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\section{Klein-Gordon Equation}     A relativistic one particle Shr\"{o}dinger Equation: (Also describes acoustic phonons.)   $$\hbar^2 (-\partial_{ct}^2 +\nabla^2) \psi = M^2 c^2 \psi$$   $$-\hbar^2\partial^\mu\partial_\mu \psi = M^2c^2 \psi$$     \begin{enumerate}     \item Problems with K-G.     Problem: $|\psi|^2$ is no longer prob. density. Not conserved.     What is conserved? $\rho = i(\psi^* \dot{\psi} - \dot{\psi}^*\psi)$. However, this can be negative.     Also, energy have negative solutions.     Classical limit: $\ddot{\phi} = (c^2\nabla^2 -\frac{(Mc^2)^2}{\hbar^2})\phi \Longrightarrow \ddot{\phi} = (c^2\nabla^2 - \omega_{min}^2)\phi$     \item K-G arises in CM physics     \begin{enumerate}   \item $M=0$ case: $\ddot{\phi} = c^2 \nabla^2 \phi$. Dispersion relation: (Note, $c$ is the wave speed, not speed of light.)   $$\omega^2 = c^2 k^2$$   Arbitratily low frequency waves $\Longrightarrow$ ``acoustic limit''     \item $M\ne 0$ case: $\ddot{\phi} = (c^2\nabla^2 - \omega_{min}^2)\phi$. Dispersion relation: (``optical'' branches)   $$\omega^2 = c^2 k^2 + \omega_{min}^2$$     \item Nomenclature: gap.     \end{enumerate}     \item Quantizing K-G equation: $\ddot{\phi} = (\nabla^2 - m^2)\phi$     \begin{enumerate}   \item Action:   $$S[\phi(\mathbf{x},t)] = \int$$     \end{enumerate}     \end{enumerate}