Conserved Charges of Stress Energy Tensor

[WarDieS]

Hello, Hi There

I am trying to obtain the relations of the conserved charges of the stress tensor, it has 4, one is the hamiltonian and the other three are the momentum components.

$\vec{P}=-\int d^3y \sum_i{(-\pi_i(y) \nabla \phi_i(y))}$

And i have to prove the conmutators

$[\phi_i(x),\vec{P}]=-i \nabla\phi(x)$ and $[\pi_i(x),\vec{P}]=i \nabla \pi_i(x)$

I got the first one just fine

$[\phi_i(x),\vec{P}]=-\int d^3 y \sum_j{[\phi_i(x),\pi_j(y)]\nabla \phi_j(y)}= -\int d^3 y \sum_j{i \delta_{ij} \delta^{(3)}(\vec{x}-\vec{y}) \nabla \phi_j(y)}=-i\nabla\phi_i(x)$



But the second one is driving me crazy

$[\pi(x),\vec{P}]=-\int d^3 y \sum_j{[\pi_i(x),\pi_j(y)]\nabla \phi_j(y)}$

That conmutator is zero, ¿what i am doing wrong? how can those don't conmute.

Also, what's the meaning of this relations

Thans for the time

 

[andrien]

Why are you taking the $\phi$ part out in the second case,in first commutation it works because $\phi$ will commute with other $\phi$ but in the second case it will be $\pi$ which will be taken outside because $\pi's$ will commute.Use by part in second commutation to shift the derivative on $\pi$ and then it's easy.

 

[WarDieS]

Yes you are right andrien, i can't believe i didnt notice it myself, many thanks!, i was just considering it as a number for not clear reasons, thanks again.