Thanksgiving

This year we give thanks for a feature of the physical world that many people grumble about rather than celebrating, but is undeniably central to how Nature works at a deep level: the speed of light. (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, the Fourier Transform and Riemannian Geometry.)

The speed of light in vacuum, traditionally denoted by c, is 299,792,458 meters per second. It’s exactly that, not just approximately; it turns out to be easier to measure intervals of time to very high precision than it is to measure distances in space, so we measure the length of a second experimentally, then define the meter to be “the distance that light travels 299,792,458 of in one second.” Personally I prefer to characterize c as “one light-year per year”; that’s equally exact, and it’s easier to remember all the significant figures that way.

There are a few great things about the speed of light. One is that it’s a fixed, universal constant, as measured by inertial (unaccelerating) observers, in vacuum (empty space). Of course light can slow down if it propagates through a medium, but that’s hardly surprising. The other great thing is that it’s an upper limit; physical particles, as far as we know in the real world, always move at speeds less than or equal to c.

That first fact, the universal constancy of c, is the startling feature that set Einstein on the road to figuring out relativity. It’s a crazy claim at first glance: if two people are moving relative to each other (maybe because one is in a moving car and one is standing on the sidewalk) and they measure the speed of a third object (like a plane passing overhead) relative to themselves, of course they will get different answers. But not with light. I can be zipping past you at 99% of c, directly at an oncoming light beam, and both you and I will measure it to be moving at the same speed. That’s only sensible if something is wonky about our conventional pre-relativity notions of space and time, which is what Einstein eventually figured out. It was his former teacher Minkowski who realized the real implication is that we should think of the world as a single four-dimensional spacetime; Einstein initially scoffed at the idea as typically useless mathematical puffery, but of course it turned out to be central in his eventual development of general relativity (which explains gravity by allowing spacetime to be curved).

Because the speed of light is universal, when we draw pictures of spacetime we can indicate the possible paths light can take through any point, in a way that will be agreed upon by all observers. Orienting time vertically and space horizontally, the result is the set of light cones — the pictorial way of indicating the universal speed-of-light limit on our motion through the universe. Moving slower than light means moving “upward through your light cones,” and that’s what all massive objects are constrained to do. (When you’ve really internalized the lessons of relativity, deep in your bones, you understand that spacetime diagrams should only indicate light cones, not subjective human constructs like “space” and “time.”)

Light Cones

The fact that the speed of light is such an insuperable barrier to the speed of travel is something that really bugs people. On everyday-life scales, c is incredibly fast; but once we start contemplating astrophysical distances, suddenly it seems maddeningly slow. It takes just over a second for light to travel from the Earth to the Moon; eight minutes to get to the Sun; over five hours to get to Pluto; four years to get to the nearest star; twenty-six thousand years to get to the galactic center; and two and a half million years to get to the Andromeda galaxy. That’s why almost all good space-opera science fiction takes the easy way out and imagines faster-than-light travel. (In the real world, we won’t ever travel faster than light, but that won’t stop us from reaching the stars; it’s much more feasible to imagine extending human lifespans by many orders of magnitude, or making cryogenic storage feasible. Not easy — but not against the laws of physics, either.)

It’s understandable, therefore, that we sometimes get excited by breathless news reports about faster-than-light signals, though they always eventually disappear. But I think we should do better than just be grumpy about the finite speed of light. Like it or not, it’s an absolutely crucial part of the nature of reality. It didn’t have to be, in the sense of all possible worlds; the Newtonian universe is a relatively sensible set of laws of physics, in which there is no speed-of-light barrier at all.

That would be a very different world indeed. Continue reading

Posted in Science | 23 Comments

Gifford Lectures on Natural Theology

In October I had the honor of visiting the University of Glasgow to give the Gifford Lectures on Natural Theology. These are a series of lectures that date back to 1888, and happen at different Scottish universities: Glasgow, Aberdeen, Edinburgh, and St. Andrews. “Natural theology” is traditionally the discipline that attempts to learn about the nature of God via our experience of the world (in contrast to by revelation or contemplation). The Gifford Lectures have always interpreted this regime rather broadly; many theologians have given the talks, but also people like Neils Bohr, Arthur Eddington, Hannah Arendt, Noam Chomsky, Carl Sagan, Richard Dawkins, and Steven Pinker.

