We use Dirac's constrained analysis to quantize the O(3) nonlinear $\sigma$ model and the ${\mathrm{CP}}^1$ model without using gauge constraints. The equivalence of the models (at the quantum level) is established provided the identification $n^a=Z^{*}{\sigma }^{a}Z$, ${\pi }^a=\frac{1}{2}(\pi {\sigma }^{a}Z+Z^{*}{\sigma }^a{\pi }^{*})$ (${\sigma }^a=\mathrm{Pauli}\mathrm{matrices}$) is used to relate the canonical variables. While the first relation has been discussed in the literature, the second relation connecting the canonical momenta is a new observation. It plays a significant role in displaying the above stated equivalence.

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