Half-density ($\psi (q)\sqrt{|d{q}_1\cdots d{q}_n|}$)

A half-density transforms by the square root of the absolute Jacobian under a coordinate change. The product of a half-density with its complex conjugate is an integrable density, so its norm is coordinate-independent without choosing a separate volume form. Half-densities provide an intrinsic configuration-space Hilbert space for a real cotangent polarization.