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The Propagator Matrix Reloaded
by João F. Melo
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Submission summary
| Authors (as registered SciPost users): | João Melo |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2112.09119v2 (pdf) |
| Date submitted: | 2022-01-13 11:04 |
| Submitted by: | Melo, João |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
The standard way to perform calculations for quantum field theories involves the S-matrix and the assumption that the theory is free at past and future infinity. However, this assumption may not hold for field theories in non-trivial backgrounds such as curved spacetimes or finite temperature. In this work we examine the situation at early times for Minkowski spacetime at finite temperature via the use of the Schwinger-Keldysh formalism. We find that there are additional cross terms between the real and imaginary time fields making our propagator matrix $3\times 3$ rather than the more familiar $2\times 2$. This suggests the theory is indeed not free at past infinity even for Minkowski spacetime at finite temperature.
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Reports on this Submission
Report #3 by Anonymous (Referee 3) on 2022-3-3 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2112.09119v2, delivered 2022-03-03, doi: 10.21468/SciPost.Report.4615
Weaknesses
1- This has already been treated at length in the literature
2- The abstract alludes to curved spacetimes but the paper only addresses finite-temperature field theory.
Report
The Author proposes to construct a Keldysh formalism to compute correlators with finite temperature initial conditions that are set at a finite time in the past. He treats, in particular, the case of a scalar φ^4 theory, first at the quadratic level, and then to one-loop order.
"We find that there are additional cross terms between the real and imaginary time fields making our propagator matrix rather than the more familiar 2×2."
"This is the main subject of this manuscript: constructing a formalism to calculate correlators
with finite temperature initial conditions set at a finite time in the past to check if the standard assumptions hold."
These two sentences from the abstract and the introduction strongly suggest that the Author is unfortunately unaware of the existing and relevant literature around the use of Keldysh formalism at finite temperature and/or with initial conditions. This topic is by now well understood.
The 3-branch contour mentioned by the Author is known as the so-called Kadanoff-Baym-Keldysh contour. The construction of this formalism and how to effectively compute correlation functions are covered by introductory textbooks to the Keldysh formalism, such as
Introduction to the Keldysh Formalism
van Leeuwen R., Dahlen N., Stefanucci G., Almbladh CO., von Barth U.
Lecture Notes in Physics, vol 706. Springer, Berlin, Heidelberg, 2006
doi:10.1007/3-540-35426-3_3.
The Author will find there all the Schwinger-Dyson equations he listed in Sec. 3 when dealing with the quadratic theory.
His one-loop attempt to a perturbative treatment of interaction in Sec. 4 is also well understood by now. See, e.g.,
Equilibrium and nonequilibrium many-body perturbation theory:
a unified framework based on the Martin-Schwinger hierarchy
R Van Leeuwen, G Stefanucci
Journal of Physics: Conference Series, 2013
doi:10.1088/1742-6596/427/1/012001
"This suggests the theory is indeed not free at past infinity even for Minkowski spacetime at finite temperature."
This statement, among many others, generates confusion where things are actually well understood. Gaussian (free) theories stay Gaussian even if they were prepared in a finite-temperature thermal equilibrium, even if they require additional branches to the Baym-Kadanoff contour. How interactions can be adiabatically turned on in the far past has already been discussed at length in the literature (see, e.g., the textbook above). Interacting theories stay interacting: integrating out the initial degrees of freedom typically yields a retarded self-energy contributions to the real-time propagators. When those contributions can be neglected has already been discussed at length in the literature.
To summarize:
The Author clearly lacks hindsight in this field. The treatment of interacting initial conditions in real-time quantum evolutions has already been worked out in great detail in textbooks and is widely known among practitioners of the Keldysh formalism. Therefore, while this is definitely a valiant effort, this work does not bring any novel understanding. Furthermore, it carries a great risk of instilling unnecessary confusion in the reader's mind.
First of all, I would like to thank you for your very insightful comments. The two references you mentioned were not familiar to me as I come from a high energy physics background and will certainly be added to the final draft of the manuscript as they complement the discussion rather nicely.
