Renormalizing Open Quantum Field Theories
Round 1
Reviewer 1 Report
The paper presents results on the renormalization flow of scalar field theories in the closed time formalism with quartic (open) interactions, Eq.(17). The Wegner-Houghton equation (26) is taken as basic and after expansion in the local potential approximation (37) is used to read off the beta functions (38) for the standard as well as open couplings. The resulting flow is discussed in detail.
The paper is much improved compared to the arXiv version,
(2012.13811), but still lacks clarity and conciseness.
1. The summary of main results (l.32-l.42) is somewhat misleading.
Points (i) and (iii) are debatable conceptual viewpoints forming the
premise of the paper, not the results. Given these premises (ii) is a feature of the beta functions (38), which constitute
the main technical result of the paper.
2. The Wegner-Houghton equation is is neither mentioned in the abstract
nor in the introduction, yet is the main tool used and is well known to
any practitioner in the field. Incidentally, t = ln k/\Lambda
is a poor choice of notation for the renormalization time, as
the Minkowski time t also features in many equations. The
\dot notation should be introduced in (26) not 3 pages later
following Eq. (43).
3. The parallelism with the Euclidean formalism in Section 3.1
renders the exposition in Section 3.3 rather confusing.
If |p| = \sqrt{p_0^2 - p_1^2 - ... p_d^2} denotes Lorentzian
signature momenta, what exactly is being suppressed by |p| < const
\Lambda. Null momenta are seemingly not constrained? The authors
presumably have an answer, but it is not shared with the reader.
For clarity's sake the instrumental Eqs (25), (26) should be spelled
out in full detail.
4. Initial conditions for the flow are specified at k =\Lambda.
As with other Exact Renormalization Group equations, the dynamics
enters solely through the choice of initial conditions, yet no
comment on the principle governing the choice is offered.
Since the issue of ultraviolet renormalizability is postponed
(Section 4.4.) one cannot allude to the boundary conditions used
in the effective action formalism.
Even if one accepts the conceptual premises (i), (iii) the
given account raises many questions on the proposed `open'
renormalization formalism. These issues may legitimately be
postponed but the character of the paper as a proposal
based on limited results should be made clear.
The paper may be suited for publication in "Universe" if
the changes in exposition 1 and 2 are implemented and
satisfactory clarifications on 3 and 4 have been provided.
Author Response
All the remarks of the Referee has been taken into account, the changes of the text are set in red.
1. "summary of main results": All three points are results presented in the paper. This is made explicit by insertions into sections 4.1, 4.2 and 4.3.
(i) It is obvious that the cutoff theories take into account particle modes within a restricted kinematic domain. Furthermore it is well known that the system-environment entanglement changes the open dynamics in a decisive manner. Hence point (i) is trivial. It is mentioned as a result only because this feature of quantum field theories has not been tracked down. It is shown here in an explicit manner in section 4.3.
(iii) The presence of a nonperturbative regime as we approach the closed theory initial conditions of the renormalization group flow has been neither expected nor hypotethesised. It is shown in Fig. 5.
2. "Wegner-Houghton equation": We fully agree with the referee that the Wegner-Houghton equation, the main tool of this paper, is well known. The Wegner-Houghton paper has actually been cited both at eqs. (21) and (26). It is now mentioned in the Introduction, as well. The logarithmic parameter of the flow is now denoted by tau and defined after (26).
3. "Parallelism with the Euclidean formalism": The notation of the Introduction using bold face characters for the momentum variable indicates that the cutoff corresponds to the three-momentum. This is now noted explicitly in the Introduction.
"(25), (26)": The derivation is now indicated.
4. "Initial conditions": It is now mentioned at the beginning of section 4 that we restrict our attention to the renormalized trajectories starting in the vicinity of the Gaussian fixed point. There is no justifiable ansatz for the action in the case of strong coupling.
"Proposal based on limited results": It is now emphasized in the Summary that the extension of these results for a larger ansatz space of action is needed.
