Allvar Gullstrand was an exceptional talent, equally adept in medicine (which he had trained in) and in math (which he learned on his own). With this rare expertise, he made important contributions to both ophthalmology and to physics.

In ophthalmology, he mathematically explained how the eye acts as an optical system and produces images. He studied astigmatism, accommodation, and aberration. The instruments he designed—slit lamp1 and reflex-free ophthalmoscope—changed the practice of ophthalmology forever. In physics, the well-known formula for the effective power of two combined lenses bears his name. For these contributions, Gullstrand had the rare honor of being the frontrunner for not one but two Nobel prizes—for physics and medicine. He went on to win the Nobel Prize in Physiology or Medicine in 1911.2

The year Gullstrand won his nobel prize, Einstein was deep in the development of general relativity, which would be complete in 1915. He had already discovered special relativity and photoelectric effect, and was a strong contender for the Nobel prize himself. But Gullstrand stood in the way.

The great optical expert never saw the appeal of Einstein’s relativity. He took a strong dislike to it. This would not have mattered ordinarily, but Gullstrand also happened to be a member (and later chair) of the Nobel committee from 1911 onwards. Einstein’s name returned again and again to Stockholm. Gullstrand was one of those who kept shooting it down.

In the initial years, perhaps he had a point. General relativity was still to pass the test of experimental confirmation that the Nobel prize required. But after the the 1919 eclipse observations confirming Einstein’s prediction of bending of light by the Sun, it was becoming impossible to deny Einstein. Still, Gullstrand remained adamant.

In 1921, the Nobel committee asked him to report on relativity. Gullstrand’s report was both highly negative and full of errors. He misunderstood the theory and dismissed the experimental confirmation. The optical lens expert had failed to appreciate the lensing effect of gravity. He also argued that Einstein’s theory had not yet met the standard required by Nobel’s will that the prize reward work of clear benefit to humanity. Because of Gullstrand and a few other members’ opposition to the most nominated candidate, the Nobel committee reached a stalemate and no one was awarded that year.

The next year, the committee solved the problem by giving Einstein the reserved 1921 prize for the photoelectric effect. Relativity, Einsten’s greatest contribution, was absent from the Nobel citation. Gullstrand was not the only one responsible, but he certainly played a major role.

There is an ironic twist in Gullstrand’s tale.

Following his report on relativity, Gullstrand was convinced that he could show that general relativity was unreliable, that its solutions depended too much on arbitrary choices of coordinates.

His chosen target was the Schwarzschild solution. First discovered by Karl Schwarzschild almost immediately after Einstein completed general relativity, it described the gravitational field outside a spherical, non-rotating mass. The solution had a problem. In the common form of the Schwarzschild metric, the mathematics misbehaved at a certain radius. Today, we know that “problematic” solution represents a black hole and call that surface the event horizon. But at the time, it was widely believed (including by Einstein) that the misbehaving maths signaled physical impossibility.

Gullstrand thought this vulnerability could be used against Einstein. He looked for a solution in a different coordinate system. If he could show that Schwarzschild’s form had a singularity at the horizon, and if another form did not, then perhaps the theory lacked physical grip.

Gullstrand found a solution in a different coordinate system, adjusted to an observer freely falling towards the black hole. His solution showed no singularities. Gullstrand took this as evidence against Einstein’s theory—it seemed to give different results for the same problem depending on the coordinate system used.

As it happened, this was exactly the opposite of the right interpretation. Gullstrand’s coordinates were related to Schwarzschild’s by a change of coordinates, under which the solutions matched. It was a case of different maps for the same territory. What Gullstrand had discovered was a new map, different from Schwarzschild’s. But they both described the same territory—gravitational field of a spherical object.

Gullstrand’s discovery carried an important lesson, one that neither him nor most of the experts of the time appreciated. But later relativists did. The fact that the solution, expressed in his coordinates, was singularity-free was significant. It showed that the problem was not with the territory, but with the map. There was nothing physically wrong with the event horizon, it was just that Schwarzschild’s coordinates had a bug there. Gullstrand’s result made it clear that the inward-falling observer could pass through the event horizon just fine. It was one of the key lessons towards establishing the reality of black holes.

Gullstrand had not been the first to those coordinates. Paul Painlevé had discovered the same coordinates in 1921. Gullstrand found it independently in 1922. Now they are called Painlevé-Gullstrand coordinates. They are taught in textbooks of general relativity and crop up regularly in papers on black holes, for which they continue to be very useful.

Perhaps the moral of the story is that if you are going to be a hater, commit to it like Gullstrand. Then even if you are wrong, you might end up leaving an important contribution. Although I am not altogether sure if the notoriously arrgant Gullstrand would have appreciated the twist of fate that linked his name forever to the theory he despised.

When I was just starting out in research, an unexpected experimental result appeared and turned physics upside down. In 2011, the OPERA collaboration reported that neutrinos fired from CERN appeared to arrive at Gran Sasso faster than light. For a few electrifying weeks, the physics world held its breath. Headlines screamed of Einstein being overturned. ArXiv, the repository of scientific preprints, was overflowing as paper after paper appeared claiming to explain this apparent lawless behavior of neutrinos. Coffee lounges at research institutions were abuzz with wild theories.

The excitement, alas, did not last. The result was eventually traced to a loose cable and a timing calibration error, putting a dampener on any dreams of achieving faster-than-light travel. But it left behind an interesting question. Would faster-than-light (FTL) signals really have meant the end of relativity?

