We are all Wilsonians now

Ken Wilson

Ken Wilson passed away on June 15 at age 77. He changed how we think about physics.

Renormalization theory, first formulated systematically by Freeman Dyson in 1949, cured the flaws of quantum electrodynamics and turned it into a precise computational tool. But the subject seemed magical and mysterious. Many physicists, Dirac prominently among them, questioned whether renormalization rests on a sound foundation.

Wilson changed that.

The renormalization group concept arose in an extraordinary paper by Gell-Mann and Low in 1954. It was embraced by Soviet physicists like Bogoliubov and Landau, and invoked by Landau to challenge the consistency of quantum electrodynamics. But it was an abstruse and inaccessible topic, as is well illustrated by the baffling discussion at the very end of the two-volume textbook by Bjorken and Drell.

Wilson changed that.

Ken Wilson turned renormalization upside down. Dyson and others had worried about the “ultraviolet divergences” occurring in Feynman diagrams. They introduced an artificial cutoff on integrations over the momenta of virtual particles, then tried to show that all the dependence on the cutoff can be eliminated by expressing the results of computations in terms of experimentally accessible quantities. It required great combinatoric agility to show this trick works in electrodynamics. In other theories, notably including general relativity, it doesn’t work.

Wilson adopted an alternative viewpoint. Take the short-distance cutoff seriously, he said, regarding it as part of the physical formulation of the field theory. Now ask what physics looks like at distances much larger than the cutoff. Wilson imagined letting the short-distance cutoff grow, while simultaneously adjusting the theory to preserve its low-energy predictions. This procedure sounds complicated, but Wilson discovered something wonderful — for the purpose of computing low-energy processes the theory becomes remarkably simple, completely characterized by just a few (renormalized) parameters. One recovers Dyson’s results plus much more, while also acquiring a rich and visually arresting physical picture of what is going on.

When I started graduate school in 1975, Wilson, not yet 40, was already a legend. Even Sidney Coleman, for me the paragon of razor sharp intellect, seemed to regard Wilson with awe. (They had been contemporaries at Caltech, both students of Murray Gell-Mann.) It enhanced the legend that Wilson had been notoriously slow to publish. He spent years pondering the foundations of quantum field theory before finally unleashing a torrent of revolutionary papers in the early 70s.Cornell had the wisdom to grant tenure despite Wilson’s unusually low productivity during the 60s.

As a student, I spent countless hours struggling through Wilson’s great papers, some of which were quite difficult. One introduced me to the operator product expansion, which became a workhorse of high-energy scattering theory and the foundation of conformal field theory. Another considered all the possible ways that renormalization group fixed points could control the high-energy behavior of the strong interactions. Conspicuously missing from the discussion was what turned out to be the correct idea — asymptotic freedom. Wilson had not overlooked this possibility; instead he “proved” it to be impossible. The proof contains a subtle error. Wilson analyzed charge renormalization invoking both Lorentz covariance and positivity of the Hilbert space metric, forgetting that gauge theories admit no gauge choice with both properties. Even Ken Wilson made mistakes.

Wilson also formulated the strong-coupling expansion of lattice gauge theory, and soon after pioneered the Euclidean Monte Carlo method for computing the quantitative non-perturbative predictions of quantum chromodynamics, which remains today an extremely active and successful program. But of the papers by Wilson I read while in graduate school, the most exciting by far was this one about the renormalization group. Toward the end of the paper Wilson discussed how to formulate the notion of the “continuum limit” of a field theory with a cutoff. Removing the short-distance cutoff is equivalent to taking the limit in which the correlation length (the inverse of the renormalized mass) is infinitely long compared to the correlation length — the continuum limit is a second-order phase transition. Wilson had finally found the right answer to the decades-old question, “What is quantum field theory?” And after reading his paper, I knew the answer, too! This Wilsonian viewpoint led to further deep insights mentioned in the paper, for example that an interacting self-coupled scalar field theory is unlikely to exist (i.e. have a continuum limit) in four spacetime dimensions.

Wilson’s mastery of quantum field theory led him to another crucial insight in the 1970s which has profoundly influenced physics in the decades since — he denigrated elementary scalar fields as unnatural. I learned about this powerful idea from an inspiring 1979 paper not by Wilson, but by Lenny Susskind. That paper includes a telltale acknowledgment: “I would like to thank K. Wilson for explaining the reasons why scalar fields require unnatural adjustments of bare constants.”

