Tag Archives: quantum mechanics

Classical Teleportation Is Easier Than Quantum Teleportation

Quantum teleportation confuses people.

Maybe you’ve heard the buzzword, and you imagine science fiction become reality: teleporting people across the galaxy, or ansibles communicating faster than light. Maybe you’ve heard a bit more, and know that quantum teleportation can’t transfer information faster than light, that it hasn’t been used on something even as complicated as a molecule…and you’re still confused, because if so, why call it teleportation in the first place?

There’s a simple way to clear up this confusion. You just have to realize that classical teleportation is easy.

What do I mean by “classical teleportation”?

Let’s start with the simplest teleporter you could imagine. It scans you on one end, then vaporizes you, and sends your information to a teleportation pad on the other end. The other end uses that information to build a copy of your body from some appropriate raw materials, and there you are!

(If the machine doesn’t vaporize you, then you end up with an army of resurrected Derek Parfits.)

Doing this with a person is, of course, absurdly difficult, and well beyond the reach of current technology.

And no, nothing about the Star Trek version changes that

Do it with a document, though, and you’ve essentially invented the fax machine.

Yes, faxes don’t copy a piece of paper atom by atom, but they don’t need to: they just send what’s written on it. This sort of “classical teleportation” is commonplace. Trade Pokémon, and your Pikachu gets “classical teleported” from one device to another. Send an email, and your laptop teleports it to someone else. The ability to “classically teleport” is essential for computers to function, the idea that you can take the “important information” about something and copy it somewhere else.

Note that under this definition, “classical teleportation” is not faster than light. You still need to send a signal, between a “scanner” and a “printer”, and that’s only as fast as your signal normally is. Note also that the “printer” needs some “ink”, you still need the right materials to build or record whatever is being teleported over.

So suppose you’re building a quantum computer, one that uses the unique properties of quantum mechanics. Naturally, you want to be able to take a quantum state and copy it somewhere else. You need “quantum teleportation”. And the first thing you realize is that it’s harder than it looks.

The problem comes when you try to “scan” your quantum state. You might have heard quantum states described as “inherently uncertain” or “inherently indeterminate”. For this post, a better way to think about them is “inherently unknown”. For any quantum state, there is something you can’t know about its behavior. You can’t know which slit the next electron will go through, you can’t know whether Schrödinger’s cat is alive or dead. If you did, the state wouldn’t be quantum: no matter how you figure it out, there isn’t a way to discover which slit the electron will go through without getting rid of the quantum diffraction pattern.

This means that if you try to just “classically teleport” a quantum state, you lose the very properties you care about. To “scan” your state, you have to figure out everything important about it. The only way to do that, for an arbitrary state on your teleportation pad, is to observe its behavior. If you do that, though, you’ll end up knowing too much: a state whose behavior you know is not a quantum state, and it won’t do what you want it to on the other end. You’ve tried to “clone” it, and there’s a theorem proving you can’t.

(Note that this description should make sense even if you believe in a “hidden variable” interpretation of quantum mechanics. Those hidden variables have to be “non-local”, they aren’t close enough for your “scanner” to measure them.)

Since you can’t “classically teleport” your quantum state, you have to do something more subtle. That’s where “quantum teleportation” comes in. Quantum teleportation uses “entanglement”, long-distance correlations between quantum states. With a set of two entangled states, you can sneak around the “scanning” part, manipulating the states on one end to compute instructions that let someone use the other entangled particle to rebuild the “teleported” state.

Those instructions still have to be transferred normally, once again quantum teleportation isn’t faster than light. You still need the right kind of quantum state at your target, your “printer” still needs ink. What you get, though, is a way to transport the “inherently unknown” behavior of a quantum state, without scanning it and destroying the “mystery”. Quantum teleportation isn’t easier than classical teleportation, it’s harder. What’s exciting is that it’s possible at all.

On an unrelated topic, KKLT have fired back at their critics, with an impressive salvo of papers. (See also this one from the same day.) I don’t have the time or expertise to write a good post about this at the moment, currently hoping someone else does!

Adversarial Collaborations for Physics

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Sometimes physics debates get ugly. For the scientists reading this, imagine your worst opponents. Think of the people who always misinterpret your work while using shoddy arguments to prop up their own, where every question at a talk becomes a screaming match until you just stop going to the same conferences at all.

Now, imagine writing a paper with those people.