Sometimes the speakers turn their lectures into short published books; in my case, I had just written a book that fit well into the topic, so I spoke about the ideas in The Big Picture. Unfortunately the first of the five lectures was not recorded, but the subsequent four were. Here are those recordings, along with a copy of my slides for the first talk. It’s not a huge loss, as many of the ideas in the first lecture can be found in previous talks I’ve given on the arrow of time; it’s about the evolution of our universe, how that leads to an arrow of time, and how that helps explain things like memory and cause/effect relations. The second lecture was on the Core Theory and why we think it will remain accurate in the face of new discoveries. The third lecture was on emergence and how different ways of talking about the world fit together, including discussions of effective field theory and why the universe itself exists. Lecture four dealt with the evolution of complexity, the origin of life, and the nature of consciousness. (I might have had to skip some details during that one.) And the final lecture was on what it all means, why we are here, and how to live in a universe that doesn’t come with any instructions. Enjoy!

(Looking at my YouTube channel makes me realize that I’ve been in a lot of videos.)

Lecture One: Cosmos, Time, Memory (slides only, no video)
Slideshare

Lecture Two: The Stuff of Which We Are Made

Lecture Three: Layers of Reality

Lecture Four: Simplicity, Complexity, Thought

Lecture Five: Our Place in the Universe

Posted in Big Picture, Philosophy, Religion, Science | 10 Comments

Talking About Dark Matter and Dark Energy

Trying to keep these occasional Facebook Live videos going. (I’ve looked briefly into other venues such as Periscope, but FB is really easy and anyone can view without logging in if they like.)

So here is one I did this morning, about why cosmologists think dark matter and dark energy are things that really exist. I talk in particular about a recent paper by Nielsen, Guffanti, and Sarkar that questioned the evidence for universal acceleration (I think the evidence is still very good), and one by Erik Verlinde suggesting that emergent gravity can modify Einstein’s general relativity on large scales to explain away dark matter (I think it’s an intriguing idea, but am skeptical it can ever fit the data from the cosmic microwave background).

Feel free to propose topics for future conversations, or make suggestions about the format.

Posted in Science | 50 Comments

Leonard Cohen

What a goddamn week. Leonard Cohen, one of the greatest singer-songwriters in living memory, has died at age 82. His music meant a lot to me personally, as it did to countless others. Usually sad, sometimes melodramatic, always thoughtful and poetic and provocative. I never met him in person (though I did go to a couple of concerts), but he lived not too far away from me in LA, and somehow felt as if I knew him. We’ll miss you, Leonard.

Let’s hope he was right about this democracy thing.

Posted in Music | 30 Comments

The Future of Democratic Values

Hey, did you know we are having an election here in the United States? I think I saw it mentioned on TV. Whatever your preferences may be, everyone eligible should try to get out and vote.

This election has, without a doubt, been somewhat unique. I’m cautiously optimistic that Hillary Clinton will win, that we will celebrate the election of the first female President in the history of the republic, and that she will do a relatively good job — although as a good Bayesian I know that empirical predictions are never certain, and in an atmosphere like this uncertainty runs relatively high.

Even if Clinton wins and the U.S. avoids complete embarrassment, I’m still very worried about what this election has revealed about the state of the country. No matter who our next President might be, there are real reasons to be concerned that the U.S. is veering away from some of the foundational principles that are necessary to a functioning democracy. That may sound alarmist, but I don’t think it’s unwarranted. Historically, democracies don’t always last forever; we’d be foolish to think that it can’t happen here.