What textbooks will tell you

Open almost any relativity textbook and you’ll find the classic argument against FTL motion. Imagine an FTL bullet. Then in certain reference frames, observers will see the target struck before the gun fires. That is, they will see effect before cause! Often, this argument is used to pronounce causality (i.e. cause-before-effect) to be incompatible with special relativity.

Is that a paradox, though? Not really. Seeing events in a different order does not automatically create a contradiction. To break physics, you need causal loops to form.

Imagine that A hits B with a FTL bullet. Angered, B fires back with a FTL bullet of their own and hits A. Depending on the relative speeds of A and B, it can so happen that B’s bullet hits A before A even drew their gun. That is a causal loop.

Causal loops raise all kinds of paradoxes, like the one above. A popular one is the “Grandfather paradox”—what if you went back in time and killed your own grandfather? These paradoxes make FTL sound like a doomed idea.

But what if one could have faster-than-light travel without creating causal loops?

How to break the speed barrier without breaking physics

While implicit in some earlier projects, the possibility of FTL without loops was first stated explicitly in a 2010 paper by Robert Geroch.

The key idea is simple. Causality holds if the past completely determines the future. This automatically rules out causal loops1. If you can show that your theory has the property that the past fully fixes the future, then you need not worry about causality violations anymore, even if your theory admits FTL signaling. In chatgptesque language: causality isn’t about speeds, it’s about the tyranny of the past on the future.

But what ensures such tyranny? The answer is in the equations that govern physical systems. Fields can vary with both space and time, so the laws governing them take the form of partial differential equations (PDEs). These equations tell us how fields evolve: how electromagnetic waves move, how fluids flow or spacetime itself bends.

Geroch shows that if the system is governed by a set of first-order PDEs that satisfy the technical condition of being “symmetric-hyperbolic”, causality issues are banished. This condition ensures two related things:

1. Initial data — a snapshot of the system at one moment — fully determines the future evolution. In math terms, a well-defined initial value formulation exists. In plain English, the past pwns the future.

2. Influences propagate at a finite speed. Just like the lightcone, these influences define their own causal cone.

Systems with these properties aren’t exotic—all actual physical systems like Maxwell’s equation for electromagnetism, Dirac’s equation for fermions etc. have this property. It is just that their causal cone is the same as the light-cone. But nothing prevents a causality-respecting system to have a different causal cone.

Democracy of Cones

A different causal cone is not a novel concept for physicists. Sound waves in a fluid propagate at the speed of sound, forming a “sound cone.” It is just that this cone typically lies inside the light cone since sound is slower than light. But mathematically, nothing prevents a system from having a wider cone.

Each physical system is free to carry its own causal structure, determined by its equations. The light-cone need not be the privileged cone. Geroch calls this “democracy of cones.”

But what if you combine systems — say, electromagnetism plus a hypothetical superluminal fluid? If the partial differential equations for the combined system can be cast into a symmetric hyperbolic form, no problem. Causality will work fine. The overall causal cone will become the combination of the individual ones.

Revisiting the grandfather paradox

So what about the grandfather paradox?

On the one hand, no causal loops means no grandfather paradox.

On the other: think back to our thought experiment of the pistol duel with superluminal bullets. Surely, if superluminal speeds are allowed, one can recreate that paradox?

The answer is again in the tyranny of hyperbolic equations. You are allowed to specify initial data at a moment of time, and the equations determine everything else. You could set up your initial condition with A shooting a bullet in the general direction of B, but that’s where your control stops and equations take over.

Since causal loops are incompatible with the equations, they will never form. Something will necessarily prevent B from shooting A back in time. We can’t say what unless we run the equations, but it has to happen.

The Takeaway

There are proposals which explicitly realize multiple causal cones, such as various bimetric gravity theories. But as of now, they remain speculative. Current evidence strongly supports the light cone as the ultimate causal boundary in nature. General relativity and quantum field theory are built around that assumption, and countless experiments confirm it.

What Geroch’s argument shows is that causality is not tied to light-cones but hard-coded into the equations that govern the system. In principle, different systems having different causal cones. If the Opera anomaly had not turned out to be a false alarm, it would not have broken physics. The universe may be more interesting than we imagine.

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:

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:

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:

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:

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:

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:

The condition for foliation dependence becomes:

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):

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:

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!

There was something decidedly odd about the box. For reasons I could not understand, it reminded me of a cat I used to have back in Osaka, a placid ginger tom whose sole notable trait was a preternatural ability to vanish. It was in one of my hunts for the cat that I first met Kimiko. I had followed a hunch to a small Italian restaurant off a side street, a place that smelled of old espresso and regret. And there she was, alone, folding linen napkins into severe, geometric shapes. The cat was nestled on her lap, soundly asleep.

I wanted to check what was inside the box, but the idea filled me with unknown anxiety. In my mind, the box had become entwined with Kimiko leaving. Somehow, I knew that the possibility of her return depended on what was inside the box

I was at a laundromat, thinking about the cat, the box and Kimiko when I first met her.

“You have the cat-box,” she said, not looking up from the shirt she was folding.

“I have a box,” I replied.

She shook her head slowly, as if I were a child who’d mispronounced a simple word. “Not a box. You have the box.” She left the shirt in a heap, and walked out into the rain without a coat. I took the shirt home, not knowing what else to do.

I returned to the apartment and thought about Kimiko. I thought about the precise, ceremonial way she folded my shirts while listening to Ella Fitzgerald. I thought about how inconsolable she was when my old ginger cat, whom she had named Schrödinger, disappeared for the last time. I could not shake off the feeling that if I opened the box and found it empty, Kimiko would never return.

