Domino Valdano

The sci-fi film we've been talking about making at work has to do with the many worlds interpretation of quantum mechanics. I can't go into details (because they have involved many hours of conversations), but basically it starts from the premise that somehow, a group of hackers tinkering with various electrical equipment accidentally stumble upon a way to communicate with other decoherent branches of the multiverse, and this opens pandora's box in a lot of ways.

I was asked to evaluate how this could be made realistic from a physics perspective, and we had many conversations about it. What my coworker pointed out, which sounds right, is that it seems like communicating between different branches of the multiverse would have to involve some kind of non-linear modifications of quantum mechanics. This led us to read up a little on how realistic such modifications would be. Most physicists assume that quantum mechanics is an exact description of the world, but many are open to the possibility that there are slight modifications to it at the scale of quantum gravity which haven't been detected yet. Unfortunately, every time someone has explored this possibility theoretically, they tend to be led to the conclusion that any kind of non-linear modification, no matter how slight, tends to lead to problems that make the whole theory inconsistent and incompatible with other more sacred laws of physics such as thermodynamics.
My advisor's advisor was one who went down this path and eventually concluded it was probably a blind alley.

It appears that there are a lot of connected things that happen when you monkey with the laws of physics as we know them. For example, if you add non-linear modifications to quantum mechanics, you tend to violate the 2nd law of thermodynamics. In creating a lot of negative energy, you also tend to violate the 2nd law of thermodynamics. And in creating a wormhole, you tend to create the possibility for closed timelike loops. David Deutsch and others have analyzed what might happen theoretically if closed timelike loops (CTC's) were possible, and the conclusion is that you'd be able to solve NP complete problems in polynomial time, ie it implies P=NP. Closed timelike loops also violate the 2nd law of thermodynamics, because entropy cannot always increase within a closed loop of time. Either entropy would have to remain constant throughout the whole loop, or increase for a while and then suddenly decrease. It's like one of those staircases from an Escher painting: the staircase that always goes up cannot connect to itself in a circle. Either it doesn't go up, or it goes up and then comes back down. The same with entropy.

So many things are connected here. Non-linear modifications of quantum mechanics, negative energy, perpetual motion, antigravity, time travel, traversable wormholes, and P=NP. The more I read about these (and especially when I read Scott Aaronson's stuff) the more it seems like either you have to accept all of them or none of them.

There are actually multiple connections between computational complexity and wormholes I've found, not just via the connection between CTC's and P=NP. For example, there is the ER=EPR conjecture which is a very exciting proposal by two of the world's greatest living theoretical physicists, Leonard Susskind and Juan Maldacena. They have found a possible way in which wormholes are the same thing as quantum entanglement. Again, I don't have time to delve into the details, but this has to do with the blackhole firewall paradox. Many physicists have been worried that if there are no modifications to quantum mechanics (ie, information is never lost in black holes) then this would imply the existence of "blackhole firewalls" where an infalling observer would reach a flaming wall of fire at the event horizon. This violates a central principle of general relativity known as "the equivalence principle" which states that physics is the same in all reference frames.

But what Susskind says is that maybe firewalls don't actually form inside a black hole, and aren't needed to resolve the information paradox. Instead, the explanation for how things like the no-cloning theorem are preserved in the context of quantum mechanics in a black hole is that the interior of the black hole is protected by an "armor of computational complexity" (as Scott Aaronson puts it). You could try to send messages from the outside to the interior non-locally via quantum entanglement (or equivalently, through a traversable wormhole), but it would require you to solve a computational problem which is so complex it's in the complexity class known as QSZK (Quantum Statistical Zero Knowledge). If I understand correctly, the only reason you cannot send such a message is because quantum computers are not powerful enough to solve problems in this class.

I mentioned in my previous post that while it seems crystal clear that there's no way an advanced civilization could ever build a macroscopic wormhole that something the size of a human could pass through, it's a lot less clear why they couldn't build a microscopic traversable wormhole and use it to send information faster than light. If they could do that, then they could also create timelike loops and hence solve P=NP problems. So maybe the only reason why they couldn't do it is also related to computational complexity. Maybe it's the same general reason Susskind suggests in the context of black hole physics: somehow, computational complexity prohibits the transmission of meaningful information through such a wormhole, even though it would otherwise appear to be possible to build one.