Susskind, channeling Wilson, clearly explains a glaring flaw in the standard model of particle physics — ensuring that the Higgs boson mass is much lighter than the Planck (i.e., cutoff) scale requires an exquisitely careful tuning of the theory’s bare parameters. Susskind proposed to banish the Higgs boson in favor of Technicolor, a new strong interaction responsible for breaking the electroweak gauge symmetry, an idea I found compelling at the time. Technicolor fell into disfavor because it turned out to be hard to build fully realistic models, but Wilson’s complaint about elementary scalars continued to drive the quest for new physics beyond the standard model, and in particular bolstered the hope that low-energy supersymmetry (which eases the fine tuning problem) will be discovered at the Large Hadron Collider. Both dark energy (another fine tuning problem) and the absence so far of new physics beyond the HIggs boson at the LHC are prompting some soul searching about whether naturalness is really a reliable criterion for evaluating success in physical theories. Could Wilson have steered us wrong?

Wilson’s great legacy is that we now regard nearly every quantum field theory as an effective field theory. We don’t demand or expect that the theory will continue working at arbitrarily short distances. At some stage it will break down and be replaced by a more fundamental description. This viewpoint is now so deeply ingrained in how we do physics that today’s students may be surprised to hear it was not always so. More than anyone else, we have Ken Wilson to thank for this indispensable wisdom. Few ideas have changed physics so much.

Quantum Matter Animated!

by Jorge Cham

What does it mean for something to be Quantum? I have to confess, I don’t know. My Ph.D was in Robotics and Kinematics, so my neurons are deeply trained to think in terms of classical dynamics. I asked my siblings (two engineers and one architect) what comes to mind for them when they hear the word Quantum, what they remember from college physics, and here is what they said:

- “Quantum Leap!” (the late 80′s TV show)

- “Quantum of Solace!” (the James Bond movie which, incidentally, was filmed in my home country of Panama, even though the movie was set in Bolivia)

- “I don’t remember anything I learned in college”

- “Light acting as a particle instead of a wave?”

The third answer came from my sister, who went to MIT. The fourth came from my brother, who went to Stanford (+1 point for Stanford!).

I also asked my spouse what comes to mind for her. She said, “Quantum Computing: it’s the next big advance in computers. Transistors the size of atoms.” Clearly, I married someone smarter than me (she also went to Stanford). When I asked if she knew how they worked, she said, “I don’t know how it works.” She also said, “Quantum is related to how time moves more slowly as you approach the speed of light, right?” Nice try, but that’s Relativity (-1 point for Stanford!).

I think the word Quantum has a special power in our collective consciousness. It’s used to convey science-iness, technology, the weirdness of the Physical world. If you Google “Quantum”, most of the top hits are for technology companies that have nothing to do with Quantum Physics (including Quantum Fishing Tackles. I suppose that half the time, you pull up a dead fish).

It’s one of those words that a lot of people have heard of, but very few really understand what it means. Which is why I was excited when Spiros Michalakis and IQIM approached me to produce a series of animations that explore and explain Quantum Information and Matter. Like my previous videos (The Higgs Boson, Dark Matter, Exoplanets), I’d have the chance to interview experts in this field and use their expertise and their voices to learn and to help others learn what amazing things lie just around the corner, beyond our classical understanding of the Universe.

This will be a big Leap for me (I’m trying to avoid the obvious pun), and a journey of exploration. The first installment goes live today, and you can watch it below. Like Schrödinger’s box, I don’t know what we’ll discover with these videos, but I know there are exciting possibilities inside. This is also going to be a BIG challenge. Understanding and putting Quantum concepts in visual form will be hard. I mean, Hair-pulling hard. Fortunately, I’ve discovered there’s a remedy for that.

Watch the first installment of this series:

Jorge Cham is the creator of Piled Higher and Deeper (www.phdcomics.com).

CREDITS:

Featuring: Amir Safavi-Naeini and Oskar Painter http://copilot.caltech.edu/

Produced in Partnership with the Institute for Quantum Information and Matter (http://iqim.caltech.edu) at Caltech with funding provided by the National Science Foundation.

