AI's First Physics Paper: Slop or Banger?

Can AI crack quantum physics? While multiple mathematicians have written about how AI assistance has been helping them solve problems, little has been heard from physicists. Is AI useless for physics? At least one physicist would disagree.

Physicist Stephen Hsu has published a paper in Physics Letters B whose key idea came from GPT 5. Hsu, who has been commendably transparent about the process, used the generator-verifier approach. In this approach, one AI generates an idea while another critiques it. You let the AIs hash it out for a few loops and then check for convergence. If the AIs come to an agreement, the resulting idea is probably a good one.

Hsu’s paper may be the first in AI-led theoretical physics research, but it certainly won’t be the last. As with all things AI, hype is unavoidable (Hsu’s announcement was retweeted by OpenAI’s Greg Brokman).

So how good is AI as a physicist, at this point? Will it put us out of jobs? Or will it drown us in a sea of slop?

I have gone back and forth on this. My current conclusion: AI has entered its graduate student arc. With careful prompting, it can work through computations and come up with useful ideas. But like most grad students, it still has some way to go before becoming a matured researcher. If you ask it to solve a nontrivial problem, it will give you slop. But with supervision and scrutiny, it can produce impressive results.

This paper shows both sides of AI. The central idea is neat. If one of my grad students came up with it, I would be quite impressed. But there certain issues, as pointed out by Jonathan Oppenheim (You should also read his take on the paper). It also shows, as we will see, AI’s blindspots about the literature.

In this post, I will describe the paper and the critique in detail, for the informed reader to make up their mind. This will get technical fast, so I have added TL/DRs for the non-expert reader.

The Background

We start from the most important equation of quantum mechanics—Schrödinger’s equation. It tells us how the physical “state”¹ of the system (denoted by |Ψ(t)>) changes with time:

\(i\hbar \frac{d}{dt} \lvert \Psi(t) \rangle \;=\; \hat H \,\lvert \Psi(t) \rangle. \)

Here H is known as the Hamiltonian operator². The meaning of this equation is that the Hamiltonian H governs how the state |Ψ(t)> changes with time. A key property of quantum mechanics is that H is a linear operator, i.e. independent of the state |Ψ(t)>.

Back in the 80s, physicist Steven Weinberg tried the intrepid move of modifying quantum mechanics and making it nonlinear. That is, he allowed Hamiltonians that did depend on the state. As far as physics ideas from one of the GOATs go, Weinberg’s nonlinear quantum mechanics was exceptionally short-lived. It took mere months for other physicists to point out fatal flaws. The most striking of the problems, discovered by physicist Nicholas Gisin, showed that such a nonlinear modification would imply faster-than-light transmission of signals.

Decades later, in 2021, Kaplan and Rajendran(K&R) resurrected the idea of a nonlinear quantum mechanics with a new model that claims to avoid causality issues. The idea is to start from quantum field theory, introduce state-dependence while maintaining causality, and then proceed to one particle quantum mechanics. This gives you Hamiltonians of the type:

\(H= \int d^4y \; G_{\text{ret}}(x,y) \;\langle \Psi | \hat O(y) | \Psi \rangle \;\hat P(x)\)

Here G_{ret} is a retarded potential. It vanishes outside the lightcone and ensures no faster-than-light signaling³.

I have not read K&R but thus far, K&R model has not been shown to have problems with causality, at least in print (I found only one quantum foundations paper that discusses this).

This where Hsu’s paper comes in. It tries to produce a criterion that can rule out nonlinear modifications like K&R.

TL/DR: Nonlinear modifications of quantum mechanics are typically ruled out by violations of causality, but a recent model by Kaplan and Rajendran (K&R) has not yet been shown to be guilty of such violations. This paper aims to rule out this (and other) nonlinear models. This is a valid unsolved problem.

The Theory of Tomonaga and Schwinger

The aim of Hsu’s paper paper is to scrutinize state-dependent modifications, but for quantum field theory instead of quantum mechanics. It approaches the problem through a rather unexpected route—a lesser known approach to quantum field theory pioneered by physicists Tomonaga and Schwinger back in the 1950s.

The usual formulation of QFT in flat spacetime is in terms of inertial frames. Inertial frames are one particular way to break up (or “foliate”) space-time into space and time, as shown in the image below:

But even for flat spacetime, we can slice up the lasagna of space-time into weird, arbitrary ways, like in this image:

Can we formulate quantum field theory (QFT) for an arbitrary slicing? That was the question QFT pioneers Tomonaga and Schwinger had independently tried to answer (Dirac too chipped in at a later point), birthing the approach now named after them.