This isn’t a worry about the specific horrible wrongness of Donald Trump — it’s a worry about the forces that propelled him to the nomination of one of our two major political parties, and the fires he so willingly stoked along the way. Just as a quick and hopelessly incomplete recap:

  • Trump built his early political notoriety via “birtherism,” explicitly working to undermine the legitimacy of our elected President.
  • He has continually vilified immigrants and foreigners generally, promoting an us-against-them mentality between people of different races and ethnicities.
  • He has pledged to violate the Constitutional principle of freedom of religion, from banning Muslims from entering the country to tracking ones that are here.
  • His campaign, and the Republican party more generally, has openly engaged in suppressing the vote from groups unlikely to support him. (“‘We have three major voter suppression operations underway,’ says a senior [Trump] official.”)
  • He has glorified violence against protesters who disagree with him.
  • He has lied at an unprecedented, astonishing rate, secure in the knowledge that his statements will be taken as true by a large fraction of his intended audience.
  • He has presented himself as a uniquely powerful strongman who can solve problems through his personal force of will, and spoke admiringly of dictators from Vladimir Putin to Kim Jong-un to Saddam Hussein.
  • He has vowed that if he wins the election, he will seek vengeance on those who opposed him, including throwing his opponent into prison.
  • He has repeatedly cast doubt on the legitimacy of the election outcome, implying that he would refuse to accept the result if he lost.
  • He has pointed fingers at a shadowy global conspiracy in charge of world finance, often with explicitly anti-Semitic overtones.
  • Several Republican politicians have broached the prospect of refusing to confirm any Supreme Court nominees from a Democratic President.
  • A government agency, the FBI, has interfered in a Presidential election.
  • Republicans have accused Democratic officeholders of being traitors.
  • A number of Trump supporters have spoken of the prospect of violent resistance if Clinton is elected.

This is not a list of “why Donald Trump is a bad person who is disastrously unqualified for the Presidency”; that would be much longer. Rather, I wanted to highlight features of the campaign that are specifically attacks on (small-“d”) democratic norms and values. The assumptions, often unspoken, by which legitimate political opponents have generally agreed to operate by over the course of the last two centuries and more. Not all of them, of course; there are glaring exceptions, authoritarians who have run roughshod over one or more of these norms in the name of personal glory. History generally looks down upon them, and we consider ourselves fortunate that they didn’t have greater success. But fortune can run out.

The most worrisome aspect of the situation is the very real prospect that these attacks on the foundations of liberal democracy will not simply disappear once Donald Trump rides off into the gold-plated sunset; that they will be seized upon and deployed by other politicians who couldn’t help but notice Trump’s success. If that’s the case, we will have a real reason to be concerned that American democracy will stop working, perhaps sooner rather than later. I don’t think it’s likely that such a disastrous scenario would come to pass, but one has to balance the small likelihood against the devastating consequences — and right now the probability seems closer to 0.05 than to 10-5.

Democracy is a curious and fragile thing. It’s not just “majority rules”; crucial to the project are the ideas that (1) minority rights are still respected, and (2) in return, losing minorities respect electoral outcomes. It’s the second of these that is under siege at the moment. Since the time of the Federalist Papers, it’s been understood that democracy is an attempt to provide common self-rule for people who don’t agree on everything, but who at least share the common values of democracy itself. Having strong, even extremely passionate, political disagreements is inevitable in a democratic system. The question is whether we cast those with whom we disagree as enemies, traitors, and cheaters who must be opposed in every measure at every turn; or as partners in a grand project with whom we can fiercely disagree and yet still work with.

I don’t claim to have a complete understanding of how we got to this precarious point, though there are a number of factors that certainly have contributed. Continue reading

Posted in Humanity, Politics | 72 Comments

Entropy and Complexity, Cause and Effect, Life and Time

Finally back from Scotland, where I gave a series of five talks for the Gifford Lectures in Glasgow. The final four, at least, were recorded, and should go up on the web at some point, but I’m not sure when.

Meanwhile, I had a very fun collaboration with Henry Reich, the wizard behind the Minute Physics videos. Henry and I have known each other for a while, and I previously joined forces with him to talk about dark energy and the arrow of time.

This time, we made a series of five videos (sponsored by Google and Audible.com) based on sections of The Big Picture. In particular, we focused on the thread connecting the arrow of time and entropy to such everyday notions of cause and effect and the appearance of complex structures, ending with the origin of life and how low-entropy energy from the Sun powers the biosphere here on Earth. Henry and I wrote the scripts together, based on the book; I read the narration, and of course he did the art.

Enjoy!