Transcription: Noel Dilworth
Thanks to: Spiros Michalakis, John Preskill and Bert Painter

Entanglement = Wormholes

One of the most enjoyable and inspiring physics papers I have read in recent years is this one by Mark Van Raamsdonk. Building on earlier observations by Maldacena and by Ryu and Takayanagi. Van Raamsdonk proposed that quantum entanglement is the fundamental ingredient underlying spacetime geometry. Since my first encounter with this provocative paper, I have often mused that it might be a Good Thing for someone to take Van Raamsdonk’s idea really seriously.

Now someone has.

I love wormholes. (Who doesn’t?) Picture two balls, one here on earth, the other in the Andromeda galaxy. It’s a long trip from one ball to the other on the background space, but there’s a shortcut:You can walk into the ball on earth and moments later walk out of the ball in Andromeda. That’s a wormhole.

I’ve mentioned before that John Wheeler was one of my heros during my formative years. Back in the 1950s, Wheeler held a passionate belief that “everything is geometry,” and one particularly intriguing idea he called “charge without charge.” There are no pointlike electric charges, Wheeler proclaimed; rather, electric field lines can thread the mouth of a wormhole. What looks to you like an electron is actually a tiny wormhole mouth. If you were small enough, you could dive inside the electron and emerge from a positron far away. In my undergraduate daydreams, I wished this idea could be true.

But later I found out more about wormholes, and learned about “topological censorship.” It turns out that if energy is nonnegative, Einstein’s gravitational field equations prevent you from traversing a wormhole — the throat always pinches off (or becomes infinitely long) before you get to the other side. It has sometimes been suggested that quantum effects might help to hold the throat open (which sounds like a good idea for a movie), but today we’ll assume that wormholes are never traversable no matter what you do.

Love in a wormhole throat: Alice and Bob are in different galaxies, but each lives near a black hole, and their black holes are connected by a wormhole. If both jump into their black holes, they can enjoy each other’s company for a while before meeting a tragic end.

Are wormholes any fun if we can never traverse them? The answer might be yes if two black holes are connected by a wormhole. Then Alice on earth and Bob in Andromeda can get together quickly if each jumps into a nearby black hole. For solar mass black holes Alice and Bob will have only 10 microseconds to get acquainted before meeting their doom at the singularity. But if the black holes are big enough, Alice and Bob might have a fulfilling relationship before their tragic end.

This observation is exploited in a recent paper by Juan Maldacena and Lenny Susskind (MS) in which they reconsider the AMPS puzzle (named for Almheiri, Marolf, Polchinski, and Sully). I wrote about this puzzle before, so I won’t go through the whole story again. Here’s the short version: while classical correlations can easily be shared by many parties, quantum correlations are harder to share. If Bob is highly entangled with Alice, that limits his ability to entangle with Carrie, and if he entangles with Carrie instead he can’t entangle with Alice. Hence we say that entanglement is “monogamous.” Now, if, as most of us are inclined to believe, information is “scrambled” but not destroyed by an evaporating black hole, then the radiation emitted by an old black hole today should be highly entangled with radiation emitted a long time ago. And if, as most of us are inclined to believe, nothing unusual happens (at least not right away) to an observer who crosses the event horizon of a black hole, then the radiation emitted today should be highly entangled with stuff that is still inside the black hole. But we can’t have it both ways without violating the monogamy of entanglement!

The AMPS puzzle invites audacious reponses, and AMPS were suitably audacious. They proposed that an old black hole has no interior — a freely falling observer meets her doom right at the horizon rather than at a singularity deep inside.

MS are also audacious, but in a different way. They helpfully summarize their key point succinctly in a simple equation:

ER = EPR

Here, EPR means Einstein-Podolsky-Rosen, whose famous paper highlighted the weirdness of quantum correlations, while ER means Einstein-Rosen (sorry, Podolsky), who discovered wormhole solutions to the Einstein equations. (Both papers were published in 1935.) MS (taking Van Raamsdonk very seriously) propose that whenever any two quantum subsystems are entangled they are connected by a wormhole. In many cases, these wormholes are highly quantum mechanical, but in some cases (where the quantum system under consideration has a weakly coupled “gravitational dual”), the wormhole can have a smooth geometry like the one ER described. That wormholes are not traversable is important for the consistency of ER = EPR: just as Alice cannot use their shared entanglement to send a message to Bob instantaneously, so she is unable to send Bob a message through their shared wormhole.

