Defending against high-frequency attacks

Posted on April 8, 2014 by shaunmaguire

It was the summer of 2008. I was 22 years old, and it was my second week working in the crude oil and natural gas options pit at the New York Mercantile Exchange (NYMEX.) My head was throbbing after two consecutive weeks of disorientation. It was like being born into a new world, but without the neuroplasticity of a young human. And then the crowd erupted. “Yeeeehawwww. YeEEEeeHaaaWWWWW. Go get ’em cowboy.”

It seemed that everyone on the sprawling trading floor had started playing Wild Wild West and I had no idea why. After at least thirty seconds, the hollers started to move across the trading floor. They moved away 100 meters or so and then doubled back towards me. After a few meters, he finally got it, and I’m sure he learned a life lesson. Don’t be the biggest jerk in a room filled with traders, and especially, never wear triple-popped pastel-colored Lacoste shirts. This young aspiring trader had been “spurred.”

In other words, someone had made paper spurs out of trading receipts and taped them to his shoes. Go get ’em cowboy.

I was one academic quarter away from finishing a master’s degree in statistics at Stanford University and I had accepted a full time job working in the algorithmic trading group at DRW Trading. I was doing a summer internship before finishing my degree, and after three months of working in the algorithmic trading group in Chicago, I had volunteered to work at the NYMEX. Most ‘algo’ traders didn’t want this job, because it was far-removed from our mental mathematical monasteries, but I knew I would learn a tremendous amount, so I jumped at the opportunity. And by learn, I mean, get ripped calves and triceps, because my job was to stand in place for seven straight hours updating our mathematical models on a bulky tablet PC as trades occurred.

I have no vested interests in the world of high-frequency trading (HFT). I’m currently a PhD student in the quantum information group at Caltech and I have no intentions of returning to finance. I found the work enjoyable, but not as thrilling as thinking about the beginning of the universe (what else is?) However, I do feel like the current discussion about HFT is lop-sided and I’m hoping that I can broaden the perspective by telling a few short stories.

What are the main attacks against HFT? Three of them include the evilness of: front-running markets, making money out of nothing, and instability. It’s easy to point to extreme examples of algorithmic traders abusing markets, and they regularly do, but my argument is that HFT has simply computerized age-old tactics. In this process, these tactics have become more benign and markets more stable.

Front-running markets: large oil producing nations, such as Mexico, often want to hedge their exposure to changing market prices. They do this by purchasing options. This allows them to lock in a minimum sale price, for a fee of a few dollars per barrel. During my time at the NYMEX, I distinctly remember a broker shouting into the pit: “what’s the price on DEC9 puts.” A trader doesn’t want to give away whether they want to buy or sell, because if the other traders know, then they can artificially move the price. In this particular case, this broker was known to sometimes implement parts of Mexico’s oil hedge. The other traders in the pit suspected this was a trade for Mexico because of his anxious tone, some recent geopolitical news, and the expiration date of these options.

Some confident traders took a risk and faded the market. They ended up making between $1-2 million dollars from these trades, relative to what the fair price was at that moment. I mention relative to the fair price, because Mexico ultimately received the better end of this trade. The price of oil dropped in 2009, and Mexico executed its options enabling it to sell its oil at a higher than market price. Mexico spent $1.5 billion to hedge its oil exposure in 2009.

This was an example of humans anticipating the direction of a trade and capturing millions of dollars in profit as a result. It really is profit as long as the traders can redistribute their exposure at the ‘fair’ market price before markets move too far. The analogous strategy in HFT is called “front-running the market” which was highlighted in the New York Times’ recent article “the wolf hunters of Wall Street.” The HFT version involves analyzing the prices on dozens of exchanges simultaneously, and once an order is published in the order book of one exchange, then using this demand to adjust its orders on the other exchanges. This needs to be done within a few microseconds in order to be successful. This is the computerized version of anticipating demand and fading prices accordingly. These tactics as I described them are in a grey area, but they rapidly become illegal.

Making money from nothing: arbitrage opportunities have existed for as long as humans have been trading. I’m sure an ancient trader received quite the rush when he realized for the first time that he could buy gold in one marketplace and then sell it in another, for a profit. This is only worth the trader’s efforts if he makes a profit after all expenses have been taken into consideration. One of the simplest examples in modern terms is called triangle arbitrage, and it usually involves three pairs of currencies. Currency pairs are ratios; such as USD/AUD, which tells you, how many Australian dollars you receive for one US dollar. Imagine that there is a moment in time when the product of ratios $\frac{USD}{AUD}\frac{AUD}{CAD}\frac{CAD}{USD}$ is 1.01. Then, a trader can take her USD, buy AUD, then use her AUD to buy CAD, and then use her CAD to buy USD. As long as the underlying prices didn’t change while she carried out these three trades, she would capture one cent of profit per trade.

After a few trades like this, the prices will equilibrate and the ratio will be restored to one. This is an example of “making money out of nothing.” Clever people have been trading on arbitrage since ancient times and it is a fundamental source of liquidity. It guarantees that the price you pay in Sydney is the same as the price you pay in New York. It also means that if you’re willing to overpay by a penny per share, then you’re guaranteed a computer will find this opportunity and your order will be filled immediately. The main difference now is that once a computer has been programmed to look for a certain type of arbitrage, then the human mind can no longer compete. This is one of the original arenas where the term “high-frequency” was used. Whoever has the fastest machines, is the one who will capture the profit.

Instability: I believe that the arguments against HFT of this type have the most credibility. The concern here is that exceptional leverage creates opportunity for catastrophe. Imaginations ran wild after the Flash Crash of 2010, and even if imaginations outstripped reality, we learned much about the potential instabilities of HFT. A few questions were posed, and we are still debating the answers. What happens if market makers stop trading in unison? What happens if a programming error leads to billions of dollars in mistaken trades? Do feedback loops between algo strategies lead to artificial prices? These are reasonable questions, which are grounded in examples, and future regulation coupled with monitoring should add stability where it’s feasible.

The culture in wealth driven industries today is appalling. However, it’s no worse in HFT than in finance more broadly and many other industries. It’s important that we dissociate our disgust in a broad culture of greed from debates about the merit of HFT. Black boxes are easy targets for blame because they don’t defend themselves. But that doesn’t mean they aren’t useful when implemented properly.

Are we better off with HFT? I’d argue a resounding yes. The primary function of markets is to allocate capital efficiently. Three of the strongest measures of the efficacy of markets lie in “bid-ask” spreads, volume and volatility. If spreads are low and volume is high, then participants are essentially guaranteed access to capital at as close to the “fair price” as possible. There is huge academic literature on how HFT has impacted spreads and volume but the majority of it indicates that spreads have lowered and volume has increased. However, as alluded to above, all of these points are subtle–but in my opinion, it’s clear that HFT has increased the efficiency of markets (it turns out that computers can sometimes be helpful.) Estimates of HFT’s impact on volatility haven’t been nearly as favorable but I’d also argue these studies are more debatable. Basically, correlation is not causation, and it just so happens that our rapidly developing world is probably more volatile than the pre-HFT world of the last Millennia.

We could regulate away HFT, but we wouldn’t be able to get rid of the underlying problems people point to unless we got rid of markets altogether. As with any new industry, there are aspects of HFT that should be better monitored and regulated, but we should have level-heads and diverse data points as we continue this discussion. As with most important problems, I believe the ultimate solution here lies in educating the public. Or in other words, this is my plug for Python classes for all children!!

I promise that I’ll repent by writing something that involves actual quantum things within the next two weeks!

IQIM Presents …”my father”

Posted on April 2, 2014 by Iram Parveen Bilal

Debaleena Nandi at Caltech

Following the IQIM teaser, which was made with the intent of creating a wider perspective of the scientist, to highlight the normalcy behind the perception of brilliance and to celebrate the common human struggles to achieve greatness, we decided to do individual vignettes of some of the characters you saw in the video.

We start with Debaleena Nandi, a grad student in Prof Jim Eisenstein’s lab, whose journey from Jadavpur University in West Bengal, India to the graduate school and research facility at the Indian institute of Science, Bangalore, to Caltech has seen many obstacles. We focus on the essentials of an environment needed to manifest the quest for “the truth” as Debaleena says. We start with her days as a child when her double-shift working father sat by her through the days and nights that she pursued her homework.

She highlights what she feels is the only way to growth; working on what is lacking, to develop that missing tool in your skill set, that asset that others might have by birth but you need to inspire by hard work.

Debaleena’s motto: to realize and face your shortcomings is the only way to achievement.

As we build Debaleena up, we also build up the identity of Caltech through its breathtaking architecture that oscillates from Spanish to Goth to modern. Both Debaleena and Caltech are revealed slowly, bit by bit.

This series is about dissecting high achievers, seeing the day to day steps, the bit by bit that adds up to the more often than not, overwhelming, impressive presence of Caltech’s science. We attempt to break it down in smaller vignettes that help us appreciate the amount of discipline, intent and passion that goes into making cutting edge researchers.

