Manuscript code: BH11160 RECVD: Mon Feb 8 16:14: Resubmission to: Physical Review B Resubmission type: resubmit

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1 1 of 6 05/17/ :32 AM Date: Mon, 8 Feb :14:55 UT From: esub-adm@aps.org To: prbtex@ridge.aps.org CC: wen@dao.mit.edu Subject: [Web] resub BH11160 Gu Subject: BH11160 Manuscript code: BH11160 RECVD: Mon Feb 8 16:14: Resubmission to: Physical Review B Resubmission type: resubmit Replaced files: lqgrv.tex Details of changes: Dear Prof : We would like to thank you for your time and energy to review two of our papers. The main purpose of our appeal is to have some further discussion on the physics accosiated with the emergent quantum gravity. We hope that we still have a chance to discuss the issue further. First, We would like to make a statement: So far, we do not know any 3D lattice Hamiltonian with a continuous time that have the following properties: 1) has gapless excitations 2) all low energy excitations are described by a single helicity +/- 2 mode (and there no other gapless excitations or other low energy modes) 3) the helicity +/- 2 mode has a linear dispersion. Actually, I think that we do not even know any 3D lattice Hamiltonian with a continuous time that have the first two properties (Correct me if I am wrong). If you agree with the above statement, then the significance of our paper can be undertsood as to propose the first lattice model (the L-type model) that have the first two properties (this is reliable result). We also propose a N-type model that may even have the above three properties. But the result on the N-type model is less reliable. Second, We would like to list our practical and minimal definition of quantum gravity as the following (as stated in the paper): Quantum gravity is (a) A quantum theory. (b) Its Hilbert space has a finite dimension. (c) The Hamiltonian is a sum of local operators. (d) A single gapless helicity $\pm 2$ mode is the only type of low energy excitations. (e) The helicity $\pm 2$ excitations have a linear dispersion. (f) The gravitons interact in the way consistent with experimental observations.

2 2 of 6 05/17/ :32 AM We very much like to hear your comments on the above minimal definition of quantum gravity. The main objection to our paper is that the paper is not related to quantum gravity. If our definition make sense, then our paper is related to quantum gravity by addressing (a-d) or (a-e). There are several published papers that proposed other 3D lattice Hamiltonians of gravity. But those papers fail to show the existence of gapless excitation. Compare to those published proposals, our models do have something to do with gravity. We completely agree with you that the property (f) is very important. But here, our main concern is to have a well defined local quantum theory (such as a 3D lattice Hamiltonian) that has gravitons. The property (f) is left for future study. The following is our detailed reply to your comments. We wish that you may reconsider your recommendation on our paper Report of the Divisional Associate Editor -- LH12204/Gu Appeal LH12204 Lattice qubit model as a quantum theory of gravity by Zheng-Cheng Gu and Xiao-Gang Wen BH11160 Emergence of helicity $\pm2$ modes or gravitons from qubit models by Zheng-Cheng Gu and Xiao-Gang Wen Note: this is a joint Editorial Board response to the appeals of two papers, LH12204 (submitted to PRL) and a longer version, BH11160 (submitted to PRB). The basic issues involved are the same for both. If gravity were merely a massless spin 2 field, life would be much easier, albeit less interesting. But gravity is much more: in this language, it is a massless spin 2 field that couples universally to all forms of energy, including its own self-energy. This universality is crucial both theoretically and experimentally. Theoretically, it is what allows us to treat gravity as a geometric phenomenon, by ensuring that all objects respond identically to a gravitational field. Observationally, it is perhaps the best-tested aspect of gravity: tests of the principle of equivalence -- including tests of the self-coupling of gravity, through searches for the Nordtvedt effect -- now typically have accuracies of parts in $10^{12}$. We totally agrees with the Editor's statement on gravity. But here we would like to stress that the Editor also agrees that the quantum gravity at least has "a massless spin 2 field". At moment, we do not have any 3D lattice model that has a helicity +/- 2 mode as the only low energy

