= Quadratic fluctuation Hamiltonian
{title2=$H_2=\frac12\int[\pi_\eta^2+\eta\mathcal H\eta],dX$}
Expanding the <Hamiltonian> about a static classical solution removes the linear term by its <Euler-Lagrange field equation>. For a canonical <real scalar field> with potential $U$, the quadratic fluctuation operator is $\mathcal H=-\partial_X^2+U''(\varphi_{\rm cl})$. Positive-frequency <normal modes> become <quantum harmonic oscillators> under <canonical quantization>, yielding the Gaussian contribution to a <one-loop soliton mass correction>. A negative <eigenvalue> signals an instability, while a physical <zero mode in field theory> must be replaced by a <collective coordinate>.
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