Shortly after writing my last entry on this, I discovered an interesting recent paper from May 2014 on traversable wormholes from a physicist at Cambridge. He wrote the paper in an attempt to construct a stable wormhole geometry using the throat of the wormhole itself to generate the required negative energy to stabilize it. He ended up finding that it was not possible to completely stabilize it, for the particular parameters he was using. But he argues that even though it isn't stable, it would collapse slowly enough that a beam of light would still have a chance to pass through it before it completely collapsed. So if you could somehow construct that geometry (a daunting task), maybe it could be considered "traversable" in that light could temporarily pass through it. He also speculates that maybe you could find other geometries with less symmetry where the whole thing could be stabilized, but this seems like wishful thinking to me. The possibility of having some kind of temporary closed timelike loop while an unstable microscopic wormhole is collapsing I find very intriguing, and unlike the large wormholes of Interstellar it's not something I would completely rule out as a possibility. However, again... if you accept that then it seems like you'd have to accept all of the above weird violations of physics, including P=NP. And to me, it seems more likely that somehow, this armor of computational complexity Susskind and Aaronson are talking about comes into play and stops the beam of light from making it all the way through. Or at least, stops there from being any meaningful information encoded in the light. It occurs to me that this may be exactly what Hawking had in mind with his Chronology Protection Conjecture.

The only somewhat serious possibility I've left out here is if somehow, Kip Thorne's original conclusion that traversable wormholes necessarily imply the ability to build time machines is flawed. If that's the case, then I will admit there is a real possibility for faster than light communication in the future; and then the armor of computational complexity, or the chronology protection conjecture, or whatever you want to call it, would only come into play when you tried to make the wormhole into a time machine. This seems far fetched to me, but less far fetched than the idea that P=NP, perpetual motion machines are possible, and the 2nd law of thermodynamics is wrong, all of which I think would need to be true in order to have an actual microscopic wormhole that could be made into a closed timelike curve.

In part 3, I mentioned there was a difference between the standard local energy conditions which were originally proposed in classical General Relativity and the "averaged" conditions. But I went off on a tangent about quantum inequalities and quantum interest, and never got around to connecting this back with the averaged conditions or defining what they are.

The local energy conditions original proposed in GR apply to every point in spacetime. Since general relativity is a theory about the large-scale structure of the universe, the definition of a "point" in spacetime can be rather loose. For the purposes of cosmology, thinking of a point as being a ball of 1km radius is plenty accurate enough. You won't find any significant curvature of spacetime that's smaller than that, so whether it's exactly 0 in size or 1km in size doesn't matter. But for quantum mechanics, it matters a lot because it's a theory of the very small scale structure of the universe. There, the difference between 0 and 1km is huge, in fact so huge that even anything the size of a millimeter is already considered macroscopic.

So if you're going to ask whether quantum field theory respects the energy conditions proposed in general relativity, you have to get more precise with your definitions of these energy conditions. The question isn't "can energy be negative at a single point in spacetime?" but "can the average energy be negative in some macroscopic region of space over some period of time long enough for anyone to notice?" The actual definition of the AWEC (averaged weak energy condition) is: energy averaged along any timelike trajectory through spacetime is always zero or positive. A timelike trajectory basically means the path that a real actual observer in space who is traveling at less than the speed of light could follow. From the reference frame of this observer, this just means the energy averaged at a single point over all time. The ANEC (averaged null energy condition) is similar but for "null" trajectories through spacetime. Null trajectories are the paths that photons and other massless particles follow--all particles that move at the speed of light. A real observer could not follow this trajectory, but you can still ask what the energy density averaged over this path would be.

From what I understand, the quantum energy inequalities are actually a bit stronger than these averaged energy conditions. The AWEC basically says that if there is a negative energy spike somewhere, then eventually there has to be a positive energy spike that cancels it out. The QEI's say that not only does this have to be true, but the positive spike has to come very soon after the negative spike--the larger the spikes are, the sooner.