  1. Why Doesn’t Time Flow Backwards?
  2. Do Cause and Effect Really Exist?
  3. Where Does Complexity Come From?
  4. How Entropy Powers the Earth
  5. What Is the Purpose of Life?
Posted in Big Picture, Science, Time | 27 Comments

Live Q&As, Past and Future

On Friday I had a few minutes free, and did an experiment: put my iPhone on a tripod, pointed it at myself, and did a live video on my public Facebook page, taking questions from anyone who happened by. There were some technical glitches, as one might expect from a short-notice happening. The sound wasn’t working when I first started, and in the recording below the video fails (replacing the actual recording with a still image of me sideways, for inexplicable reasons) just when the sound starts working. (I don’t think this happened during the actual event, but maybe it did and everyone was too polite to mention it.) And for some reason the video keeps going long after the 20-some minutes for which I was actually recording.

But overall I think it was fun and potentially worth repeating. If I were to make this an occasional thing, how best to do it? This time around I literally just read off a selection of questions that people were typing into the Facebook comment box. Alternatively, I could just talk on some particular topic, or I could solicit questions ahead of time and pick out some good ones to answer in detail.

What do you folks think? Also — is Facebook Live the right tool for this? I know the kids these days use all sorts of different technologies. No guarantees that I’ll have time to do this regularly, but it’s worth contemplating.

What makes the most sense to talk about in live chats?
Posted in Internet, Personal | 41 Comments

Consciousness and Downward Causation

For many people, the phenomenon of consciousness is the best evidence we have that there must be something important missing in our basic physical description of the world. According to this worry, a bunch of atoms and particles, mindlessly obeying the laws of physics, can’t actually experience the way a conscious creature does. There’s no such thing as “what it is to be like” a collection of purely physical atoms; it would lack qualia, the irreducibly subjective components of our experience of the world. One argument for this conclusion is that we can conceive of collections of atoms that behave physically in exactly the same way as ordinary humans, but don’t have those inner experiences — philosophical zombies. (If you think about it carefully, I would claim, you would realize that zombies are harder to conceive of than you might originally have guessed — but that’s an argument for another time.)

The folks who find this line of reasoning compelling are not necessarily traditional Cartesian dualists who think that there is an immaterial soul distinct from the body. On the contrary, they often appreciate the arguments against “substance dualism,” and have a high degree of respect for the laws of physics (which don’t seem to need or provide evidence for any non-physical influences on our atoms). But still, they insist, there’s no way to just throw a bunch of mindless physical matter together and expect it to experience true consciousness.

People who want to dance this tricky two-step — respect for the laws of physics, but an insistence that consciousness can’t reduce to the physical — are forced to face up to a certain problem, which we might call the causal box argument. It goes like this. (Feel free to replace “physical particles” with “quantum fields” if you want to be fastidious.)

  1. Consciousness cannot be accounted for by physical particles obeying mindless equations.
  2. Human beings seem to be made up — even if not exclusively — of physical particles.
  3. To the best of our knowledge, those particles obey mindless equations, without exception.
  4. Therefore, consciousness does not exist.

Nobody actually believes this argument, let us hasten to add — they typically just deny one of the premises.

But there is a tiny sliver of wiggle room that might allow us to salvage something special about consciousness without giving up on the laws of physics — the concept of downward causation. Here we’re invoking the idea that there are different levels at which we can describe reality, as I discussed in The Big Picture at great length. We say that “higher” (more coarse-grained) levels are emergent, but that word means different things to different people. So-called “weak” emergence just says the obvious thing, that higher-level notions like the fluidity or solidity of a material substance emerge out of the properties of its microscopic constituents. In principle, if not in practice, the microscopic description is absolutely complete and comprehensive. A “strong” form of emergence would suggest that something truly new comes into being at the higher levels, something that just isn’t there in the microscopic description.

Downward causation is one manifestation of this strong-emergentist attitude. It’s the idea that what happens at lower levels can be directly influenced (causally acted upon) by what is happening at the higher levels. The idea, in other words, that you can’t really understand the microscopic behavior without knowing something about the macroscopic.

There is no reason to think that anything like downward causation really happens in the world, at least not down to the level of particles and forces. While I was writing The Big Picture, I grumbled on Twitter about how people kept talking about it but how I didn’t want to discuss it in the book; naturally, I was hectored into writing something about it.