Adversarial collaborations, subject of a recent a contest on the blog Slate Star Codex, are a proposed method for resolving scientific debates. Two scientists on opposite sides of an argument commit to writing a paper together, describing the overall state of knowledge on the topic. For the paper to get published, both sides have to sign off on it: they both have to agree that everything in the paper is true. This prevents either side from cheating, or from coming back later with made-up objections: if a point in the paper is wrong, one side or the other is bound to catch it.

This won’t work for the most vicious debates, when one (or both) sides isn’t interested in common ground. But for some ongoing debates in physics, I think this approach could actually help.

One advantage of adversarial collaborations is in preventing accusations of bias. The debate between dark matter and MOND-like proposals is filled with these kinds of accusations: claims that one group or another is ignoring important data, being dishonest about the parameters they need to fit, or applying standards of proof they would never require of their own pet theory. Adversarial collaboration prevents these kinds of accusations: whatever comes out of an adversarial collaboration, both sides would make sure the other side didn’t bias it.

Another advantage of adversarial collaborations is that they make it much harder for one side to move the goalposts, or to accuse the other side of moving the goalposts. From the sidelines, one thing that frustrates me watching string theorists debate whether the theory can describe de Sitter space is that they rarely articulate what it would take to decisively show that a particular model gives rise to de Sitter. Any conclusion of an adversarial collaboration between de Sitter skeptics and optimists would at least guarantee that both parties agreed on the criteria. Similarly, I get the impression that many debates about interpretations of quantum mechanics are bogged down by one side claiming they’ve closed off a loophole with a new experiment, only for the other to claim it wasn’t the loophole they were actually using, something that could be avoided if both sides were involved in the experiment from the beginning.

It’s possible, even likely, that no-one will try adversarial collaboration for these debates. Even if they did, it’s quite possible the collaborations wouldn’t be able to agree on anything! Still, I have to hope that someone takes the plunge and tries writing a paper with their enemies. At minimum, it’ll be an interesting read!

Epistemology, Not Metaphysics, Justifies Experiments

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While I was visiting the IAS a few weeks back, they had a workshop on Quantum Information and Black Holes. I didn’t see many of the talks, but I did get to see Leonard Susskind talk about his new slogan, GR=QM.

For some time now, researchers have been uncovering deep connections between gravity and quantum mechanics. Juan Maldacena jump-started the field with the discovery of AdS/CFT, showing that theories that describe gravity in a particular curved space (Anti-de Sitter, or AdS) are equivalent to non-gravity quantum theories describing the boundary of that space (specifically, Conformal Field Theories, or CFTs). The two theories contain the same information and, with the right “dictionary”, describe the same physics: in our field’s vernacular, they’re dual. Since then, physicists have found broader similarities, situations where properties of quantum mechanics, like entanglement, are closely linked to properties of gravity theories. Maldacena’s ER=EPR may be the most publicized of these, a conjectured equivalence between Einstein-Rosen bridges (colloquially known as wormholes) and entangled pairs of particles (famously characterized by Einstein, Podolsky, and Rosen).

GR=QM is clearly a riff on ER=EPR, but Susskind is making a more radical claim. Based on these developments, including his own work on quantum complexity, Susskind is arguing that the right kind of quantum mechanical system automatically gives rise to quantum gravity. What’s more, he claims that these systems will be available, using quantum computers, within roughly a decade. Within ten years or so, we’ll be able to do quantum gravity experiments.

That sounds ridiculous, until you realize he’s talking about dual theories. What he’s imagining is not an experiment at the absurdly high energies necessary to test quantum gravity, but rather a low-energy quantum mechanics experiment that is equivalent, by something like AdS/CFT, to a quantum gravity experiment.

Most people would think of that as a simulation, not an actual test of quantum gravity. Susskind, though, spends quite a bit of time defending the claim that it really is gravity, that literally GR=QM. His description of clever experiments and overarching physical principles is aimed at piling on evidence for that particular claim.

What do I think? I don’t think it matters much.

The claim Susskind is making is one of metaphysics: the philosophy of which things do and do not “really” exist. Unlike many physicists, I think metaphysics is worth discussing, that there are philosophers who make real progress with it.

But ultimately, Susskind is proposing a set of experiments. And what justifies experiments isn’t metaphysics, it’s epistemology: not what’s “really there”, but what we can learn.