AMPS imagined that Alice could distill qubit C from the black hole’s early radiation and carry it back to the black hole, successfully verifying its entanglement with another qubit B distilled from the recent radiation. Monogamy then ensures that qubit B cannot be entangled with qubit A behind the horizon. Hence when Alice falls through the horizon she will not observe the quiescent vacuum state in which A and B are entangled; instead she encounters a high-energy particle. MS agree with this conclusion.

AMPS go on to say that Alice’s actions before entering the black hole could not have created that energetic particle; it must have been there all along, one of many such particles constituting a seething firewall.

Here MS disagree. They argue that the excitation encountered by Alice as she crosses the horizon was actually created by Alice herself when she interacted with qubit C. How could Alice’s actions, executed far, far away from the black hole, dramatically affect the state of the black hole’s interior? Because C and A are connected by a wormhole!

The ER = EPR conjecture seems to allow us to view the early radiation with which the black hole is entangled as a complementary description of the black hole interior. It’s not clear yet whether this picture works in detail, and even if it does there could still be firewalls; maybe in some sense the early radiation is connected to the black hole via a wormhole, yet this wormhole is wildly fluctuating rather than a smooth geometry. Still, MS provide a promising new perspective on a deep problem.

As physicists we often rely on our sense of smell in judging scientific ideas, and earlier proposed resolutions of the AMPS puzzle (like firewalls) did not smell right. At first whiff, ER = EPR may smell fresh and sweet, but it will have to ripen on the shelf for a while. If this idea is on the right track, there should be much more to say about it. For now, wormhole lovers can relish the possibilities.

Eventually, Wheeler discarded “everything is geometry” in favor of an ostensibly deeper idea: “everything is information.” It would be a fitting vindication of Wheeler’s vision if everything in the universe, including wormholes, is made of quantum correlations.

A Public Lecture on Quantum Information

Sooner or later, most scientists are asked to deliver a public lecture about their research specialties. When successful, lecturing about science to the lay public can give one a feeling of deep satisfaction. But preparing the lecture is a lot of work!

Caltech sponsors the Earnest C. Watson lecture series (named after the same Earnest Watson mentioned in my post about Jane Werner Watson), which attracts very enthusiastic audiences to Beckman Auditorium nine times a year. I gave a Watson lecture on April 3 about Quantum Entanglement and Quantum Computing, which is now available from iTunes U and also on YouTube:

I did a Watson lecture once before, in 1997. That occasion precipitated some big changes in my presentation style. To prepare for the lecture, I acquired my first laptop computer and learned to use PowerPoint. This was still the era when a typical physics talk was handwritten on transparencies and displayed using an overhead projector, so I was sort of a pioneer. And I had many anxious moments in the late 1990s worrying about whether my laptop would be able to communicate with the projector — that can still be a problem even today, but was a more common problem then.

I invested an enormous amount of time in preparing that 1997 lecture, an investment still yielding dividends today. Aside from figuring out what computer to buy (an IBM ThinkPad) and how to do animation in PowerPoint, I also learned to draw using Adobe Illustrator under the tutelage of Caltech’s digital media expert Wayne Waller. And apart from all that technical preparation, I had to figure out the content of the lecture!

That was when I first decided to represent a qubit as a box with two doors, which contains a ball that can be either red or green, and I still use some of the drawings I made then.

Entanglement, illustrated with balls in boxes.

This choice of colors was unfortunate, because people with red-green color blindness cannot tell the difference. I still feel bad about that, but I don’t have editable versions of the drawings anymore, so fixing it would be a big job …

I also asked my nephew Ben Preskill (then 10 years old, now a math PhD candidate at UC Berkeley), to make a drawing for me illustrating weirdness.

The desire to put weirdness to work has driven the emergence of quantum information science.

I still use that, for sentimental reasons, even though it would be easier to update.

The turnout at the lecture was gratifying (you can’t really see the audience with the spotlight shining in your eyes, but I sensed that the main floor of the Auditorium was mostly full), and I have gotten a lot of positive feedback (including from the people who came up to ask questions afterward — we might have been there all night if the audio-visual staff had not forced us to go home).

I did make a few decisions about which I have had second thoughts. I was told I had the option of giving a 45 minute talk with a public question period following, or a 55 minute talk with only a private question period, and I opted for the longer talk. Maybe I should have pushed back and insisted on allowing some public questions even after the longer talk — I like answering questions. And I was told that I should stay in the spotlight, to ensure good video quality, so I decided to stand behind the podium the whole time to curb my tendency to pace across the stage. But maybe I would have seemed more dynamic if I had done some pacing.