Presenting the emotional alongside the rational is something this series aspires to achieve. It honors and celebrates human limitations surrounding limitless boundaries, discoveries and possibilities.

Stay tuned for more vignettes in the IQIM Presents “My _______” Series.

But for now, here is the video. Watch, like and share!

Inflation on the back of an envelope

Posted on March 23, 2014 by preskill

Last Monday was an exciting day!

After following the BICEP2 announcement via Twitter, I had to board a transcontinental flight, so I had 5 uninterrupted hours to think about what it all meant. Without Internet access or references, and having not thought seriously about inflation for decades, I wanted to reconstruct a few scraps of knowledge needed to interpret the implications of r ~ 0.2.

I did what any physicist would have done … I derived the basic equations without worrying about niceties such as factors of 3 or $2 \pi$ . None of what I derived was at all original — the theory has been known for 30 years — but I’ve decided to turn my in-flight notes into a blog post. Experts may cringe at the crude approximations and overlooked conceptual nuances, not to mention the missing references. But some mathematically literate readers who are curious about the implications of the BICEP2 findings may find these notes helpful. I should emphasize that I am not an expert on this stuff (anymore), and if there are serious errors I hope better informed readers will point them out.

By tradition, careless estimates like these are called “back-of-the-envelope” calculations. There have been times when I have made notes on the back of an envelope, or a napkin or place mat. But in this case I had the presence of mind to bring a notepad with me.

Notes from a plane ride

According to inflation theory, a nearly homogeneous scalar field called the inflaton (denoted by $\phi$ ) filled the very early universe. The value of $\phi$ varied with time, as determined by a potential function $V(\phi)$ . The inflaton rolled slowly for a while, while the dark energy stored in $V(\phi)$ caused the universe to expand exponentially. This rapid cosmic inflation lasted long enough that previously existing inhomogeneities in our currently visible universe were nearly smoothed out. What inhomogeneities remained arose from quantum fluctuations in the inflaton and the spacetime geometry occurring during the inflationary period.

Gradually, the rolling inflaton picked up speed. When its kinetic energy became comparable to its potential energy, inflation ended, and the universe “reheated” — the energy previously stored in the potential $V(\phi)$ was converted to hot radiation, instigating a “hot big bang”. As the universe continued to expand, the radiation cooled. Eventually, the energy density in the universe came to be dominated by cold matter, and the relic fluctuations of the inflaton became perturbations in the matter density. Regions that were more dense than average grew even more dense due to their gravitational pull, eventually collapsing into the galaxies and clusters of galaxies that fill the universe today. Relic fluctuations in the geometry became gravitational waves, which BICEP2 seems to have detected.

Both the density perturbations and the gravitational waves have been detected via their influence on the inhomogeneities in the cosmic microwave background. The 2.726 K photons left over from the big bang have a nearly uniform temperature as we scan across the sky, but there are small deviations from perfect uniformity that have been precisely measured. We won’t worry about the details of how the size of the perturbations is inferred from the data. Our goal is to achieve a crude understanding of how the density perturbations and gravitational waves are related, which is what the BICEP2 results are telling us about. We also won’t worry about the details of the shape of the potential function $V(\phi)$ , though it’s very interesting that we might learn a lot about that from the data.

Exponential expansion

Einstein’s field equations tell us how the rate at which the universe expands during inflation is related to energy density stored in the scalar field potential. If a(t) is the “scale factor” which describes how lengths grow with time, then roughly

$\left(\frac{\dot a}{a}\right)^2 \sim \frac{V}{m_P^2}$ .

Here $\dot a$ means the time derivative of the scale factor, and $m_P = 1/\sqrt{8 \pi G} \approx 2.4 \times 10^{18}$ GeV is the Planck scale associated with quantum gravity. (G is Newton’s gravitational constant.) I’ve left our a factor of 3 on purpose, and I used the symbol ~ rather than = to emphasize that we are just trying to get a feel for the order of magnitude of things. I’m using units in which Planck’s constant $\hbar$ and the speed of light c are set to one, so mass, energy, and inverse length (or inverse time) all have the same dimensions. 1 GeV means one billion electron volts, about the mass of a proton.

(To persuade yourself that this is at least roughly the right equation, you should note that a similar equation applies to an expanding spherical ball of radius a(t) with uniform mass density V. But in the case of the ball, the mass density would decrease as the ball expands. The universe is different — it can expand without diluting its mass density, so the rate of expansion $\dot a / a$ does not slow down as the expansion proceeds.)

During inflation, the scalar field $\phi$ and therefore the potential energy $V(\phi)$ were changing slowly; it’s a good approximation to assume $V$ is constant. Then the solution is

$a(t) \sim a(0) e^{Ht},$

where $H$ , the Hubble constant during inflation, is

$H \sim \frac{\sqrt{V}}{m_P}.$

To explain the smoothness of the observed universe, we require at least 50 “e-foldings” of inflation before the universe reheated — that is, inflation should have lasted for a time at least $50 H^{-1}$ .

Slow rolling

During inflation the inflaton $\phi$ rolls slowly, so slowly that friction dominates inertia — this friction results from the cosmic expansion. The speed of rolling $\dot \phi$ is determined by

$H \dot \phi \sim -V'(\phi).$

Here $V'(\phi)$ is the slope of the potential, so the right-hand side is the force exerted by the potential, which matches the frictional force on the left-hand side. The coefficient of $\dot \phi$ has to be $H$ on dimensional grounds. (Here I have blown another factor of 3, but let’s not worry about that.)

Density perturbations

The trickiest thing we need to understand is how inflation produced the density perturbations which later seeded the formation of galaxies. There are several steps to the argument.

Quantum fluctuations of the inflaton

As the universe inflates, the inflaton field is subject to quantum fluctuations, where the size of the fluctuation depends on its wavelength. Due to inflation, the wavelength increases rapidly, like $e^{Ht}$ , and once the wavelength gets large compared to $H^{-1}$ , there isn’t enough time for the fluctuation to wiggle — it gets “frozen in.” Much later, long after the reheating of the universe, the oscillation period of the wave becomes comparable to the age of the universe, and then it can wiggle again. (We say that the fluctuations “cross the horizon” at that stage.) Observations of the anisotropy of the microwave background have determined how big the fluctuations are at the time of horizon crossing. What does inflation theory say about that?

Well, first of all, how big are the fluctuations when they leave the horizon during inflation? Then the wavelength is $H^{-1}$ and the universe is expanding at the rate $H$ , so $H$ is the only thing the magnitude of the fluctuations could depend on. Since the field $\phi$ has the same dimensions as $H$ , we conclude that fluctuations have magnitude

$\delta \phi \sim H.$

From inflaton fluctuations to density perturbations

Reheating occurs abruptly when the inflaton field reaches a particular value. Because of the quantum fluctuations, some horizon volumes have larger than average values of $\phi$ and some have smaller than average values; hence different regions reheat at slightly different times. The energy density in regions that reheat earlier starts to be reduced by expansion (“red shifted”) earlier, so these regions have a smaller than average energy density. Likewise, regions that reheat later start to red shift later, and wind up having larger than average density.

When we compare different regions of comparable size, we can find the typical (root-mean-square) fluctuations $\delta t$ in the reheating time, knowing the fluctuations in $\phi$ and the rolling speed $\dot \phi$ :

$\delta t \sim \frac{\delta \phi}{\dot \phi} \sim \frac{H}{\dot\phi}.$

Small fractional fluctuations in the scale factor $a$ right after reheating produce comparable small fractional fluctuations in the energy density $\rho$ . The expansion rate right after reheating roughly matches the expansion rate $H$ right before reheating, and so we find that the characteristic size of the density perturbations is

$\delta_S\equiv\left(\frac{\delta \rho}{\rho}\right)_{hor} \sim \frac{\delta a}{a} \sim \frac{\dot a}{a} \delta t\sim \frac{H^2}{\dot \phi}.$

The subscript hor serves to remind us that this is the size of density perturbations as they cross the horizon, before they get a chance to grow due to gravitational instabilities. We have found our first important conclusion: The density perturbations have a size determined by the Hubble constant $H$ and the rolling speed $\dot \phi$ of the inflaton, up to a factor of order one which we have not tried to keep track of. Insofar as the Hubble constant and rolling speed change slowly during inflation, these density perturbations have a strength which is nearly independent of the length scale of the perturbation. From here on we will denote this dimensionless scale of the fluctuations by $\delta_S$ , where the subscript $S$ stands for “scalar”.

Perturbations in terms of the potential

Putting together $\dot \phi \sim -V' / H$ and $H^2 \sim V/{m_P}^2$ with our expression for $\delta_S$ , we find

$\delta_S^2 \sim \frac{H^4}{\dot\phi^2}\sim \frac{H^6}{V'^2} \sim \frac{1}{{m_P}^6}\frac{V^3}{V'^2}.$

The observed density perturbations are telling us something interesting about the scalar field potential during inflation.

