= Translational dynamics of a phi-four kink
{title2=$L_{\rm eff}=-M+\tfrac12M\dot X^2$}
For $\phi_K(x-X)=c\tanh(c(x-X))$ in the unit kinetic normalization, $\int(\phi_K')^2dx=M=4c^3/3$. Substitution of $X(t)$ therefore gives the free-particle <collective-coordinate effective Lagrangian> $-M+M\dot X^2/2$. Its quantization has <momentum> $p$ and energy $M+p^2/(2M)$ to this order. The exact uniformly moving classical kink is a <Lorentz boost> of the static one, with energy $M/\sqrt{1-v^2}$.
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