= Collective-coordinate integration measure
A collective coordinate labels a family of physically distinct saddle configurations, such as the center $\tau_0$ of an <instanton>. Differentiating the saddle with respect to that coordinate gives a zero mode of the quadratic fluctuation operator. An ordinary Gaussian integral over its coefficient would be divergent, because moving along the family costs no quadratic <action>.
Instead, change variables from the zero-mode coefficient to the collective coordinate and include the associated Jacobian. For a one-dimensional instanton with $m\int\dot q_I^2d\tau=S_I$, the standard normalized measure is $\sqrt{S_I/(2\pi\hbar)}\,d\tau_0$. The remaining fluctuations contribute a primed <functional determinant>, excluding this zero mode. Integrating over the allowed center positions supplies the physical observation-time factor rather than deleting the mode and losing that factor.
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