Gravitational waves and the meaning of r

The gravitational field as well as the inflaton field is subject to quantum fluctuations during inflation. We call these tensor fluctuations to distinguish them from the scalar fluctuations in the energy density. The tensor fluctuations have an effect on the microwave anisotropy which can be distinguished in principle from the scalar fluctuations. We’ll just take that for granted here, without worrying about the details of how it’s done.

While a scalar field fluctuation with wavelength $\lambda$ and strength $\delta \phi$ carries energy density $\sim \delta\phi^2 / \lambda^2$ , a fluctuation of the dimensionless gravitation field $h$ with wavelength $\lambda$ and strength $\delta h$ carries energy density $\sim m_P^2 \delta h^2 / \lambda^2$ . Applying the same dimensional analysis we used to estimate $\delta \phi$ at horizon crossing to the rescaled field $m_P h$ , we estimate the strength $\delta_T$ of the tensor fluctuations (the fluctuations of $h$ ) as

$\delta_T^2 \sim \frac{H^2}{m_P^2}\sim \frac{V}{m_P^4}.$

From observations of the CMB anisotropy we know that $\delta_S\sim 10^{-5}$ , and now BICEP2 claims that the ratio

$r = \frac{\delta_T^2}{\delta_S^2}$

is about $r\sim 0.2$ at an angular scale on the sky of about one degree. The conclusion (being a little more careful about the O(1) factors this time) is

$V^{1/4} \sim 2 \times 10^{16}~GeV \left(\frac{r}{0.2}\right)^{1/4}.$

This is our second important conclusion: The energy density during inflation defines a mass scale, which turns our to be $2 \times 10^{16}~GeV$ for the observed value of $r$ . This is a very interesting finding because this mass scale is not so far below the Planck scale, where quantum gravity kicks in, and is in fact pretty close to theoretical estimates of the unification scale in supersymmetric grand unified theories. If this mass scale were a factor of 2 smaller, then $r$ would be smaller by a factor of 16, and hence much harder to detect.

Rolling, rolling, rolling, …

Using $\delta_S^2 \sim H^4/\dot\phi^2$ , we can express $r$ as

$r = \frac{\delta_T^2}{\delta_S^2}\sim \frac{\dot\phi^2}{m_P^2 H^2}.$

It is convenient to measure time in units of the number $N = H t$ of e-foldings of inflation, in terms of which we find

$\frac{1}{m_P^2} \left(\frac{d\phi}{dN}\right)^2\sim r;$

Now, we know that for inflation to explain the smoothness of the universe we need $N$ larger than 50, and if we assume that the inflaton rolls at a roughly constant rate during $N$ e-foldings, we conclude that, while rolling, the change in the inflaton field is

$\frac{\Delta \phi}{m_P} \sim N \sqrt{r}.$

This is our third important conclusion — the inflaton field had to roll a long, long, way during inflation — it changed by much more than the Planck scale! Putting in the O(1) factors we have left out reduces the required amount of rolling by about a factor of 3, but we still conclude that the rolling was super-Planckian if $r\sim 0.2$ . That’s curious, because when the scalar field strength is super-Planckian, we expect the kind of effective field theory we have been implicitly using to be a poor approximation because quantum gravity corrections are large. One possible way out is that the inflaton might have rolled round and round in a circle instead of in a straight line, so the field strength stayed sub-Planckian even though the distance traveled was super-Planckian.

Spectral tilt

As the inflaton rolls, the potential energy, and hence also the Hubble constant $H$ , change during inflation. That means that both the scalar and tensor fluctuations have a strength which is not quite independent of length scale. We can parametrize the scale dependence in terms of how the fluctuations change per e-folding of inflation, which is equivalent to the change per logarithmic length scale and is called the “spectral tilt.”

To keep things simple, let’s suppose that the rate of rolling is constant during inflation, at least over the length scales for which we have data. Using $\delta_S^2 \sim H^4/\dot\phi^2$ , and assuming $\dot\phi$ is constant, we estimate the scalar spectral tilt as

$-\frac{1}{\delta_S^2}\frac{d\delta_S^2}{d N} \sim - \frac{4 \dot H}{H^2}.$

Using $\delta_T^2 \sim H^2/m_P^2$ , we conclude that the tensor spectral tilt is half as big.

From $H^2 \sim V/m_P^2$ , we find

$\dot H \sim \frac{1}{2} \dot \phi \frac{V'}{V} H,$

and using $\dot \phi \sim -V'/H$ we find

$-\frac{1}{\delta_S^2}\frac{d\delta_S^2}{d N} \sim \frac{V'^2}{H^2V}\sim m_P^2\left(\frac{V'}{V}\right)^2\sim \left(\frac{V}{m_P^4}\right)\left(\frac{m_P^6 V'^2}{V^3}\right)\sim \delta_T^2 \delta_S^{-2}\sim r.$

Putting in the numbers more carefully we find a scalar spectral tilt of $r/4$ and a tensor spectral tilt of $r/8$ .

This is our last important conclusion: A relatively large value of $r$ means a significant spectral tilt. In fact, even before the BICEP2 results, the CMB anisotropy data already supported a scalar spectral tilt of about .04, which suggested something like $r \sim .16$ . The BICEP2 detection of the tensor fluctuations (if correct) has confirmed that suspicion.

Summing up

If you have stuck with me this far, and you haven’t seen this stuff before, I hope you’re impressed. Of course, everything I’ve described can be done much more carefully. I’ve tried to convey, though, that the emerging story seems to hold together pretty well. Compared to last week, we have stronger evidence now that inflation occurred, that the mass scale of inflation is high, and that the scalar and tensor fluctuations produced during inflation have been detected. One prediction is that the tensor fluctuations, like the scalar ones, should have a notable spectral tilt, though a lot more data will be needed to pin that down.

I apologize to the experts again, for the sloppiness of these arguments. I hope that I have at least faithfully conveyed some of the spirit of inflation theory in a way that seems somewhat accessible to the uninitiated. And I’m sorry there are no references, but I wasn’t sure which ones to include (and I was too lazy to track them down).

It should also be clear that much can be done to sharpen the confrontation between theory and experiment. A whole lot of fun lies ahead.

Added notes (3/25/2014):

Okay, here’s a good reference, a useful review article by Baumann. (I found out about it on Twitter!)

From Baumann’s lectures I learned a convenient notation. The rolling of the inflaton can be characterized by two “potential slow-roll parameters” defined by

$\epsilon = \frac{m_p^2}{2}\left(\frac{V'}{V}\right)^2,\quad \eta = m_p^2\left(\frac{V''}{V}\right).$

Both parameters are small during slow rolling, but the relationship between them depends on the shape of the potential. My crude approximation ( $\epsilon = \eta$ ) would hold for a quadratic potential.

We can express the spectral tilt (as I defined it) in terms of these parameters, finding $2\epsilon$ for the tensor tilt, and $6 \epsilon - 2\eta$ for the scalar tilt. To derive these formulas it suffices to know that $\delta_S^2$ is proportional to $V^3/V'^2$ , and that $\delta_T^2$ is proportional to $H^2$ ; we also use

$3H\dot \phi = -V', \quad 3H^2 = V/m_P^2,$

keeping factors of 3 that I left out before. (As a homework exercise, check these formulas for the tensor and scalar tilt.)

It is also easy to see that $r$ is proportional to $\epsilon$ ; it turns out that $r = 16 \epsilon$ . To get that factor of 16 we need more detailed information about the relative size of the tensor and scalar fluctuations than I explained in the post; I can’t think of a handwaving way to derive it.

We see, though, that the conclusion that the tensor tilt is $r/8$ does not depend on the details of the potential, while the relation between the scalar tilt and $r$ does depend on the details. Nevertheless, it seems fair to claim (as I did) that, already before we knew the BICEP2 results, the measured nonzero scalar spectral tilt indicated a reasonably large value of $r$ .

Once again, we’re lucky. On the one hand, it’s good to have a robust prediction (for the tensor tilt). On the other hand, it’s good to have a handle (the scalar tilt) for distinguishing among different inflationary models.

One last point is worth mentioning. We have set Planck’s constant $\hbar$ equal to one so far, but it is easy to put the powers of $\hbar$ back in using dimensional analysis (we’ll continue to assume the speed of light c is one). Since Newton’s constant $G$ has the dimensions of length/energy, and the potential $V$ has the dimensions of energy/volume, while $\hbar$ has the dimensions of energy times length, we see that

$\delta_T^2 \sim \hbar G^2V.$

Thus the production of gravitational waves during inflation is a quantum effect, which would disappear in the limit $\hbar \to 0$ . Likewise, the scalar fluctuation strength $\delta_S^2$ is also $O(\hbar)$ , and hence also a quantum effect.