The laundromat lady appeared again at the convenience store. She was holding a carton of milk to her ear, gently shaking it as if listening for a distant tide. “It’s not fresh,” she announced, to no one in particular, then placed it back. Her eyes met mine. She seemed to have been expecting me. “We need to talk,” she said, “You should make spaghetti tonight.” She left without waiting for my response.

I made spaghetti for dinner, as I was told. I filled a pot with water and watched it go from still to simmer to boil. *Waltz for Debby* played in the background. As Bill Evans’s piano notes filled the kitchen, I remembered the first time I took Kimiko for a dance in a club in Osaka.

She appeared in my doorway just as the spaghetti was ready.

“How did you know about the cat box?” I asked her over a glass of pinot noir that tasted of cherries.

“I dreamt of it,” She said. “I can access the other branches,” she added, as if that explained everything.
“Anyway, why didn’t you open it yet?”

“I am worried that if I find nothing Kimiko will never return.” I replied.

She tilted her head and considered my response for a moment.

“You know, in most branches, she does not return. There are some where you become a famous author,” She said, “But in all the branches, you and Kimiko meet the exact same way, through the cat. ”

“What happens in this branch?” I asked.

“The box is a choice.” She said, and left.

That night, I went to the box. I thought about the multiple branches. I thought not of Kimiko, but of all the Kimikos—the one who left, the one who might have stayed in Osaka, the one who was maybe, right now, listening to Ella Fitzgerald and folding someone’s shirts. Which branch was I in? Which branch did I want to be in?

I opened the lid.

Inside, on a bed of yesterday’s newsprint, sat a small, breathing, ginger kitten. A surge of relief washed through me. Then I saw the note next to it, written in a familiar handwriting:

“His name is Schrödinger. Bring him when you find me. —K.”

But if you only know of the multiverse from Marvel movies, it will come as a surprise that there isn’t just one multiverse in physics. When physicists speak of the multiverse, they might be talking about one of three completely different concepts.

Let’s meet the Multiverses.

The Quantum Multiverse

Quantum Mechanics is notoriously cagey. It refuses to give definite answers till a measurement is made. Before you make your observation, be it on a radioactive atom, a spinning electron or Schrödinger’s cat, nature exists in a superposition of possible outcomes. The Cat is both dead and alive.

When you make your measurement, you get only one outcome. The cat either comes out purring, or it has joined the choir invisible and become an ex-cat. The orthodox interpretation of quantum mechanics says that the superposition collapses into a single state—the one we observe. But what if it did not?

In 1957, physicist Hugh Everett III1 came up with a brilliant new way of looking at quantum mechanics. He proposed that superpositions don’t collapse, they just pull the observers themselves into the superposition. After Schrödinger opens the box to check on his cat, reality continues to exist in a superposition of the two outcomes: one with Schrödinger playing with his cat and the other with Schrödinger writing its obit.

Everett’s radical unorthodoxy led to what is now known as the Many Worlds Interpretation (MWI). In this interpretation, observations do not choose a possibility out of many. The wavefunction never collapses; it just proliferates into multiple branches. It’s the ultimate choose-your-adventure show where every possible ending happens simultaneously.

When movies and shows imagine forking timelines and infinite doppelgängers, it is the MWI that they are inspired by. Where fiction differs from fact: it is impossible to detect the other branches, much less go on adventures with your multiversal counterparts. After all, the MWI is an interpretation of quantum mechanics, not a separate theory with its own predictions.

The Cosmic Multiverse

This version of the multiverse has nothing to do with infinite copies of you, and everything to do with the infancy of the universe. The theory of inflation suggests that the infant universe had a growth spurt to end all growth spurts: stretching space flat like a pancake. But here’s the kicker: inflation might not have stopped everywhere at once.

As per a large class of inflationary models, inflation is eternal once it starts. Quantum jitters keep it rolling in some patches, creating a never-ending churn of “bubble universes.” Self-contained cosmoses with their own cosmic histories, bubble universes are separated by horizons across which no light can ever pass. Our visible universe would be just one bubble in this infinite champagne.

The cosmic multiverse is neither an interpretation of a known theory nor fanciful speculation, it arises naturally from many versions of inflation. And while inflation remains to be proven beyond reasonable doubt, it is still the theory that best explains our cosmological observations.

If the prevailing models of inflation get it right, the cosmic multiverse is very likely real. But just like in the quantum multiverse although for totally different reasons, it is impossible to detect the existence of our cousin universes2.

The String Multiverse

If the cosmic multiverse expands space itself, the string multiverse multiplies the possible laws that govern it.

String theory is the leading candidate for a Theory of Everything: the fundamental theory that unifies all interactions. In the early days of string theory, it was expected that our observable universe would pop out of the equations of string theory. That expectation hit a hitch when it was discovered that the theory’s equations admit an enormous number of stable configurations.

Each corresponds to a different way of curling up extra dimensions and threading them with fields, leading to different effective laws of physics. Each could embody a distinct set of physical constants. Gravity might be stronger in one, electromagnetism weaker in another.

This is the string landscape. While the recent Swampland program has been pruning the landscape out of candidates that don’t fit with quantum gravity, it is still an unreasonably large number.

Combining the string landscape with eternal inflation introduces a new twist: cosmic vacuum bubbles that each stabilize into a different string vacuum. In this picture, the multiverse becomes populated with regions that realize distinct string vacua—each with its own physical laws, constants, and particles. That’s the string landscape multiverse.
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Three concepts, all with the moniker “multiverse.” The first multiplies outcomes, the second space, the third laws. Together, they hint that reality may be far larger and stranger than the universe we see.