The starting point of this approach is the Tomonaga-Schwinger equation:

\(i\hbar\,\frac{\delta}{\delta \sigma(x)} \bigl|\Psi,\Sigma\bigr\rangle \;=\; \hat H(x)\,\bigl|\Psi,\Sigma\bigr\rangle \)

This looks a lot like the Schrödinger equation above but there are a few differences. First, unlike a particle which is at one point in space, a field is everywhere in space. So the wave function gets replaced by a wave functional⁴ over fields. |Ψ,Σ> is the state or wave functional over fields in some spatial slice Σ. Second, we see that the LHS has a derivative with σ(x) instead of time. Here σ(x) is a deformation of the spatial slice at a point x, in a direction normal to the slice. This is what you would expect for a theory of fields—the state now evolves from one spatial slice to another.

This equation would tell us how to evolve the state once you fix a way to slice up spacetime. But what if we choose to carve up spacetime in a different way? If arbitrary slicings are to be allowed, all results should remain unchanged under any such change of foliation. This is expressed by following condition of foliation independence:

\(\Big[\frac{\delta}{\delta\sigma(x)},\frac{\delta}{\delta\sigma(y)}\Big]=0\)

This is a central principle in the Tomonaga-Schwinger approach. Hsu’s paper takes foliation independence as its starting point.

TL/DR: In its attempt to weed out nonlinear modifications of quantum field theory, the paper adopts a relatively obscure approach to quantum field theory. A key principle of this approach, known as “foliation independence” is taken as the central principle of the paper.

A Concrete Criteria

Now let’s see what changes in nonlinear quantum field theory. Suppose that the time evolution is state-dependent. This is achieved by adding a state-dependent term to the Hamiltonian:

\(i\hbar\,\frac{\delta}{\delta \sigma(x)} \bigl|\Psi,\Sigma\bigr\rangle \;=\; \bigl[\hat H(x) + \hat N_x[\Psi]\bigr] \,\bigl|\Psi,\Sigma\bigr\rangle, \)

Here N_x[Ψ] is the state-dependent modification. It is an operator that itself depends on the state at Σ.

The key result comes from asking: what does the principle of foliation dependence imply for such a theory? It turns out that:

\(\Bigl[ \; \frac{\delta}{\delta \sigma(x)}, \; \frac{\delta}{\delta \sigma(y)} \Bigr] \bigl|\Psi,\Sigma\bigr\rangle \;=\; -\frac{1}{\hbar^2}\Bigl( [\hat H(x) + \hat N_x,\; \hat H(y) + \hat N_y] \;+\; i\hbar \bigl(\delta_y \hat N_x - \delta_x \hat N_y\bigr) \Bigr)\bigl|\Psi,\Sigma\bigr\rangle \)

The condition for foliation dependence becomes:

\( -\frac{1}{\hbar^2}\Bigl( [\hat H(x) + \hat N_x,\; \hat H(y) + \hat N_y] \;+\; i\hbar \bigl(\delta_y \hat N_x - \delta_x \hat N_y\bigr) \Bigr)= 0 \)

They call this the TS integrability criterion (eqn 5 of the paper). This the key criterion they derive in the paper. Any theory that fails to satisfy it is not foliation independent (and therefore destined for the bin).

Cognoscenti would want to know at this point what the weird looking δN terms in the TS criterion mean? The idea is this: when the foliation is changed via a deformation, it changes the state. Changing the state in turn changes the state-dependent operator N. The δ_yN_x terms essentially track how state-dependent operator changes due to a deformation.

Here’s a point that will be important later: to satisfy the TS integrability criterion, you would generically need both the commutator part and the state-dependent derivative part δN to vanish.

TL/DR: By demanding that the principle of foliation independence be satisfied, the paper derives a criterion for nonlinear modifications. This criterion, which they call the TS integrability criterion, is the paper’s novel contribution.

Putting Models to TS Test

With the criterion derived, the next step is to test it. They do this for three nonlinear models (equations 7, 17, 20 of the paper):

\(\hat N^W_x \;= \lambda \langle \Psi | \hat O(x) | \Psi \rangle\; \hat O(x), \)

\(\hat N^2_x[\Psi] \;=\; \int d^3z \; K(x,z)\;\langle \Psi | \hat O(z) | \Psi \rangle \;\hat O(x) \)

\(\hat N^{KR}_x[\Psi] \;=\; \int d^4y \; G_{\text{ret}}(x,y) \;\langle \Psi | \hat O(y) | \Psi \rangle \;\hat P(x), \)

The first one resembles the original model of Weinberg (we will call it the Weinberg model) that was ruled out by Gisin and others. The second one we need not worry about at this point. The third is of the type introduced by K&R.