What can we learn from the sorts of experiments Susskind is proposing?

Let’s get this out of the way first: we can’t learn which theory describes quantum gravity in our own world.

That’s because every one of these experiments relies on setting up a quantum system with particular properties. Every time, you’re choosing the “boundary theory”, the quantum mechanical side of GR=QM. Either you choose a theory with a known gravity partner, and you know how the inside should behave, or you choose a theory with an unknown partner. Either way, you have no reason to expect the gravity side to resemble the world we live in.

Plenty of people would get suspicious of Susskind here, and accuse him of trying to mislead people. They’re imagining headlines, “Experiment Proves String Theory”, based on a system intentionally set up to have a string theory dual, a system that can’t actually tell us whether string theory describes the real world.

That’s not where I’m going with this.

The experiments that Susskind is describing can’t prove string theory. But we could still learn something from them.

For one, we could learn whether these pairs of theories really are equivalent. AdS/CFT, ER=EPR, these are conjectures. In some cases, they’re conjectures with very good evidence. But they haven’t been proven, so it’s still possible there’s a problem people overlooked. One of the nice things about experiments and simulations is that they’re very good at exposing problems that were overlooked.

For another, we could get a better idea of how gravity behaves in general. By simulating a wide range of theories, we could look for overarching traits, properties that are common to most gravitational theories. We wouldn’t be sure that those properties hold in our world…but with enough examples, we could get pretty confident. Hopefully, we’d stumble on things that gravity has to do, in order to be gravity.

Susskind is quite capable of making these kinds of arguments, vastly more so than I. So it frustrates me that every time I’ve seen him talk or write about this, he hasn’t. Instead, he keeps framing things in terms of metaphysics, whether quantum mechanics “really is” gravity, whether the experiment “really” explores a wormhole. If he wants to usher in a new age of quantum gravity experiments, not just as a buzzword but as real, useful research, then eventually he’s going to have to stop harping on metaphysics and start talking epistemology. I look forward to when that happens.

The Quantum Kids

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I gave a pair of public talks at the Niels Bohr International Academy this week on “The Quest for Quantum Gravity” as part of their “News from the NBIA” lecture series. The content should be familiar to long-time readers of this blog: I talked about renormalization, and gravitons, and the whole story leading up to them.

(I wanted to title the talk “How I Learned to Stop Worrying and Love Quantum Gravity”, like my blog post, but was told Danes might not get the Doctor Strangelove reference.)

I also managed to work in some history, which made its way into the talk after Poul Damgaard, the director of the NBIA, told me I should ask the Niels Bohr Archive about Gamow’s Thought Experiment Device.

“What’s a Thought Experiment Device?”

This, apparently

If you’ve heard of George Gamow, you’ve probably heard of the Alpher-Bethe-Gamow paper, his work with grad student Ralph Alpher on the origin of atomic elements in the Big Bang, where he added Hans Bethe to the paper purely for an alpha-beta-gamma pun.

As I would learn, Gamow’s sense of humor was prominent quite early on. As a research fellow at the Niels Bohr Institute (essentially a postdoc) he played with Bohr’s kids, drew physics cartoons…and made Thought Experiment Devices. These devices were essentially toy experiments, apparatuses that couldn’t actually work but that symbolized some physical argument. The one I used in my talk, pictured above, commemorated Bohr’s triumph over one of Einstein’s objections to quantum theory.

Learning more about the history of the institute, I kept noticing the young researchers, the postdocs and grad students.

Lev Landau, George Gamow, Edward Teller. The kids are Aage and Ernest Bohr. Picture from the Niels Bohr Archive.

We don’t usually think about historical physicists as grad students. The only exception I can think of is Feynman, with his stories about picking locks at the Manhattan project. But in some sense, Feynman was always a grad student.

This was different. This was Lev Landau, patriarch of Russian physics, crowning name in a dozen fields and author of a series of textbooks of legendary rigor…goofing off with Gamow. This was Edward Teller, father of the Hydrogen Bomb, skiing on the institute lawn.

These were the children of the quantum era. They came of age when the laws of physics were being rewritten, when everything was new. Starting there, they could do anything, from Gamow’s cosmology to Landau’s superconductivity, spinning off whole fields in the new reality.

On one level, I envy them. It’s possible they were the last generation to be on the ground floor of a change quite that vast, a shift that touched all of physics, the opportunity to each become gods of their own academic realms.

