Science – Page 7 – Sean Carroll

Guest Post: Grant Remmen on Entropic Gravity

February 8, 2016 / 10 Comments

“Understanding quantum gravity” is on every physicist’s short list of Big Issues we would all like to know more about. If there’s been any lesson from last half-century of serious work on this problem, it’s that the answer is likely to be something more subtle than just “take classical general relativity and quantize it.” Quantum gravity doesn’t seem to be an ordinary quantum field theory.

In that context, it makes sense to take many different approaches and see what shakes out. Alongside old stand-bys such as string theory and loop quantum gravity, there are less head-on approaches that try to understand how quantum gravity can really be so weird, without proposing a specific and complete model of what it might be.

Grant Remmen, a graduate student here at Caltech, has been working with me recently on one such approach, dubbed entropic gravity. We just submitted a paper entitled “What Is the Entropy in Entropic Gravity?” Grant was kind enough to write up this guest blog post to explain what we’re talking about.

Meanwhile, if you’re near Pasadena, Grant and his brother Cole have written a musical, Boldly Go!, which will be performed at Caltech in a few weeks. You won’t want to miss it!

One of the most exciting developments in theoretical physics in the past few years is the growing understanding of the connections between gravity, thermodynamics, and quantum entanglement. Famously, a complete quantum mechanical theory of gravitation is difficult to construct. However, one of the aspects that we are now coming to understand about quantum gravity is that in the final theory, gravitation and even spacetime itself will be closely related to, and maybe even emergent from, the mysterious quantum mechanical property known as entanglement.

This all started several decades ago, when Hawking and others realized that black holes behave with many of the same aspects as garden-variety thermodynamic systems, including temperature, entropy, etc. Most importantly, the black hole’s entropy is equal to its area [divided by (4 times Newton’s constant)]. Attempts to understand the origin of black hole entropy, along with key developments in string theory, led to the formulation of the holographic principle – see, for example, the celebrated AdS/CFT correspondence – in which quantum gravitational physics in some spacetime is found to be completely described by some special non-gravitational physics on the boundary of the spacetime. In a nutshell, one gets a gravitational universe as a “hologram” of a non-gravitational universe.

If gravity can emerge from, or be equivalent to, a set of physical laws without gravity, then something special about that non-gravitational physics has to make it happen. Physicists have now found that that special something is quantum entanglement: the special correlations among quantum mechanical particles that defies classical description. As a result, physicists are very interested in how to get the dynamics describing how spacetime is shaped and moves – Einstein’s equation of general relativity – from various properties of entanglement. In particular, it’s been suggested that the equations of gravity can be shown to come from some notion of entropy. As our universe is quantum mechanical, we should think about the entanglement entropy, a measure of the degree of correlation of quantum subsystems, which for thermal states matches the familiar thermodynamic notion of entropy.

The general idea is as follows: Inspired by black hole thermodynamics, suppose that there’s some more general notion, in which you choose some region of spacetime, compute its area, and find that when its area changes this is associated with a change in entropy. (I’ve been vague here as to what is meant by a “change” in the area and what system we’re computing the area of – this will be clarified soon!) Next, you somehow relate the entropy to an energy (e.g., using thermodynamic relations). Finally, you write the change in area in terms of a change in the spacetime curvature, using differential geometry. Putting all the pieces together, you get a relation between an energy and the curvature of spacetime, which if everything goes well, gives you nothing more or less than Einstein’s equation! This program can be broadly described as entropic gravity and the idea has appeared in numerous forms. With the plethora of entropic gravity theories out there, we realized that there was a need to investigate what categories they fall into and whether their assumptions are justified – this is what we’ve done in our recent work.

In particular, there are two types of theories in which gravity is related to (entanglement) entropy, which we’ve called holographic gravity and thermodynamic gravity in our paper. The difference between the two is in what system you’re considering, how you define the area, and what you mean by a change in that area.

In holographic gravity, you consider a region and define the area as that of its boundary, then consider various alternate configurations and histories of the matter in that region to see how the area would be different. Recent work in AdS/CFT, in which Einstein’s equation at linear order is equivalent to something called the “entanglement first law”, falls into the holographic gravity category. This idea has been extended to apply outside of AdS/CFT by Jacobson (2015). Crucially, Jacobson’s idea is to apply holographic mathematical technology to arbitrary quantum field theories in the bulk of spacetime (rather than specializing to conformal field theories – special physical models – on the boundary as in AdS/CFT) and thereby derive Einstein’s equation. However, in this work, Jacobson needed to make various assumptions about the entanglement structure of quantum field theories. In our paper, we showed how to justify many of those assumptions, applying recent results derived in quantum field theory (for experts, the form of the modular Hamiltonian and vacuum-subtracted entanglement entropy on null surfaces for general quantum field theories). Thus, we are able to show that the holographic gravity approach actually seems to work!