3 3 of 6 05/17/ :32 AM modes. Our paper is the first paper that proposed a 3D lattice Hamiltonian that has a helicity +/- 2 mode as the only low energy modes. Only after having a quamtum model that has a helicity +/- 2 mode, can we start to consider the conditions that make gravitons to have a proper form of interaction that is consistant with general equivalence principle. The authors of this paper attempt to derive gravity as an emergent phenomenon on a spatial lattice. Their model seems somewhat artificial, in the sense that they get spin 2 out only by putting spin 2 in (i.e., placing symmetric tensors on the lattice); this is not quite what people usually mean when they talk about emergent gravity. Still, if the result was a good lattice model of gravity, the work would be of some interest. Although, our model is motivated by put symmetric tensors on the lattice, our model is actually a qbit model on lattice. Each site only have some quantum states (ie qbits). Strictly speaking, there is no symmetric tensor in our lattice model. Unfortunately, the authors only consider the linearized level, where the issue of self-coupling is absent, and they do not discuss the coupling to general matter fields at all. Moreover, even at this linear level, they state in the PRB submission that the dispersion relation for the model that most closely resembles linearized gravity "is not a reliable result" because of the possibility of uncontrolled quantum fluctuations at higher orders. (I did not see a similar caveat in the PRL submission.) The authors state explicitly that the higher order terms key to a genuine theory of gravity are nonuniversal in their model. They express a hope that the standard (experimentally tested) result might be obtained by fine-tuning the lattice model, but offer no evidence for this hope, and no reason to expect that Nature would do such fine-tuning. In particular, they offer no reason to believe that such a tuning at one scale would be preserved under renormalization group flow. But without the right universal coupling, this is very clearly not a theory of gravity. Here Editor raised a very important issue about the proper self interaction of gravitons. Our paper only proposed a 3D lattice Hamiltonian with a helicity +/- 2 mode as the only low energy modes. The issue of graviton self interaction is not really addressed in our paper. The graviton self interaction will be a future research direction, after this paper about the emergence of gravition get published. We agree that without the right universal coupling, a theory cannot not be a full theory of gravity. But we like to stress that without gapless helicity +/- 2 modes, a theory is even less a theory of gravity.

4 4 of 6 05/17/ :32 AM There are other basic issues as well. While the authors discuss the idea that "geometry is an emergent phenomenon that appears only at long distances," in practice they assume a fixed, a priori geometry -- their cubic lattice, with fixed edge lengths, gives a flat Euclidean background. There was once a time that one could simply assume Euclidean geometry and proceed from there, but that time has long passed; now that we understand that geometry can be dynamical, a starting assumption of a flat background is no longer self-evident. If the authors want to explain gravity as an emergent phenomenon on a flat lattice, they also need field equations that determine why the lattice is flat. The distance defined by lattice is not the same as the physical distance. The physical distance depends on the fluctuations $h_{ij}$ (the helicity +/- 2 modes at low energies). So the emergent geometry can be curved if the collective mode $h_{ij}$ is excitated. The equation for $h_{ij}$ is the equation for the dynamical geometry. This is not just a philosophical issue. The choice of a flat background makes it very hard to compare the theory to general relativity. As the referees have stressed, diffeomorphism invariance is not a local gauge transformation: diffeomorphisms do not act at a single point, but move points. The metric, for example, transforms under $x^a \rightarrow x^a + \xi^a$ as $$g_{ab} \rightarrow g_{ac}\partial_b\xi^c + g_{bc}\partial_a\xi^c + \xi^c\partial_c g_{ab}$$ The last term is crucial. Unlike an ordinary gauge transformation, an infinitesimal diffeomorphism involves derivatives. Unfortunately, if one linearizes around a flat background, this last term vanishes at lowest order. (This is not the case, even at lowest order, if one expands around a curved background.) But unless this term is present, the symmetry of the model is not the symmetry of general relativity. The fact that diffeomorphisms relate fields at different points is the basic feature that makes it so hard to quantize gravity, implying, for instance, that there are no local observables. If one wants to quantize gravity by simply ignoring this feature, one has a heavy burden: one must prove that the resulting theory really can reproduce the observationally tested results of general relativity. This is again a very good point. But this is about the issue of self interaction of gravitons. Trying to understand under what condition can the self interaction of gravitons naturally take a form that is consistant with general equivalence principle is a future research direction. Our paper is just about how to have massless graviton emerge from a 3D lattice Hamiltonian. The issue of self interaction of gravitons is not really addressed.