Therefore the detection of primordial gravitational waves by BICEP2, if correct, confirms that gravity is quantized just like the other fundamental forces. That shouldn’t be a surprise, but it’s nice to know.

My 10 biggest thrills

Posted on March 21, 2014 by preskill

Wow!

BICEP2 results for the ratio r of gravitational wave perturbations to density perturbations, and the density perturbation spectral tilt n.

Evidence for gravitational waves produced during cosmic inflation. BICEP2 results for the ratio r of gravitational wave perturbations to density perturbations, and the density perturbation spectral tilt n.

Like many physicists, I have been reflecting a lot the past few days about the BICEP2 results, trying to put them in context. Other bloggers have been telling you all about it (here, here, and here, for example); what can I possibly add?

The hoopla this week reminds me of other times I have been really excited about scientific advances. And I recall some wise advice I received from Sean Carroll: blog readers like lists. So here are (in chronological order)…

My 10 biggest thrills (in science)

This is a very personal list — your results may vary. I’m not saying these are necessarily the most important discoveries of my lifetime (there are conspicuous omissions), just that, as best I can recall, these are the developments that really started my heart pounding at the time.

1) The J/Psi from below (1974)

I was a senior at Princeton during the November Revolution. I was too young to appreciate fully what it was all about — having just learned about the Weinberg-Salam model, I thought at first that the Z boson had been discovered. But by stalking the third floor of Jadwin I picked up the buzz. No, it was charm! The discovery of a very narrow charmonium resonance meant we were on the right track in two ways — charm itself confirmed ideas about the electroweak gauge theory, and the narrowness of the resonance fit in with the then recent idea of asymptotic freedom. Theory triumphant!

2) A magnetic monopole in Palo Alto (1982)

By 1982 I had been thinking about the magnetic monopoles in grand unified theories for a few years. We thought we understood why no monopoles seem to be around. Sure, monopoles would be copiously produced in the very early universe, but then cosmic inflation would blow them away, diluting their density to a hopelessly undetectable value. Then somebody saw one …. a magnetic monopole obediently passed through Blas Cabrera’s loop of superconducting wire, producing a sudden jump in the persistent current. On Valentine’s Day!

According to then current theory, the monopole mass was expected to be about 10^16 GeV (10 million billion times heavier than a proton). Had Nature really been so kind as the bless us with this spectacular message from an staggeringly high energy scale? It seemed too good to be true.

It was. Blas never detected another monopole. As far as I know he never understood what glitch had caused the aberrant signal in his device.

3) “They’re green!” High-temperature superconductivity (1987)

High-temperature superconductors were discovered in 1986 by Bednorz and Mueller, but I did not pay much attention until Paul Chu found one in early 1987 with a critical temperature of 77 K. Then for a while the critical temperature seemed to be creeping higher and higher on an almost daily basis, eventually topping 130K …. one wondered whether it might go up, up, up forever.

It didn’t. Today 138K still seems to be the record.

My most vivid memory is that David Politzer stormed into my office one day with a big grin. “They’re green!” he squealed. David did not mean that high-temperature superconductors would be good for the environment. He was passing on information he had just learned from Phil Anderson, who happened to be visiting Caltech: Chu’s samples were copper oxides.

4) “Now I have mine” Supernova 1987A (1987)

What was most remarkable and satisfying about the 1987 supernova in the nearby Large Magellanic Cloud was that the neutrinos released in a ten second burst during the stellar core collapse were detected here on earth, by gigantic water Cerenkov detectors that had been built to test grand unified theories by looking for proton decay! Not a truly fundamental discovery, but very cool nonetheless.

Soon after it happened some of us were loafing in the Lauritsen seminar room, relishing the good luck that had made the detection possible. Then Feynman piped up: “Tycho Brahe had his supernova, Kepler had his, … and now I have mine!” We were all silent for a few seconds, and then everyone burst out laughing, with Feynman laughing the hardest. It was funny because Feynman was making fun of his own gargantuan ego. Feynman knew a good gag, and I heard him use this line at a few other opportune times thereafter.

5) Science by press conference: Cold fusion (1989)

The New York Times was my source for the news that two chemists claimed to have produced nuclear fusion in heavy water using an electrochemical cell on a tabletop. I was interested enough to consult that day with our local nuclear experts Charlie Barnes, Bob McKeown, and Steve Koonin, none of whom believed it. Still, could it be true?

I decided to spend a quiet day in my office, trying to imagine ways to induce nuclear fusion by stuffing deuterium into a palladium electrode. I came up empty.

My interest dimmed when I heard that they had done a “control” experiment using ordinary water, had observed the same excess heat as with heavy water, and remained just as convinced as before that they were observing fusion. Later, Caltech chemist Nate Lewis gave a clear and convincing talk to the campus community debunking the original experiment.

6) “The face of God” COBE (1992)

I’m often too skeptical. When I first heard in the early 1980s about proposals to detect the anisotropy in the cosmic microwave background, I doubted it would be possible. The signal is so small! It will be blurred by reionization of the universe! What about the galaxy! What about the dust! Blah, blah, blah, …

The COBE DMR instrument showed it could be done, at least at large angular scales, and set the stage for the spectacular advances in observational cosmology we’ve witnessed over the past 20 years. George Smoot infamously declared that he had glimpsed “the face of God.” Overly dramatic, perhaps, but he was excited! And so was I.

7) “83 SNU” Gallex solar neutrinos (1992)

Until 1992 the only neutrinos from the sun ever detected were the relatively high energy neutrinos produced by nuclear reactions involving boron and beryllium — these account for just a tiny fraction of all neutrinos emitted. Fewer than expected were seen, a puzzle that could be resolved if neutrinos have mass and oscillate to another flavor before reaching earth. But it made me uncomfortable that the evidence for solar neutrino oscillations was based on the boron-beryllium side show, and might conceivably be explained just by tweaking the astrophysics of the sun’s core.

The Gallex experiment was the first to detect the lower energy pp neutrinos, the predominant type coming from the sun. The results seemed to confirm that we really did understand the sun and that solar neutrinos really oscillate. (More compelling evidence, from SNO, came later.) I stayed up late the night I heard about the Gallex result, and gave a talk the next day to our particle theory group explaining its significance. The talk title was “83 SNU” — that was the initially reported neutrino flux in Solar Neutrino Units, later revised downward somewhat.

8) Awestruck: Shor’s algorithm (1994)

I’ve written before about how Peter Shor’s discovery of an efficient quantum algorithm for factoring numbers changed my life. This came at a pivotal time for me, as the SSC had been cancelled six months earlier, and I was growing pessimistic about the future of particle physics. I realized that observational cosmology would have a bright future, but I sensed that theoretical cosmology would be dominated by data analysis, where I would have little comparative advantage. So I became a quantum informationist, and have not regretted it.

9) The Higgs boson at last (2012)

The discovery of the Higgs boson was exciting because we had been waiting soooo long for it to happen. Unable to stream the live feed of the announcement, I followed developments via Twitter. That was the first time I appreciated the potential value of Twitter for scientific communication, and soon after I started to tweet.

10) A lucky universe: BICEP2 (2014)

Many past experiences prepared me to appreciate the BICEP2 announcement this past Monday.

I first came to admire Alan Guth‘s distinctive clarity of thought in the fall of 1973 when he was the instructor for my classical mechanics course at Princeton (one of the best classes I ever took). I got to know him better in the summer of 1979 when I was a graduate student, and Alan invited me to visit Cornell because we were both interested in magnetic monopole production in the very early universe. Months later Alan realized that cosmic inflation could explain the isotropy and flatness of the universe, as well as the dearth of magnetic monopoles. I recall his first seminar at Harvard explaining his discovery. Steve Weinberg had to leave before the seminar was over, and Alan called as Steve walked out, “I was hoping to hear your reaction.” Steve replied, “My reaction is applause.” We all felt that way.

I was at a wonderful workshop in Cambridge during the summer of 1982, where Alan and others made great progress in understanding the origin of primordial density perturbations produced from quantum fluctuations during inflation (Bardeen, Steinhardt, Turner, Starobinsky, and Hawking were also working on that problem, and they all reached a consensus by the end of the three-week workshop … meanwhile I was thinking about the cosmological implications of axions).

I also met Andrei Linde at that same workshop, my first encounter with his mischievous grin and deadpan wit. (There was a delegation of Russians, who split their time between Xeroxing papers and watching the World Cup on TV.) When Andrei visited Caltech in 1987, I took him to Disneyland, and he had even more fun than my two-year-old daughter.

During my first year at Caltech in 1984, Mark Wise and Larry Abbott told me about their calculations of the gravitational waves produced during inflation, which they used to derive a bound on the characteristic energy scale driving inflation, a few times 10^16 GeV. We mused about whether the signal might turn out to be detectable someday. Would Nature really be so kind as to place that mass scale below the Abbott-Wise bound, yet high enough (above 10^16 GeV) to be detectable? It seemed unlikely.