I’m glad to know about them too, though, to see them as rambunctious grad students. It’s all too easy to feel like there’s an unbridgeable gap between postdocs and professors, to worry that the only people who make it through seem to have always been professors at heart. Seeing Gamow and Landau and Teller as “quantum kids” dispels that: these are all-too-familiar grad students and postdocs, joking around in all-too-familiar ways, who somehow matured into some of the greatest physicists of their era.

The Way You Think Everything Is Connected Isn’t the Way Everything Is Connected

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I hear it from older people, mostly.

“Oh, I know about quantum physics, it’s about how everything is connected!”

“String theory: that’s the one that says everything is connected, right?”

“Carl Sagan said we are all stardust. So really, everything is connected.”

It makes Connect Four a lot easier anyway

I always cringe a little when I hear this. There’s a misunderstanding here, but it’s not a nice clean one I can clear up in a few sentences. It’s a bunch of interconnected misunderstandings, mixing some real science with a lot of confusion.

To get it out of the way first, no, string theory is not about how “everything is connected”. String theory describes the world in terms of strings, yes, but don’t picture those strings as links connecting distant places: string theory’s proposed strings are very, very short, much smaller than the scales we can investigate with today’s experiments. The reason they’re thought to be strings isn’t because they connect distant things, it’s because it lets them wiggle (counteracting some troublesome wiggles in quantum gravity) and wind (curling up in six extra dimensions in a multitude of ways, giving us what looks like a lot of different particles).

(Also, for technical readers: yes, strings also connect branes, but that’s not the sort of connection these people are talking about.)

What about quantum mechanics?

Here’s where it gets trickier. In quantum mechanics, there’s a phenomenon called entanglement. Entanglement really does connect things in different places…for a very specific definition of “connect”. And there’s a real (but complicated) sense in which these connections end up connecting everything, which you can read about here. There’s even speculation that these sorts of “connections” in some sense give rise to space and time.

You really have to be careful here, though. These are connections of a very specific sort. Specifically, they’re the sort that you can’t do anything through.

Connect two cans with a length of string, and you can send messages between them. Connect two particles with entanglement, though, and you can’t send messages between them…at least not any faster than between two non-entangled particles. Even in a quantum world, physics still respects locality: the principle that you can only affect the world where you are, and that any changes you make can’t travel faster than the speed of light. Ansibles, science-fiction devices that communicate faster than light, can’t actually exist according to our current knowledge.

What kind of connection is entanglement, then? That’s a bit tricky to describe in a short post. One way to think about entanglement is as a connection of logic.

Imagine someone takes a coin and cuts it along the rim into a heads half and a tails half. They put the two halves in two envelopes, and randomly give you one. You don’t know whether you have heads or tails…but you know that if you open your envelope and it shows heads, the other envelope must have tails.

Unless they’re a spy. Then it could contain something else.

Entanglement starts out with connections like that. Instead of a coin, take a particle that isn’t spinning and “split” it into two particles spinning in different directions, “spin up” and “spin down”. Like the coin, the two particles are “logically connected”: you know if one of them is “spin up” the other is “spin down”.

What makes a quantum coin different from a classical coin is that there’s no way to figure out the result in advance. If you watch carefully, you can see which coin gets put in to which envelope, but no matter how carefully you look you can’t predict which particle will be spin up and which will be spin down. There’s no “hidden information” in the quantum case, nowhere nearby you can look to figure it out.

That makes the connection seem a lot weirder than a regular logical connection. It also has slightly different implications, weirdness in how it interacts with the rest of quantum mechanics, things you can exploit in various ways. But none of those ways, none of those connections, allow you to change the world faster than the speed of light. In a way, they’re connecting things in the same sense that “we are all stardust” is connecting things: tied together by logic and cause.

So as long as this is all you mean by “everything is connected” then sure, everything is connected. But often, people seem to mean something else.

Sometimes, they mean something explicitly mystical. They’re people who believe in dowsing rods and astrology, in sympathetic magic, rituals you can do in one place to affect another. There is no support for any of this in physics. Nothing in quantum mechanics, in string theory, or in big bang cosmology has any support for altering the world with the power of your mind alone, or the stars influencing your day to day life. That’s just not the sort of connection we’re talking about.