Reviewer 2 Report
According to the title and abstract of this manuscript, the aim of the authors is to study the renormalization of a scalar quantum field model with quadratic self-interaction as an open system. However, from section 2 onward, it becomes clear that authors have mixed and confused two different topics and corresponding methodology, namely the technique of
Closed Time Path (CTP) integral used for studying non-equilibrium quantum field theories and the concept of open quantum systems, in which part of quantum degrees of freedom are not accessible to the observer and are traced out. The authors introduce what they call doubling of the scalar field, without explaining (or realizing) that they present the same field on the two branches of the CTP. These fields are rather treated like a complex scalar field. In section 2.2. the authors introduce an environment field, which is traced out to make the system quantum mechanically mixed. But, their formulation does not show what is the consequence of such nontrivial operation, which is indeed the main subject of this manuscript according to its title. The rest of the work is an inconsistent ensemble of formulations, which mostly can be found in the literature on CTP.
In addition to these problems, the manuscript contains meaningless and undefined expressions such as: "open interaction vertices" (the title of section 2.5); "the bra and ket sectors" (ket and bra are the name of symbols used to indicate a pure quantum state and its conjugate); etc.
Finally, the subject of renormalization, which according to the title is the main goal of this work, is discussed only briefly and qualitatively in section 4.
In conclusion, considering many short comings in this manuscript, it should not be accepted for publication.
Author Response
"Authors have mixed and confused two different topics and corresponding methodology": The topics covered by this work are the extension of the renormalization group method for open quantum system and the CTP formalism of quantum field theory. This latter is more general than a tool for non-equilibrium quantum field theories. Therefore the superficial view of relating the CTP formalism and non-equilibrium physics is not relevant here. The former has not been developed in a systematical manner, we rely here on our previous work [17] as the starting point to develop here further.
"Doubling of the scalar field": This is a by now a widespread name of the CTP doublet and this concept is used by us just in that sense. The doublet has been properly introduced after (2). We consider a hermitian scalar field theory, no non-hermitian field is mentioned or used.
"Tracing out the environment field": We agree with the Referee that the consequences of this step is the main subject of this paper. However the "rest" is a consistent application of the functional evolution equation of the renormalization group, projected onto a limited ansatz space defined within the framework of the cluster expansion. The need of checking the stability of our results against the extension of this space is mentioned in the Summary. The list of the existing works about the implementation of the renormalization group program within the CTP formalism is given in the Introduction, the present approach and results are new.
"Inconsistent ensemble of formulations": While we believe that our presentation is consistent and clear we would be glad to improve it, it is a common interest of authors and editors to optimize the way the message gets across. However one needs more concrete referee report for this purpose.
"Meaningless and undefined expressions": The meaning of "open interaction vertices" is now explained in the Introduction and the word "sectors" is removed.
"Subject of renormalization": The derivation of the equations whose numerical solution is discussed in section 4 makes up sections 2 and 3.
Reviewer 3 Report
Open systems are ubiquitous both in quantum and classical domain.Starting with a Caldeira-Leggett system+environment+interaction model one constructs or at least attempts to construct reduced with respect to environmental degrees of freedom state of the system in order to study its dynamics or thermodynamics. The problem becomes highly subtle if the system -environment interaction fails to be perturbative. Path integrals ten come to succour. One utilizes Keldysh approach and Feynman-Vernon approach as two yet closely related techniques. cf U. Weiss, “Dissipative quantum systems” or Zaikin’s & Golubev’s “Dissipative quantum mechanics of naostructores” or, in more field-oriented context, in Calzetta and Hu, “Nonequilibrium Quantum Field Theory”. Situation becomes even more complicated if one is going to describe non-equilibrium phenomena in quantum many-body systems with interactions.
The Authors of the paper under consideration consider \phi^4 model (and its renormalization group equations) as an open system where the openness arises from cut-offs both in IR and UV regimes. In other words, the modes which are cut for an environment for that what is observed and the dynamics becomes open. It is stated also that since the cutof is necessary for a proper description of the model closed theories become then inconsistent according to the applied renormalization group methods.