We heard the music of black holes for the first time when they gently brushed against the sensors of LIGO observatory ten years ago. The prediction of that sound, though, goes back decades. In 1969, Indian physicist C.V. Vishveshwara first uncovered black holes’ hidden talent for percussion.

Vishveshwara was a PhD student at Maryland university at the time. His thesis problem (assigned by his adviser Charles Misner) was to figure out if you can destroy a black hole by striking them hard enough. As is typical of physicists, he started by asking the simplest version of the question—what happens when you hit a black hole really softly, say by a light object plunging into it?

Visheveshwara discovered that a black hole thus pinged doesn’t just distort silently—it radiates gravitational waves with a unique, telltale pattern. Almost like a struck drum, except it radiates gravitational and not sound waves.

I say almost, because there is a subtle but important difference. If there was no damping due to air resistance or other effects, a struck drum (or bell, or guitar string) would ring on forever. In physics terms, pure notes that would ideally ring on till eternity are called “normal modes.” The black hole’s beats are quasinormal modes.

Quasinormal modes are notes that are born to fade. Unlike the everlasting tones of normal modes, they have built-in decays. They die out as the black hole stabilizes. The black hole is like a drum submerged in molasses—it gives a few, definitive *boings* before being swallowed by silence. It rings down.

The Black Hole Beatniks

The next breakthrough came from a young South African prodigy, Saul Teukolsky. A non-spinning black hole, as Visheveshwara had considered, was already tricky enough. But real black holes spin. That makes the maths nightmarishly complicated.

Teukolsky pulled it off. He boiled all the complexity down to a single master equation—the Teukolsky equation. Physicists now had a way to predict the ringdown of a rotating black hole, no matter how wild the spin.

Then, in the late 1970s, Subrahmanyan Chandrasekhar1, already a legend in astrophysics, turned his formidable mathematical mind to black hole perturbations. He systematized and refined the mathematics, bringing out the deep symmetry in the equations (often working with Detweiler and others). This culminated in his magnum opus, The Mathematical Theory of Black Holes, which quickly became the bible of the field.2

The First Record

For decades, this was all theoretical brilliance. We could read the score, but we had no access to the music itself.

That changed on September 14, 2015. The LIGO observatory finally heard the black hole’s beats.

Two black holes had collided in a galaxy far, far away. The collision sent a ripple through spacetime and LIGO had detected that ripple. The signal started with a rapid *chirp* as the black holes spiraled inward. Then came the violent crash of the two black holes merging into one. Finally, as the newborn, larger black hole stabilized: the ringdown. In that signal’s final moments, physicists identified the clear, fading signature of quasinormal modes. The birth cry of the baby black hole.

It was the direct proof of gravitational waves and the first-ever recording of the phenomenon predicted decades prior. Vishveshwara, who passed away in 2017, lived to witness the symphony of black holes that he had uncovered come to life.

The Beats Whisper Secrets

These cosmic notes are not just interesting theoretically. They are a powerful new tool for black hole spectroscopy. Just as a chemist can identify an element by the light it emits, astronomers can now study the pitch and decay of these gravitational waves to determine the properties of the black hole that made them. Is it massive? Is it spinning wildly? The quasinormal modes reveal all.

They are also a crucial test for Einstein’s theory of general relativity. Does the black hole ring exactly as Einstein’s equations? Or is there a discordant note that points to new physics? Every time we discover a new ringdown, Einstein is put on trial. So far, he keeps getting away with it. But many physicists are hopeful that the music of the black holes may help guide us to a new era of physics.

It’s a joke on how physicists’ reliance on wildly improbable idealizations. From zero friction surfaces to gas molecules that pass through each other, the best known physics models resemble the real world as much as a 5 year old’s crayon painting resembles the Mona Lisa. But wacky though they may sound, idealized models are the lifeblood of physics.

The Model Zoo

Reality is messy. Cows are lumpy, the air is turbulent and quantum fields buzz with incredible complexity. A model is not meant to be a perfect picture of reality—it’s a way to strip the problem down to something tractable.

In this post, we take a walk through some of physicist’s most beloved models and see what makes them so powerful.

The point particle

The first physics model students encounter takes ‘stripping the problem down’ to its limit, modelling objects as a point of the same weight. Taken at face value, the point particle is an absurdity—such an entity would be infinitely dense.

But when it comes to situations when the dimensions of the object are irrelevant, the point particle provides a powerful approximation. Consider, for example, the problem: how long does it take for the earth to complete one turn around the Sun? The Earth, a giant lumpy ball hurtling through space, is pretty far from a point. But approximating it as one returns the time period of 365.26 days. Boom! Ignoring mountains, oceans, and cows (spherical or not) gets us within 0.001% of the right answer. Mind slightly blown?

Of course, one has to know the limits of idealizations. The point particle becomes less and less effective the more and more the shape and size of the object begins to matter. For example, when an object is falling through the air, the air drag depends on its surface area, shape and flow conditions. A point particle would be a terrible approximation for someone falling with a parachute.

The ideal gas

Molecules imagined as non-interacting point particles. Computationally, it’s baby's first gas model. But the payoff is massive: many gases under ordinary pressures and temperatures follow “the ideal gas law” (PV=nRT). From speed of sound to hot air balloons, the ideal gas model explains all.

It begins to fray in situations where interactions and size of molecules becomes important, such as low temperatures and high pressures, or during phase transitions. Even in the places where it fails, ideal gas serves as a reference point: deviations highlight the importance of intermolecular forces. This shows another function of models: to serve as benchmarks.

The Black Body

Every object absorbs some of the light and heat that it receives and radiates back some of it. The black body is an impossible perfect absorber and emitter of heat—it absorbs and emits back all radiation that comes its way. Think of it as a vacuum cleaner for radiation.