Sometimes, “everything is connected” means something a bit more loose, the idea that someone’s desires guide their fate, that you could “know” something happened to your kids the instant it happens from miles away. This has the same problem, though, in that it’s imagining connections that let you act faster than light, where people play a special role. And once again, these just aren’t that sort of connection.

Sometimes, finally, it’s entirely poetic. “Everything is connected” might just mean a sense of awe at the deep physics in mundane matter, or a feeling that everyone in the world should get along. That’s fine: if you find inspiration in physics then I’m glad it brings you happiness. But poetry is personal, so don’t expect others to find the same inspiration. Your “everyone is connected” might not be someone else’s.

What’s in a Conjecture? An ER=EPR Example

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A few weeks back, Caltech’s Institute of Quantum Information and Matter released a short film titled Quantum is Calling. It’s the second in what looks like will become a series of pieces featuring Hollywood actors popularizing ideas in physics. The first used the game of Quantum Chess to talk about superposition and entanglement. This one, featuring Zoe Saldana, is about a conjecture by Juan Maldacena and Leonard Susskind called ER=EPR. The conjecture speculates that pairs of entangled particles (as investigated by Einstein, Podolsky, and Rosen) are in some sense secretly connected by wormholes (or Einstein-Rosen bridges).

The film is fun, but I’m not sure ER=EPR is established well enough to deserve this kind of treatment.

At this point, some of you are nodding your heads for the wrong reason. You’re thinking I’m saying this because ER=EPR is a conjecture.

I’m not saying that.

The fact of the matter is, conjectures play a very important role in theoretical physics, and “conjecture” covers a wide range. Some conjectures are supported by incredibly strong evidence, just short of mathematical proof. Others are wild speculations, “wouldn’t it be convenient if…” ER=EPR is, well…somewhere in the middle.

Most popularizers don’t spend much effort distinguishing things in this middle ground. I’d like to talk a bit about the different sorts of evidence conjectures can have, using ER=EPR as an example.

Our friendly neighborhood space octopus

The first level of evidence is motivation.

At its weakest, motivation is the “wouldn’t it be convenient if…” line of reasoning. Some conjectures never get past this point. Hawking’s chronology protection conjecture, for instance, points out that physics (and to some extent logic) has a hard time dealing with time travel, and wouldn’t it be convenient if time travel was impossible?

For ER=EPR, this kind of motivation comes from the black hole firewall paradox. Without going into it in detail, arguments suggested that the event horizons of older black holes would resemble walls of fire, incinerating anything that fell in, in contrast with Einstein’s picture in which passing the horizon has no obvious effect at the time. ER=EPR provides one way to avoid this argument, making event horizons subtle and smooth once more.

Motivation isn’t just “wouldn’t it be convenient if…” though. It can also include stronger arguments: suggestive comparisons that, while they could be coincidental, when put together draw a stronger picture.

In ER=EPR, this comes from certain similarities between the type of wormhole Maldacena and Susskind were considering, and pairs of entangled particles. Both connect two different places, but both do so in an unusually limited way. The wormholes of ER=EPR are non-traversable: you cannot travel through them. Entangled particles can’t be traveled through (as you would expect), but more generally can’t be communicated through: there are theorems to prove it. This is the kind of suggestive similarity that can begin to motivate a conjecture.

(Amusingly, the plot of the film breaks this in both directions. Keanu Reeves can neither steal your cat through a wormhole, nor send you coded messages with entangled particles.)

Nor live forever as the portrait in his attic withers away

Motivation is a good reason to investigate something, but a bad reason to believe it. Luckily, conjectures can have stronger forms of evidence. Many of the strongest conjectures are correspondences, supported by a wealth of non-trivial examples.

In science, the gold standard has always been experimental evidence. There’s a reason for that: when you do an experiment, you’re taking a risk. Doing an experiment gives reality a chance to prove you wrong. In a good experiment (a non-trivial one) the result isn’t obvious from the beginning, so that success or failure tells you something new about the universe.

In theoretical physics, there are things we can’t test with experiments, either because they’re far beyond our capabilities or because the claims are mathematical. Despite this, the overall philosophy of experiments is still relevant, especially when we’re studying a correspondence.

“Correspondence” is a word we use to refer to situations where two different theories are unexpectedly computing the same thing. Often, these are very different theories, living in different dimensions with different sorts of particles. With the right “dictionary”, though, you can translate between them, doing a calculation in one theory that matches a calculation in the other one.