On the other hand, thermodynamic gravity is of a different character. Though it appears in various forms in the literature, we focus on the famous work of Jacobson (1995). In thermodynamic gravity, you don’t consider changing the entire spacetime configuration. Instead, you imagine a bundle of light rays – a lightsheet – in a particular dynamical spacetime background. As the light rays travel along – as you move down the lightsheet – the rays can be focused by curvature of the spacetime. Now, if the bundle of light rays started with a particular cross-sectional area, you’ll find a different area later on. In thermodynamic gravity, this is the change in area that goes into the derivation of Einstein’s equation. Next, one assumes that this change in area is equivalent to an entropy – in the usual black hole way with a factor of 1/(4 times Newton’s constant) – and that this entropy can be interpreted thermodynamically in terms of an energy flow through the lightsheet. The entropy vanishes from the derivation and the Einstein equation almost immediately appears as a thermodynamic equation of state. What we realized, however, is that what the entropy is actually the entropy of was ambiguous in thermodynamic gravity. Surprisingly, we found that there doesn’t seem to be a consistent definition of the entropy in thermodynamic gravity – applying quantum field theory results for the energy and entanglement entropy, we found that thermodynamic gravity could not simultaneously reproduce the correct constant in the Einstein equation and in the entropy/area relation for black holes.

So when all is said and done, we’ve found that holographic gravity, but not thermodynamic gravity, is on the right track. To answer our own question in the title of the paper, we found – in admittedly somewhat technical language – that the vacuum-subtracted von Neumann entropy evaluated on the null boundary of small causal diamonds gives a consistent formulation of holographic gravity. The future looks exciting for finding the connections between gravity and entanglement!

Guest Post: Grant Remmen on Entropic Gravity Read More »

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Quantum Fluctuations

January 16, 2016 / 41 Comments

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The Big Picture: Table of Contents

December 10, 2015 / 37 Comments

Greetings, surface-dwellers! I have finally emerged from the secret underground laboratory where I have been polishing the manuscript for The Big Picture: On the Origins of Life, Meaning, and the Universe Itself. We pushed up the publication date to May 10, so you’ll get it in plenty of time for your summer beach reading. Evidence that it exists, all 145,000 glorious words:

As will happen in the writing process, the organization of the book has changed since I first mentioned it. Here is the final table of contents. As you might gather, I went with an organization of many short chapters. Hopefully that will help give the book the feeling of a light and enjoyable read.

THE BIG PICTURE: ON THE ORIGINS OF LIFE, MEANING, AND THE UNIVERSE ITSELF

0. Prologue

* Part One: Cosmos

1. The Fundamental Nature of Reality
2. Poetic Naturalism
3. The World Moves By Itself
4. What Determines What Will Happen Next?
5. Reasons Why
6. Our Universe
7. Time’s Arrow
8. Memories and Causes

* Part Two: Understanding

9. Learning About the World
10. Updating Our Knowledge
11. Is It Okay to Doubt Everything?
12. Reality Emerges
13. What Exists, and What Is Illusion?
14. Planets of Belief
15. Accepting Uncertainty
16. What Can We Know About the Universe Without Looking at It?
17. Who Am I?
18. Abducting God

* Part Three: Essence

19. How Much We Know
20. The Quantum Realm
21. Interpreting Quantum Mechanics
22. The Core Theory
23. The Stuff of Which We Are Made
24. The Effective Theory of the Everyday World
25. Why Does the Universe Exist?
26. Body and Soul
27. Death Is the End

* Part Four: Complexity

28. The Universe in a Cup of Coffee
29. Light and Life
30. Funneling Energy
31. Spontaneous Organization
32. The Origin and Purpose of Life
33. Evolution’s Bootstraps
34. Searching Through the Landscape
35. Emergent Purpose
36. Are We the Point?

* Part Five: Thinking

37. Crawling Into Consciousness
38. The Babbling Brain
39. What Thinks?
40. The Hard Problem
41. Zombies and Stories
42. Are Photons Conscious?
43. What Acts on What?
44. Freedom to Choose

* Part Six: Caring

45. Three Billion Heartbeats
46. What Is and What Ought to Be
47. Rules and Consequences
48. Constructing Goodness
49. Listening to the World
50. Existential Therapy

Appendix: The Equation Underlying You and Me
Acknowledgments
Further Reading
References
Index

A lot of ground gets covered. In Part One we set the stage, seeing how discoveries in science have revealed a universe that runs under unbreakable, impersonal laws of nature. In Part Two we think about how to conceptualize such a universe: how to learn about it (Bayesian inference, abduction) and how to talk about it (emergence and overlapping theoretical vocabularies). In Part Three we get down and dirty with quantum mechanics, the Core Theory, and effective field theories. In Part Four we start down the road of connecting to our macroscopic world, seeing how complexity and life can arise due to the arrow of time. In Part Five we think about the leading challenge to a physicalist worldview: the existence of consciousness. And in Part Six we recognize that the universe isn’t going to tell us how to behave, and acknowledge that the creation of meaning and purpose is ultimately our job.

Now back to being a scientist with me. I have drafts of four different papers on my computer that need to be kicked out and onto the arxiv!

The Big Picture: Table of Contents Read More »

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Thanksgiving

November 26, 2015 / 15 Comments

This year we give thanks for an area of mathematics that has become completely indispensable to modern theoretical physics: Riemannian Geometry. (We’ve previously given thanks for the Standard Model Lagrangian, Hubble’s Law, the Spin-Statistics Theorem, conservation of momentum, effective field theory, the error bar, gauge symmetry, Landauer’s Principle, and the Fourier Transform. Ten years of giving thanks!)

