The quantum dynamics of Lagrange multipliers

When implementing a non-linear constraint in quantum field theory by means of a Lagrange multiplier, $\text{ł}(x)$, it is often the case that quantum dynamics induce quadratic and even higher order terms in $\text{ł}(x)$, which then does not enforce the constraint anymore. This is illustrated in the case of Unimodular Gravity, where the constraint is that the metric tensor has to be unimodular ($g(x)\equiv \det g_{\text{m}\text{n}}(x)=-1$).