Path integral formulation for Green's functions

Question

In the first place, I am struggling when trying to derive the path integral formulation of the Green function for non-interacting particles

$$G_{ij}(\tau)=-\frac{1}{Z}\int D(\bar{\psi},\psi) \psi_i(\tau)\bar{\psi}_j(0)e^{-iS(\bar{\psi},\psi)}$$

with the action $$S=\sum_i\int^{\beta}_{0}d\tau \bar{\psi}_i(\tau)(i\partial_{\tau}+\epsilon_i-\mu)\psi_i(\tau),$$ from the definition

$$G_{ij}(\tau)=-\langle \text{T}_\tau \psi_i(\tau)\psi_j^\dagger(0)\rangle$$

with $\text{T}_\tau$ being the time-ordering operator and $\psi_i(\tau)=e^{\hat{H}\tau}\psi_i(0)e^{-\hat{H}\tau}$. I am told to use the fact that $\langle \hat{A} \rangle =-\frac{1}{Z}\text{tr}(e^{-\beta\hat{H}}\hat{A})$ and I know the general construction allowing to express traces in terms of path integrals, but the latter requires the operators in the argument of the trace to be normal ordered. If I try to rearrenge them in order to achieve this, the calculations get really messy. Do you have any ideas on what I am missing?

Secondly, how can I rewrite the path integral formulation of $G_{ij}(\tau)$ as a Matsubara summation via Gaussian integrals? In fact, the integral in the exponent prevents me to have a prototypical Gaussian integral which I can rewrite right away. It seems I am missing something here as well.

This is a standard calculation in QFT. Related: https://physics.stackexchange.com/q/201482/2451 — Qmechanic, Nov 24 '20 at 20:12
this is addressed either in chapter 2 of Negele,Orland or your favourite condensed matter qft book — tbt, Nov 24 '20 at 20:22
@Qmechanic if someone asks me what 2+2 is, as it seems this question sounds like this for you, I answer 4 rather than pointing out that the question is standard. — Milarepa, Nov 24 '20 at 21:05
@tbt I havent found exactly this derivation neither in Altland's nor in Coleman's book and I didn't manage to access Negele's one — Milarepa, Nov 24 '20 at 21:06
@Adam 1) I thought that in the process of turning the trace into a path integral inserting $N$ resolutions of the identity using coherent states and letting $N \rightarrow \infty$ required having all annihilation operators on the left and all creation operators on the right such that these can act on bra and ket respectively as on their eigenstates. In my case, performing such a reordering would be very messy. But more generally I would really appreciate seeing a clear derivation of the result from scratch. 2) to cut it short: I am told to evaluate a Gaussian integral, but I can't find it — Milarepa, Nov 27 '20 at 14:36
this exists in altland. for matsubara just apply fourier transform since time is periodic frequency will be discrete. — physshyp, Nov 27 '20 at 15:06
first derive everything in pages 102- 105, then derive everything in 158-168. trust me after really deriving everything at those pages you will get your answer. but do not just read just derive. — physshyp, Nov 27 '20 at 15:10
on top of that, i am sure you have some misunderstanding of path integrals, you may first learn path integrals better, use peskin ch 9 for that. — physshyp, Nov 27 '20 at 15:12
and finally path integral formulation of greens function is automatically time ordered. you do not need to think about it, just use the gaussian integral formula of grassman numbers in those pages of altland. never forget, path integral always gives time ordered stuff by definition. — physshyp, Nov 27 '20 at 15:20
@Milarepa 1) Indeed. Assume $\tau>0$, write the green function explicitly in terms of $\exp(\tau \hat H)$ and split operator as usual, etc. Insert the identity, being careful when you insert it between the evolution operators and $\psi$ and $\psi^\dagger$. And voila. 2) Change variables from $\psi(\tau)$ to $\psi_n$, its fourier series. The action becomes a sum of quadratic terms in $\psi_n$, thus a product of Gaussian integrals. — Adam, Nov 27 '20 at 15:23
@Adam regarding 2): I can't perform the Gaussian integrals after performing the Fourier transform as the integrand of the path integral consists not only of the exponential (which alone would make Gaussian integrals work) but is multiplied by $\psi_i(\tau)$ and $\bar{\psi}_j(0)$, themselves Fourier-transformed — Milarepa, Nov 28 '20 at 17:10
@Milarepa 2) Starting from a gaussian integral, it is quite easy to compute its moments. The different Matsubara modes are independent (that's the point of using them), so that's no problem. — Adam, Nov 28 '20 at 17:32
@Adam so you mean that after transforming $\psi_i(\tau) \bar{\psi}j(0)= \sum{n,m} \psi_{i,n} \bar{\psi}_{j,m} e^{-i\omega_n \tau}$ I can take all of these factors out of the path integral? — Milarepa, Nov 28 '20 at 17:47
I think eq (1.275) and the discussion below will answer both questions https://www.lptmc.jussieu.fr/files/chap_fi.pdf — Adam, Nov 28 '20 at 18:06
@Adam As I am supposed to deal with non_interacting bosons/fermions I'm still not sure from where the $\delta_{ij}$ in $G_{ij}$ stems from — Milarepa, Nov 28 '20 at 20:02
@Adam and how is eq. 1.306 in your reference derived? That was my question from the beginning. They refer to some appendix sections regarding Gaussian integrals but these are not very helpful as well — Milarepa, Nov 30 '20 at 19:26
@Milarepa Well, at some point, you need to learn how to compute gaussian integrals if you want to do QFT... That's pretty much the only thing we know how to do without approximation ;-) — Adam, Dec 01 '20 at 12:48
@Milarepa They are the simplest generalization of gaussian integrals, and the results can be computed easily from the standard gaussian integrals. Or from the appendix of the reference I gave. Have you looked at? Have you try anything? — Adam, Dec 01 '20 at 14:37
I would suggest that you write what you tried in the question, so we can help you. Your problem seems to be quite basic, and once we know where the problem is, it should be straightforward to solve it. — Adam, Dec 02 '20 at 08:27

Path integral formulation for Green's functions

0 Answers0