= Zero mode
{title2=$Af=0,\quad f\ne0$}
A <zero mode> is a nonzero eigenvector of an operator with eigenvalue zero. For a differential fluctuation operator it must also satisfy the specified <boundary conditions>. A zero mode prevents ordinary inversion of the operator and makes its unmodified <functional determinant> vanish. Gauge directions can produce zero modes of a kinetic operator, while stationary-path focusing can produce them in a semiclassical boundary-value problem. Removing, separately integrating or otherwise treating these directions is necessary before using a Gaussian determinant formula.
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