But you can see why the concept of downward causation might be attractive to someone who doesn’t think that consciousness can be accounted for by the fields and equations of the Core Theory. Sure, the idea would be, maybe electrons and nuclei act according to the laws of physics, but those laws need to include feedback from higher levels onto that microscopic behavior — including whether or not those particles are part of a conscious creature. In that way, consciousness can play a decisive, causal role in the universe, without actually violating any physical laws.

One person who thinks that way is John Searle, the extremely distinguished philosopher from Berkeley (and originator of the Chinese Room argument). I recently received an email from Henrik Røed Sherling, who took a class with Searle and came across this very issue. He sent me this email, which he was kind enough to allow me to reproduce here:

Hi Professor Carroll,

I read your book and was at the same time awestruck and angered, because I thought your entire section on the mind was both well-written and awfully wrong — until I started thinking about it, that is. Now I genuinely don’t know what to think anymore, but I’m trying to work through it by writing a paper on the topic.

I took Philosophy of Mind with John Searle last semester at UC Berkeley. He convinced me of a lot of ideas of which your book has now disabused me. But despite your occasionally effective jabs at Searle, you never explicitly refute his own theory of the mind, Biological Naturalism. I want to do that, using an argument from your book, but I first need to make sure that I properly understand it.

Searle says this of consciousness: it is caused by neuronal processes and realized in neuronal systems, but is not ontologically reducible to these; consciousness is not just a word we have for something else that is more fundamental. He uses the following analogy to visualize his description: consciousness is to the mind like fluidity is to water. It’s a higher-level feature caused by lower-level features and realized in a system of said lower-level features. Of course, for his version of consciousness to escape the charge of epiphenomenalism, he needs the higher-level feature in this analogy to act causally on the lower-level features — he needs downward causation. In typical fashion he says that “no one in their right mind” can say that solidity does not act causally when a hammer strikes a nail, but it appears to me that this is what you are saying.

So to my questions. Is it right to say that your argument against the existence of downward causation boils down to the incompatible vocabularies of lower-level and higher-level theories? I.e. that there is no such thing as a gluon in Fluid Dynamics, nor anything such as a fluid in the Standard Model, so a cause in one theory cannot have an effect in the other simply because causes and effects are different things in the different theories; gluons don’t affect fluidity, temperaturs and pressures do; fluids don’t affect gluons, quarks and fields do. If I have understood you right, then there couldn’t be any upward causation either. In which case Searle’s theory is not only epiphenomenal, it’s plain inaccurate from the get-go; he wants consciousness to both be a higher-level feature of neuronal processes and to be caused by them. Did I get this right?

Best regards,
Henrik Røed Sherling

Here was my reply:

Dear Henrik–

Thanks for writing. Genuinely not knowing what to think is always an acceptable stance!

I think your summary of my views are pretty accurate. As I say on p. 375, poetic naturalists tend not to be impressed by downward causation, but not by upward causation either! At least, not if your theory of each individual level is complete and consistent.

Part of the issue is, as often happens, an inconsistent use of a natural-language word, in this case “cause.” The kinds of dynamical, explain-this-occurrence causes that we’re talking about here are a different beast than inter-level implications (that one might be tempted to sloppily refer to as “causes”). Features of a lower level, like conservation of energy, can certainly imply or entail features of higher-level descriptions; and indeed the converse is also possible. But saying that such implications are “causes” is to mean something completely different than when we say “swinging my elbow caused the glass of wine to fall to the floor.”

So, I like to think I’m in my right mind, and I’m happy to admit that solidity acts causally when a hammer strikes a nail. But I don’t describe that nail as a collection of particles obeying the Core Theory *and* additionally as a solid object that a hammer can hit; we should use one language or the other. At the level of elementary particles, there’s no such concept as “solidity,” and it doesn’t act causally.

To be perfectly careful — all this is how we currently see things according to modern physics. An electron responds to the other fields precisely at its location, in quantitatively well-understood ways that make no reference to whether it’s in a nail, in a brain, or in interstellar space. We can of course imagine that this understanding is wrong, and that future investigations will reveal the electron really does care about those things. That would be the greatest discovery in physics since quantum mechanics itself, perhaps of all time; but I’m not holding my breath.

