my question is about some steps in the book "Finite Temperature Field Theory" by Kapusta and Gale. To explain the setting, I how to evaluate the functional integrals in the expression $$ \ln Z_1=\frac{-\lambda\int d\tau\int d^3 x \int [d\phi]e^{S_0}\phi^4(\bf{x},\tau)}{\int [d\phi]e^{S_0}} $$ where $S_0$ is the non-interacting Lagrangian. $\tau$ is imaginary time. We assume a finite volume V, such that both energies and momenta are quantized, so we can Fourier transform $$ \phi(x,\tau)=\sqrt\frac{\beta}{V}\sum_n\sum_{\bf{p}}e^{i(\bf{p} x+\omega_n \tau)}\tilde\phi_n(\bf{p}), $$ where the energies are $\omega_n=2\pi nT$. Using this Fourier series we can reexpress $$ \ln Z=-\lambda \int d\tau \int d^3x \sum_{n_1,\ldots,n_4}\sum_{\bf{p}_1,\ldots,\bf{p}_4}\frac{\beta^2}{V^2}e^{i(\bf{p}_1+\ldots+\bf{p}_4)\cdot\bf{x}}e^{i(\omega_{n_1}+\ldots+\omega_{n_4})\tau}\frac{A}{B}, $$ where $$ A=\prod_l\prod_\bf{q}\int d\tilde{\phi}_l(\bf{q})\;e^{-\frac{1}{2}\beta^2(\omega_l^2+\bf{q}^2+m^2)\tilde\phi_l(\bf{q})\tilde\phi_{-l}(-\bf{q})}\;\tilde\phi_{n_1}(\bf{p}_1)\ldots\tilde\phi_{n_4}(\bf{p}_4), $$ and B is just the same without the four $\tilde\phi$ in the end coming from the $\phi^4$ term. The integrals over $\bf{x}$ and $\tau$ amount to energy-momentum conservation.

Now, using $$ \frac{\int_{-\infty}^\infty dx\;x^2 e^{-ax^2/2}}{\int_{-\infty}^\infty dx\; e^{-ax^2/2}}=\frac{1}{a} $$ I am supposed to get $$ \ln Z_1=-3\beta V \left(T\sum_n \int\frac{d^3p}{(2\pi)^3}\frac{1}{\omega_n^2+\bf{p}^2+m^2}\right)^2. $$

I understand that the factors in A where $l\not\in\{n_1,\ldots,n_4\}$ and $\bf{q}\not\in\{\bf{p}_1,\ldots,\bf{p}_1\}$ simply cancel with the corresponding ones in B.

Because of $$ \int_{-\infty}^{\infty} dx\; xe^{-ax^2}=0, $$ we are supposed zero if not $n_3=-n1$, $\bf{p}_3=-\bf{p}_1$, $n_4=-n2$, $\bf{p}_4=-\bf{p}_2$ or permutations thereof. I don't understand that argument, lets say none of these cases is true, then we get a factor in the integral of the form $$ \int d\tilde\phi_{n_1}(\bf{p_1})\;e^{-\frac{1}{2}\beta^2(\omega_l^2+\bf{p}_1^2+m^2)\tilde\phi_{n_1}(\bf{p}_1)\tilde\phi_{-n_1}(-\bf{p}_1)}\;\tilde\phi_{n_1}(\bf{p}_1) $$ I don't see where the quadratic term in the exponential comes from as this is linear in $\phi_{n_1}(\bf{p}_1)$.

I guess this is my general problem, also for the case that actually $n_3=-n1$, $\bf{p}_3=-\bf{p}_1$, $n_4=-n_2$, $\bf{p}_4=-\bf{p}_2$ or a permutation thereof: How come there is a Gaussian integral appearing when $\phi_{l}(\bf{q})$ and $\phi_{-l}(-\bf{q})$ are independent variables to be integrated over?



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.