Even when we can’t do non-trivial experiments, then, we can still have non-trivial examples. When the result of a calculation isn’t obvious from the beginning, showing that it matches on both sides of a correspondence takes the same sort of risk as doing an experiment, and gives the same sort of evidence.

Some of the best-supported conjectures in theoretical physics have this form. AdS/CFT is technically a conjecture: a correspondence between string theory in a hyperbola-shaped space and my favorite theory, N=4 super Yang-Mills. Despite being a conjecture, the wealth of nontrivial examples is so strong that it would be extremely surprising if it turned out to be false.

ER=EPR is also a correspondence, between entangled particles on the one hand and wormholes on the other. Does it have nontrivial examples?

Some, but not enough. Originally, it was based on one core example, an entangled state that could be cleanly matched to the simplest wormhole. Now, new examples have been added, covering wormholes with electric fields and higher spins. The full “dictionary” is still unclear, with some pairs of entangled particles being harder to describe in terms of wormholes. So while this kind of evidence is being built, it isn’t as solid as our best conjectures yet.

I’m fine with people popularizing this kind of conjecture. It deserves blog posts and press articles, and it’s a fine idea to have fun with. I wouldn’t be uncomfortable with the Bohemian Gravity guy doing a piece on it, for example. But for the second installment of a star-studded series like the one Caltech is doing…it’s not really there yet, and putting it there gives people the wrong idea.

I hope I’ve given you a better idea of the different types of conjectures, from the most fuzzy to those just shy of certain. I’d like to do this kind of piece more often, though in future I’ll probably stick with topics in my sub-field (where I actually know what I’m talking about 😉 ). If there’s a particular conjecture you’re curious about, ask in the comments!

Have You Given Your Kids “The Talk”?

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If you haven’t seen it yet, I recommend reading this delightful collaboration between Scott Aaronson (of Shtetl-Optimized) and Zach Weinersmith (of Saturday Morning Breakfast Cereal). As explanations of a concept beyond the standard popular accounts go, this one is pretty high quality, correcting some common misconceptions about quantum computing.

I especially liked the following exchange:

ontology

I’ve complained before about people trying to apply ontology to physics, and I think this gets at the root of one of my objections.

People tend to think that the world should be describable with words. From that perspective, mathematics is just a particular tool, a system we’ve created. If you look at the world in that way, mathematics looks unreasonably effective: it’s ability to describe the real world seems like a miraculous coincidence.

Mathematics isn’t just one tool though, or just one system. It’s all of them: not just numbers and equations, but knots and logic and everything else. Deep down, mathematics is just a collection of all the ways we’ve found to state things precisely.

Because of that, it shouldn’t surprise you that we “put complex numbers in our ontologies”. Complex numbers are just one way we’ve found to make precise statements about the world, one that comes in handy when talking about quantum mechanics. There doesn’t need to be a “correct” description in words: the math is already stating things as precisely as we know how.

That doesn’t mean that ontology is a useless project. It’s worthwhile to develop new ways of talking about things. I can understand the goal of building up a philosophical language powerful enough to describe the world in terms of words, and if such a language was successful it might well inspire us to ask new scientific questions.

But it’s crucial to remember that there’s real work to be done there. There’s no guarantee that the project will work, that words will end up sufficient. When you put aside our best tools to make precise statements, you’re handicapping yourself, making the problem harder than it needed to be. It’s your responsibility to make sure you’re getting something worthwhile out of it.

Pi in the Sky Science Journalism

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You’ve probably seen it somewhere on your facebook feed, likely shared by a particularly wide-eyed friend: pi found hidden in the hydrogen atom!

From the headlines, this sounds like some sort of kabbalistic nonsense, like finding the golden ratio in random pictures.

Read the actual articles, and the story is a bit more reasonable. The last two I linked above seem to be decent takes on it, they’re just saddled with ridiculous headlines. As usual, I blame the editors. This time, they’ve obscured an interesting point about the link between physics and mathematics.

So what does “pi found hidden in the hydrogen atom” actually mean?

It doesn’t mean that there’s some deep importance to the number pi in nature, beyond its relevance in mathematics in general. The reason that pi is showing up here isn’t especially deep.

It isn’t trivial either, though. I’ve seen a few people whose first response to this article was “of course they found pi in the hydrogen atom, hydrogen atoms are spherical!” That’s not what’s going on here. The connection isn’t about the shape of the hydrogen atom, it’s about one particular technique for estimating its energy.