Last week I caught up with the rumors about the BICEP2 results by scanning my Twitter feed on my iPad, while still lying in bed during the early morning. I immediately leapt up and stumbled around the house in the dark, mumbling to myself over and over again, “Holy Shit! … Holy Shit! …” The dog cast a curious glance my way, then went back to sleep.

Like millions of others, I was frustrated Monday morning, trying to follow the live feed of the discovery announcement broadcast from the hopelessly overtaxed Center for Astrophysics website. I was able to join in the moment, though, by following on Twitter, and I indulged in a few breathless tweets of my own.

Many of his friends have been thinking a lot these past few days about Andrew Lange, who had been the leader of the BICEP team (current senior team members John Kovac and Chao-Lin Kuo were Caltech postdocs under Andrew in the mid-2000s). One day in September 2007 he sent me an unexpected email, with the subject heading “the bard of cosmology.” Having discovered on the Internet a poem I had written to introduce a seminar by Craig Hogan, Andrew wrote:

“John,

just came across this – I must have been out of town for the event.

l love it.

it will be posted prominently in our lab today (with “LISA” replaced by “BICEP”, and remain our rallying cry till we detect the B-mode.

have you set it to music yet?

a”

I lifted a couplet from that poem for one of my tweets (while rumors were swirling prior to the official announcement):

We’ll finally know how the cosmos behaves
If we can detect gravitational waves.

Assuming the BICEP2 measurement r ~ 0.2 is really a detection of primordial gravitational waves, we have learned that the characteristic mass scale during inflation is an astonishingly high 2 X 10^16 GeV. Were it a factor of 2 smaller, the signal would have been far too small to detect in current experiments. This time, Nature really is on our side, eagerly revealing secrets about physics at a scale far, far beyond what we will every explore using particle accelerators. We feel lucky.

We physicists can never quite believe that the equations we scrawl on a notepad actually have something to do with the real universe. You would think we’d be used to that by now, but we’re not — when it happens we’re amazed. In my case, never more so than this time.

The BICEP2 paper, a historic document (if the result holds up), ends just the way it should:

“We dedicate this paper to the memory of Andrew Lange, whom we sorely miss.”

Summer of Science: Caltech InnoWorks 2013

Posted on March 4, 2014 by evgiarta

The following post is a collaboration between visiting undergraduates Evan Giarta from Stanford University and Joy Hui from Harvard University. As mentors for the 2013 Caltech InnoWorks Academy, Evan and Joy agreed to share their experience with the audience of this blog.

All throughout modern history, science and mathematics have been the foundation for engineering new, world-advancing technologies. From the wheel and sailboat to the automobile and jumbo jet, the fields of science, technology, engineering and math (STEM) have helped our world and its people move faster, farther, and forward. Now, more than ever, products that were unimaginable a century ago are used every day in households and businesses all over the earth, products made possible by generations of scientists, mathematicians, engineers and technologists.

There is, however, some troublesome news regarding the state of our nation’s math and science education. For the past few years, education and news reports have ranked the United States behind other developed countries in science and mathematics. In fact, the proportion of students that score in the most advanced levels of math and science in the United States is significantly lower than that of several other nations. This reality brings to light a stark concern: If proficiency in science and math is necessary for the engineering of novel technologies and breakthrough discoveries, but is something the next generation will lack, what will become of the production industry, our economy, and global security? While the answer to this question might range from complete and utter chaos to little or no effect, it seems reasonable to try and avoid finding out. Rather, we–the community of collective scientists, technologists, engineers, mathematicians–should seek to solve this problem: What can we do to restore the integrity and substance of the educational system, especially in science and math?

Although there is no easy answer to this question, there are ongoing efforts to address this important issue, one of which is the InnoWorks Academy. Founded in 2003 by a group of Duke undergraduates, the United InnoWorks Academy has developed summer programs that encourage underprivileged middle school students to explore the fields of science, technology, engineering, math, and medicine, (STEM^2) free of charge. The academy is sponsored by a variety of businesses and organizations, including GlaxoSmithKline, Cisco, Project Performance Corporation, Do Something, St. Andrew’s School, University of Pennsylvania, University of California, Los Angeles, and many others. InnoWorks Academy was a winner of the 2007 BRICK Awards and received both the MIT Global Community Choice Award and the MIT IDEAS Challenge Grand Prize in 2011. In ten short years, the United InnoWorks Academy has successfully conducted over 50 summer programs for more than 2,000 students through the contributions of over 1,000 volunteers. Currently, InnoWorks has chapters at about a dozen universities and is scheduled to add three more chapters this coming summer.

In early August of 2013, the Caltech InnoWorks chapter hosted its second annual summer camp, and Joy and I (Evan) were privileged to be invited as program mentors. As people on the inside, we got a first hand look at how the organization prepares and runs the week-long event and what the students themselves experience in the hands-on, interactive and collaborative one-of-a-kind opportunity that is InnoWorks.

Since Joy and I kept a diary of each day’s activities, we thought you may like to see what our middle-schoolers experienced during that week. Here is a play-by-play of the first few days from Joy’s perspective.

Monday, August 5, 2013 was the first day of our camp! Before I go on with talking about the cool things we did that day, I’d like to introduce my team. The self-titled YOLOSWAG consisted of Michael, Chase, Evan, Phaelan and myself. Michael was the oldest, and a little shy at first, but he definitely started talking when he got comfortable. Chase was respectful and polite, and we hit it off immediately. Evan loved science and had lots of questions and knew a TON. Phaelan was the only other girl on the team, but she was very nice and friendly to the other students, and eager to help with anything she could. We all seem different, right? But wait, here’s the best part: we were all die-hard Percy Jackson (property of Rick Riordan) fans! We were definitely the best team.

Anyway, back to the actual stuff we did. One of the first things we saw was a cloud demo, essentially the creation of a cloud in a fish tank, with lots of dry ice and water. The demonstrator, Rob Usiskin, stuck a lot of dry ice in the (empty) fish tank, and poured some water into the tank, which caused the water to turn into a fog, which turns out to be the exact form of a cloud! Add a bubble maker, a question of whether bubbles will float or sink on the cloud, and a room full of InnoWorks campers (about 40 of them), and you will get an hour of general excitement. See the pictures for yourself! (The bubbles, float, by the way, even when the cloud is invisible!)

Following the demonstration, we made Soap Boats. We were apparently supposed to cut index cards into the shape of boats, and dab a little bit of soap on the bottom of the boat, and set the boat in the water. These “Soap Boats” were supposed to be propelled forward by the soap’s ability to decrease the surface tension of the water it touched. Ours, appropriately named “Titanic,” however, simply sat in the water until the water soaked through, and the Titanic sank for the second time in history. Many other teams’ boats fared about the same, but we certainly had a blast designing and naming our Soap Boat!

The last activity of the day was a secret message decoded with bio-luminescence. Each team was given a vial of dried up ostracods, which are sea creatures found glowing in the darkness of the deep sea. Then, each team crushed the ostracods and mixed the resulting powder with water to catalyze the bio-luminescence. Every mentor had written a secret message on a slip of paper, folded it, and handed it to their teams to decipher in the dark (Mine said: YOLOSWAG for the win). The convenient cleaning closet provided said darkness–Spiros, our faculty mentor, suggested that it might also provide passage to Narnia. I don’t think we lost any of our students that day, so no Narnia-traveling was done; by students. Nevertheless, it was a fun-filled and action-packed day, and a great start to an eventful week.

Tuesday was full of fun, hands-on activities. After a short lesson on the effects of air resistance and gravity on free-falling objects, we demonstrated the concept with a thin sheet of paper and a large textbook. As expected, when placed side by side and dropped, air resistance caused the sheet of paper to hit the ground much later than the textbook. But when the sheet of paper was placed directly above the textbook, to the shock of students, both items fell at the exact same rate. Though thorough in their knowledge of physical laws, the connection between their conceptual understanding and real life application had yet to be established. And as a result, when asked to explain what happened and why, the best answer one could muster was simply, “SCIENCE!”.

Following an exercise consisting of blow dryers and floating ping pong balls, the kids received a brief tutorial on how tornadoes are formed by air moving through high and low pressure regions and gusts of vertically rising winds. Due to the forces it is producing and acting upon, the tornado would then be able to more or less sustain itself. To explore this concept further, students constructed water tornado machines by taping two soda bottles together at their openings. Laughter and wetness ensued. Some groups added small trinkets in their tornado machine to observed the water tornado’s effect on “debris”. One team in particular inserted duct tape sharks, and aptly renamed themselves as Sharknado.