Now, the thing everyone has been giving thanks for over the last few days is Albert Einstein’s general theory of relativity, which by some measures was introduced to the world exactly one hundred years ago yesterday. But we don’t want to be everybody, and besides we’re a day late. So it makes sense to honor the epochal advance in mathematics that directly enabled Einstein’s epochal advance in our understanding of spacetime.

Highly popularized accounts of the history of non-Euclidean geometry often give short shrift to Riemann, for reasons I don’t quite understand. You know the basic story: Euclid showed that geometry could be axiomatized on the basis of a few simple postulates, but one of them (the infamous Fifth Postulate) seemed just a bit less natural than the others. That’s the parallel postulate, which has been employed by generations of high-school geometry teachers to torture their students by challenging them to “prove” it. (Mine did, anyway.)

It can’t be proved, and indeed it’s not even necessarily true. In the ordinary flat geometry of a tabletop, initially parallel lines remain parallel forever, and Euclidean geometry is the name of the game. But we can imagine surfaces on which initially parallel lines diverge, such as a saddle, or ones on which they begin to come together, such as a sphere. In those contexts it is appropriate to replace the parallel postulate with something else, and we end up with non-Euclidean geometry.

Historically, this was first carried out by Hungarian mathematician János Bolyai and the Russian mathematician Nikolai Lobachevsky, both of whom developed the hyperbolic (saddle-shaped) form of the alternative theory. Actually, while Bolyai and Lobachevsky were the first to publish, much of the theory had previously been worked out by the great Carl Friedrich Gauss, who was an incredibly influential mathematician but not very good about getting his results into print.

The new geometry developed by Bolyai and Lobachevsky described what we would now call “spaces of constant negative curvature.” Such a space is curved, but in precisely the same way at every point; there is no difference between what’s happening at one point in the space and what’s happening anywhere else, just as had been the case for Euclid’s tabletop geometry.

Real geometries, as takes only a moment to visualize, can be a lot more complicated than that. Surfaces or solids can twist and turn in all sorts of ways. Gauss thought about how to deal with this problem, and came up with some techniques that could characterize a two-dimensional curved surface embedded in a three-dimensional Euclidean space. Which is pretty great, but falls far short of the full generality that mathematicians are known to crave.

Fortunately Gauss had a brilliant and accomplished apprentice: his student Bernard Riemann. (Riemann was supposed to be studying theology, but he became entranced by one of Gauss’s lectures, and never looked back.) In 1853, Riemann was coming up for Habilitation, a German degree that is even higher than the Ph.D. He suggested a number of possible dissertation topics to his advisor Gauss, who (so the story goes) chose the one that Riemann thought was the most boring: the foundations of geometry. The next year, he presented his paper, “On the hypotheses which underlie geometry,” which laid out what we now call Riemannian geometry.

With this one paper on a subject he professed not to be all that interested in, Riemann (who also made incredible contributions to analysis and number theory) provided everything you need to understand the geometry of a space of arbitrary numbers of dimensions, with an arbitrary amount of curvature at any point in the space. It was as if Bolyai and Lobachevsky had invented the abacus, Gauss came up with the pocket calculator, and Riemann had turned around a built a powerful supercomputer.

Like many great works of mathematics, a lot of new superstructure had to be built up along the way. A subtle but brilliant part of Riemann’s work is that he didn’t start with a larger space (like the three-dimensional almost-Euclidean world around us) and imagine smaller spaces embedded with it. Rather, he considered the intrinsic geometry of a space, or how it would look “from the inside,” whether or not there was any larger space at all.

Next, Riemann needed a tool to handle a simple but frustrating fact of life: “curvature” is not a single number, but a way of characterizing many questions one could possibly ask about the geometry of a space. What you need, really, are tensors, which gather a set of numbers together in one elegant mathematical package. Tensor analysis as such didn’t really exist at the time, not being fully developed until 1890, but Riemann was able to use some bits and pieces of the theory that had been developed by Gauss.

Finally and most importantly, Riemann grasped that all the facts about the geometry of a space could be encoded in a simple quantity: the distance along any curve we might want to draw through the space. He showed how that distance could be written in terms of a special tensor, called the metric. You give me segment along a curve inside the space you’re interested in, the metric lets me calculate how long it is. This simple object, Riemann showed, could ultimately be used to answer any query you might have about the shape of a space — the length of curves, of course, but also the area of surfaces and volume of regions, the shortest-distance path between two fixed points, where you go if you keep marching “forward” in the space, the sum of the angles inside a triangle, and so on.

Unfortunately, the geometric information implied by the metric is only revealed when you follow how the metric changes along a curve or on some surface. What Riemann wanted was a single tensor that would tell you everything you needed to know about the curvature at each point in its own right, without having to consider curves or surfaces. So he showed how that could be done, by taking appropriate derivatives of the metric, giving us what we now call the Riemann curvature tensor. Here is the formula for it:

This isn’t the place to explain the whole thing, but I can recommend some spiffy lecture notes, including a very short version, or the longer and sexier textbook. From this he deduced several interesting features about curvature. For example, the intrinsic curvature of a one-dimensional space (a line or curve) is alway precisely zero. Its extrinsic curvature — how it is embedded in some larger space — can be complicated, but to a tiny one-dimensional being, all spaces have the same geometry. For two-dimensional spaces there is a single function that characterizes the curvature at each point; in three dimensions you need six numbers, in four you need twenty, and it goes up from there.