How does the TS integrability criterion fare against these models? For the second and the third case, the δN terms are non-zero. So these models fall afoul of TS integrability and must be banished. The striking thing is that although K&R’s model is designed to be causal, it still falls to TS integrability/foliation independence. I consider this to be their potentially most important result.

For the Weinberg model, however, the δN terms vanish. Only the commutator terms could be non-zero. The upshot is that TS criterion is satisfied *provided* that all operators at spacelike separation commute with each other. This is known as the microcausality condition in QFT and essentially means no faster than light travel.

The paper argues that microcausality would fail for a state-dependent time evolution and therefore TS integrability still rules the model out. But a conclusive proof was missing⁵. In fact, I don’t think microcausality is violated⁶. It seems to me that TS integrability is unable to rule out the first kind of nonlinear modification, even though it is known to violate causality. We will see shortly where exactly TS fails.

TL/DR: The TS integrability condition is able to rule out certain nonlinear models, most notably the K&R model. This is a novel, non-obvious result and impressive (but wait till the end). However, it does not rule out all nonlinear modifications—some modifications that are known to violate faster-than-light signaling satisfy TS.

What did AI do

How much of this was Hsu, an accomplished and experienced physicist, and how much was AI? According to a picture shared by Hsu, this is what AI did:

Note that Hsu gave AI a pretty big pointer: look at the whole wavefunctional. AI then suggested working in the Tomonaga-Schwinger approach. This was probably not a big stretch, TS being a natural formalism for working with wavefunctionals in quantum field theory.

But taking the foliation independence condition as the starting point and deriving what it says about nonlinear Hamiltonians does not seem trivial to me. No deep, magical insight for sure but still a neat idea to approach the problem. What impressed me most here is that this did not seem like a rehash of a strategy that had been tried before, as one might expect from a llm, because the problem was entirely novel. But I could be wrong.

TL/DR: AI was given a direction to think about and it came up with the idea of taking foliation independence as a starting point and deriving its implication for a nonlinear system (The TS integrability criterion). Its approach seems both clever and genuinely novel (i.e not an obvious rehash of a tried idea) .

So far, so impressive. But let’s look at some of the potential flaws in the paper.

The Inconclusive Proof

We have already encountered this one. The paper claims that the TS integrability criteria does rule out the Weinberg term by arguing that microcausality must fail, but the argument falls short of conclusive. This is a case where the paper makes a claim that it hasn’t satisfactorily proven. The right conclusion would have been to acknowledge the gap in the proof when it comes to ruling out the Weinberg term.

Not sure whether AI takes the blame for this one, but I feel this bit needed a little more work and caution.

TL/DR: One of the claims was not satisfactorily proven (and possibly wrong). Needed a little more caution in the claim.

Nonlinear or Nonlocal?

This crucial observation is due to Jonathan Oppenheim: what the TS integrability criteria catches is non-locality and not nonlinearity. Non-locality is when your Hamiltonian requires information of more than one point. So if we modified the Hamiltonian with a nonlocal but linear term like this one:

\(\hat{N}^{NLo}=\int dx\,K(x)\hat{O}(x)\)

TS integrability criteria would still fail. Likewise, the two nonlinear modifications which fail TS both had some form of nonlocality built in via the integrals. Whereas the one term which had no integral/non-locality anywhere—the Weinberg term—satisfied TS.

I am now going to add some technical pedantry to Jonathan’s astute observation.

We can distinguish between two types of locality⁷: operator locality and what I will call “functional locality”.

Operator locality holds when N_x is local as an operator in Hilbert space i.e its support is confined to a single point x. All three examples of nonlinear N_x that we saw were local as operators, N^{Nlo} is not (since it involves an integration over the operator O(x)).

Functional locality, on the other hand, holds when the coefficient of the operator is a local state-dependent functional. That is, the term in N_x multiplying the operator has to be supported on a single point x. Functional locality holds for N^W and N^{NLO} (in the latter case because the coefficient is not state dependent) but not for the other two examples. For N^2 and N^KR , you can see that the coefficients involve integrals.