Carl Hagen is a physicist at the University of Rochester who was teaching a quantum mechanics class in which he taught a well-known approximation technique called the variational principle. Specifically, he had his students apply this technique to the hydrogen atom. The nice thing about the hydrogen atom is that it’s one of the few atoms simple enough that it’s possible to find its energy levels exactly. The exact calculation can then be compared to the approximation.

What Hagen noticed was that this approximation was surprisingly good, especially for high energy states for which it wasn’t expected to be. In the end, working with Rochester math professor Tamar Friedmann, he figured out that the variational principle was making use of a particular identity between a type of mathematical functions, called Gamma functions, that are quite common in physics. Using those Gamma functions, the two researchers were able to re-derive what turned out to be a 17th century formula for pi, giving rise to a much cleaner proof for that formula than had been known previously.

So pi isn’t appearing here because “the hydrogen atom is a sphere”. It’s appearing because pi appears all over the place in physics, and because in general, the same sorts of structures appear again and again in mathematics.

Pi’s appearance in the hydrogen atom is thus not very special, regardless. What is a little bit special is the fact that, using the hydrogen atom, these folks were able to find a cleaner proof of an old approximation for pi, one that mathematicians hadn’t found before.

That, if anything, is the interesting part of this news story, but it’s also part of a broader trend, one in which physicists provide “physics proofs” for mathematical results. One of the more famous accomplishments of string theory is a class of “physics proofs” of this sort, using a principle called mirror symmetry.

The existence of “physics proofs” doesn’t mean that mathematics is secretly constrained by the physical world. Rather, they’re a result of the fact that physicists are interested in different aspects of mathematics, and in general are a bit more reckless in using approximations that haven’t been mathematically vetted. A physicist can sometimes prove something in just a few lines that mathematicians would take many pages to prove, but usually they do this by invoking a structure that would take much longer for a mathematician to define. As physicists, we’re building on the shoulders of other physicists, using concepts that mathematicians usually don’t have much reason to bother with. That’s why it’s always interesting when we find something like the Amplituhedron, a clean mathematical concept hidden inside what would naively seem like a very messy construction. It’s also why “physics proofs” like this can happen: we’re dealing with things that mathematicians don’t naturally consider.

So please, ignore the pi-in-the-sky headlines. Some physicists found a trick, some mathematicians found it interesting, the hydrogen atom was (quite tangentially) involved…and no nonsense needs to be present.

What’s so Spooky about Action at a Distance?

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With Halloween coming up, it’s time once again to talk about the spooky side of physics. And what could be spookier than action at a distance?

Pictured here.

Ok, maybe not an obvious contender for spookiest concept of the year. But physicists have struggled with action at a distance for centuries, and there are deep reasons why.

It all dates back to Newton. In Newton’s time, all of nature was expected to be mechanical. One object pushes another, which pushes another in turn, eventually explaining everything that every happens. And while people knew by that point that the planets were not circling around on literal crystal spheres, it was still hoped that their motion could be explained mechanically. The favored explanations of the time were vortices, whirlpools of celestial fluid that drove the planets around the Sun.

Newton changed all that. Not only did he set down a law of gravitation that didn’t use a fluid, he showed that no fluid could possibly replicate the planets’ motions. And while he remained agnostic about gravity’s cause, plenty of his contemporaries accused him of advocating “action at a distance”. People like Leibniz thought that a gravitational force without a mechanical cause would be superstitious nonsense, a betrayal of science’s understanding of the world in terms of matter.

For a while, Newton’s ideas won out. More and more, physicists became comfortable with explanations involving a force stretching out across empty space, using them for electricity and magnetism as these became more thoroughly understood.

Eventually, though, the tide began to shift back. Electricity and Magnetism were explained, not in terms of action at a distance, but in terms of a field that filled the intervening space. Eventually, gravity was too.

The difference may sound purely semantic, but it means more than you might think. These fields were restricted in an important way: when the field changed, it changed at one point, and the changes spread at a speed limited by the speed of light. A theory composed of such fields has a property called locality, the property that all interactions are fundamentally local, that is, they happen at one specific place and time.

Nowadays, we think of locality as one of the most fundamental principles in physics, on par with symmetry in space and time. And the reason why is that true action at a distance is quite a spooky concept.

