How generalized hydrodynamics time evolution arises from a form factor expansion

1- Interesting and timely topic.
2- Systematic approach.

1- Many assumptions are hard to test.
2- Some passages of the paper need clearer explanation.
3- Some parts of the summary of known results are too hasted.
4- The discussion of corrections is incomplete.

This paper proposes a generalisation of the Quench Action approach that is able treat a certain class of inhomogeneous initial states (inhomogeneous GGEs). This approach is then used to recover the leading- order-in-time description of Generalised Hydrodynamics and a to propose a systematic identification of the leading corrections. On the technical level, the author uses form factor expansions and, in particular, some results from the “Thermodynamic Boostrap Program”. The latter is an axiomatic approach recently proposed by the current author and a collaborator to treat form factors on states with finite density.
I think that the paper is very interesting. It studies an interesting problem in an innovative way. A particular strength of the method proposed is that it could lead to a systematic classification of all corrections to the leading-order GHD. On the other hand I think that a substantial weakness of the method lies in the impracticality of testing many of the assumptions by independent means (other calculations, numerics, ...). Also considering this weakness, however, I think that the paper is certainly worth publication.
That being said I think that the presentation in the paper can be substantially improved in three main aspects (see the requested changes below). First, I think that some passages in the summary of known results are not are too hasted. Second, I think that some of the assumptions in the derivation of the main result deserve a more expanded explanation. Finally, the last section (the one about the corrections) is incomplete.

1- The discussion about the dressing is misleading. In the GHD literature there are two conventions for introducing the “dressing” dating back to the original works ([4] and [5]). The author here (Eq. 5) is choosing the one of [4]. The charge dressed with this convention, however, does not correspond to the charge of "a particle excitation of rapidity $\theta$, created on top of the thermodynamic state" as the author states before equation 5.

2- After Eq. 29 the author writes: "For such linear quench actions, it is know that there is only one saddle point (see for instance the computation in [44]), so all of the quench action logic follows through.". Has this fact been proven at least in some example? How is the calculation in [44] showing this point? Please clarify.

3- When discussing Eq. 31 it could be useful to recall that the approximation performed there is equivalent to the Local Density Approximation.

4- Why is the solution 35 explicit? As far as I see one still needs to determine u(x, t, θ) from equation 36. Please give more details.

5- The discussion following 45 (i.e. the one involving the introduction of the operator SˆK ) is not very QA
clear to me.

6- I agree with the author’s point of this paper not being about the corrections. To substantiate the method (and the many assumptions involved in it), however, it would be very useful to identify at least the class of terms giving the $1/\sqrt t$ correction to the expectation value of local operators.

7- Some typos:

Second paragraph on the second page: concern → concerns

There are some typos in Eqs. 44 and 45 and in the inline equations nearby. The sum over i disappeared and that over K became over k.

Before Eq. 49: ρi → σi

Eq. 55: = sign missing

1. Timely subject.
2. Establishing a link between different approaches (GHD vs. form factor methods).
3. Generally applicable framework.
4. Proposal for the leading quantum corrections to the GHD approach.

1. It is not made clear how much of the thermodynamic form factor framework is relevant to the derivation.
2. In some points the author overlooks some known issues (cf. requested changes).
3. The derivation of leading quantum corrections is unfinished.

This paper treats an important subject, and represent a significant and interesting advance in understanding Generalised Hydrodynamics (GHD). Based on these, I do consider that the results merit publication.

However, before the eventual publication I find it necessary that the author addresses some questions (listed in requested changes).

1. By linking the thermodynamic FF program to the GHD, the author provides an interesting piece of evidence for its validity. However, it is important to clarify how much the thermodynamic FF as prescribed by recently introduced bootstrap program eventually plays a role in the derivation.

2. The finite part defined in (51) is eventually ambiguous, as the limit depends on the direction i.e. the ratio of the kappa parameters. This issue is known since long and is only resolved in the finite volume FF formalism, cf. B. Pozsgay and G. Takacs, Nucl.Phys.B788:167-208,2008; Nucl.Phys.B788:209-251,2008. For a proper presentation this fact must be duly pointed out. It is a fortunate set of circumstances that in the derivation of GHD the kappa-regularisation is applied with a single kappa, where this ambiguity does not arise. However, this issue may very well affect the computation of quantum corrections, similarly to the case of finite temperature correlators (cf. B. Pozsgay and G. Takács J. Stat. Mech. (2010) P11012).