I was in particular completely unaware of the book mentioned as it concerns non-relativistic field theory. It had previously come to my attention that in this case the 3x3 formalism is more well known, https://iopscience.iop.org/article/10.3367/UFNe.0185.201512b.1271 is another good example of this discussion. In the end I decided against including these references as the physical situation is fundamentally different. The propagator equations are first order in time (c.f. (3.40) from https://link.springer.com/chapter/10.1007/3-540-35426-3_3) which fundamentally changes their character. In particular, one no longer needs to fix the time derivatives at the joining points, allowing for more standard applications of Fourier methods. I can nevertheless add a small comment to the introduction regarding the non-relativistic case.
Apologies if I was unclear and seemed to claim the time contour with three segments is a new development. This is certainly not true, and several of the references in my introduction use or mention it. The same can be said of the equations in section 3. The reason I presented them in such detail was to be clear about all the assumptions and make it easier for a reader who is not familiar with the literature to read and follow the discussion. Further, since they are usually dropped in the end most references do not give a full account on how to calculate these terms. However, I could make it clearer in these sections that these ideas aren’t entirely new.
The central difference between the usual treatments found in the literature and mine is that I keep $t_0$ finite throughout and never make any assumptions about whether or not the theory is approximately free at any time.
The standard treatments (as can be evidenced by the references found in the introduction) take $t_0\to-\infty$ and then make some assumption that makes the initial state be that of a free theory. For example, in https://iopscience.iop.org/article/10.1088/1742-6596/427/1/012001 the authors add a term to the Hamiltonian so that the interactions are adiabatically turned off at very early times. Another popular alternative is to use a version of the $i \epsilon$-prescription and projecting to the free theory at early times.
This is a different physical situation to the one I am considering. In my case, I set a thermal initial condition at time $t_0$ using the full Hamiltonian, including interactions, but without including any time dependence in the interactions to forcefully turn them off at early times. The central statement of the paper is that if I do this calculation with no approximations and keeping $t_0$ finite throughout, the final answer is in fact independent of $t_0$ and it differs from what we would get had we assumed the theory was free at $t_0$.
Further, there are some mathematical inconsistencies if one attempts to do renormalisation without using the full 3x3 matrix, we obtain a different answer depending on whether we put the counterterm in the interaction or quadratic Hamiltonian.
The reason behind the comment "This suggests the theory is indeed not free at past infinity even for Minkowski spacetime at finite temperature." is that the final answer is completely independent of $t_0$. Let me expand on this.
At time $t_0$ the density matrix using the full interacting Hamiltonian is certainly not Gaussian, as we have explicitly included a quartic term. This yields different answers than the ones we get from a free Hamiltonian. Given the final answer is completely independent of $t_0$, we can take $t_1,t_2\to-\infty$ (keeping $t_1-t_2$ finite for the time being) and conclude that the answer we have got is different to that we would have obtained if the theory was free at past infinity.
The only caveat to this statement is that perturbation theory is badly behaved for $t_1-t_2$ very big and the correct resummation appears to be a mass shift. For reasons expanded upon in the conclusion of my manuscript, this shift is the same as what is found assuming the theory was indeed free at past infinity. Therefore there could be an issue of order of limits rather than the theory actually staying completely interacting at past infinity. To resolve this question one would need to check if the resummation indeed works to all orders in perturbation theory with the 3x3 formalism. Going to higher order in perturbation theory was left for future work.
Finally, regarding the abstract, I mentioned curved spacetime as that was the original motivation. I chose to consider the Minkowski case separately to make the discussion cleaner but it is still of interest to curved spacetime practitioners. I felt it was important to explicitly have those words to attract their attention.
I hope this clarifies all of the points that were raised.
Report #1 by Anonymous (Referee 1) on 2022-2-27 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2112.09119v2, delivered 2022-02-27, doi: 10.21468/SciPost.Report.4538
Report
The paper revisits formal aspects of the Schwinger-Keldysh (SK) formalism, especially at finite temperature, where the complex time contour splits into two Lorentzian segments and one Euclidean part. Most discussions of the formalism consider a 2x2 matrix of two-point propagators with operator insertions on the real time segments. The author argues that generic two-point functions also include those with insertions on the Euclidean part, and that these are important for a consistent renormalisation scheme (illustrated at 1-loop), especially when the thermal initial state is located at a finite time in the past.