The black body accurately describes thermal radiation across a wide range of systems — from incandescent filaments to the cosmic microwave background. Assuming the Sun to be a black body gets the temperature of the Sun to a 0.4% error. That’s the power of a good lie.

Oppenheimer-Snyder model for black hole formation

Models are not just good as first approximations, they also illuminate the essence of processes by cropping away inessential details.

A great example is Oppenheimer-Snyder’s model for black hole formation. They considered the collapse of a perfectly spherical star with no internal forces or pressure. This reduced a complex problem to one that could be solved exactly. In the solution, you could see the story of the star collapsing into a black hole unfold.

The alternate, adding realistic details, would have resulted in complicated equations that could only be worked out numerically. If one did not know what one was looking for, the big picture would be lost in the haze of numerical simulations. But once we know the core narrative, it is much easier to interpret the results.

The Ising model

The GOAT of models. A minimalist grid of spins that can only point up or down, interacting solely with their nearest neighbours.

Born to explain ferromagnets, it ditches realism—no quantum weirdness, no long-distance interactions, no disorder. Yet this minimalist masterpiece nails phase transitions in magnets—even the exact numbers!1 But the Ising model has transcended its origin. Today, this grid of up/down arrows explains everything from stock market crashes to social media trends. GOAT for a reason.

The subtle art of modelling

Coming up with a good model then is as much art as it is science. Pick the wrong simplification? You just erased a hurricane. Modelling the atmosphere as an ideal gas works for many purposes—but ignore moisture content and you miss the latent heat driving weather systems. Treating electrons in a metal as a free gas gives you good conductivity predictions—but without considering electron–phonon interactions, you can’t explain superconductivity.

The tradeoff between realism and insight is a tightrope walk. Too much reality? You are lost in the weeds. Too little? Your model is useless. A good model builder needs to have a keen instinct for which aspects are essential for a specific problem and which can be thrown away. The above models are essential to physics because they tread the path perfectly.

As George Box famously said: all models are wrong, but some are useful. The useful ones are worth their weight in gold. So the next time you hear about a physicist's spherical cow, tip your hat a little. She is not just a joke at physicist’s expense, she is our mascot!

What was the debate around black holes and what was Penrose’s beautiful idea that resolved it?

Black Holes before Penrose

The first discovery of black holes goes back to fifteen years before Penrose was born. In 1916, Karl Schwarzschild had shown that Einstein’s theory of general relativity predicted areas of no-return now known as black holes. Moreover, at the center of the black hole sits a “singularity”—a point where gravity becomes infinitely strong. In 1939, Robert Oppenheimer and Hartland Snyder took the next step and showed that the collapse1 of a spherical star would create a black hole. Penrose would have been eight at the time.

Yet, Penrose’s Nobel citation says that it was "for the discovery that black hole formation is a robust prediction of the general theory of relativity." Didn’t Oppenheimer and Snyder already show that?

The key to this mystery is in the word “robust.” Oppenheimer and Snyder had assumed a perfectly spherical star. But real stars are never that symmetrical. A robust prediction would need to show that black holes could form from any old collapsing star. Most physicists were convinced that they could not.

The opposition to black holes went back to Einstein. From the moment he encountered them in Schwarzschild’s work, Einstein was steadfast in his belief that black holes were mathematical bugs and not physical features. The physics community tended to agree.

For the few black hole pilled physicists who did not, the challenge of proving their existence was enormous. To track the collapse of a realistic star, they would have to solve a set of equations known as Einstein’s equations. No joke even in the idealized “spherical cow” cases, at the time they were impossible to deal with for realistic cases.

The field remained stagnant between Oppenheimer and Snyder’s 1939 result to the 1960s. The conventional methods were failing.

Enter Penrose

Roger Penrose was nothing if not unconventional. Physicist Abhay Ashtekar described his approach: "(Penrose) had this way of dreaming, looking into the future, groping in the dark and coming up with completely unbelievable ideas."

Penrose’s dreamlike approach to physics was based on his unbounded visual imagination. Right from childhood, he had been fascinated by geometry. His diagrams are legendary among physicists. As a PhD student, Penrose had been inspired by the paintings of M.C. Escher to invent his own impossible shape which in turn inspired Escher himself2.

The idea he conjured up while crossing the road in 1964 was stunningly visual. It allowed him to completely bypass the nasty partial differential equations and find a far simpler route to show that black hole formation is inevitable. This was the idea of a trapped surface.

Trapping the Singularity

Imagine you’re standing on the surface of a soap bubble. You fire two flashes of light: one inward, one outward. In ordinary space, the outward flash spreads away from you. Penrose imagined a situation where the pull of gravity is so intense that both flashes get dragged inward. This was his idea of a trapped surface.

Now imagine a star collapsing and shrinking. No matter whether it starts as a perfect sphere or not, as it becomes smaller and smaller, gravity keeps getting stronger and outward flashes pulled more and more inwards. Eventually, gravity will get so intense that both flashes will spread inwards. Even for a generically shaped star, a trapped surface is bound to form at some point in its collapse3.

In his 1965 paper, Penrose showed that once a trapped surface forms (and matter does not do anything too weird), there is no escape from singularities.

To prove this he tracked the rays of light inwards using a tool known as Raychaudhuri’s equation4 and proved that it was inevitable that the trajectory of certain light rays ended abruptly. The implication: they had encountered a singularity5.

