Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,873 answers , 20,701 comments
1,470 users with positive rep
502 active unimported users
More ...

Retarded thermal Green function

+ 2 like - 0 dislike
1688 views

I'm working with finite temperature field theory, but I'm having problems understanding the retarded Green's function in this formalism.

I'm reading Niemi and Semenoff's article "Finite Temperature Quantum Field Theory in Minkowski Space", but you can find this on many other articles. In the Schwinger Keldysh formalism, the "same theory propagator" for a scalar field is \begin{equation} i D_{11}(k)=-\frac{i}{k^2+m^2-i\varepsilon}-\frac{2 \pi}{e^{\beta |k_0|}-1}\delta(k^2+m^2) \end{equation}

Now, this is the Fenyman propagator. If I go to position space (working in 1+1 dimensions) the propagator will be a sum of Bessel $K_0$ function, where I have a sum over images for the periodicity in time. I have to change my epsilon prescription to obtain the retarded Green function, which I can do with no problem for the first term. What happens with the second term, however? How can I obtain a quantity that vanishes for $t >0$?

Can I just do the $T=0$ case, and after I have this just add a sum over images for imaginary time? However, this "feels" euclidean, and what I do in euclidean space will give me the Feynman propagator, not the retarded one (besides, it will also diverge).

I can also think of taking the canonically quantized scalar field and then compute $D_R=-i \theta (t) \frac{Tr(e^{-\beta H} [\phi(x,t), \phi(0)])}{Tr(e^{- \beta H})}$, but I have troubles with this computation. If I take $x=0$, I get the same result I would get for the $T=0$ case (but then I cannot restore $x$ since Lorentz invariance is broken at finite $T$), while if I set $x=0$ I just get $0$ (due to some squares and square roots that give me an answer which doesn't change if I do $t \to - t$ )

Any suggestions? Thanks

This post imported from StackExchange Physics at 2014-10-11 09:50 (UTC), posted by SE-user user22710
asked Oct 10, 2014 in Theoretical Physics by user22710 (15 points) [ no revision ]

Your candidate for $D_R$, the one written in terms of the trace, cannot evidently work (if you expect to obtain a result different from the one at $T=0$): $[\phi(x,t), \phi(0)] = E(x,t)I$ where $E$ is a $c$-number and thus it can be extracted form the trace producing the same result as for $T=0$.

As a matter of fact the Green functions constructed out of the commutator do not depend on the state, so this result is expected: The retarded and advanced Green functions do not see the state and if there is a temperature or not. They are, in a sense, classical objects as they only depend on the equation of motion. Instead, if I understand correctly, you are interested in the Green functions depending on the state, like Feynman propagator. They are  constructed using  Wightman functions...

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsOv$\varnothing$rflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...