A lattice non-perturbative definition of an SO(10) chiral gauge theory and its induced standard model

Question

Referee this paper: arXiv:1305.1045

Please use comments to point to previous work in this direction, and reviews to referee the accuracy of the paper. Feel free to edit this submission to summarise the paper (just click on edit, your summary will then appear under the horizontal line)

(Is this your paper?)

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Abstract

The standard model is a chiral gauge theory where the gauge fields couple to the right-hand and the left-hand fermions differently. The standard model is defined perturbatively and describes all elementary particles (except gravitons) very well. However, for a long time, we do not know if we can have a non-perturbative definition of standard model as a Hamiltonian quantum mechanical theory. In this paper, we propose a way to give a modified standard model (with 48 two-component Weyl fermions) a non-perturbative definition by embedding the modified standard model into a SO(10) chiral gauge theory and then putting the SO(10) chiral gauge theory on a 3D spatial lattice with a continuous time. Such a non-perturbatively defined standard model is a Hamiltonian quantum theory with a finite-dimensional Hilbert space for a finite space volume. Using the defining connection between gauge anomalies and the symmetry-protected topological orders, we show that any chiral gauge theory can be non-perturbatively defined by putting it on a lattice in the same dimension, as long as the chiral gauge theory is free of all anomalies.

Comments

This paper was not very well received, and here's a PRL referee review posted on Xiao-Gang Wen's blog(the blog is partially Chinese, but all the review information is in English). The reason I'm more interested in a review for this one is precisely that it was not well received, since I somewhat take for granted the values of other highly applauded work by Xiao-Gang Wen.  

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The background of the paper:

A well defined quantum system (on a space of a finite volume) is described by (1) a finite dimensional Hilbert space and (2) a local Hamiltonian. People try to define the 123 standard model as a quantum system defined above, but all previous attempts fail. So the 123 standard model may not satisfy the above two conditions, and thus it may not even be a well defined quantum system. The standard model is only known to be defined via a perturbative expansion, which is known to be divergent. This is the famous chiral fermion problem.

The result:

In this paper, we solved the chiral fermion problem. We show that a modified standard model with 16 Weyl fermions per family can be well defined as a finite quantum system (ie satisfy (1) and (2)). This result is obtained by simply allowing fermions in lattice gauge model to directly interact. We show that such interacting lattice gauge model can produce the modified standard model at low energies.

The precise statement of results:

A chiral fermion theory in d-dimensional space-time with a gauge group $G$ can appear as the low energy effective theory of a well defined lattice quantum system with a finite degrees of freedom per site if (1) there exist (possibly symmetry breaking) mass terms that make all the fermions massive, and (2) $\pi_n (G/G_{grnd} ) = 0$ for $n \leq d + 1$, where $G_{grnd}$ is the unbroken symmetry group.

An application of the above result: The SO(10) chiral fermion theory in the SO(10) grand unification can appear as a low energy effective theory of a lattice gauge model in 3D space with a continuous time, which has a finite number of degrees of freedom per site.

The logic flow:

(a) A chiral fermion theory can be gapped without breaking the symmetry and without ground state degeneracy

(b) A chiral fermion theory is free of all anomalies.

(c) A chiral fermion theory can be non-perturbatively defined (ie put on a lattice of the same dimension), without breaking the symmetry.

Logic relations: (a) <-> (b) <-> (c). In this paper, we show (a)  for a particular model which imply (b) and (c) for that model.

Remarks to answer some comments:

The above results are motivated by the previous long papers on SPT, gauge anomalies, and group cohomology. But this paper presents a self contained arguement, independent of group cohomology approach. In other words, to understand the big picture and the background, one needs to read the previous long papers. But to just understand the above two results, this paper is self contained, and enough.

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@Xiao-GangWen The thing you say above is clear, this is not the part that was confusing when sitting to write a review. The thing that makes the paper very difficult to evaluate with confidence is the reliance on a much longer (very interesting) previous paper of yours that gave the general classification of anomalies, including global anomalies, using several completely distinct and (for me) sophisticated homology constructions, not all of which I was able to understand.

I voted up the accuracy first and foremost because what you are saying is for sure possible, and necessary, 't Hooft gave the unrealistic construction which does it which was referenced here somewhere, by interpolating the lattice fields into a continuum field and then defining the anomaly cancelling Fermions' determinants using the interpolating continuum field. This is an easy but practically useless construction which definitely works, because one can understand it very easily, and your construction seems to be the correct lattice version, but it's harder to understand. Your construction looked more and more correct the more I looked at it. But I had a million questions regarding the homology in the longer paper, I wasn't sure if it was exhaustive, and the analysis in the new paper wasn't 100% self-contained because of this, and it just left questions.

If you don't mind, at some point, could you answer some elementary questions regarding the longer paper in chat? This work, I believe is extremely important.

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I wonder also whether this still makes sense in light of all the ``beyond" group cohomology phases we now know about ( and for which the classification has recently been somewhat understood ;) ).

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@RonMaimon    I will be very happy to answer questions.

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@RyanThorngren   I updated my comment to address your concern.

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@Xiao-GangWen: I think I asked the question to you once but I still can't get my head around it: why is condition (1)  "a finite dimensional Hilbert space" nontrivial?  I mean. if you start with a finite lattice and consider only fermions, how can you not end up with a finite dimensional Hilbert space?

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@Jia Yiyang: Many field theories give rise to infinite dimensional Hilbert space even when the total volume of space is finite. Even for lattice, you might put an infinite dimensional Hilbert space per site. Condition (1) rule out this situation.

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@Xiao-GangWen: Not sure if we are on the same page: whenever I see "fermions on lattice", I think about lattice fermions, i.e. finite difference operators, and on finite-volume lattice, they are just finite dimensional matrices, how can we get a infinite-dim vector space out of them?(by the way, to properly @ a user, you need to eliminate all the spaces in the user's name)

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@Jia Yiyang: One might put an inifnite number of orbitals per site. Condition (1) rule out this situation.

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@Xiao-GangWen, I fail to understand what you meant by putting orbitals on a site. It sounds like you are not using conventional lattice fermion(difference operator) construction, can you elaborate how are you defining fermion on lattice， given a particular Hamiltonian/Lagrangian? If you want, I can start a new post for this question.