Much of horror boils down to fear of the unknown. From what might lurk in the dark to the depths of the ocean, we fear that which we cannot know. And true action at a distance would mean that our knowledge might forever be incomplete. As long as everything is mediated by some field that changes at the speed of light, we can limit our search for causes. We can know that any change must be caused by something only a limited distance away, something we can potentially observe and understand. By contrast, true action at a distance would mean that forces from potentially anywhere in the universe could alter events here on Earth. We might never know the ultimate causes of what we observe; they might be stuck forever out of reach.

Some of you might be wondering, what about quantum mechanics? The phrase “spooky action at a distance” was famous because Einstein used it as an accusation against quantum entanglement, after all.

The key thing about quantum mechanics is that, as J. S. Bell showed, you can’t have locality…unless you throw out another property, called realism. Realism is the idea that quantum states have definite values for measurements before those measurements are taken. And while that sounds important, most people find getting rid of it much less scary than getting rid of locality. In a non-realistic world, at least we can still predict probabilities, even if we can’t observe certainties. In a non-local world, there might be aspects of physics that we just can’t learn. And that’s spooky.

When to Look under the Bed

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Last week, Sabine Hossenfelder blogged about a rather interesting experiment, designed to test the quantum properties of gravity. Normally, quantum gravity is essentially unobservable: quantum effects are typically only relevant for very small systems, where gravity is extremely weak. However, there has been a lot of progress in putting larger and larger systems into interesting quantum states, and a team of experimentalists has recently proposed a setup. The experiment wouldn’t have enough detail to, for example, distinguish between rival models of quantum gravity, but it would provide evidence as to whether or not gravity is quantum at all.

Lubos Motl, meanwhile, argues that such an experiment is utterly pointless, because there is no possible way that gravity could not be quantum. I won’t blame you if you don’t read his argument since it’s written in his trademark…aggressive…style, but the gist is that it’s really hard to make sense of the idea that there are non-quantum things in an otherwise quantum world. It causes all sorts of issues with pretty much every interpretation of quantum mechanics, and throws the differences between those interpretations into particularly harsh and obvious light. From this perspective, checking to see if gravity might not actually be quantum (an idea called semi-classical gravity) is a bit like checking for a monster under the bed.

You might find semi-classical gravity!

In general, I share Motl’s reservations about semi-classical gravity. As I mentioned back when journalists were touting the BICEP2 results as evidence of quantum gravity, the idea that gravity could not be quantum doesn’t really make much sense. (Incidentally, Hossenfelder makes a similar point in her post.)

All that said, sometimes in science it’s absolutely worth looking under the bed.

Take another unlikely possibility, that of cell phone radiation causing cancer. Things that cause cancer do it by messing with the molecular bonds in DNA. In order to mess with molecular bonds, you need high-frequency light. That’s how UV light from the sun can cause skin cancer. Cell phones emit microwaves, which are very low-frequency light. It’s what allows them to be useful inside of buildings, where normal light wouldn’t reach. It also means it’s impossible for them to cause cancer.

Nevertheless, if nobody had ever studied whether cell phones cause cancer, it would probably be worth at least one study. If that study came back positive, it would say something interesting, either about the study’s design or about other possible causes of cancer. If negative, the topic could be put to bed more convincingly. As it happens, those studies have been done, and overall confirm the expectations we have from basic science.

Another important point here is that experimentalists and theorists have different priorities, due to their different specializations. Theorists are interested in confirmation for particular theories: they want not just an unknown particle, but a gluino, and not just a gluino, but the gluino predicted by their particular model of supersymmetry. By contrast, experimentalists typically aren’t very interested in proving or disproving one theory or another. Rather, they look for general signals that indicate broad classes of new physics. For example, experimentalists might use the LHC to look for a leptoquark, a particle that allows quarks and leptons to interact, without caring what theory might produce them. Experimentalists are also very interested in improving their techniques. Much like theorists, a lot of interesting work in the field involves pushing the current state-of-the-art as far as it will go.

So, when should we look under the bed?

Well, if nobody has ever looked under this particular bed before, and if seeing something strange under this bed would at least be informative, and if looking under the bed serves as a proving ground for the latest in bed-spelunking technology, then yes, we should absolutely look under this bed.

Just don’t expect to see any monsters.

4 gravitons

Stories about physics from someone who's been there