 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 $m\int {\dot{q}}_I^{2}d\tau =S_I$, the standard normalized measure is $\sqrt{S_I/(2\pi \hslash )}d{\tau }_0$. 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.