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.
[itex]\vec{P}=-\int d^3y \sum_i{(-\pi_i(y) \nabla \phi_i(y))}[/itex]
And i have to prove the conmutators
[itex][\phi_i(x),\vec{P}]=-i \nabla\phi(x)[/itex] and [itex][\pi_i(x),\vec{P}]=i \nabla \pi_i(x)[/itex]
I got the first one just fine
[itex][\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) [/itex]
But the second one is driving me crazy
[itex][\pi(x),\vec{P}]=-\int d^3 y \sum_j{[\pi_i(x),\pi_j(y)]\nabla \phi_j(y)} [/itex]
That conmutator is zero, ¿what i am doing wrong? how can those don't conmute.
Also, whats the meaning of this relations
Thans for the time
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.
[itex]\vec{P}=-\int d^3y \sum_i{(-\pi_i(y) \nabla \phi_i(y))}[/itex]
And i have to prove the conmutators
[itex][\phi_i(x),\vec{P}]=-i \nabla\phi(x)[/itex] and [itex][\pi_i(x),\vec{P}]=i \nabla \pi_i(x)[/itex]
I got the first one just fine
[itex][\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) [/itex]
But the second one is driving me crazy
[itex][\pi(x),\vec{P}]=-\int d^3 y \sum_j{[\pi_i(x),\pi_j(y)]\nabla \phi_j(y)} [/itex]
That conmutator is zero, ¿what i am doing wrong? how can those don't conmute.
Also, whats the meaning of this relations
Thans for the time
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