5 5 of 6 05/17/ :32 AM This same problem shows up in the authors' constraints. As they point out in the PRB submission (eqn. (26)), their constraints all commute. This is most certainly not the case in conventional gravity, where the constraints form a "surface deformation algebra." One would not write down a theory with three commuting U(1) gauge symmetries and call it an SU(3) theory; why should one claim that a theory with four commuting constraints has anything to do with general relativity? We made the claim because our 3D lattice model has helicity +/- 2 modes as the only low energy excitation. There are several published papers that proposed other 3D lattice Hamiltonians of gravity. But those papers fail to show the existence of gapless excitation. Compare to those published proposals, our model does have something to do with gravity. I also concur with the referees that these papers have a distressing tendency to ignore previous work. The authors write in one of their replies that their approach is different because it uses a spatial lattice but continuous time. But this is not a new idea; I believe it was introduced first by Renteln and Smolin, Class. Quantum Grav. 6 (1989) 275, using Ashtekar variables; continuous time Regge calculus has been studied by Collins and Williams, Khatsymovsky, Brewin, and a number of others. I also agree with one referee's suggestion of Hamber's monograph. There is also a huge literature on emergent gravity; a starting point might be the paper by Barcelo et al. in Living Reviews in Relativity. A greater familiarity with this work would have alerted the authors to the crucial problem of universality and the equivalence principle. Indeed, quantum gravity is not our main research field. But we did try very hard to find papers on quantum gravity that start with a well defined local quantum model (a 3D lattice Hamiltonian). Although, there are may space-time lattice based approach, we find only a few papers which proposed a 3D lattice Hamiltonian with a continuous time. (Those references 33,34,35 are included in the new version.) Non of those paper discussed the low energy excitations of the propose Hamiltonian. We do not even know if the proposed Hamiltonians have gapless excitation or not. Our paper did discuss low energy excitations and show the emergence of gapless helicity +/- 2 mode. So we feel that our paper does have something to do with gravity, and deserve to be published. In summary: the authors have shown, perhaps, a way to give masses to the spin 0 and spin 1 pieces of a symmetric field on a lattice, while keeping the spin 2 piece massless (although they admit that for the most interesting model, their reported low-energy behavior is unreliable).

6 6 of 6 05/17/ :32 AM We agree with Editor's above characterization of our paper. (If we replace "spin-m" by "helicity-m", the above statement will be more accurate.) It is highly highly non-trival to design a 3D lattice Hamiltonian that give energy gap to the helicity-0 and helicity-1 modes while keeping the mode-2 as the only gapless excitation. We believe that our model is the first and only proposal to achieve this. This is the reason why we think our paper deserve to be published. They have not, however, connected the resulting theory to gravity, and I see no sign that such a connection can be made. It is certainly not true that a massless spin 2 field in itself implies gravity, especially with no evidence for a universal coupling to energy. Much more needs to be done to even start to make such a connection. But without such a relationship to gravity, these papers seem to be of very limited interest. We respectfully disagree with Editor's conclusion here. We believe that our model is the first and only 3D lattice Hamiltonian that has helicity +/- 2 modes as the only low energy excitation. Thus, we feel that our paper deserve publication. We agree with the Editor that the self intercation of gravitons and universal coupling to energy are very important, and they are our future research directions. Only after having a quamtum model that has a helicity +/- 2 mode, can we start to consider the conditions that make gravitons to have a proper form of interaction and universal coupling to energy. Here we try to learn how to walk before we run. We feel that the first paper about "how to walk" deserve publication. I concur with the referees' recommendations that these papers should not be published. We sincerely wish the Editor to reconsider. Maybe the key points are (a) does our paper propose the first and only 3D lattice Hamiltonian that has helicity +/- 2 modes as the only low energy excitation. (b) does the emergence of gapless helicity +/- 2 modes have some relation to gravity and deserve publication? Many papers on quantum gravity did not show the presence gapless helicity +/- 2 modes. Without massless gravitons, one cannot even start to discuss the proper self interaction of gravitons at low energies. We feel that stressing the presence gapless helicity +/- 2 modes is important for development of the field of quantum gravity. Only after showing the presence of gapless helicity +/- 2 modes, can we consider the proper self interaction between those modes.