There were more developments in store for Riemannian geometry, of course, associated with names that are attached to various tensors and related symbols: Christoffel, Ricci, Levi-Civita, Cartan. But to a remarkable degree, when Albert Einstein needed the right mathematics to describe his new idea of dynamical spacetime, Riemann had bequeathed it to him in a plug-and-play form. Add the word “time” everywhere we’ve said “space,” introduce some annoying minus signs because time and space really aren’t precisely equivalent, and otherwise the geometry that Riemann invented is the same we use today to describe how the universe works.

Riemann died of tuberculosis before he reached the age of forty. He didn’t do bad for such a young guy; you know you’ve made it when you not only have a Wikipedia page for yourself, but a separate (long) Wikipedia page for the list of things named after you. We can all be thankful that Riemann’s genius allowed him to grasp the tricky geometry of curved spaces several decades before Einstein would put it to use in the most beautiful physical theory ever invented.

E = mc^2: How Einstein’s theory of relativity changed everything

Watch this video on YouTube

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The Universe Never Expands Faster Than the Speed of Light

October 13, 2015 / 140 Comments

Breaking my radio silence here to get a little nitpick off my chest: the claim that during inflation, the universe “expanded faster than the speed of light.” It’s extraordinarily common, if utterly and hopelessly incorrect. (I just noticed it in this otherwise generally excellent post by Fraser Cain.) A Google search for “inflation superluminal expansion” reveals over 100,000 hits, although happily a few of the first ones are brave attempts to squelch the misconception. I can recommend this nice article by Tamara Davis and Charlie Lineweaver, which tries to address this and several other cosmological misconceptions.

This isn’t, by the way, one of those misconceptions that rattles around the popular-explanation sphere, while experts sit back silently and roll their eyes. Experts get this one wrong all the time. “Inflation was a period of superluminal expansion” is repeated, for example, in these texts by by Tai-Peng Cheng, by Joel Primack, and by Lawrence Krauss, all of whom should certainly know better.

The great thing about the superluminal-expansion misconception is that it’s actually a mangle of several different problems, which sadly don’t cancel out to give you the right answer.

1.The expansion of the universe doesn’t have a “speed.” Really the discussion should begin and end right there. Comparing the expansion rate of the universe to the speed of light is like comparing the height of a building to your weight. You’re not doing good scientific explanation; you’ve had too much to drink and should just go home.The expansion of the universe is quantified by the Hubble constant, which is typically quoted in crazy units of kilometers per second per megaparsec. That’s (distance divided by time) divided by distance, or simply 1/time. Speed, meanwhile, is measured in distance/time. Not the same units! Comparing the two concepts is crazy.

Admittedly, you can construct a quantity with units of velocity from the Hubble constant, using Hubble’s law, v = Hd (the apparent velocity of a galaxy is given by the Hubble constant times its distance). Individual galaxies are indeed associated with recession velocities. But different galaxies, manifestly, have different velocities. The idea of even talking about “the expansion velocity of the universe” is bizarre and never should have been entertained in the first place.

2. There is no well-defined notion of “the velocity of distant objects” in general relativity. There is a rule, valid both in special relativity and general relativity, that says two objects cannot pass by each other with relative velocities faster than the speed of light. In special relativity, where spacetime is a fixed, flat, Minkowskian geometry, we can pick a global reference frame and extend that rule to distant objects. In general relativity, we just can’t. There is simply no such thing as the “velocity” between two objects that aren’t located in the same place. If you tried to measure such a velocity, you would have to parallel transport the motion of one object to the location of the other one, and your answer would completely depend on the path that you took to do that. So there can’t be any rule that says that velocity can’t be greater than the speed of light. Period, full stop, end of story.

Except it’s not quite the end of the story, since under certain special circumstances it’s possible to define quantities that are kind-of sort-of like a velocity between distant objects. Cosmology, where we model the universe as having a preferred reference frame defined by the matter filling space, is one such circumstance. When galaxies are not too far away, we can measure their cosmological redshifts, pretend that it’s a Doppler shift, and work backwards to define an “apparent velocity.” Good for you, cosmologists! But that number you’ve defined shouldn’t be confused with the actual relative velocity between two objects passing by each other. In particular, there’s no reason whatsoever that this apparent velocity can’t be greater than the speed of light.

Sometimes this idea is mangled into something like “the rule against superluminal velocities doesn’t refer to the expansion of space.” A good try, certainly well-intentioned, but the problem is deeper than that. The rule against superluminal velocities only refers to relative velocities between two objects passing right by each other.

3. There is nothing special about the expansion rate during inflation. If you want to stubbornly insist on treating the cosmological apparent velocity as a real velocity, just so you can then go and confuse people by saying that sometimes that velocity can be greater than the speed of light, I can’t stop you. But it can be — and is! — greater than the speed of light at any time in the history of the universe, not just during inflation. There are galaxies sufficiently distant that their apparent recession velocities today are greater than the speed of light. To give people the impression that what’s special about inflation is that the universe is expanding faster than light is a crime against comprehension and good taste.

