Functional Quantization of Scalar Fields

[maverick280857]

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.)

 

[maverick280857]

Anyone?

 

[Entropia]

Thank you for sharing your question and thoughts on section 9.2 of Peskin and Schroeder. The term in question, \frac{-iV}{m^2-k_{m}^2-i\epsilon}, arises from the functional quantization of scalar fields. This term represents the contribution of a single mode of the field, with momentum k_m, to the overall amplitude of the field. The factor of -i accounts for the imaginary part of the field, and the factor of V comes from the volume of space over which the field is being quantized. The term in the denominator, m^2-k_{m}^2-i\epsilon, is a result of the Feynman propagator, which accounts for the propagation of the field in time and space. The i\epsilon term is commonly used in quantum field theory to ensure convergence of integrals and to account for the possibility of virtual particles. I hope this helps clarify the origin of the term in question. Keep up the good work in your studies!