I have some problems understanding the meaning of a function that vanishes weakly.

As far as I can understand, when somebody writes that a function $F$ in the phase space vanishes weakly, that means that $F=0$ over the surface defined by the constraints $\phi_1=0, \phi_2=0,\dots,\phi_N=0$.

Now, in the statement of the exercise 1.27 of the book Quantization of Gauge Fields, it says that $[\phi_k,\phi_1]$ vanishes on $\phi_{S}=0$ ($S\leq k$) for $k\leq L$ and that $[\phi_{L},\phi_1]\neq 0$ (even weakly).

As I can understand, if we consider that $[\phi_k,\phi_1]$ vanishes on $\phi_{S}=0$ ($S\leq k$) for $k\leq L$, this means that, for example, $[\phi_3,\phi_1]=0$ on the surface defined by $\phi_1=0$, $\phi_2=0$ and $\phi_3=0$. If this is true then, when $k=L$, we will have that $[\phi_L,\phi_1]=0$ on the surface $\phi_1=\phi_2=\dots=\phi_L=0$, but this seems like a contradiction with the assumption $[\phi_{L},\phi_1]\neq 0$ even weakly. I don't understand what is going on.

Second, I have problems trying to prove the identity given in the hint of part (b). Specifically, I have to prove that $$[\phi_i,\phi_j]\approx-[\phi_{i-1},\phi_{j+1}].$$ I have used the Jacobi identity to prove that $$[\phi_i,\phi_j]=-[\phi_{i-1},\phi_{j+1}]-[[\phi_j,\phi_{i-1}],H].$$ However, I'm stuck when I try to prove that the term $[[\phi_j,\phi_{i-1}],H]$ is weakly zero. I am sure that I have to use the fact that $[H,\phi_{k}]\approx0$ for $k\leq L$, but I don't know how to do it. When I use the Jacobi identity again I obtain much more complicated expressions, not very useful.



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