I got some gentle criticism from my wife, Roberta, who suggested I could modulate my voice more. I have heard that before, particularly in teaching evaluations that complain about my “soporific” tone. I recall that Mike Freedman once commented after watching a video of a public lecture I did at the KITP in Santa Barbara — he praised its professionalism and “newscaster quality”. But that cuts two ways, doesn’t it? Paul Ginsparg listened to a podcast of that same lecture while doing yardwork, and then sent me a compliment by email, with a characteristic Ginspargian twist. Noting that my sentences were clear, precise, and grammatical, Paul asked: “is this something that just came naturally at some early age, or something that you were able to acquire at some later stage by conscious design (perhaps out of necessity, talks on quantum computing might not go over as well without the reassuring smoothness)?”

Another criticism stung more. To illustrate the monogamy of entanglement, I used a slide describing the frustration of Bob, who wants to entangle with both Alice and Carrie, but finds that he can increase his entanglement with Carrie only my sacrificing some of his entanglement with Alice.

Entanglement is monogamous. Bob is frustrated to find that he cannot be fully entangled with both Alice and Carrie.

This got a big laugh. But I used the same slide in a talk at the APS Denver meeting the following week (at a session celebrating the 100th anniversary of Niels Bohr’s atomic model), and a young woman came up to me after that talk to complain. She suggested that my monogamy metaphor was offensive and might discourage women from entering the field!

After discussing the issue with Roberta, I decided to address the problem by swapping the gender roles. The next day, during the question period following Stephen Hawking’s Public Lecture, I spoke about Betty’s frustration over her inability to entangle fully with both Adam and Charlie. But is that really an improvement, or does it reflect negatively on Betty’s morals? I would appreciate advice about this quandary in the comments.

In case you watch the video, there are a couple of things you should know. First, in his introduction, Tom Soifer quotes from a poem about me, but neglects to name the poet. It is former Caltech postdoc Patrick Hayden. And second, toward the end of the lecture I talk about some IQIM outreach activities, but neglect to name our Outreach Director Spiros Michalakis, without whose visionary leadership these things would not have happened.

The most touching feedback I received came from my Caltech colleague Oskar Painter. I joked in the lecture about how mild mannered IQIM scientists can unleash the superpower of quantum information at a moment’s notice.

Mild mannered professor unleashes the superpower of quantum information.

After watching the video, Oskar shot me an email:

“I sent a link to my son [Ewan, age 11] and daughter [Quinn, age 9], and they each watched it from beginning to end on their iPads, without interruption.  Afterwards, they had a huge number of questions for me, and were dreaming of all sorts of “quantum super powers” they imagined for the future.”

Project X Squared

Alicia Hardesty: full-time fashion designer, part-time nerd.

Have you seen the movie Frankenweenie? It’s a black and white cartoon (an experiment in itself these days) with a very important message:

Don’t be afraid to do what you love and don’t be afraid to be good at it.

The main character is a smart, sensitive kid who is ostracized for his science experiments. Like the teacher says, people don’t understand science so they are afraid of it. Ironically, artists often deal with the same kind of misunderstandings from the public.

I’m not technically a scientist, but I do love to experiment and try stuff. I’m a fashion designer, which requires it’s own level of scientific conviction. I create, combine unlikely variables, hypothesize, and work within my own scientific method throughout my process.

How does this relate to you?

Project X Squared. Where art, science, and technology meet fashion to create a clothing line, much like an experiment, with the underlying hypothesis being that a quantum physicist, a neuroscientist and a fashion designer can create something tangible together.

Largest prime number found?

Over the past few months, I have been inundated with tweets about the largest prime number ever found. That number, according to Nature News, is $2^{57,885,161}-1$. This is certainly a very large prime number and one would think that we would need a supercomputer to find a prime number larger than this one. In fact, Nature mentions that there are infinitely many prime numbers, but the powerful prime number theorem doesn’t tell us how to find them!
Well, I am here to tell you of the discovery of the new largest prime number ever found, which I will call $P_{euclid}$. Here it is:

$P_{euclid} = 2\cdot 3\cdot 5\cdot 7\cdot 11 \cdot \cdots \cdot (2^{57,885,161}-1) +1.$

This number, the product of all prime numbers known so far plus one, is so large that I can’t even write it down on this blog post. But it is certainly (proof left as an exercise…!) a prime number (see Problem 4 in The allure of elegance) and definitely larger than the one getting all the hype. Finally, I will be getting published in Nature!