I really do think that enormous confusion is caused in many areas — not just consciousness, but free will and even more purely physical phenomena — by the simple mistake of starting sentences in one language or layer of description (“I thought about summoning up the will power to resist that extra slice of pizza…”) but then ending them in a completely different vocabulary (“… but my atoms obeyed the laws of the Standard Model, so what could I do?”) The dynamical rules of the Core Theory aren’t just vague suggestions; they are absolutely precise statements about how the quantum fields making up you and me behave under any circumstances (within the “everyday life” domain of validity). And those rules say that the behavior of, say, an electron is determined by the local values of other quantum fields at the position of the electron — and by nothing else. (That’s “locality” or “microcausality” in quantum field theory.) In particular, as long as the quantum fields at the precise position of the electron are the same, the larger context in which it is embedded is utterly irrelevant.

It’s possible that the real world is different, and there is such inter-level feedback. That’s an experimentally testable question! As I mentioned to Henrik, it would be the greatest scientific discovery of our lifetimes. And there’s basically no evidence that it’s true. But it’s possible.

So I don’t think downward causation is of any help to attempts to free the phenomenon of consciousness from arising in a completely conventional way from the collective behavior of microscopic physical constituents of matter. We’re allowed to talk about consciousness as a real, causally efficacious phenomenon — as long as we stick to the appropriate human-scale level of description. But electrons get along just fine without it.

Posted in Big Picture, Philosophy, Science | 421 Comments

Maybe We Do Not Live in a Simulation: The Resolution Conundrum

Greetings from bucolic Banff, Canada, where we’re finishing up the biennial Foundational Questions Institute conference. To a large extent, this event fulfills the maxim that physicists like to fly to beautiful, exotic locations, and once there they sit in hotel rooms and talk to other physicists. We did manage to sneak out into nature a couple of times, but even there we were tasked with discussing profound questions about the nature of reality. Evidence: here is Steve Giddings, our discussion leader on a trip up the Banff Gondola, being protected from the rain as he courageously took notes on our debate over “What Is an Event?” (My answer: an outdated notion, a relic of our past classical ontologies.)

stevegiddings

One fun part of the conference was a “Science Speed-Dating” event, where a few of the scientists and philosophers sat at tables to chat with interested folks who switched tables every twenty minutes. One of the participants was philosopher David Chalmers, who decided to talk about the question of whether we live in a computer simulation. You probably heard about this idea long ago, but public discussion of the possibility was recently re-ignited when Elon Musk came out as an advocate.

At David’s table, one of the younger audience members raised a good point: even simulated civilizations will have the ability to run simulations of their own. But a simulated civilization won’t have access to as much computing power as the one that is simulating it, so the lower-level sims will necessarily have lower resolution. No matter how powerful the top-level civilization might be, there will be a bottom level that doesn’t actually have the ability to run realistic civilizations at all.

This raises a conundrum, I suggest, for the standard simulation argument — i.e. not only the offhand suggestion “maybe we live in a simulation,” but the positive assertion that we probably do. Here is one version of that argument:

  1. We can easily imagine creating many simulated civilizations.
  2. Things that are that easy to imagine are likely to happen, at least somewhere in the universe.
  3. Therefore, there are probably many civilizations being simulated within the lifetime of our universe. Enough that there are many more simulated people than people like us.
  4. Likewise, it is easy to imagine that our universe is just one of a large number of universes being simulated by a higher civilization.
  5. Given a meta-universe with many observers (perhaps of some specified type), we should assume we are typical within the set of all such observers.
  6. A typical observer is likely to be in one of the simulations (at some level), rather than a member of the top-level civilization.
  7. Therefore, we probably live in a simulation.

Of course one is welcome to poke holes in any of the steps of this argument. But let’s for the moment imagine that we accept them. And let’s add the observation that the hierarchy of simulations eventually bottoms out, at a set of sims that don’t themselves have the ability to perform effective simulations. Given the above logic, including the idea that civilizations that have the ability to construct simulations usually construct many of them, we inevitably conclude:

  • We probably live in the lowest-level simulation, the one without an ability to perform effective simulations. That’s where the vast majority of observers are to be found.

Hopefully the conundrum is clear. The argument started with the premise that it wasn’t that hard to imagine simulating a civilization — but the conclusion is that we shouldn’t be able to do that at all. This is a contradiction, therefore one of the premises must be false.