However, you may notice that the QEI's (and the averaged energy conditions) just refer to averaging over time. What about space? Personally, I don't fully understand why Kip Thorne and others focused on whether the average over time is violated but didn't seem to care about the average over space. Because the average over space seems important for constructing wormholes too--if you can't generate negative energy more than a few Planck lengths in width, then how would you ever expect to get enough macroscopic negative energy to support and stabilize a wormhole that someone could actually travel through?

I haven't mentioned the Casimir Effect yet, which is a big omission as it's one of the first things people will cite as soon as you ask them how they think someone could possibly build a traversable wormhole. Do the quantum inequalities apply to the Casimir Effect? Yes and no.

As I understand them, the quantum inequalities don't actually limit the actual absolute energy density, they limit the difference between the energy density and the vacuum energy density. Ordinarily, vacuum energy density is zero or very close to it. (It's actually very slightly positive because of dark energy, also known as the cosmological constant, but this is so small it doesn't really matter for our purposes.) The vacuum energy is pretty much the same everywhere in the universe on macroscopic scales. So ordinarily, if a quantum energy inequality tells you that you can't have an energy density less than minus (some extremely small number) then this also places a limit on the absolute energy density. But this is not true in the case of the Casimir Effect. Because the Casimir Effect lowers the vacuum energy in a very thin region of space below what it normally is. This lowered value of the energy (which is slightly negative) can persist for as long as you want in time. But energy fluctuations below that slightly lowered value are still limited by the QEI's.

This seems like really good news for anyone hoping to build a traversable wormhole--it's a way of getting around the quantum energy inequalities, as they are usually formulated. However, if you look at how the Casimir Effect actually works you see a very similar limitation on the negative energy density--it's just that it is limited in space instead of limited in time.

The Casimir Effect is something that happens when you place 2 parallel plates extremely close to each other. It produces a very thin negative vacuum energy density in the region of space between these plates. To get any decent amount of negative energy, the plates have to be enormous but extremely close together. It's worth mentioning that this effect has been explained without any reference to quantum field theory (just as the relativistic version of the van der Waals force). As far as I understand, both explanations are valid they are just two different ways of looking at the same effect. The fact that there is a valid description that doesn't make any reference to quantum field theory lends weight to the conclusion that despite it being a little weird there is no way to use it to do very weird things that you couldn't do classically like build wormholes. However, I admit that I'm not sure what happens to the energy density in the relativistic van der Waals description--I'm not sure there is even a notion of vacuum energy in that way of looking at it, as vacuum energy itself is a concept that exists only in quantum field theory (it's the energy of the ground state of the quantum fields).

Most of what I've read on quantum inequalities has come from Ford and Roman. They seem very opposed to the idea that traversable wormholes would be possible. I've also read a bit by Matt Visser, who seems more open to the possibility. The three of them, as well as Thorne, Morris, and Hawking seem to be the most important people who have written papers on this subject. Most other people writing on it write just a few papers here or there, citing one of them. Visser, Ford, and Roman seem to have all dedicated most of their careers to understanding what the limits on negative energy densities are and what their implications are for potentially building wormholes, time machines, or other strange things (like naked singularities--"black holes" that don't have an event horizon).

There are a few more things I'd like to wrap up in the next (and I think--final) part. One is to give some examples of the known limitations on how small and how short lived these negative energy densities can be, and what size of wormhole that would allow you to build. Another is to mention Alcubierre drives (a concept very similar to a wormhole that has very similar limitations). Another is to try to enumerate which averaged energy conditions are known for sure to hold in quantum field theory and in which situations, comparing this with which conditions would need to be violated to make various kinds of wormholes. And finally, to try to come up with any remotely realistic scenario for how this might be possible and give a sense for the extremely ridiculous nature of things that an infinitely advanced civilization would need to be able to do in order for that to happen practically, from a technological perspective.

I started my review of wormholes by reading Kip Thorne's famous paper on them from 1989. Thorne is the T in "MTW" a book by Misner, Thorne, and Wheeler called Gravitation, written in 1973 and still one of the most widely used textbooks on general relativity.