In the afternoon, the campers were presented a lesson on the transportation of sailboats and aircraft. Contrary to what most people intuit, the fastest way to control a boat is not to flow in the direction of the wind, but to place the sail at the heading which produces a net force, which can be explained by Bernoulli’s Principle. It states that faster moving fluid has less pressure than slower moving fluid, therefore producing a force from the slower moving side to the faster moving side. Though in sailing this force is initially barely noticeable, over time it creates a large impulse to move the craft at considerable speeds. The same principle can also explain the way a plane takes flight.

With the importance of good design in mind, students were tasked with prototyping the fastest water-bottle boat. Given a solar panel, electric motor, various propellers, empty bottle, tape, and other construction essentials, kids started with basic designs, then diversified in order to gain an edge against other teams. Some teams boasted two-bottle designs, and others used one, each type having trade-offs in speed and stability. Few implemented style upgrades with graphics and colors, and even fewer leveraged performance modifications with ballast and crude control systems. But with a tough deadline to meet, not all boats met their intended specifications. Nonetheless, the races commenced and each team’s innovation was tested in the torrents of Caltech’s Beckman Fountain. Some failed, but those that survived were rewarded accordingly.

By the end of the second day, many of the campers’ initial shyness had been replaced with conversation and budding new friendships. Lunch hour and break times allowed time for kids and mentors alike to hang out and enjoy themselves in California’s summer sun in-between discovering the applications of science and math to engineering, medicine, and technology. These moments of discovery, no matter how rare, are the reasons why we do what we do as we continue our research, studies, and work to improve the world we live in.

Oh, the Places You’ll Do Theoretical Physics!

Posted on March 2, 2014 by Nicole Yunger Halpern

I won’t run lab tests in a box.

I won’t run lab tests with a fox.

But I’ll prove theorems here or there.

Yes, I’ll prove theorems anywhere…

Physicists occupy two camps. Some—theorists—model the world using math. We try to predict experiments’ outcomes and to explain natural phenomena. Others—experimentalists—gather data using supermagnets, superconductors, the world’s coldest atoms, and other instruments deserving of superlatives. Experimentalists confirm that our theories deserve trashing or—for this we pray—might not model the world inaccurately.

Theorists, people say, can work anywhere. We need no million-dollar freezers. We need no multi-pound magnets.* We need paper, pencils, computers, and coffee. Though I would add “quiet,” colleagues would add “iPods.”

Theorists’ mobility reminds me of the book Green Eggs and Ham. Sam-I-am, the antagonist, drags the protagonist to spots as outlandish as our workplaces. Today marks the author’s birthday. Since Theodor Geisel stimulated imaginations, and since imagination drives physics, Quantum Frontiers is paying its respects. In honor of Oh, the Places You’ll Go!, I’m spotlighting places you can do theoretical physics. You judge whose appetite for exotica exceeds whose: Dr. Seuss’s or theorists’.

I’ve most looked out-of-place doing physics by a dirt road between sheep-populated meadows outside Lancaster, UK. Lancaster, the War of the Roses victor, is a city in northern England. The year after graduating from college, I worked in Lancaster University as a research assistant. I studied a crystal that resembles graphene, a material whose superlatives include “superstrong,” “supercapacitor,” and “superconductor.” From morning to evening, I’d submerse in math till it poured out my ears. Then I’d trek from “uni,” as Brits say, to the “city centre,” as they write.

The trek wound between trees; fields; and, because I was in England, puddles. Many evenings, a rose or a sunset would arrest me. Other evenings, physics would. I’d realize how to solve an equation, or that I should quit banging my head against one. Stepping off the road, I’d fish out a notebook and write. Amidst the puddles and lambs. Cyclists must have thought me the queerest sight since a cloudless sky.

A colleague loves doing theory in the sky. On planes, he explained, hardly anyone interrupts his calculations. And who minds interruptions by pretzels and coffee?

“A mathematician is a device for turning coffee into theorems,” some have said, and theoretical physicists live down the block from mathematicians in the neighborhood of science. Turn a Pasadena café upside-down and shake it, and out will fall theorists. Since Hemingway’s day, the romanticism has faded from the penning of novels in cafés. But many a theorist trumpets about an equation derived on a napkin.

Trumpeting filled my workplace in Oxford. One of Clarendon Lab’s few theorists, I neighbored lasers, circuits, and signs that read “DANGER! RADIATION.” Though radiation didn’t leak through our walls (I hope), what did contributed more to that office’s eccentricity more than radiation would. As early as 9:10 AM, the experimentalists next door blasted “Born to Be Wild” and Animal House tunes. If you can concentrate over there, you can concentrate anywhere.

One paper I concentrated on had a Crumple-Horn Web-Footed Green-Bearded Schlottz of an acknowledgements section. In a physics paper’s last paragraph, one thanks funding agencies and colleagues for support and advice. “The authors would like to thank So-and-So for insightful comments,” papers read. This paper referenced a workplace: “[One coauthor] is grateful to the Half Moon Pub.” Colleagues of the coauthor confirmed the acknowledgement’s aptness.

Though I’ve dwelled on theorists’ physical locations, our minds roost elsewhere. Some loiter in atoms; others, in black holes; some, on four-dimensional surfaces; others, in hypothetical universes. I hobnob with particles in boxes. As Dr. Seuss whisks us to a Bazzim populated by Nazzim, theorists tell of function spaces populated by Rényi entropies.

The next time you see someone standing in a puddle, or in a ditch, or outside Buckingham Palace, scribbling equations, feel free to laugh. You might be seeing a theoretical physicist. You might be seeing me. To me, physics has relevance everywhere. Scribbling there and here should raise eyebrows no more than any setting in a Dr. Seuss book.

The author would like to thank this emporium of Seussoria. And Java & Co.

*We need for them to confirm that our theories deserve trashing, but we don’t need them with us. Just as, when considering quitting school to break into the movie business, you need for your mother to ask, “Are you sure that’s a good idea, dear?” but you don’t need for her to hang on your elbow. Except experimentalists don’t say “dear” when crushing theorists’ dreams.

Guns versus butter in quantum information

Posted on February 17, 2014 by Nicole Yunger Halpern

From my college’s computer-science club, I received a T-shirt that reads:

while(not_dead){

sleep--;

time--;

awesome++;

}

/*There’s a reason we can’t hang out with you…*/

The message is written in Java, a programming language. Even if you’ve never programmed, you likely catch the drift: CS majors are the bees’ knees because, at the expense of sleep and social lives, they code. I disagree with part of said drift: CS majors hung out with me despite being awesome.

The rest of the drift—you have to give some to get some—synopsizes the physics I encountered this fall. To understand tradeoffs, you needn’t study QI. But what trades off with what, according to QI, can surprise us.

The T-shirt haunted me at the University of Nottingham, where researchers are blending QI with Einstein’s theory of relativity. Relativity describes accelerations, gravity, and space-time’s curvature. In other sources, you can read about physicists’ attempts to unify relativity and quantum mechanics, the Romeo and Tybalt of modern physics, into a theory of quantum gravity. In this article, relativity tangos with quantum mechanics in relativistic quantum information (RQI). If I move my quantum computer, RQIers ask, how do I change its information processing? How does space-time’s curvature affect computation? How can motion affect measurements?

Answers to these questions involve tradeoffs.

Nottingham researchers kindly tolerating a seminar by me

For example, acceleration entangles particles. Decades ago, physicists learned that acceleration creates particles. Say you’re gazing into a vacuum—not empty space, but nearly empty space, the lowest-energy system that can exist. Zooming away on a rocket, I accelerate relative to you. From my perspective, more particles than you think—and higher-energy particles—surround us.

Have I created matter? Have I violated the Principle of Conservation of Energy (and Mass)? I created particles in a sense, but at the expense of rocket fuel. You have to give some to get some:

Fuel--;
Particles++;

The math that describes my particles relates to the math that describes entanglement.* Entanglement is a relationship between quantum systems. Say you entangle two particles, then separate them. If you measure one, you instantaneously affect the other, even if the other occupies another city.

Say we encode information in quantum particles stored in a box.** Just as you encode messages by writing letters, we write messages in the ink of quantum particles. Say the box zooms off on a rocket. Just as acceleration led me to see particles in a vacuum, acceleration entangles the particles in our box. Since entanglement facilitates computation, you can process information by shaking a box. And performing another few steps.

When an RQIer told me so, she might as well have added that space-time has 106 dimensions and the US would win the World Cup. Then my T-shirt came to mind. To get some, you have to give some. When you give something, you might get something. Giving fuel gets you entanglement. To prove that statement, I need to do and interpret math. Till I have time to,

Fuel--;
Entanglement++;

offers intuition.

After cropping up in Nottingham, my T-shirt reared its head (collar?) in physics problem after physics problem. By “consuming entanglement”—forfeiting that ability to affect the particle in another city—you can teleport quantum information.

Entanglement--;
Quantum teleportation++;

My research involves tradeoffs between information and energy. As the Hungarian physicist Leó Szilárd showed, you can exchange information for work. Say you learn which half of a box*** a particle occupies, and you trap the particle in that half. Upon freeing the particle—forfeiting your knowledge about its location—you can lift a weight, charge a battery, or otherwise store energy.

