It turns out that yesterday there was another AI result coming out of OpenAI, quite different than the slopocalypse and of more interest to fundamental physics. Nathaniel Craig, a particle theorist now on a one-year visiting position at OpenAI, put on the arXiv a preprint entitled A gauge-invariant measure for lattice chiral fermions. This was produced with heavy use of OpenAI’s latest model, GPT-6 Astra:
OpenAI’s GPT-6 Astra model was essential to this result. I proposed a strategy to extend Lüscher’s abelian result using refinement methods that underwent several rounds of iteration with Astra and outside experts before arriving at the current Astra-led proof. The manuscript combines author-written sections with extensive rewrites of the Astra-composed proof. My colleagues and I proposed tests of the method, including Lean verification, that were carried out by Astra; supplementary materials documenting some of these checks were written by Astra.
Some observations about the manuscript, including ways this is different than the products of yesterday’s math dump:
- A great deal more human effort went into writing this. It’s 123 pages long and large sections are similar to the pure Astra math manuscripts, but also clearly have gone through some editing to make them more readable by human beings. Other large sections are obviously written largely by Craig: they’re very different in style than the Astra ones, comparable to a usual high-quality human written HEP theory paper.
- As far as I know, Craig has never before written a paper about lattice gauge theory. That he has identified an important problem outside his usual range of expertise and chosen to try to use AI to provide some of the replacement expertise is quite interesting. Will other physicists and mathematicians try to break out of the constraints of their current expertise using AI in this way?
- Craig is not at all a mathematician or mathematical physicist. In this manuscript, Astra is playing the role of a mathematician collaborator, producing large amounts of rigorous argument, with proofs, even formalizable in Lean. This is another interesting choice: will theoretical physicists in the future adopt usage of an AI agent to add this sort of material to their papers?
As background for those not much aware of the underlying problem, here’s a brief summary. In order to make the Standard Model mathematically well-defined non-perturbatively, one needs to do some sort of regularization, then study the limit as the regulator is removed. The most successful version of this is lattice gauge theory, which provides a beautiful and simple gauge invariant regularization of the gauge field degrees of freedom. This does a great job in allowing non-perturbative calculations for the strong interaction sector of the Standard Model
Introducing fermionic spinor degrees of freedom is much trickier, with the nature of the Dirac equation causing difficult problems with regularizing the theory. These can be handled in various ways when the gauge field interactions are non-chiral (as in the EM and strong sectors), but chiral interactions such as those of the electroweak sector introduce new problems of principle. We’ve been for fifty years now in a situation where in principle (and often in practice), the lattice provides a way to consistently do non-perturbative calculations in the EM and strong sectors, but not in the electroweak sector.
There have been a bafflingly large number of proposals about how to address this (Craig lists nearly 60 of them as references 25-84). He’s chosen one particular one, due to Martin Lüscher, who worked out the details for the abelian gauge field case back in 1998. With Astra’s help he’s extending this to some non-abelian cases including the Standard Model case. The only references I see for anyone else trying to do this in the past 25 years are three papers by Kikukawa and Kadoh/Kikukawa.
I haven’t tried to follow the details of the paper, especially not the complicated ones provided by Astra. The results of this work are a modest advance over Lüscher, still a long ways from providing a full definition in principle of a theory with the desired features in a continuum limit. As far as I can tell, this is not something that will allow actual non-perturbative electroweak calculations in practice.
It’s very interesting to see this sort of human/AI collaboration on an important question in fundamental theory. I’m afraid though that it illustrates the basic problem that afflicts attempts to use AI to make progress on such deep issues in HEP theory. What Craig is doing is following a program for how to solve this problem that started in the 1980s, was developed further by a small number of researchers in the 1990s, then largely abandoned as unpromising. Bringing AI firepower to bear on pushing forward a complicated proposal that was stalled for good reason isn’t likely to lead anywhere interesting.
Personally I believe that our lack of understanding of the mysteries of non-perturbative chiral spinor gauge theories is at the heart of the decades-long stagnation of the field. Changing this situation though will require something much deeper and more insightful than the program studied in this paper. Helping with that appears at the moment to be something far beyond the capabilities of current AI frontier models.
There is in fact some overlap between Luscher’s construction and some of the issues you’ve been discussing on the blog. The key idea behind the construction is to let $\psi$ and $\bar\psi$ transform differently. This allows one to escape the Nielsen-Ninomiya theorem, but, at least in the presence of gauge fields, makes the theory non reflection positive. This is not necessarily the end of the world, as for example some improvement schemes also have the same drawback and it’s not a problem in practice. One can hope for example that reflection positivity is recovered in the continuum limit. But it does seem that breaking reflection positivity explicitly is important to avoid no-go theorems in the Hamiltonian picture.
An even more puzzling aspect of this story is that free overlap fermions (no gauge fields) are in fact reflection positive (arXiv:1005.3751). On the other hand, they are not ultralocal and it looks hopeless to try to obtain a transfer matrix. Then what is the corresponding Hamiltonian theory? Or perhaps, some other subtlety in the reconstruction theorem, such as some of the issues you’ve been discussing, prevents its application to this case?
For those of us who haven’t been following this topic, can you explain why this construction doesn’t by itself define a theory with the desired properties in the continuum limit? What are the remaining gaps that would need to be filled?
4gravitons,
You don’t even know that the continuum limit has the properties that you want for pure Yang-Mills.
For some comments about finite lattice spacing problems, see Alex above or Section 13 of the paper. Maybe someone expert in these issues (not me…) can comment on what this leaves open (as well as what the open issues are in the many other approaches).
My main worry about this kind of construction is that, whether or not it solves problems of principle, it doesn’t seem to provide a way forward to actually do calculations. In pure gauge theory (and for non-chiral fermions) we don’t have rigorous control of the continuum limit, but we can do calculations and try and approach the continuum limit numerically.
Relatedly, it looks like the recent physics –> openAI squad are co-opting the KITP for an entire week: https://www.kitp.ucsb.edu/activities/physicsai-m26
That’s a take, anonymous commentator.