I'm not actually sure whether Kip Thorne believes that wormholes are possible--I assume he would at least lean towards "no" but I have no idea. You might think that because he has written important papers on them and because he consulted on a movie that depicts one, he believes they are. But that doesn't follow, because theoretical physicists often explore ideas that they don't think will work out, to see where they will lead and find the limits of existing theories and uncover new questions or problems with them. I didn't search for comments from him so I don't know what his present take on them is or if it has changed any, but in his 1989 paper he doesn't say they are possible, he just outlines what the conditions would have to be in order for them to be possible.

In his paper, he does three main things. The first is to construct a simple example of a stable traversable wormhole geometrically. In other words, he describes what the shape of space and time would have to be, and what distribution of energy would be needed in order to create this shape. (Remember, the basic idea of general relativity is that matter and energy warp space and time; given any distribution of energy you get a well defined shape of space and time.) Unfortunately he finds that the distribution would have to be quite "exotic", meaning it would require a lot of negative energy, a substance which is very different from ordinary matter and energy. The main question is: could such a substance exist or be created somehow, and if so could it exist in large enough quantities to make a wormhole? At the time, little was known about the answer to this question but a lot more work has been done since which is the topic I focused most of the rest of my reading on.

The second important thing he does in his 1989 paper is to show that if it is possible to create even the simplest kind of wormhole that just connects point A to point B in space, then it is also possible to build a time machine out of the wormhole, that could be used for traveling backwards in time.

So while the fact that a prominent very respectable physicist was even discussing the possibility of wormholes must have been very exciting to the sci fi community, what they may not have realized is that both of these results make wormholes less likely, not more likely. The first because he demonstrated that they depend on a substance not known to exist. And the second because time travel has a whole set of causality and consistency problems that come with it. If it were possible to build a wormhole that couldn't be made into a time machine, that would be much more believable than a wormhole that could be made into a time machine. But sadly, it doesn't seem like the first scenario is possible, at least according to Kip Thorne's 1989 results. However, there is some encouraging news here: in 1992 Stephen Hawking conjectured that there may be weird as of yet unknown effects in physics which act to protect against time travel. (He called this the "Chronology Protection Conjecture".) It seems like pure speculation to me, but if Hawking's suggestion is right then there might plausibly be some mechanism that prevents someone traveling through a wormhole if they plan to travel backwards in time. Like, maybe the wormhole suddenly closes up or becomes unstable. However, I don't think he has much reason to believe this is true other than wishful thinking--it would be nice if some kind of wormhole were possible, without having to face all of the obviously troublesome inconsistencies that time travel brings (grandfather paradox, etc.) So he tried to think of any way in which it could be. This is one way of avoiding that problem, but seems unlikely and doesn't do anything to solve the main problem which is a lack of negative energy.

In particle physics, everything is based on symmetry. Symmetry magazine is the main industry newsletter for particle physics. All of the fundamental laws of physics can be expressed as symmetries. Other than symmetry, the only rule is basically "anything that can happen, will happen".

There are two general categories of symmetries in particle physics, internal symmetries and spacetime symmetries. I'm only going to discuss spacetime symmetries here.

Within the category of spacetime symmetries there are continuous symmetries like rotational symmetry (responsible for the conservation of angular momentum), translational symmetry (responsible for regular conservation of momentum), time translation (responsible for conservation of energy), and Lorentz boosts (responsible for Einstein's theory of relativity).

But then there is also another kind of spacetime symmetry--discrete symmetries. There are 2 important discrete spacetime symmetries and they are pretty simple to explain. The first is called time reversal symmetry, usually denoted by the symbol T. As an operator, T represents the operation of flipping the direction of time from forwards to backwards--basically, hitting the rewind button. Parts of physics are symmetric with respect to T and other parts are not. The other important one is P (parity), which flips space instead of time--it's basically what you see when you look in the mirror and left and right are reversed, everything is backwards.