Information--;
Energy++;

If you expend energy, Rolf Landauer showed, you can gain knowledge.

Energy--;
Information++;

No wonder my computer-science friends joked about sleep deprivation. But information can energize. For fuel, I forage in the blending of fields like QI and relativity, and in physical intuitions like those encapsulated in the pseudo-Java above. Much as Szilard’s physics enchants me, I’m glad that the pursuit of physics contradicts his conclusion:

while(not_dead){

Information++;

Energy++;

}

The code includes awesome++ implicitly.

*Bogoliubov transformations, to readers familiar with the term.

**In the fields in a cavity, to readers familiar with the terms.

***Physicists adore boxes, you might have noticed.

With thanks to Ivette Fuentes and the University of Nottingham for their hospitality and for their introduction to RQI.

Making predictions in the multiverse

Image

I am a theoretical physicist at University of California, Berkeley. Last month, I attended a very interesting conference organized by Foundamental Questions Institute (FQXi) in Puerto Rico, and presented a talk about making predictions in cosmology, especially in the eternally inflating multiverse. I very much enjoyed discussions with people at the conference, where I was invited to post a non-technical account of the issue as well as my own view of it. So here I am.

I find it quite remarkable that some of us in the physics community are thinking with some “confidence” that we live in the multiverse, more specifically one of the many universes in which low-energy physical laws take different forms. (For example, these universes have different elementary particles with different properties, possibly different spacetime dimensions, and so on.) This idea of the multiverse, as we currently think, is not simply a result of random imagination by theorists, but is based on several pieces of observational and theoretical evidence.

Observationally, we have learned more and more that we live in a highly special universe—it seems that the “physical laws” of our universe (summarized in the form of standard models of particle physics and cosmology) takes such a special form that if its structure were varied slightly, then there would be no interesting structure in the universe, let alone intelligent life. It is hard to understand this fact unless there are many universes with varying “physical laws,” and we simply happen to emerge in a universe which allows for intelligent life to develop (which seems to require special conditions). With multiple universes, we can understand the “specialness” of our universe precisely as we understand the “specialness” of our planet Earth (e.g. the ideal distance from the sun), which is only one of the many planets out there.

Perhaps more nontrivial is the fact that our current theory of fundamental physics leads to this picture of the multiverse in a very natural way. Imagine that at some point in the history of the universe, space is exponentially expanding. This expansion—called inflation—occurs when space is filled with a “positive vacuum energy” (which happens quite generally). We knew, already in 80’s, that such inflation is generically eternal. During inflation, various non-inflating regions called bubble universes—of which our own universe could be one—may form, much like bubbles in boiling water. Since ambient space expands exponentially, however, these bubbles do not percolate; rather, the process of creating bubble universes lasts forever in an eternally inflating background. Now, recent progress in string theory suggests that low energy theories describing phyics in these bubble universes (such as the elementary particle content and their properties) may differ bubble by bubble. This is precisely the setup needed to understand the “specialness” of our universe because of the selection effect associated with our own existence, as described above.

A schematic depiction of the eternally inflating multiverse. The horizontal and vertical directions correspond to spatial and time directions, respectively, and various regions with the inverted triangle or argyle shape represent different universes. While regions closer to the upper edge of the diagram look smaller, it is an artifact of the rescaling made to fit the large spacetime into a finite drawing—the fractal structure near the upper edge actually corresponds to an infinite number of large universes.

This particular version of the multiverse—called the eternally inflating multiverse—is very attractive. It is theoretically motivated and has a potential to explain various features seen in our universe. The eternal nature of inflation, however, causes a serious issue of predictivity. Because the process of creating bubble universes occurs infinitely many times, “In an eternally inflating universe, anything that can happen will happen; in fact, it will happen an infinite number of times,” as phrased in an article by Alan Guth. Suppose we want to calculate the relative probability for (any) events $A$ and $B$ to happen in the multiverse. Following the standard notion of probability, we might define it as the ratio of the numbers of times events $A$ and $B$ happen throughout the whole spacetime

$P = \frac{N_A}{N_B}$ .

In the eternally inflating multiverse, however, both $A$ and $B$ occur infinitely many times: $N_A, N_B = \infty$ . This expression, therefore, is ill-defined. One might think that this is merely a technical problem—we simply need to “regularize” to make both $N_{A,B}$ finite, at a middle stage of the calculation, and then we get a well-defined answer. This is, however, not the case. One finds that depending on the details of this regularization procedure, one can obtain any “prediction” one wants, and there is no a priori preferred way to proceed over others—predictivity of physical theory seems lost!

Over the past decades, some physicists and cosmologists have been thinking about many aspects of this so-called measure problem in eternal inflation. (There are indeed many aspects to the problem, and I’m omitting most of them in my simplified presentation above.) Many of the people who contributed were in the session at the conference, including Aguirre, Albrecht, Bousso, Carroll, Guth, Page, Tegmark, and Vilenkin. My own view, which I think is shared by some others, is that this problem offers a window into deep issues associated with spacetime and gravity. In my 2011 paper I suggested that quantum mechanics plays a crucial role in understanding the multiverse, even at the largest distance scales. (A similar idea was also discussed here around the same time.) In particular, I argued that the eternally inflating multiverse and quantum mechanical many worlds a la Everett are the same concept:

Multiverse = Quantum Many Worlds

in a specific, and literal, sense. In this picture, the global spacetime of general relativity appears only as a derived concept at the cost of overcounting true degrees of freedom; in particular, infinitely large space associated with eternal inflation is a sort of “illusion.” A “true” description of the multiverse must be “intrinsically” probabilistic in a quantum mechanical sense—probabilities in cosmology and quantum measurements have the same origin.

To illustrate the basic idea, let us first consider an (apparently unrelated) system with a black hole. Suppose we drop some book $A$ into the black hole and observe subsequent evolution of the system from a distance. The book will be absorbed into (the horizon of) the black hole, which will then eventually evaporate, leaving Hawking radiation. Now, let us consider another process of dropping a different book $B$ , instead of $A$ , and see what happens. The subsequent evolution in this case is similar to the case with $A$ , and we will be left with Hawking radiation. However, this final-state Hawking radiation arising from $B$ is (believed by many to be) different from that arising from $A$ in its subtle quantum correlation structure, so that if we have perfect knowledge about the final-state radiation then we can reconstruct what the original book was. This property is called unitarity and is considered to provide the correct picture for black hole dynamics, based on recent theoretical progress. To recap, the information about the original book will not be lost—it will simply be distributed in final-state Hawking radiation in a highly scrambled form.

A puzzling thing occurs, however, if we observe the same phenomenon from the viewpoint of an observer who is falling into the black hole with a book. In this case, the equivalence principle says that the book does not feel gravity (except for the tidal force which is tiny for a large black hole), so it simply passes through the black hole horizon without any disruption. (Recently, this picture was challenged by the so-called firewall argument—the book might hit a collection of higher energy quanta called a firewall, rather than freely fall. Even if so, it does not affect our basic argument below.) This implies that all the information about the book (in fact, the book itself) will be inside the horizon at late times. On the other hand, we have just argued that from a distant observer’s point of view, the information will be outside—first on the horizon and then in Hawking radiation. Which is correct?

One might think that the information is simply duplicated: one copy inside and the other outside. This, however, cannot be the case. Quantum mechanics prohibits faithful copying of full quantum information, the so-called no-cloning theorem. Therefore, it seems that the two pictures by the two observers cannot both be correct.

The proposed solution to this puzzle is interesting—both pictures are correct, but not at the same time. The point is that one cannot be both a distant observer and a falling observer at the same time. If you are a distant observer, the information will be outside, and the interior spacetime must be viewed as non-existent since you can never access it even in principle (because of the existence of the horizon). On the other hand, if you are a falling observer, then you have the interior spacetime in which the information (the book itself) will fall, but this happens only at the cost of losing a part of spacetime in which Hawking radiation lies, which you can never access since you yourself are falling into the black hole. There is no inconsistency in either of these two pictures; only if you artificially “patch” the two pictures, which you cannot physically do, does the apparent inconsistency of information duplication occurs. This somewhat surprising aspect of a system with gravity is called black hole complementarity, pioneered by ‘t Hooft, Susskind, and their collaborators.

What does this discussion of black holes have to do with cosmology, and, in particular the eternally inflating multiverse? In cosmology our space is surrounded by a cosmological horizon. (For example, imagine that space is expanding exponentially; this makes it impossible for us to obtain any signal from regions farther than some distance because objects in these regions recede faster than speed of light. The definition of appropriate horizons in general cases is more subtle, but can be made.) The situation, therefore, is the “inside out” version of the black hole case viewed from a distant observer. As in the case of the black hole, quantum mechanics requires that spacetime on the other side of the horizon—in this case the exterior to the cosmological horizon—must be viewed as non-existent. (In the paper I made this claim based on some simple supportive calculations.) In a more technical term, a quantum state describing the system represents only the region within the horizon—there is no infinite space in any single, consistent description of the system!