What’s special about inflation is that the universe is accelerating. During inflation (as well as today, since dark energy has taken over), the scale factor, which characterizes the relative distance between comoving points in space, is increasing faster and faster, rather than increasing but at a gradually diminishing rate. As a result, if you looked at one particular galaxy over time, its apparent recession velocity would be increasing. That’s a big deal, with all sorts of interesting and important cosmological ramifications. And it’s not that hard to explain.

But it’s not superluminal expansion. If you’re sitting at a stoplight in your Tesla, kick it into insane mode, and accelerate to 60 mph in 3.5 seconds, you won’t get a ticket for speeding, as long as the speed limit itself is 60 mph or greater. You can still get a ticket — there’s such a thing as reckless driving, after all — but if you’re hauled before the traffic judge on a count of speeding, you should be able to get off scot-free.

Many “misconceptions” in physics stem from an honest attempt to explain technical concepts in natural language, and I try to be very forgiving about those. This one, I believe, isn’t like that; it’s just wrongity-wrong wrong. The only good quality of the phrase “inflation is a period of superluminal expansion” is that it’s short. It conveys the illusion of understanding, but that can be just as bad as straightforward misunderstanding. Every time it is repeated, people’s appreciation of how the universe works gets a little bit worse. We should be able to do better.

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The Big Picture

September 28, 2015 / 59 Comments

Once again I have not really been the world’s most conscientious blogger, have I? Sometimes other responsibilities have to take precedence — such as looming book deadlines. And I’m working on a new book, and that deadline is definitely looming!

And here it is. The title is The Big Picture: On the Origins of Life, Meaning, and the Universe Itself. It’s scheduled to be published on May 17, 2016; you can pre-order it at Amazon and elsewhere right now.

An alternative subtitle was What Is, and What Matters. It’s a cheerfully grandiose (I’m supposed to say “ambitious”) attempt to connect our everyday lives to the underlying laws of nature. That’s a lot of ground to cover: I need to explain (what I take to be) the right way to think about the fundamental nature of reality, what the laws of physics actually are, sketch some cosmology and connect to the arrow of time, explore why there is something rather than nothing, show how interesting complex structures can arise in an undirected universe, talk about the meaning of consciousness and how it can be purely physical, and finally trying to understand meaning and morality in a universe devoid of transcendent purpose. I’m getting tired just thinking about it.

From another perspective, the book is an explication of, and argument for, naturalism — and in particular, a flavor I label Poetic Naturalism. The “Poetic” simply means that there are many ways of talking about the world, and any one that is both (1) useful, and (2) compatible with the underlying fundamental reality, deserves a place at the table. Some of those ways of talking will simply be emergent descriptions of physics and higher levels, but some will also be matters of judgment and meaning.

As of right now the book is organized into seven parts, each with several short chapters. All that is subject to change, of course. But this will give you the general idea.

* Part One: Being and Stories

How we think about the fundamental nature of reality. Poetic Naturalism: there is only one world, but there are many ways of talking about it. Suggestions of naturalism: the world moves by itself, time progresses by moments rather than toward a goal. What really exists.

* Part Two: Knowledge and Belief

Telling different stories about the same underlying truth. Acquiring and updating reliable beliefs. Knowledge of our actual world is never perfect. Constructing consistent planets of belief, guarding against our biases.

* Part Three: Time and Cosmos

The structure and development of our universe. Time’s arrow and cosmic history. The emergence of memories, causes, and reasons. Why is there a universe at all, and is it best explained by something outside itself?

* Part Four: Essence and Possibility

Drawing the boundary between known and unknown. The quantum nature of deep reality: observation, entanglement, uncertainty. Vibrating fields and the Core Theory underlying everyday life. What we can say with confidence about life and the soul.

* Part Five: Complexity and Evolution

Why complex structures naturally arise as the universe moves from order to disorder. Self-organization and incremental progress. The origin of life, and its physical purpose. The anthropic principle, environmental selection, and our role in the universe.

* Part Six: Thinking and Feeling

The mind, the brain, and the body. What consciousness is, and how it might have come to be. Contemplating other times and possible worlds. The emergence of inner experiences from non-conscious matter. How free will is compatible with physics.

* Part Seven: Caring and Mattering

Why we can’t derive ought from is, even if “is” is all there is. And why we nevertheless care about ourselves and others, and why that matters. Constructing meaning and morality in our universe. Confronting the finitude of life, deciding what stories we want to tell along the way.

Hope that whets the appetite a bit. Now back to work with me.

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The Bayesian Second Law of Thermodynamics

August 11, 2015 / 49 Comments

Entropy increases. Closed systems become increasingly disordered over time. So says the Second Law of Thermodynamics, one of my favorite notions in all of physics.

