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On a smooth manifold, a half-density has local expression $\sigma =\psi (q)|d{q}_1\cdots d{q}_n{|}^{1/2}$. Under a change from coordinates $q$ to $Q$,Consequently $\overline{\sigma }\sigma$ is a nonnegative density on a manifold. It can be integrated without choosing an orientation or a volume form, but the integral need not be finite. Square-integrable half-densities are precisely those satisfyingTheir inner product is $⟨\sigma ,\tau ⟩={\int }_M\overline{\sigma }\tau$. Completing the smooth compactly supported half-densities gives an intrinsic Hilbert space; the Jacobian transformation makes its norm independent of coordinates.^[ref]

For instance, $|dq{|}^{1/2}$ on the real line is a smooth half-density with infinite squared norm, whereas $e^{-q^2}|dq{|}^{1/2}$ has finite squared norm. In geometric quantization, half-densities let configuration-space states carry a coordinate-independent inner product.