This isn’t such an unusual outcome in these quasi-anthropic “we are typical observers” kinds of arguments. The measure on all such observers often gets concentrated on some particular subset of the distribution, which might not look like we look at all. In multiverse cosmology this shows up as the “youngness paradox.”

Personally I think that premise 1. (it’s easy to perform simulations) is a bit questionable, and premise 5. (we should assume we are typical observers) is more or less completely without justification. If we know that we are members of some very homogeneous ensemble, where every member is basically the same, then by all means typicality is a sensible assumption. But when ensembles are highly heterogeneous, and we actually know something about our specific situation, there’s no reason to assume we are typical. As James Hartle and Mark Srednicki have pointed out, that’s a fake kind of humility — by asserting that “we are typical” in the multiverse, we’re actually claiming that “typical observers are like us.” Who’s to say that is true?

I highly doubt this is an original argument, so probably simulation cognoscenti have debated it back and forth, and likely there are standard responses. But it illustrates the trickiness of reasoning about who we are in a very big cosmos.

Posted in Philosophy, Science | 102 Comments

You Should Love (or at least respect) the Schrödinger Equation

Over at the twitter dot com website, there has been a briefly-trending topic #fav7films, discussing your favorite seven films. Part of the purpose of being on twitter is to one-up the competition, so I instead listed my #fav7equations. Slightly cleaned up, the equations I chose as my seven favorites are:

  1. {\bf F} = m{\bf a}
  2. \partial L/\partial {\bf x} = \partial_t ({\partial L}/{\partial {\dot {\bf x}}})
  3. {\mathrm d}*F = J
  4. S = k \log W
  5. ds^2 = -{\mathrm d}t^2 + {\mathrm d}{\bf x}^2
  6. G_{ab} = 8\pi G T_{ab}
  7. \hat{H}|\psi\rangle = i\partial_t |\psi\rangle

In order: Newton’s Second Law of motion, the Euler-Lagrange equation, Maxwell’s equations in terms of differential forms, Boltzmann’s definition of entropy, the metric for Minkowski spacetime (special relativity), Einstein’s equation for spacetime curvature (general relativity), and the Schrödinger equation of quantum mechanics. Feel free to Google them for more info, even if equations aren’t your thing. They represent a series of extraordinary insights in the development of physics, from the 1600’s to the present day.

Of course people chimed in with their own favorites, which is all in the spirit of the thing. But one misconception came up that is probably worth correcting: people don’t appreciate how important and all-encompassing the Schrödinger equation is.

I blame society. Or, more accurately, I blame how we teach quantum mechanics. Not that the standard treatment of the Schrödinger equation is fundamentally wrong (as other aspects of how we teach QM are), but that it’s incomplete. And sometimes people get brief introductions to things like the Dirac equation or the Klein-Gordon equation, and come away with the impression that they are somehow relativistic replacements for the Schrödinger equation, which they certainly are not. Dirac et al. may have originally wondered whether they were, but these days we certainly know better.

As I remarked in my post about emergent space, we human beings tend to do quantum mechanics by starting with some classical model, and then “quantizing” it. Nature doesn’t work that way, but we’re not as smart as Nature is. By a “classical model” we mean something that obeys the basic Newtonian paradigm: there is some kind of generalized “position” variable, and also a corresponding “momentum” variable (how fast the position variable is changing), which together obey some deterministic equations of motion that can be solved once we are given initial data. Those equations can be derived from a function called the Hamiltonian, which is basically the energy of the system as a function of positions and momenta; the results are Hamilton’s equations, which are essentially a slick version of Newton’s original {\bf F} = m{\bf a}.

There are various ways of taking such a setup and “quantizing” it, but one way is to take the position variable and consider all possible (normalized, complex-valued) functions of that variable. So instead of, for example, a single position coordinate x and its momentum p, quantum mechanics deals with wave functions ψ(x). That’s the thing that you square to get the probability of observing the system to be at the position x. (We can also transform the wave function to “momentum space,” and calculate the probabilities of observing the system to be at momentum p.) Just as positions and momenta obey Hamilton’s equations, the wave function obeys the Schrödinger equation,

\hat{H}|\psi\rangle = i\partial_t |\psi\rangle.

