The Klein-Gordon equation! Here is the simplest field theory: a scalar field $ \phi(t, x) $ that obeys the equation of motion\[\partial_t^2 \psi - \nabla^2 \psi + m^2 \psi = 0,\]where $ \nabla $ is our usual gradient operator, and $ \partial_t = \frac{\partial}{\partial t} $ is a nice short-hand. Plug the Ansatz\[\psi(t, x) = A(k) e^{-iEt + ik \cdot x}\]into the equation and find $ E(k) $ such that the Ansatz solves the equation! Surprise!

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Answered: The Klein-Gordon equation! Here is the…

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