The paper under consideration is very interesting and important. First, because of all the reasons mentioned by the Authors themselves, but second because its potential influence on the studies of many-body nanosystems studied from the field-theoretic perspective. In particular, field theoretic models can serve as can supplement (or vice versa) for recent approaches for quantum transport in wires coupled to leads serving as environments cf. Rev. Mod. Phys. 93, 025003. I recommend to include a short comment on that potential link. I also recommend to include a short comment concerning a relation of the CTP or FV approach to the canonical language as it is done in Calzetta and Hu, “Nonequilibrium Quantum Field Theory”. I understand that it does not push forward the theory itself but potentially can enlarge a group of readers by attracting people from nano-, solid state or even optic community. In particular, the IR and UV entanglement discussed in the paper in standard terms is qualified for a density matrix obtained by tracing irrelevant modes. The effect of decoherence can also be derived (at least formally) via the Feynman-Vernon approach. I recommend to include a comment which clarifies that we are roughly talking about the same phenomena.
In summary: the paper is well written and sound. I expect it to be influential and I recommend its publication after minor amendments implemeting my recommendations.
Author Response
All the remarks of the Referee has been taken into account, the changes of the text are set in red.
We thank for the excellent references, they all have been inserted in the Introduction, sections 2.2 and 2.6. They improve a lot the visibility of our work.
Round 2
Reviewer 1 Report
The authors address some of the issues raised but the exposition is still wanting.
- Abstract: It *must* be mentioned in the abstract which of
the many renormalization group formalisms is being used.
Why not refer to a "variant of the Wegner-Houghton" equation
(it's a well established technical term and Eq. (27) won't
become known as the Nagy-Polonyi equation). The abstract should
also state that a "sharp cutoff on the *spatial* momenta"
is being used.
2. The introduction still needs to be improved:
-- the terms "open dynamics" and "open channels" should come
with pointers to Section 2 and the literature.
-- the object of the study should be made fully explicit.
Presumably, it is a "scalar field theory in four
dimensional Minkowski space at zero temperature with at least
quartic selfinteractions"? Please state this clearly.
-- This system is subjected to a "sharp cutoff on the spatial momenta".
Please state this clearly. The paragraph after l.63 could be misread
as general musings about cutoffs and regulators. How is the $k$ used
later in the text related to $\delta_{\pm}$?
-- The paragraph following l.88 could be merged with the one
starting at l.44.
3. The authors still conflate conceptual points they wish to
stress (like l.15,16) with results. Point (i) in l.33 is according
to the author's reply a logical triviality entailed by their tacit
definition of the terms "open" and "closed". As such it is not a
result. Point (ii) requires further specification: it cannot
hold in a strictly UV renormalizable theory (like selfinteracting
scalars in 1+1 or 1+2 dimensions). In 1+3 dimensions, should the
knowledgeable reader think of it as a reinterpretation of the known
non-perturbative triviality of the theory?
4. Since section 3.1 discusses finite temperature QFT
it should be specified that section 3.2 (presumably) refers
to zero temperature.
5. Equation (27) is a functional differential equation.
It *must* be accompanied by (at least) a conceptual discussion
of initial conditions at ($k=\Lambda)$ before turning to specific
truncations, etc. Presumably, (27) is exactly soluble in the case
without selfinteractions (and it would be instructive to see
this solution). However, it is clearly not the solution aimed
at, hence the specification of the initial condition. Further,
how precisely does the cryptic l.284 determine the correct
interpretation of (27)?
6. What precisely does the use of the Wegner-Houghton equation
gain compared to standard one-loop perturbation theory, e.g. along the
lines of ref.37? Would application of the latter also give Eq.(39)?
The paper may be suited for publication in "Universe" if
the changes in exposition 1. - 4. are implemented and
satisfactory clarifications on 5 and 6 have been provided.
Author Response
The changes, made along the demand of the referee are shown in red.
1. The required information is now presented in the abstract.
2. The Introduction has been modified.
3. While point (i) is trivial it has not been emphasized or followed up in the literature. Its explicit demonstration is a new result. The issue of renormalization is taken up in section 4.4 where it is clearly stated that the renormalizability is indeed a more difficult question than for closed theories. The argument makes clear that the difficulties created by the open interaction channels are not related to triviality.
4. The remark is added to Section 3.2.
5. The explanations were added.
6. Ref. [37] uses the multiplicative renormalization group scheme where the contributions which are small for large cutoff are neglected. It is mentioned in point (iv) of section 4.4 that the functional renormalization group scheme keeps all contributions. Hence the functional evolution equation cannot be reproduced by the multiplicative renormalization group method.