Putting it all together, Penrose had shown that general relativity predicted that the formation of a singularity did not need the star to be a perfect sphere to start with, it could happen even for generic stars He had found just the right diagnostic for the singularity—the trapped surface—making it unnecessary to track how the star’s entire journey unfolds. The proof was ironclad. Physicists were finally convinced that black holes must be real. In one stroke, black holes stopped being mathematical curiosities and became unavoidable predictions of Einstein’s theory.

It was incredible triumph of imagination from the man who turned 94 this week. Happy Birthday, Sir Roger!

When protons smash together at the LHC, they produce sprays of new particles—jets, photons, sometimes even a Higgs. To test the Standard Model and hunt for hidden new physics, theorists must predict those outcomes with exquisite precision.

The traditional tool for such computations was Feynman diagrams, invented by Richard Feynman in 1948. At each level of precision (or “order” in perturbation theory), one has to compute a certain number of these Feynman diagrams and add them up. At the time of their invention, the Feynman diagram technique was a revolutionary simplification that was said to have brought quantum field theory “to the masses”. But for making precise predictions at the LHC, the Feynman diagram technique is just too complex.

LHC predictions require calculations at NNLO QCD (“next-to-next-to-leading order” in the strong force). If you try to do these predictions by adding up all Feynman diagrams one by one, the task quickly becomes impossible. The number of diagrams explodes exponentially as you add more particles and higher levels of precision. Even for the simplest processes, we are looking at hundreds of thousands of pieces to add. Despite all the computing power at our disposal, we simply can not compute.

Over the past two decades, physicists have found a radically better way. It grew out of string theory and led to new methods with names like BCFW and CHY. These aren’t speculative additions to the Standard Model; they are a different approach to compute the same physics the LHC measures. Today, they’re a staple behind the state-of-the-art predictions.

Strings, surfaces, and a tidier sum

String theory is a theory of strings where fields emerge as vibrational modes. Like particles, strings too interact with each other and we can compute what happens when strings collide. But there is a difference.

Particles are 0-dimensional, so their trajectory in spacetime (the graph of their location at different times) is a line. Strings are one-dimensional. They sweep out a two-dimensional surface known as a worldsheet.

Why does this help? Because the worldsheet organizes the calculation geometrically. Instead of summing over a number of separate diagrams, you integrate over shapes of one smooth surface. Thus strings provide a different way to manage the same kind of order-by-order computations.

Crucially, there’s a “field-theory limit”: if you let the string become infinitesimally small, you recover ordinary particle physics. That means tools developed in the stringy picture can be ported back to the world of quarks and gluons. And that is exactly what happened.

New methods in town

The two main modern methods of computing scattering amplitudes (i.e what happens particles collide) are named BCFW and CHY. Both grew out of insights from string theory.

BCFW (named for Britto, Cachazo, Feng, and Witten) is a clever, recursive method to compute scattering without drawing all the Feynman diagrams. You slightly “shift” the momenta of two particles into the complex numbers and then use the rules of analyticity and factorization to reconstruct a big amplitude from smaller ones.

The birth of BCFW was inspired by string theorist Edward Witten’s work on twistor strings. Witten’s paper suggested that amplitudes might be far simpler than computing thousands of Feynman-diagrams makes them look. BCFW(which included Witten) made that simplicity concrete and algorithmic.

CHY (for Cachazo, He, and Yuan) presents many particle-physics amplitudes as integrals over points on a sphere, tied together by the “scattering equations”. An astonishing fact is that the same setup yields amplitudes for different theories—Yang–Mills, gravity, and scalar fields. CHY originates from another string variant known as the ambitwistor string. In the ambitwistor string picture, the sphere with marked points is literally the string worldsheet while the scattering equations emerge naturally as well. In short: CHY uses string-style geometry to produce field-theory answers.

To be clear, BCFW and CHY can be derived independently in field theory. They do not need string theory to be proven. But historically, it was stringy ideas that led us to these field theory methods. One needed to look at field theories from a stringy perspective to discover them.

Beyond practicality: a new field reshaping old puzzles

As we already mentioned, these modern amplitude methods are essential for predictions in the LHC. They tame complexity and keep the growth of computational effort manageable, making precision predictions feasible where diagram-by-diagram methods stall.

But the practical payoffs are only part of the story. The methods provide deep insights to structures hidden in quantum field theories which we are only beginning to unravel.

An example is the Parke–Taylor formula, a simple one-line expression for a class of gluon scatterings that required mountains of Feynman diagrams. The contributions of all the diagrams just cancelled in the right way to leave behind a simple formula. No one knew why. Discovered in the 1980s, the Parke-Taylor formula remained a bit of a puzzle (or just an odd coincidence, depending on whom you asked) for decades.

In modern amplitude methods, the Parke-Taylor formula pops out directly. These methods explain why Parke-Taylor is as simple as it is. And not just this one formula, they reveal symmetries and geometric objects hidden within field theories. In the last decade, the scattering amplitudes program has blossomed into one of the most exciting research areas in theoretical high energy physics. We thought we knew quantum field theory, but we are realizing there is a hidden depth to it that we never knew existed.

Final Thoughts

When you read that the LHC confirmed the Standard Model yet again, you can tip your hat to string theory, even if just by a little. Achieving NNLO-level precision would be intractable if we stuck to the old recipe of summing every Feynman diagram. String theory played a key role in the development of BCFW, CHY and related methods that make those predictions possible.

To reiterate, BCFW and CHY stand on their own as derivations within quantum field theory. You don’t need to believe in the existence of vibrating strings to trust their outputs at the LHC. But string-theoretic thinking—worldsheets, twistors, ambitwistors—pointed the way to structures and symmetries that the jungle of Feynman diagrams had kept obscured.

