- #1
maverick280857
- 1,774
- 5
Hi everyone,
I'm reading section 9.2 of Peskin and Schroeder, and have trouble understanding the origin of a term in the transition from equation 9.26 to 9.27. Specifically, equation 9.26 is
\frac{1}{V^2}\sum_{m,l}e^{-(k_m\cdot x_1 + k_l\cdot x_2)}\left(\prod_{k_{n}^{0}>0}\int d \Re \phi_{n}d \Im \phi_{n}\right)\times (\Re \phi_m + i\Im \phi_m)(\Re \phi_n + i\Im \phi_n)\times\exp{\left[-\frac{i}{V}\sum_{k_{n}^{0}>0}(m^2-k_{n}^2)\left[(\Re \phi_n)^2 + (\Im \phi_n)^2\right]\right]}
This splits into two cases:
Case 1: k_l = k_m, when the integral is zero.
Case 2: k_l = -k_m, when the integral is nonzero.
So, evaluating the Gaussian integral one gets
\mbox{Numerator} = \frac{1}{V^2}\sum_{m,l}e^{-(k_m\cdot x_1 + k_l\cdot x_2)}\left(\prod_{k_{n}^{0}>0}\frac{-i\pi V}{m^2 - k_{n}^2}\right)\frac{-iV}{m^2-k_{m}^2-i\epsilon}
Where does the factor
\frac{-iV}{m^2-k_{m}^2-i\epsilon}
come from? I know this is like \int x^2 e^{-x^2}dx, but I can't seem to get this factor.)
I'm reading section 9.2 of Peskin and Schroeder, and have trouble understanding the origin of a term in the transition from equation 9.26 to 9.27. Specifically, equation 9.26 is
\frac{1}{V^2}\sum_{m,l}e^{-(k_m\cdot x_1 + k_l\cdot x_2)}\left(\prod_{k_{n}^{0}>0}\int d \Re \phi_{n}d \Im \phi_{n}\right)\times (\Re \phi_m + i\Im \phi_m)(\Re \phi_n + i\Im \phi_n)\times\exp{\left[-\frac{i}{V}\sum_{k_{n}^{0}>0}(m^2-k_{n}^2)\left[(\Re \phi_n)^2 + (\Im \phi_n)^2\right]\right]}
This splits into two cases:
Case 1: k_l = k_m, when the integral is zero.
Case 2: k_l = -k_m, when the integral is nonzero.
So, evaluating the Gaussian integral one gets
\mbox{Numerator} = \frac{1}{V^2}\sum_{m,l}e^{-(k_m\cdot x_1 + k_l\cdot x_2)}\left(\prod_{k_{n}^{0}>0}\frac{-i\pi V}{m^2 - k_{n}^2}\right)\frac{-iV}{m^2-k_{m}^2-i\epsilon}
Where does the factor
\frac{-iV}{m^2-k_{m}^2-i\epsilon}
come from? I know this is like \int x^2 e^{-x^2}dx, but I can't seem to get this factor.)