In the meantime, if you are looking for a real challenge, calculate how many digits my prime number has in base 10. Whoever gets it right (within an order of magnitude), will be my co-author in the shortest Nature paper ever written.

Update 2: I read somewhere that in order to get attention to your blog posts, you should sprinkle them with grammatical errors and let the commenters do the rest for you. I wish I was mastermind-y enough to engineer this post in this fashion. Instead, I get the feeling that someone will run a primality test on $P_{euclid}$ just to prove me wrong. Well, what are you waiting for? In the meantime, another challenge: What is the smallest number (ballpark it using Prime Number Theorem) of primes we need to multiply together before adding one, in order to have a number with a larger prime factor than $2^{57,885,161}-1$?

Update: The number $P_{euclid}$ given above may not be prime itself, as pointed out quickly by Steve Flammia, Georg and Graeme Smith. But, it does contain within it the new largest prime number ever known, which may be the number itself. Now, if only we had a quantum computer to factor numbers quickly…Wait, wasn’t there a polynomial time primality test?

Note: The number mentioned is the largest known Mersenne prime. That Mersenne primes are crazy hard to find is an awesome problem in number theory.

Post-Quantum Cryptography

As an undergraduate, I took Introduction to Algorithms from Ron Rivest. One of the topics he taught was the RSA public-key cryptosystem which he had created with Adi Shamir and Leonard Adleman. At the time, RSA was only about a decade old, yet already one of the banner creations of computer science. Today many of us rely on it routinely for the security of banking transactions. The internet would not be the same without it and its successors (such as elliptic curve cryptography, ECC). However, as you may have heard, quantum computation spells change for cryptography. Today I’ll tell a little of this story and talk about prospects for the future.

Ron Rivest

What is public-key cryptography (PKC)? The basic notion is due to Ralph Merkle in 1974 and (in a stronger form) to Whitfield Diffie and Martin Hellman in 1976. Their remarkable proposal was that two parties, “Alice” and “Bob”, could cooperate in cryptographic protocols, even if they had never met before. All prior cryptography, from the ancients up through and after the cryptographic adventures of WWII, had relied on the cooperating parties sharing in advance some “secret key” that gave them an edge over any eavesdropper “Eve”.

De divina proportione

As a mathematician, I often wonder if life would have been easier were I born 2,400 years ago. Back then, all you had to do to become eternally famous was to show that $3^2+4^2=5^2$ ( I am looking at you Πυθαγόρα!) OK, maybe I am not giving the Ancients enough credit. I mean, they didn’t have an iPhone back then, so they probably had to do $3^2+4^2$ by hand. All kidding aside, they did generalize the previous equality to other large numbers like $12^2+5^2=13^2$ (I am feeling a little sassy today, I guess.) Still, back then, mathematics did not start as an abstract subject about relations between numbers. It grew from a naive attempt to control elements of design that were essential to living, like building airplanes and plasma TVs. The Greeks didn’t succeed then and, if I am not mistaken, they still haven’t succeeded in making either airplanes, or plasma TVs. But, back then at least, my ancestors made some beautiful buildings. Like the Parthenon.

The temple of Athina (the Goddess of Wisdom, which gave her name to the city we now call Athens after a fierce contest with Poseidon – imagine flying into Poseidonia every time you visited Greece had she lost) was designed to be seen from far away and inspire awe in those who wished to conquer the city-state of Athens. But, those who were granted access to the space behind the doric columns, came face to face with the second divine woman to ever make Zeus stand in attention, whenever she met her dad on legendary Mt. Olympus: Αθηνά. And so, Φειδίας (Phidias), that most famous of ancient Greek sculptors, decided to immortalize Athina’s power with a magnificent statue, a tribute to the effortless grace with which she personified the wisdom of an ancient culture in harmony with the earth’s most precious gift – feta cheese.

Here she is, playing an invisible electric cello next to Yo-Yo Ma (also invisible). And yes, she liked to work out.

