Commutation relations inconsistent with constraints

Question

In section $9.5$ of Weinberg's Lectures on Quantum Mechanics, he uses an example to explain the clasification of constraints. The Lagrangian for a non-relativistic particle that is constrained to remain on a surface described by

$$f(\vec x)=0\tag{1}$$

can be taken as

$$L=\frac 12 m \dot{\vec x}^2-V(\vec x)+\lambda f(\vec x).\tag{2}$$

Apart from the primary constraint $(1)$ there is also a secondary one, arising from the imposition that $(1)$ is satisfied during the dynamics $$\dot {\vec x}\cdot\vec \nabla f(\vec x)=0.\tag{3}$$

Then he states that imposing $[x_i,p_j]=i\hbar\delta_{ij}$ would be inconsistent with the constraints $(1)$ and $(3)$ (which reads $\vec{p}\cdot\vec{\nabla}f=0$ in the Hamiltonian formalism). How can I see this inconsistency?

Answer 1

Would an example suffice? If so, consider the case $f(\vec x) = x_1$. Then (1) says $x_1=0$, which is already inconsistent with the commutation relation, and (3) says $p_1=0$, which is again inconsistent with the commutation relation. If $x_1$ or $p_1$ is zero, then we can't have $[x_1,p_1]\neq 0$.

Answer 2

On one hand, $$ 0~=~[0,0]~=~[f(x),\vec{p}\cdot\vec{\nabla}f]~=~i\hbar (\vec{\nabla}f)^2.$$ On the other hand, a constraint function $f$ typically satisfies a regularity condition $$ \left .\vec{\nabla}f \right|_{f=0}~\neq~\vec{0}.$$