# A problem understanding primary constraints meaning

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.