If a quantum state represents only space within the horizon, then where is the multiverse, which we thought exists in an eternally inflating space further away from our own horizon? The answer is—probability! The process of creating bubble universes is a probabilistic process in the quantum mechanical sense—it occurs through quantum mechanical tunneling. This implies that, starting from some initially inflating space, we could end up with different universes probabilistically. All different universes—including our own—live in probability space. In a more technical term, a state representing eternally inflating space evolves into a superposition of terms—or branches—representing different universes, but with each of them representing only the region within its own horizon. Note that there is no concept of infinitely large space here, which led to the ill-definedness of probability. The picture of initially large multiverse, naively suggested by general relativity, appears only after “patching” pictures based on different branches together; but this vastly overcounts true degrees of freedom as was the case if we include both the interior spacetime and Hawking radiation in our description of a black hole.

The description of the multiverse presented here provides complete unification of the eternally inflating multiverse and the many worlds interpretation in quantum mechanics. Suppose the multiverse starts from some initial state $|\Psi(t_0)\rangle$ . This state evolves into a superposition of states in which various bubble universes nucleate in various locations. As time passes, a state representing each universe further evolves into a superposition of states representing various possible cosmic histories, including different outcomes of “experiments” performed within that universe. (These “experiments” may, but need not, be scientific experiments—they can be any physical processes.) At late times, the multiverse state $|\Psi(t)\rangle$ will thus contain an enormous number of terms, each of which represents a possible world that may arise from $|\Psi(t_0)\rangle$ consistently with the laws of physics. Probabilities in cosmology and microscopic processes are then both given by quantum mechanical probabilities in the same manner. The multiverse and quantum many worlds are really the same thing—they simply refer to the same phenomenon occurring at (vastly) different scales.

A schematic picture for the evolution of the multiverse state. As t increases, the state evolves into a superposition of states in which various bubble universes nucleate in various locations. Each of these states then evolves further into a superposition of states representing various possible cosmic histories, including different outcomes of experiments performed within that universe.

The picture presented here does not solve all the problems in eternally inflating cosmology. What is the actual quantum state of the multiverse? What is its “initial conditions”? What is time? How does it emerge? The picture, however, does provide a framework to address these further, deep questions, and I have recently made some progress: the basic idea is that the state of the multiverse (which may be selected uniquely by the normalizability condition) never changes, and yet time appears as an emergent concept locally in branches as physical correlations among objects (along the lines of an old idea by DeWitt). Given the length already, I will not elaborate on this new development here. If you are interested, you might want to read my paper.

It is fascinating that physicists can talk about big and deep questions like the ones discussed here based on concrete theoretical progress. Nobody really knows where these explorations will finally lead us to. It seems, however, clear that we live in an exciting era in which our scientific explorations reach beyond what we thought to be the entire physical world, our universe.

Navajo Preparatory High Visit: Reflections by Ana Brown

Posted on February 12, 2014 by Jackie O'Sullivan

Evan Miyazono and I recently visited Navajo Preparatory High School in Farmington, New Mexico, the second half of the first exchange of visitors between Caltech and Navajo Prep. Two students and two teachers from the high school visited our campus last summer, spending time touring labs, working on small science and technology projects, and sharing their background and thoughts on science education in the Navajo Nation with us. Evan and I were happy to return the favor and take a trip out to Farmington.

We spent the school days giving lectures about light and solar power to the underclassmen and having discussions with the seniors about college applications, college life, and graduate school. We took over physics classes for the entire freshmen class, and spent an hour and a half with each class teaching them the basic E&M they needed to know to understand the wave nature of light and walking them through some simple experiments with lenses and diffraction gratings. When Evan put a piece of paper on which he had cut two very thin slits in front of a green laser-pointer, “oh”s and “ah”s bubbled up from the students as they saw the diffraction pattern appear on the other side of the room. During a math class where Evan was demonstrating how the area and circumference of a circle are related in an intuitive way by breaking up the area of the circle into a rectangle with sides r and pi*r, from where I stood to the side of the classroom, I could hear a student exclaim to his friend “Math is amazing!”.

The next day, after I gave a presentation to the freshmen physics classes on solar power and photovoltaic cells, four students asked for help or resources for their science projects. Two students were making a working model of a “modernized Hogan.” The school has a hogan in the middle of campus where important ceremonies and traditional Navajo gatherings take place. The students want to build a table-top model hogan with working solar water heating and photovoltaic panel to provide warm water and lighting. Another student is working on making a smart phone app for visitors to the Navajo Nation as her senior project. Her app will inform tourists about Navajo landmarks and the significance they hold to the Navajo people. She wants to inform visitors about her culture in an effort to replace the many stereotypes held about native people and Navajos in particular with factual information. These are just a few examples of how students are honoring the traditions of their culture while also taking advantage of new technology and empowering themselves through science education.

Ana with Navajo Prep Class

I shared with the students stories of friends of mine who had grown up on the Navajo Reservation and the unique issues they confronted in college and afterward. Students who come from tribal reservations often experience homesickness to a much greater degree than other groups when they go away to college. Many of these young adults are accustomed to a very tight community and strong sense of belonging and support that they feel in their hometown, so leaving that community to join a huge group of strangers with cultures foreign from their own can be a big challenge. I gave an informal seminar to the senior class where I discussed college in general as well as issues specific to native students. I gave them advice gleaned from my own experiences as well as the experiences of some of my Navajo friends. Evan and I also answered tons of questions that the students had about all aspects of college—getting in, financial aid, being successful, and finding balance. They asked how to approach a professor about doing research in their lab and how to budget their finances, what kind of summer job they should look for, and how to balance family life and ties with college life. The seniors had so many questions that together we spent more than two hours discussing these topics and answering general and specific questions. I was really happy we could provide so much useful and desired information for these young scholars.

In our free time Evan and I got to spend more time with teachers and students, sharing meals, information, and ideas. It was great to get to know the students and teachers better and learn more about their perspectives and experiences, and how education is fitting into their lives. I look forward to continuing to develop these relationships from afar. We already have plans to mentor a couple of students with their science and senior projects and hope to recruit more Caltech grad students to serve as mentors. Supporting and encouraging these native students in their participation in math and science fields, and higher education in general, as well as helping them to maintain a strong commitment and connection to their culture and families is something that I want to continue to foster as a mentor and ally of the school. I want to thank IQIM for providing the funding for this trip that made these connections possible.

A Quantum Adventure

Posted on February 10, 2014 by phdcomics

by Jorge Cham

How do you make something that has never existed before?

I often get suggestions for comics I should draw, which I welcome because A) I like to think of PHD Comics as a global collaborative effort and B) after 17 years, I’m almost out of ideas. This particular suggestion came from Chen-Lung Hung, a postdoc in Physics at Caltech:

PANEL 1 – Ask a scientist: “What motivates you to do the research you do?”

PANEL 2 – What people expect them to answer: “This can lead to real-life applications such as A, B, C, D, etc.”

PANEL 3 – How a real scientist would answer: “Because it’s cool.”

Ok, granted, the punchline needs work. Chen-Lung also asked me to make it clear that his research has important real-life applications, should someone from NSF, who funds his work, happen to be reading this blog.

Chen-Lung’s work with Prof. Jeff Kimble of Caltech’s IQIM is the subject of the third installment in our animated series of explanations of Quantum concepts and devices.

“The problem with atoms,” Prof. Kimble said at one point during our 3-4 hour conversation, “is that they exist in three dimensional space.” I didn’t know that was a problem (unless you expect them to exist in more than 3 dimensions), but Jeff explained that it means it’s very hard to control Quantum systems because the world is wide open, and information can leak and be corrupted from any direction. After a entire academic career making breakthroughs with one type of Quantum System, he’s now directing his group towards a new, experimental type which they believe has more potential for building devices with many Quantum Objects. As Jeff says in the video, “It’s a privilege to be able to explore.”

Shaping light, trapping atoms, alligator waveguides… The goal, Jeff and Chen-lung explained, is to make systems that are “surprising.” Not surprisingly, it was really hard to draw this video. How do you depict something that has never existed before? And more importantly, do you draw alligators differently from crocodiles? (Did you know alligators only exist in two places in the world: the Southern part of the United States, and in China?).

Hopefully, those of you watching will get some understanding of some key Quantum concepts and what it takes to build and manipulate Quantum systems, but to be honest, I make these videos because I think the work is really cool.

Jeff and Chen-Lung: thanks for taking us along on this adventure of yours, the privilege is all ours.

Watch the third installment of this series:

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

CREDITS:

Featuring: Jeff Kimble and Chen-Lung Hung
Animated by Jorge Cham

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 and the Betty and Gordon Moore Foundation.

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