CGH
Jan20-11, 12:39 PM
Hi there,
I have little question: reading zee 2nd edition, I.8 (pag 64) i came up with this:
start with
<k_1 k_2| e^{-iHT}| k_3 k_4>
and
H=H_0 +u
u=\lambda \int \phi^4
where H_0 is the usual hamiltonian for the free scalar field.
Then, zee says that "expanding in \lambda, we obtain -i\lambda \int <k_1k_2|\phi^4|k_3k_4>"
my question is: how did he get that?
You cannot do an expansion around H, because \lambda is the small term, so, i do the following,
\exp(-iH_0T -iu T)=\exp(-iH_0 T)\exp(-iu)\exp(-T^2[H_0,u]/2)
then, by working a little, i find that [H_0,u]=0, so, i can perform a safe expansion around [itex]\lambda[\itex],
<k_1 k_2| e^{-iHT}| k_3 k_4>=<k_1 k_2| e^{-iH_0T}(1-iT u+O(\lambda^2))| k_3 k_4>
now, i still get something different: instead of -i\lambda \int <k_1k_2|\phi^4|k_3k_4>, i get -i\lambda \int <k_1k_2|e^{-iH_0 T}\phi^4|k_3k_4> (note: T=\int dx^0, i omit the dx in the integration),
my question: what happened to the e^{-iH_0T} term?, i think that it is just a phase and you get -i\lambda \int <k_1k_2|\phi^4|k_3k_4>e^{-iET}, with E a real number, and that why it doesn't matter.
another option is, by looking (15) you realize that the term [itex]\lambda^0[/tex] is normalize as 1, so, i guess, the S-matrix is normalize, and then, you don't get the factor outside.
which one is the answer?
Saludos!
I have little question: reading zee 2nd edition, I.8 (pag 64) i came up with this:
start with
<k_1 k_2| e^{-iHT}| k_3 k_4>
and
H=H_0 +u
u=\lambda \int \phi^4
where H_0 is the usual hamiltonian for the free scalar field.
Then, zee says that "expanding in \lambda, we obtain -i\lambda \int <k_1k_2|\phi^4|k_3k_4>"
my question is: how did he get that?
You cannot do an expansion around H, because \lambda is the small term, so, i do the following,
\exp(-iH_0T -iu T)=\exp(-iH_0 T)\exp(-iu)\exp(-T^2[H_0,u]/2)
then, by working a little, i find that [H_0,u]=0, so, i can perform a safe expansion around [itex]\lambda[\itex],
<k_1 k_2| e^{-iHT}| k_3 k_4>=<k_1 k_2| e^{-iH_0T}(1-iT u+O(\lambda^2))| k_3 k_4>
now, i still get something different: instead of -i\lambda \int <k_1k_2|\phi^4|k_3k_4>, i get -i\lambda \int <k_1k_2|e^{-iH_0 T}\phi^4|k_3k_4> (note: T=\int dx^0, i omit the dx in the integration),
my question: what happened to the e^{-iH_0T} term?, i think that it is just a phase and you get -i\lambda \int <k_1k_2|\phi^4|k_3k_4>e^{-iET}, with E a real number, and that why it doesn't matter.
another option is, by looking (15) you realize that the term [itex]\lambda^0[/tex] is normalize as 1, so, i guess, the S-matrix is normalize, and then, you don't get the factor outside.
which one is the answer?
Saludos!