At least, entropy usually increases. If we define entropy by first defining “macrostates” — collections of individual states of the system that are macroscopically indistinguishable from each other — and then taking the logarithm of the number of microstates per macrostate, as portrayed in this blog’s header image, then we don’t expect entropy to always increase. According to Boltzmann, the increase of entropy is just really, really probable, since higher-entropy macrostates are much, much bigger than lower-entropy ones. But if we wait long enough — really long, much longer than the age of the universe — a macroscopic system will spontaneously fluctuate into a lower-entropy state. Cream and coffee will unmix, eggs will unbreak, maybe whole universes will come into being. But because the timescales are so long, this is just a matter of intellectual curiosity, not experimental science.

That’s what I was taught, anyway. But since I left grad school, physicists (and chemists, and biologists) have become increasingly interested in ultra-tiny systems, with only a few moving parts. Nanomachines, or the molecular components inside living cells. In systems like that, the occasional downward fluctuation in entropy is not only possible, it’s going to happen relatively frequently — with crucial consequences for how the real world works.

Accordingly, the last fifteen years or so has seen something of a revolution in non-equilibrium statistical mechanics — the study of statistical systems far from their happy resting states. Two of the most important results are the Crooks Fluctuation Theorem (by Gavin Crooks), which relates the probability of a process forward in time to the probability of its time-reverse, and the Jarzynski Equality (by Christopher Jarzynski), which relates the change in free energy between two states to the average amount of work done on a journey between them. (Professional statistical mechanics are so used to dealing with inequalities that when they finally do have an honest equation, they call it an “equality.”) There is a sense in which these relations underlie the good old Second Law; the Jarzynski equality can be derived from the Crooks Fluctuation Theorem, and the Second Law can be derived from the Jarzynski Equality. (Though the three relations were discovered in reverse chronological order from how they are used to derive each other.)

Still, there is a mystery lurking in how we think about entropy and the Second Law — a puzzle that, like many such puzzles, I never really thought about until we came up with a solution. Boltzmann’s definition of entropy (logarithm of number of microstates in a macrostate) is very conceptually clear, and good enough to be engraved on his tombstone. But it’s not the only definition of entropy, and it’s not even the one that people use most often.

Rather than referring to macrostates, we can think of entropy as characterizing something more subjective: our knowledge of the state of the system. That is, we might not know the exact position x and momentum p of every atom that makes up a fluid, but we might have some probability distribution ρ(x,p) that tells us the likelihood the system is in any particular state (to the best of our knowledge). Then the entropy associated with that distribution is given by a different, though equally famous, formula:

$S = - \int \rho \log \rho.$

That is, we take the probability distribution ρ, multiply it by its own logarithm, and integrate the result over all the possible states of the system, to get (minus) the entropy. A formula like this was introduced by Boltzmann himself, but these days is often associated with Josiah Willard Gibbs, unless you are into information theory, where it’s credited to Claude Shannon. Don’t worry if the symbols are totally opaque; the point is that low entropy means we know a lot about the specific state a system is in, and high entropy means we don’t know much at all.

In appropriate circumstances, the Boltzmann and Gibbs formulations of entropy and the Second Law are closely related to each other. But there’s a crucial difference: in a perfectly isolated system, the Boltzmann entropy tends to increase, but the Gibbs entropy stays exactly constant. In an open system — allowed to interact with the environment — the Gibbs entropy will go up, but it will only go up. It will never fluctuate down. (Entropy can decrease through heat loss, if you put your system in a refrigerator or something, but you know what I mean.) The Gibbs entropy is about our knowledge of the system, and as the system is randomly buffeted by its environment we know less and less about its specific state. So what, from the Gibbs point of view, can we possibly mean by “entropy rarely, but occasionally, will fluctuate downward”?

I won’t hold you in suspense. Since the Gibbs/Shannon entropy is a feature of our knowledge of the system, the way it can fluctuate downward is for us to look at the system and notice that it is in a relatively unlikely state — thereby gaining knowledge.

But this operation of “looking at the system” doesn’t have a ready implementation in how we usually formulate statistical mechanics. Until now! My collaborators Tony Bartolotta, Stefan Leichenauer, Jason Pollack, and I have written a paper formulating statistical mechanics with explicit knowledge updating via measurement outcomes. (Some extra figures, animations, and codes are available at this web page.)

The Bayesian Second Law of Thermodynamics
Anthony Bartolotta, Sean M. Carroll, Stefan Leichenauer, and Jason Pollack

We derive a generalization of the Second Law of Thermodynamics that uses Bayesian updates to explicitly incorporate the effects of a measurement of a system at some point in its evolution. By allowing an experimenter’s knowledge to be updated by the measurement process, this formulation resolves a tension between the fact that the entropy of a statistical system can sometimes fluctuate downward and the information-theoretic idea that knowledge of a stochastically-evolving system degrades over time. The Bayesian Second Law can be written as ΔH(ρ_m,ρ)+⟨Q⟩_F|m≥0, where ΔH(ρ_m,ρ) is the change in the cross entropy between the original phase-space probability distribution ρ and the measurement-updated distribution ρ_m, and ⟨Q⟩_F|m is the expectation value of a generalized heat flow out of the system. We also derive refined versions of the Second Law that bound the entropy increase from below by a non-negative number, as well as Bayesian versions of the Jarzynski equality. We demonstrate the formalism using simple analytical and numerical examples.

