mathbf{x}, t) - \mathcal{L}\]

   Quantization: (the rest are \(0\)) \[[\hat{\pi}(\mathbf{x}), \hat{\phi}(\mathbf{y})] = - i \hbar \delta^{(3)} (\mathbf{x} - \mathbf{y})\]

   Example: For \(\mathcal{L} = \frac{1}{2}\dot{\phi}^2 - \frac{1}{2}(\nabla \phi)^2 - \frac{1}{2}m^2 \phi^2\): \[\mathcal{H} = \frac{1}{2} \pi^2 + \frac{1}{2} (\nabla \phi)^2 + \frac{1}{2} m^2 \phi^2\] Note: this is not Lorentz inv.