OK, I may be biased on this one. For Greeks, virgin olive oil and φέτα cheese go like peanut butter and jelly (I didn’t even know the last two went so well together, until I left Greece for the country of America!) Oh yeah, you are probably wondering what feta cheese and olive oil have to do with the Goddess of Wisdom. Well, how do you think she won over the Athenians, against Zeus’ almighty brother, Poseidon? The olive branch, of course. The sea is good and all (actually, the sea is pretty freakin’ amazing in Greece), but you can’t eat it with feta – you can preserve feta in brine (salt water), which is why Poseidon had a fighting chance in the first place – but, yeah, not good enough. Which brings us to the greatest rival, nay – nemesis, of the first letter with an identity crisis, $\pi$: The letter $\phi$. You are most likely familiar with the letter-number $\pi = 3.1415925123\ldots$ (you may have even seen the modern classic, American Pie, a tour-de-force, honest look at the life of Pi. No pun intended.) But, what about the number $1.618033\ldots$? Well, I could tell you all about this number, $\phi$, named after the sculptor dude above, but I ‘d rather you figure out its history on your own through this simple math problem:

The divine proportion: Does there exist a function $f: \mathbb{N} \rightarrow \mathbb{N}$, such that $f(1)=2$, $f(f(n)) = f(n)+n$ and $f(n) < f(n+1)$ for all $n \in \mathbb{N}$?

Καλή τύχη, μικροί μου Φιμπονάτσι!

Unsolvable

Why go swimming, if you can do math instead inside a room with no windows?

Back in 1997, during my visit to beautiful Mar del Plata in Argentina, I was asked to solve a math problem that I soon realized was close to being unsolvable. The setting was the Banquet for the 38th International Math Olympiad. I was 17 and there was delicious, free food in front of me, so it was pretty impossible to get my attention. Still, the coach of the Romanian team decided to drop by the Greek table to challenge us with the following problem:

Infinite Power: If $x^{x^{x^{x^{\dots }}}} = 2$, find $x$.

I was pretty hungry and a bit annoyed at the interruption (it is rude to eat and do math at the same time!), so I looked at the problem for a moment and then challenged him back with this one:

Infiniter Power: If $x^{x^{x^{x^{\dots }}}} = 4$, find $x$.

After a few moments, he looked at me with a puzzled look. He knew I had solved his problem within a few seconds, he was annoyed that I had somehow done it in my head and he was even more annoyed that I had challenged him back with a problem that confounded him.

Your mission, if you choose to accept it, is to figure out why the coach of the Romanian IMO team left our table alone for the rest of the competition.

Good luck!

PS: The rest of the Romanian team (the kids) were really cool. In fact, a few years later, I would hang out with a bunch of them at MIT’s Department of Mathematics, or as my older brother put it, The Asylum.

A poll on the foundations of quantum theory

Erwin Schrödinger. Discussions of quantum foundations often seem to involve his much abused cat.

The group of physicists seriously engaged in studies of the “foundations” or “interpretation” of quantum theory is a small sliver of the broader physics community (perhaps a few hundred scientists among tens of thousands). Yet in my experience most scientists doing research in other areas of physics enjoy discussing foundational questions over coffee or beer.

The central question concerns quantum measurement. As often expressed, the axioms of quantum mechanics (see Sec. 2.1 of my notes here) distinguish two different ways for a quantum state to change. When the system is not being measured its state vector rotates continuously, as described by the Schrödinger equation. But when the system is measured its state “collapses” discontinuously. The Measurement Problem (or at least one version of it) is the challenge to explain why the mathematical description of measurement is different from the description of other physical processes.

My own views on such questions are rather unsophisticated and perhaps a bit muddled:

1) I know no good reason to disbelieve that all physical processes, including measurements, can be described by the Schrödinger equation.

2) But to describe measurement this way, we must include the observer as part of the evolving quantum system.

3) This formalism does not provide us observers with deterministic predictions for the outcomes of the measurements we perform. Therefore, we are forced to use probability theory to describe these outcomes.

4) Once we accept this role for probability (admittedly a big step), then the Born rule (the probability is proportional to the modulus squared of the wave function) follows from simple and elegant symmetry arguments. (These are described for example by Zurek – see also my class notes here. As a technical aside, what is special about the L2 norm is its rotational invariance, implying that the probability measure picks out no preferred basis in the Hilbert space.)

5) The “classical” world arises due to decoherence, that is, pervasive entanglement of an observed quantum system with its unobserved environment. Decoherence picks out a preferred basis in the Hilbert space, and this choice of basis is determined by properties of the Hamiltonian, in particular its spatial locality.