Whether or not strings are the ultimate building blocks of reality, they have already changed how we do particle physics. Both in day-to-day predictions and in our understanding of why scattering amplitudes are as simple and beautiful as they are. I am sure that further study of scattering amplitudes will reveal more startling connections between string and field theories!

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“Have a seat” I told her. But she fidgeted like a cat on a hot tin roof, never staying still at one place.

“I have lost valuable information, Mr. Marlowe,” she said, “at a joint they call the Black Hole. I need you to get it back for me.”

A chill went down my spine like a trickle of liquid Helium.

“Nobody crawls out of there alive. No chance.”

“You can’t be certain. Please Mr. Marlowe, this is a matter of life and death to me.” she said, “I am dead if I lose information.”

There was a real mystery to her.

“Got a name, sister?” I asked.

She tossed her calling card at me and vanished. I did not even see the door close.

The card had two letters: QM.

I staked out the Black Hole joint. Something was fishy. Identical joes in shabby overcoats kept spilling out of the place like rats from a sewer. I collared one.

“Hand me the data, Mac. Who are these pals of yours?”

“Not a byte on me, mister,” he croaked. “Name's Hawking Radiation. We all are.”

He wasn’t lying. He had nothing. None of them did.

Things weren’t adding up. Hawking Radiation mugs were seeping out of the black hole, but not one had got any information.

There was more to this caper than I had expected. I hit the streets, asking around about Hawking.

Back in my alley-side office I barely made it through the door before a fist hit me in the face like a jet from the LHC. I saw supernovae and went down like a dying star.

Two goons stepped from the gloom.

“The boss sends his regards,” one of them growled, “He says some information don’t wanna be found.”

“Hawking’s boys?” I spat blood.

The gorillas grinned. “Boss has a bet riding on information staying lost. So don’t go trying to find any.”

They left me in a heap like a spontaneously collapsed wave-function in GRW theory. I limped to Al’s bar.

Before he bought the bar, Al used to be a gumshoe like me. He listened to my sorry tale and slid me a shot of bourbon.

“I believe QM’s story is correct, but I refuse to believe it’s the complete story.” he said when I finished.

“Why is that?”

“God don’t play dice, Marlowe,” Al grinned, sliding me another shot.

Maybe Al had it right and I was being a sucker. But my gut told me it was QM who was right: the data wasn’t gone. I had a hunch it was being smuggled out bit by bit by those slippery radiation mugs.

It was time to step back into the night. I was gonna tail every last Hawking Radiation till one of them sang.

It was gonna to be a long night.

Fin.

I am going to tell you all about it.

Black Holes from Hot Air

As you all know, water has three possible states: ice, water and vapour. The temperature decides which state water prefers to be in. When a system suddenly prefers a new state and switches to it, like water -> steam it is known as a “phase transition.” A similar phase transition takes place for black holes and hot gases.

It does not happen in the universe we live in, but in a hypothetical universe known as Anti-de Sitter space or AdS in short. The AdS space is built like a box, meaning it reflects heat back and maintains a fixed temperature. This makes thermodynamics in AdS much simpler to study than our expanding universe.

In Anti-de Sitter space, the gravitational field has two stable configurations to choose from: a diffuse hot gas of radiation or a black hole. At low temperatures, hot gas is the preferred state. But if you keep cranking up the heat, a phase transition takes place. There is a critical temperature (which depends on the size of the AdS “box”) at which the hot gas “boils” into a black hole.

The reason for black hole winning at high temperatures is that entropy always wins. Black holes are the pound-for-pound champions of entropy—they pack the most entropy in a given volume of all objects in the universe. Hot air was never going to be a match.

This hot radiation-to-black hole phase transition was first discovered in 1983 by Stephen Hawking and Don Page using a method known as “Euclidean Quantum Gravity” and is now called Hawking-Page transition.

Enter Duality

This is fascinating gravity physics, but an even deeper layer is revealed by “AdS/CFT duality.”

A duality in physics is when you have two seemingly completely different systems which are secretly equivalent to each other. It’s like they are the same story, just with different characters. AdS/CFT, first discovered in 1997 via string theory, is the most well-known example.

According to AdS/CFT, every story in AdS space has a twin story in a different theory called N=4 Super Yang Mills(SYM) theory. The Hawking-Page phase transition is precisely dual to the “Confinement-Deconfinement” phase transition in N=4 SYM

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Million Dollar Quark Soup

Quarks are the tiny particles that make up protons and neutrons. Yet we have never observed an isolated quark. The strong nuclear force keeps them imprisoned within protons and neutrons. This phenomenon is known as confinement. Understanding confinement is one of the deepest puzzles in particle physics.

In fact, it is intimately connected to one of the seven 'Millennium Prize Problems' recognized by the Clay Mathematics Institute that carry a $1 million reward for its solution. The Yang-Mills Mass Gap problem asks why the particles we observe (like protons) have mass, even though their constituent quarks and gluons themselves are massless(No, Higgs does not give them mass). This mass gap is a direct mathematical consequence of confinement. Solve confinement, win a million dollars!

There is a way of breaking quarks out of their proton jails— turn up the heat. At large enough temperatures, protons turn into a hot soup of quarks and gluons, known as the quark-gluon plasma.

This is the confinement-deconfinement phase transition in Quantum Chromodynamics(QCD), the theory that describes the strong nuclear force . Quarks bound in protons/neutrons is the confined phase, the quark gluon soup is the deconfined phase.