TS criteria is tripped up by both these types of nonlocalities, but in different ways. For operator nonlocality, the commutators fail to vanish but the state dependent δN term is zero. For functional nonlocality, the commutators vanish, but δN survives.

The Weinberg term which had neither kind of non-locality satisfies TS integrability (modulo the microcausality issue we discussed).

This does not imply that TS is wrong or useless, just that TS is of limited power as a criterion of covariance/causality—some otherwise pathological theories can still be foliation independent. What TS does catch are generic nonlocalities—whether of the operator type or the functional type.

Imo it is still pretty useful. If you demand that sensible theories follow TS, and rule out nontrivial theories that violate it, that’s still a win. Now theories with operator nonlocality are typically problematic because they generically violate microcausality. But when the nonlocality hides in the state dependence, as in the K&R model, it is not so obvious that it violates a physical principle. TS integrability gives a potential way to rule out such theories and that’s a good thing.

TL/DR: TS integrability seems to catch non localities in the Hamiltonian, whether or not it is nonlinear. For local nonlinear modifications, TS integrability is satisfied. So the criterion can’t rule out all pathological models, but it is still potentially useful in ruling out a class of them.

A flawed starting point

There is another, perhaps more serious, objection. This has to do with the Tomonaga-Schwinger approach itself. Turns out it has been dead for a while.

In 1994, Torre and Varadarajan published a paper that effectively ended the Tomonaga-Schwinger program. They showed that in higher than two dimensions, evolving quantum fields through arbitrary slices simply does not work. The time evolution between different slices does not generally correspond to a unitary transformation in a Fock space. Torre-Varadarajan showed this for the simplest possible theory—free scalar field. Things can only get worse for more complicated fields.

Torre-Varadarajan’s result implies that the Tomonaga-Schwinger equation—the starting point of this paper— is not well defined to begin with. Whether the principle of foliation dependence/ TS integrability can be saved in some way from Torre-Varadarajan, I do not know. It is plausible there’s an algebraic QFT version that survives. If so, some of the results like ruling out the K&R model, would be useful. But if not, the paper is doomed.

The Torre-Varadarajan paper has 100+ citations so it is not unknown, but GPT5 was not not able to connect it with this problem. This is one of the problems with AI in its current state—it can forget the literature unless prompted to check. Like I said, still in its grad student arc.

TL/DR: The Tomonaga-Schwinger approach, on which the paper is based, was shown to fail in 1994. So the paper as it stands is based on a flawed premise and all the results are possibly invalid. The invalidation of Tomonaga-Schwinger is not a well known result, so it is not so surprising that AI missed it.

Final Word

Apart from their contribution to physics, papers like this are valuable in kickstarting a conversation around the strengths and weaknesses of AI in solving physics problems. This paper demonstrated both AI’s usefulness in assisting physicists and its potential pitfalls.

AI may still be a beginning grad student, but this paper shows it to be a promising one. Let’s hope it turns into a mature colleague soon!

1. In quantum mechanics, everything that can be known about a system is encoded in the state.

2. You can think of all the terms in this equation as matrices. Psi is like a row matrix and H is like a square matrix.

3. This is not enough to avoid Gisin-type problems, so they also get rid of collapse.

4. A functional is a function of functions. Since fields are themselves functions of space, a “wave function over fields” will be a functional.

5. One argument is this: Unitary time evolution preserves commutators between local operators:so if microcausality holds at one point of time, it will always hold. But state-dependent time evolution is not unitary and does not necessarily preserve commutators. Hence even if two space like separated operators commute at one point in time, they may not commute at a later point. The argument only shows microcausality *need not* hold. Another argument, which was harder to follow, seemed to show microcausality violation to be a necessary condition for generating instantaneous entanglement.

6. The proof would go like this: we want to compute the commutator [phi(x,0),phi(y,t)]. We can write phi(y,t) as a Dyson series of nested commutators between phi(y,0) and H. Now dividing H into linear and nonlinear parts H_0+N, we can write phi(y,t)=phi_0(y,t)+\delta phi(y,t) where phi_0 is how it would have evolved under H_0 and \delta phi contains all the rest of the terms: a bunch of nested commutators between H_0, N and phi. Substituting back in the commutator, we get (assuming H_0 was causal):[phi(x,0),\delta phi(y,t)]. For local N, I strongly suspect all the commutators between local operators involved in this should vanish.

7. None of these are the same as the type of locality we speak of when we say “quantum mechanics is not local.”