(Non) equilibrium dynamics: a (broken) symmetry of the Keldysh generating functional

1-Innovative formalism
2-Detailed discussion of the latter
3- Interesting results on the characterisation of irreversibility

1- Hard to read
2- Too technical for non-experts.

The manuscript by C. Aron et al. presents a generalization of the standard Schwinger-Keldysh field theoretic formalism which involves a re-parametrization of Keldysh contours (including the equilibrium branches) to generic contours in the complex plane. While similar analytic continuations have been historically used for example to derive various relations among Green’s functions and their convolutions, this idea is here exploited to its full power to derive universal symmetry relations satisfied by equilibrium theories and how taking the system out of equilibrium breaks them.

I think that overall the content of the paper and its results are very interesting and the paper should be published with minor revisions.

In particular, the paper is extremely technical and its style makes it at the moment hardly readable to non-experts. It took me some time to navigate through it and digest its various parts. In particular the main physical result, which is in my opinion contained in Eq.(88)-(91) (and the following fluctuation theorems), is very interesting. It is however only briefly announced in Sec.1.3 (referring to Eq.(102) though a lot of physics is contained much before that) as a “general expression for entropy production”, and later on as a “quantum generalisation of dissipated work” ( introduction of Sec.4), while it is very hard to see either one or the other in the concrete expressions, apart of course for the “slow” limit Eq.(95). The physical meaning of $\Sigma$ is still obscure the connection with other quantities and their fluctuation theorems appears unclear (e.g. Physical Review E 75, 050102(R) (2007) and generalisations).

In order to make the paper more readable, I would suggest to significantly revise the section 1.3 about “Main Results” which is now just an extended summary of the paper focusing for the greatest part on the formalism. I would suggest to rather put significant stress on the section on non equilibrium dynamics and on $S^{irr}$ or $\Sigma$ discussing its context, why it is important (including references), clarifying its physical meaning or at least its meaning in the various limits in some detail.

1- Revision of Sec.1.3 as detailed in my report

1- very clear and precise derivations;
2- effort in making a technical, hard topic understandable to the readers.

1- lacking of a concrete (even simple) application to a model of interest.

This work extends previous lore on symmetry properties exhibited by the Keldysh action for systems at thermodynamic equilibrium. The topic has a long tradition, as also pointed out by the authors, and its main practical importance stands in the fact that non-equilibrium conditions break this symmetry and novel symmetry-breaking terms appearing in the action can help in enlarging physical insight.
This is achieved by the authors, extending to time-dependent quantum systems pre-existent expertise on the subject.

A nice feature of the work is its readability by field-theoretical experts that are not familiar with Keldysh formalism: the authors have done some effort in making their derivations clear step-by-step. This, however, has the downside that the community of quantum thermodynamics (which should be one of the main beneficiary of their findings) might have an hard time in understanding the core results. In view of this, and of the fact that ultimately such technical works should always present a spectrum of interesting applications, I would recommend the authors to go after the following suggestions:

1) a summary on physical findings and outreach of applications should be substantially enlarged in the introduction;

2) they should find and add to the manuscript a concrete application of Eqs. (92-93-94), even a simple one (for example: one body, quantum, coupled to a thermal bath and to a time-dependent external field), where the predictions of their main results can become of relevance for the debate (in the quantum thermodynamics community) concerning a proper, suitable definition of work and produced entropy in non-equilibrium settings.

This would help their work and their results to be easier spread and employed by the communities of interest.

Finally, as a speculative point, one might wonder whether the equilibrium symmetry can be spontaneously broken in some particular circumstances. This comes naturally to the mind after reading their work, since they realise an explicit breaking by adding an external time-dependent drive. Do the authors have some insight in this direction? Would they like to add some comments in the conclusions about it?

Minor observations:
- in (15) the ‘;’ should be a typo;
- what do the authors mean by ‘thermal rotation’ in section 2.3?
- in 3.2 they mention the ‘thermo-field’ formalism, which is always contrasted with the Keldysh one. Since the authors show an understanding about the differences between the two, they have a chance to clarify for the community this point, which to my understanding has not been yet well inspected; for instance, they can contextualise it, when they compare their main finding in the two formalisms (between Eqs. 42 and 43).

1- extend summary on physical contents;
2- provide a workable, interesting example.