A very similar temperature-driven confinement/deconfinement transition also happens in N=4 SYM. So, through the magic of AdS/CFT: Confined SYM (bound states) = Thermal Gas in AdS. Deconfined SYM (hot soup) = Black Hole in AdS. This is not just analogy, it's an exact equivalence in a solvable setting. This is a breathtaking revelation of unity in physics: collapse into a black hole and the liberation of quarks are two faces of the same fundamental process.

What about the Real world?

Cool as the story is, N=4 SYM is not real world QCD. So why care about any of this, since it does not apply to the real world? There is an ongoing area of research where physicists build simplified dual models to approximate aspects of QCD, and this has resulted in some useful insights. But that is not the reason I want to emphasize.

The true power of this story isn't that AdS/CFT solves QCD confinement, but that it is a starting point for understanding the world through the lens of duality. The Hawking-Page / Deconfinement duality is a stunning proof-of-principle that dualities provide powerful and unexpected perspectives on deeply thorny problems like confinement.

The duality between AdS gravity and N=4 SYM is a “spherical cow” toy model for dualities in the real world—one where both sides are simple enough for us to neatly identify dual phenomena and use one to understand the other. The real world is far more complex, but this provides us a blueprint for understanding quark confinement.

Real-world QCD is far messier than its highly symmetric cousin, N=4 SYM. But there are very good reasons to believe that it too has a stringy dual. If we can discover this dual theory, we may find that the imprisonment of quarks translates into a phenomenon that's easier to understand and calculate. Someone might even end up winning the million dollar prize!

It's this dazzling possibility that makes the Hawking-Page transition more than just a cool story.

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I have experienced the pull of this critique myself. When I was entering grad school, Smolin’s “Trouble With Physics” was one of the key reasons I ditched string theory and choose loop quantum gravity instead. Yet here I am today, won over by string theory and defending it.

Critics often point to the fact that string theory does not produce predictions that we know how to test. There's undeniable validity to this concern, and I will deal with string theory and its predictions in another post. For this post, I want to talk about something that these narratives consistently leave out: the remarkable success of string theory in guiding us to profound new physics.

Gravity=Gauge

My first example is gauge1/gravity duality(aka holography aka AdS/CFT). Some of you will have heard of the duality in the context of a holographic universe. In this post, I will not get into its usefulness for quantum gravity, but highlight a simple yet under-emphasized fact.

First, some background. Gauge theories (actually all field theories) come with a parameter -- a "coupling constant" which we will call g which tells us how strongly the fields are interacting. When g is small (aka “weak coupling”), we know how to solve the theory. The physicist’s usual trick of expanding everything in powers of g works perfectly.

When g is large(“strong coupling”), gauge theories turn into a dark forest. We are pretty much lost.

Enter gauge-gravity duality. Derived from string theory by Juan Maldacena in 1998, gauge-gravity duality says that certain gauge theories are secretly equivalent to a gravitational theory in one higher dimension. Moreover, strong coupling in gauge theory maps to weak coupling in gravity.

The duality turns the dark forest into a neat, geometric path.

Example: the KSS viscosity bound popped out of the gravity side, later independently derived in field theory.

Here is what I wanted to highlight: field theorists never figured this path on their own before string theory showed us. Even now, 27 years after the discovery of gauge/gravity duality, we still don't have a first principles proof from inside gauge theory. And we have been studying gauge theories for a very long time now.

It's not just the usefulness of gauge-gravity duality that is impressive, it's the fact that gauge theory secretly contains geometry and we can’t yet carve it out with gauge‑theory tools alone.

Whether you label that math or physics, the takeaway is that well-studied quantum theories still hide deep structure from us. But string theory sees these hidden structures. Without string theory, we would never have discovered gauge-gravity duality.

Gravity=Gauge*Gauge

The strings that string theory describes can be closed or open. In 1986, Kawai, Lewellen & Tye noticed that any closed‑string "scattering amplitude"2 can be written as a product of two open‑string amplitudes. This is now known as the KLT relations.

The KLT formula is nothing more mysterious than the fact that you can have vibrations moving both left and right on a closed string, and each of those behave like vibrations of an open string. From the string POV, it is intuitive.

But if you zoom out i.e look at energies far below the string scale where everyday physics lives, you discover something surprising. In that field‑theory limit, the same trick says that the product of two gauge theory scattering amplitudes, after some reformatting involving swapping color factors, gives a gravitational amplitude. In other words: Gravity ≈ (Gauge Field) × (Gauge Field).

These are the double copy relations, so named in the seminal 2008 paper of Bern, Carrasco and Johansson(BCJ).

The double copy relations are totally wild. It says that you can take results from one theory (gauge) and do some stuff with it and end up with results from a completely different theory (gravity).

Since the BCJ paper, double‑copy has become a highly active field. Many results have been uncovered, going beyond its stringy origins. Maps between classical gravity solutions(like Kerr-Schild) and classical gauge field configurations have been found. As it is easier to compute scattering amplitudes in gauge theories than gravity, double copy even helps crunch numbers for gravitational‑wave signals.

While one can derive the double copy relations within gauge theories no one fully understands why the prescription works so broadly. Why does nature organize itself so that gravity = gauge^2? String theory still gives the only intuitive picture of why it must be true.

Bottom Line

No other candidate for quantum gravity or TOE has led us to previously unknown physics results, let alone results as powerful as gauge/gravity or double copy relations.

This isn’t to claim that the gauge/gravity duality or the double‑copy are smoking guns, but that they are powerful circumstantial evidences for string theory. What they demonstrate is string theory’s ability to see the deep structures hidden wihin the theories that describe real world physics.

Understanding how it achieves this seems a pursuit worth our effort.