Indeed, the \hat{H} that appears in the Schrödinger equation is just the quantum version of the Hamiltonian.

The problem is that, when we are first taught about the Schrödinger equation, it is usually in the context of a specific, very simple model: a single non-relativistic particle moving in a potential. In other words, we choose a particular kind of wave function, and a particular Hamiltonian. The corresponding version of the Schrödinger equation is

\displaystyle{\left[-\frac{1}{\mu^2}\frac{\partial^2}{\partial x^2} + V(x)\right]|\psi\rangle = i\partial_t |\psi\rangle}.

If you don’t dig much deeper into the essence of quantum mechanics, you could come away with the impression that this is “the” Schrödinger equation, rather than just “the non-relativistic Schrödinger equation for a single particle.” Which would be a shame.

What happens if we go beyond the world of non-relativistic quantum mechanics? Is the poor little Schrödinger equation still up to the task? Sure! All you need is the right set of wave functions and the right Hamiltonian. Every quantum system obeys a version of the Schrödinger equation; it’s completely general. In particular, there’s no problem talking about relativistic systems or field theories — just don’t use the non-relativistic version of the equation, obviously.

What about the Klein-Gordon and Dirac equations? These were, indeed, originally developed as “relativistic versions of the non-relativistic Schrödinger equation,” but that’s not what they ended up being useful for. (The story is told that Schrödinger himself invented the Klein-Gordon equation even before his non-relativistic version, but discarded it because it didn’t do the job for describing the hydrogen atom. As my old professor Sidney Coleman put it, “Schrödinger was no dummy. He knew about relativity.”)

The Klein-Gordon and Dirac equations are actually not quantum at all — they are classical field equations, just like Maxwell’s equations are for electromagnetism and Einstein’s equation is for the metric tensor of gravity. They aren’t usually taught that way, in part because (unlike E&M and gravity) there aren’t any macroscopic classical fields in Nature that obey those equations. The KG equation governs relativistic scalar fields like the Higgs boson, while the Dirac equation governs spinor fields (spin-1/2 fermions) like the electron and neutrinos and quarks. In Nature, spinor fields are a little subtle, because they are anticommuting Grassmann variables rather than ordinary functions. But make no mistake; the Dirac equation fits perfectly comfortably into the standard Newtonian physical paradigm.

For fields like this, the role of “position” that for a single particle was played by the variable x is now played by an entire configuration of the field throughout space. For a scalar Klein-Gordon field, for example, that might be the values of the field φ(x) at every spatial location x. But otherwise the same story goes through as before. We construct a wave function by attaching a complex number to every possible value of the position variable; to emphasize that it’s a function of functions, we sometimes call it a “wave functional” and write it as a capital letter,

\Psi[\phi(x)].

The absolute-value-squared of this wave functional tells you the probability that you will observe the field to have the value φ(x) at each point x in space. The functional obeys — you guessed it — a version of the Schrödinger equation, with the Hamiltonian being that of a relativistic scalar field. There are likewise versions of the Schrödinger equation for the electromagnetic field, for Dirac fields, for the whole Core Theory, and what have you.

So the Schrödinger equation is not simply a relic of the early days of quantum mechanics, when we didn’t know how to deal with much more than non-relativistic particles orbiting atomic nuclei. It is the foundational equation of quantum dynamics, and applies to every quantum system there is. (There are equivalent ways of doing quantum mechanics, of course, like the Heisenberg picture and the path-integral formulation, but they’re all basically equivalent.) You tell me what the quantum state of your system is, and what is its Hamiltonian, and I will plug into the Schrödinger equation to see how that state will evolve with time. And as far as we know, quantum mechanics is how the universe works. Which makes the Schrödinger equation arguably the most important equation in all of physics.

While we’re at it, people complained that the cosmological constant Λ didn’t appear in Einstein’s equation (6). Of course it does — it’s part of the energy-momentum tensor on the right-hand side. Again, Einstein didn’t necessarily think of it that way, but these days we know better. The whole thing that is great about physics is that we keep learning things; we don’t need to remain stuck with the first ideas that were handed down by the great minds of the past.

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