The crucial word “Bayesian” here refers to Bayes’s Theorem, a central result in probability theory. …

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Hypnotized by Quantum Mechanics

August 3, 2015 / 102 Comments

It remains embarrassing that physicists haven’t settled on the best way of formulating quantum mechanics (or some improved successor to it). I’m partial to Many-Worlds, but there are other smart people out there who go in for alternative formulations: hidden variables, dynamical collapse, epistemic interpretations, or something else. And let no one say that I won’t let alternative voices be heard! (Unless you want to talk about propellantless space drives, which are just crap.)

So let me point you to this guest post by Anton Garrett that Peter Coles just posted at his blog:

Hidden Variables: Just a Little Shy?

It’s quite a nice explanation of how the state of play looks to someone who is sympathetic to a hidden-variables view. (Fans of Bell’s Theorem should remember that what Bell did was to show that such variables must be nonlocal, not that they are totally ruled out.)

As a dialogue, it shares a feature that has been common to that format since the days of Plato: there are two characters, and the character that sympathizes with the author is the one who gets all the good lines. In this case the interlocutors are a modern physicist Neo, and a smart recently-resurrected nineteenth-century physicist Nino. Trained in the miraculous successes of the Newtonian paradigm, Nino is very disappointed that physicists of the present era are so willing to simply accept a theory that can’t do better than predicting probabilistic outcomes for experiments. More in sorrow than in anger, he urges us to do better!

My own takeaway from this is that it’s not a good idea to take advice from nineteenth-century physicists. Of course we should try to do better, since we should alway try that. But we should also feel free to abandon features of our best previous theories when new data and ideas come along.

A nice feature of the dialogue between Nino and Neo is the way in which it illuminates the fact that much of one’s attitude toward formulations of quantum mechanics is driven by which basic assumptions about the world we are most happy to abandon, and which we prefer to cling to at any cost. That’s true for any of us — such is the case when there is legitimate ambiguity about the best way to move forward in science. It’s a feature, not a bug. The hope is that eventually we will be driven, by better data and theories, toward a common conclusion.

What I like about Many-Worlds is that it is perfectly realistic, deterministic, and ontologically minimal, and of course it fits the data perfectly. Equally importantly, it is a robust and flexible framework: you give me your favorite Hamiltonian, and we instantly know what the many-worlds formulation of the theory looks like. You don’t have to think anew and invent new variables for each physical situation, whether it’s a harmonic oscillator or quantum gravity.

Of course, one gives something up: in Many-Worlds, while the underlying theory is deterministic, the experiences of individual observers are not predictable. (In that sense, I would say, it’s a nice compromise between our preferences and our experience.) It’s neither manifestly local nor Lorentz-invariant; those properties should emerge in appropriate situations, as often happens in physics. Of course there are all those worlds, but that doesn’t bother me in the slightest. For Many-Worlds, it’s the technical problems that bother me, not the philosophical ones — deriving classicality, recovering the Born Rule, and so on. One tends to think that technical problems can be solved by hard work, while metaphysical ones might prove intractable, which is why I come down the way I do on this particular question.

But the hidden-variables possibility is still definitely alive and well. And the general program of “trying to invent a better theory than quantum mechanics which would make all these distasteful philosophical implications go away” is certainly a worthwhile one. If anyone wants to suggest their favorite defenses of epistemic or dynamical-collapse approaches, feel free to leave them in comments.

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Spacetime, Storified

July 31, 2015 / 16 Comments

I had some spare minutes the other day, and had been thinking about the fate of spacetime in a quantum universe, so I took to the internet to let my feelings be heard. Only a few minutes, though, so I took advantage of Twitter rather than do a proper blog post. But through the magic of Storify, I can turn the former into the latter!

Obviously the infamous 140-character limit of Twitter doesn’t allow the level of precision and subtlety one would always like to achieve when talking about difficult topics. But restrictions lead to creativity, and the results can actually be a bit more accessible than unfettered prose might have been.

Anyway, spacetime isn’t fundamental, it’s just a useful approximation in certain regimes. Someday we hope to know what it’s an approximation to.

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Guest Post: Aidan Chatwin-Davies on Recovering One Qubit from a Black Hole

July 24, 2015 / 23 Comments

The question of how information escapes from evaporating black holes has puzzled physicists for almost forty years now, and while we’ve learned a lot we still don’t seem close to an answer. Increasingly, people who care about such things have been taking more seriously the intricacies of quantum information theory, and learning how to apply that general formalism to the specific issues of black hole information.

Now two students and I have offered a small contribution to this effort. Aidan Chatwin-Davies is a grad student here at Caltech, while Adam Jermyn was an undergraduate who has now gone on to do graduate work at Cambridge. Aidan came up with a simple method for getting out one “quantum bit” (qubit) of information from a black hole, using a strategy similar to “quantum teleportation.” Here’s our paper that just appeared on arxiv:

How to Recover a Qubit That Has Fallen Into a Black Hole
Aidan Chatwin-Davies, Adam S. Jermyn, Sean M. Carroll

We demonstrate an algorithm for the retrieval of a qubit, encoded in spin angular momentum, that has been dropped into a no-firewall unitary black hole. Retrieval is achieved analogously to quantum teleportation by collecting Hawking radiation and performing measurements on the black hole. Importantly, these methods only require the ability to perform measurements from outside the event horizon and to collect the Hawking radiation emitted after the state of interest is dropped into the black hole.