The paper is well written and nicely structured. The results clarify some basic properties of a well known formalism in a concise way. While I don't fully agree that the insights are completely new and previously unnoticed, the focus is on some interesting aspects with relevance to modern applications. The paper is therefore relevant and worth publishing. I have two points that would be nice to address at least briefly:
- Could the author comment on other prescriptions for the Keldysh contour? Most generically, one can consider a contour with two real-time segments separated by two imaginary time parts of length $a\beta$ and $(1-a)\beta$ for any $0<a<1$. Do all conclusions go through (presumably, after using a 4x4 matrix of propagators)?
- At the end of section 3.2, it might be worth mentioning that there are more relations between the 2-point functions. In particular, the author doesn't seem to use the KMS relation explicitly. Similar to the relations given at the end of this section, the KMS relation states:
\[ G_\text{dif,ave} (t_1, t_2) - G_\text{ave,dif} (t_1, t_2) = -2 \, \text{tanh}\left(\frac{i\beta\partial_{t_1}}{2} \right) G_{\text{ave,ave}}(t_1,t_2) \]
(There might be further relations involving the Euclidean segments. ) One might expect that there could be an efficient way to impose such relations to further reduce the number of independent propagators. For instance, standard discussions of the formalism perform a further basis rotation from the "average/difference" basis to the "retarded/advanced" basis, in order to manifestly reduce the number of independent correlators. Roughly speaking, this would eliminate the ave-ave propagator (3.19a) in a similar way as the dif-dif propagator has already been removed. If this is not useful here, I would suggest commenting why.
First of all, thank you very much for your kind and insightful comments.
Regarding the first point, this is partly addressed in the conclusion when I mention the thermofield double. In short, it is perfectly consistent to consider this 4 segment contour but the problems are somewhat multiplied as they now include statements about interactions in the far future. I did not consider these generalizations to keep the discussion focused on the initial conditions. However, I can add a small remark in the conclusion regarding these more general contours.
Regarding the second point. I chose not to use the KMS explicitly for two reasons: firstly, at a practical level it made the discussion cleaner without the necessity of a detour into its derivation in the case at hand and how it use it; secondly, at a philosophical level I prefer to think of the KMS condition as an output of my calculations rather than as an input. I could nevertheless add a comment clarifying that indeed there are even further relations between the different propagators.
However, I am not familiar with the "retarded/advanced" basis as something distinct from the "average/difference" basis. I recall seeing this name in the literature but used to denote what I have called "average/difference" but perhaps I have misunderstood. Could you point me towards a reference where the distinction is made clear so that I can gauge its applicability in this case?
Anonymous on 2022-02-28 [id 2250]
(in reply to João Melo on 2022-02-28 [id 2249])
Thanks for the suggested improvements, which will be enough for me to recommend publication.
Regarding the "retarded/advanced" basis: a detailed reference that comes to mind is https://arxiv.org/abs/hep-ph/9406214. Section 4 describes this basis, which achieves that the propagator matrix (42) has only two non-trivial entries. More recent discussions include your reference [37], where the transformation between "ave-dif" and "ret-adv" is given in (4.22). The point is that in this basis the "ret-ret" correlators vanish whenever the KMS condition is satisfied. It is therefore a convenient basis in the context of thermal states. If one doesn't want to impose the KMS from the start, then one can still work in the "ret-adv" basis such that checking whether KMS holds simply reduces to confirming that $G_{\text{ret,ret}}=0$. Again, while all this seems natural in general, I have not checked whether it will actually simplify your particular analysis. If you think it doesn't, then a simple comment will be enough, since there is certainly nothing wrong with your calculations. (It is worth noting that related recent studies such as https://arxiv.org/abs/2011.07039 seem to gain some mileage from using this basis.)
Anonymous on 2022-03-01 [id 2256]
Thank you for the references.
Indeed I think I have understood what is the issue. In essence, all these are relations between propagators evaluated at $t$ and propagators evaluated at $t\pm i \beta$. However, if we do not go to frequency space then these shifts are not represented my mere multiplication. Therefore, in order to use these relations we would need to invert a differential operator which is technically much more complicated. Further, I do want to avoid frequency space as I am dealing with a compact time interval, trying to use frequency space without ruining temporal boundary conditions leads to other complications.
There may be a way to get around these issues and have a useful definition of the "retarded/advanced" basis for this formalism. But I think that at this point it is better to just add a comment explaining these issues and maybe tackle them in some future work.