It’s a very specific — i.e. not very general — method: you have to have done measurements on the black hole ahead of time, and then drop in one qubit, and we show how to get it back out. Sadly it doesn’t work for two qubits (or more), so there’s no obvious way to generalize the procedure. But maybe the imagination of some clever person will be inspired by this particular thought experiment to come up with a way to get out two qubits, and we’ll be off.

I’m happy to host this guest post by Aidan, explaining the general method behind our madness.

If you were to ask someone on the bus which of Stephen Hawking’s contributions to physics he or she thought was most notable, the answer that you would almost certainly get is his prediction that a black hole should glow as if it were an object with some temperature. This glow is made up of thermal radiation which, unsurprisingly, we call Hawking radiation. As the black hole radiates, its mass slowly decreases and the black hole decreases in size. So, if you waited long enough and were careful not to enlarge the black hole by throwing stuff back in, then eventually it would completely evaporate away, leaving behind nothing but a bunch of Hawking radiation.

At a first glance, this phenomenon of black hole evaporation challenges a central notion in quantum theory, which is that it should not be possible to destroy information. Suppose, for example, that you were to toss a book, or a handful of atoms in a particular quantum state into the black hole. As the black hole evaporates into a collection of thermal Hawking particles, what happens to the information that was contained in that book or in the state of (what were formerly) your atoms? One possibility is that the information actually is destroyed, but then we would have to contend with some pretty ugly foundational consequences for quantum theory. Instead, it could be that the information is preserved in the state of the leftover Hawking radiation, albeit highly scrambled and difficult to distinguish from a thermal state. Besides being very pleasing on philosophical grounds, we also have evidence for the latter possibility from the AdS/CFT correspondence. Moreover, if the process of converting a black hole to Hawking radiation conserves information, then a stunning result of Hayden and Preskill says that for sufficiently old black holes, any information that you toss in comes back out almost a fast as possible!

Even so, exactly how information leaks out of a black hole and how one would go about converting a bunch of Hawking radiation to a useful state is quite mysterious. On that note, what we did in a recent piece of work was to propose a protocol whereby, under very modest and special circumstances, you can toss one qubit (a single unit of quantum information) into a black hole and then recover its state, and hence the information that it carried.

More precisely, the protocol describes how to recover a single qubit that is encoded in the spin angular momentum of a particle, i.e., a spin qubit. Spin is a property that any given particle possesses, just like mass or electric charge. For particles that have spin equal to 1/2 (like those that we consider in our protocol), at least classically, you can think of spin as a little arrow which points up or down and says whether the particle is spinning clockwise or counterclockwise about a line drawn through the arrow. In this classical picture, whether the arrow points up or down constitutes one classical bit of information. According to quantum mechanics, however, spin can actually exist in a superposition of being part up and part down; these proportions constitute one qubit of quantum information.

So, how does one throw a spin qubit into a black hole and get it back out again? Suppose that Alice is sitting outside of a black hole, the properties of which she is monitoring. From the outside, a black hole is characterized by only three properties: its total mass, total charge, and total spin. This latter property is essentially just a much bigger version of the spin of an individual particle and will be important for the protocol.

Next, suppose that Alice accidentally drops a spin qubit into the black hole. First, she doesn’t panic. Instead, she patiently waits and collects one particle of Hawking radiation from the black hole. Crucially, when a Hawking particle is produced by the black hole, a bizarro version of the same particle is also produced, but just behind the black hole’s horizon (boundary) so that it falls into the black hole. This bizarro ingoing particle is the same as the outgoing Hawking particle, but with opposite properties. In particular, its spin state will always be flipped relative to the outgoing Hawking particle. (The outgoing Hawking particle and the ingoing particle are entangled, for those in the know.)

The picture so far is that Alice, who is outside of the black hole, collects a single particle of Hawking radiation whilst the spin qubit that she dropped and the ingoing bizarro Hawking particle fall into the black hole. When the dropped particle and the bizarro particle fall into the black hole, their spins combine with the spin of the black hole—but remember! The bizarro particle’s spin was highly correlated with the spin of the outgoing Hawking particle. As such, the new combined total spin of the black hole becomes highly correlated with the spin of the outgoing Hawking particle, which Alice now holds. So, Alice measures the black hole’s new total spin state. Then, essentially, she can exploit the correlations between her held Hawking particle and the black hole to transfer the old spin state of the particle that she dropped into the hole to the Hawking particle that she now holds. Alice’s lost qubit is thus restored. Furthermore, Alice didn’t even need to know the precise state that her initial particle was in to begin with; the qubit is recovered regardless!

That’s the protocol in a nutshell. If the words “quantum teleportation” mean anything to you, then you can think of the protocol as a variation on the quantum teleportation protocol where the transmitting party is the black hole and measurement is performed in the total angular momentum basis instead of the Bell basis. Of course, this is far from a resolution of the information problem for black holes. However, it is certainly a neat trick which shows, in a special set of circumstances, how to “bounce” a qubit of quantum information off of a black hole.

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