Here is a video of me doing a cartwheel, an every day process which by itself would appear to break both P and T. The animation shows the forward-in-time process first which is a right-handed cartwheel, followed by the time reverse which then looks like a left-handed cartwheel. Because applying T in this case accomplishes exactly the same thing as P (if you ignore the background), this means that this process breaks both P symmetry and T symmetry, but it preserves the combination of the 2, PT:



And now for the front handspring. Unlike the cartwheel, this process respects P symmetry. If you flip left and right, it still looks the same. However, if you time reverse it, it looks like a back handspring instead of a front handspring! So the handspring respects P symmetry but not T symmetry.



Of the 4 fundamental forces of nature--gravity, electromagnetism, the strong force, and the weak force--the first 3 respect time-reversal symmetry while the fourth, the weak force, does not. Because the other 3 are symmetric, it was assumed for a long time (until the 1960's) that all laws of physics had to be symmetric under T. Only in 1964 did the first indirect evidence that the weak force does not respect T symmetry emerge, and more direct proof came in the late 90's and still more interesting examples have piled on within the past decade.

( Lots more explanation behind the cut! )

I gave 3 examples of things that dissipate in part 2: friction, electrical resistance, and hurricanes. I feel like I understand fairly well why we call these dissipative, although I've always felt or hoped that there is some unifying principle that sheds more light on the subject and explains why them and not other things. But there's a forth example that is far more interesting, and for that example I still don't feel like I really understand why exactly it's dissipative: computation.

Now, you might first think--maybe computation is dissipative because it involves the flow of electricity through circuits (whether those circuits be wires or be microchips), but that's beside the point. First, as I understand it, any kind of physical irreversible computational process must necessarily dissipate heat and increase entropy. So this applies not just to electrical circuits but to anything we could conceivably use to compute an answer to something, including for example, an abacus (of course the amount of computation that can be performed by an abacus is presumably so tiny that you wouldn't notice.) Second, it's not just the electrical resistance because supposedly, computers actually draw *more* electricity while they are involved in some intense computation, not when they are just idling. There are many circuits which are on while the computer is doing nothing, but it's not being on that creates the entropy I'm worried about... it's the entropy created specifically from irreversible computation, from switching those circuits on and off in just such a way that it computes a simple answer to a more complex question fed to it. Beforehand, there are many possible answers, but afterwards, there is only one... for example, 42. This reduces the available microstates of the system from many to one, and therefore represents a reduction of entropy (which remember, counts the number of available microstates). Because of the 2nd law, this cannot happen by itself without producing heat... it needs to produce heat in order to cancel out that entropy loss by a gain in entropy due to the heat, for exactly the same reason that the earth must dump heat into its environment if evolution is to result in more highly organized organisms. So even a perfectly efficient computer which caused no net entropy gain for the universe would still produce heat!

The only exception to the above process is if, instead of taking a large set of inputs and reducing them to one output, all of the inputs and outputs correspond exactly in a 1-to-1 fashion, in other words, you use all reversible logic gates to build the computer. An example of an irreversible logic gate is and AND gate. It takes 2 inputs and has 1 output, it outputs "Yes" if both the inputs are on, and "No" if either one of them is off. Another example is an OR gate, which outputs "Yes" if either input is on, and "No" if both are off. To build a reversible gate, you need 2 inputs and 2 outputs, so that if you ran the computation backwards, you could recover the question from the answer. For example, if you put 42 into the computer, it should be able to spit out what the ultimate question is, just as easily as going the other direction. This is the meaning of reversibility.

Maxwell's demon is a thought experiment that James Clerk Maxwell came up with which illustrates how weird this connection between entropy and information is. If there were a little demon who were watching the individual molecules in a box, and he had a switch that could slide in a divider instantly in the middle of the box, then he could sit there and watch for the moment when each gas particle (normally, bouncing around randomly in the box) was about to cross the boundary from one side of the box to the other. If he presses the switch at just the right time, he can deflect the gas particle back into the left side of the box without expending any energy. If he keeps doing this for hours and hours, eventually all of the gas particles will randomly wander into the left side of the box and get stuck there, because he will put in the partition just as they try to cross the boundary back over to the right. Because entropy is connected to volume (smaller volumes have a smaller # of microstates), the final state has less entropy than the initial state, due to having half the volume. And yet, no work was done and no heat was expended in the process! This seems to be a blatent violation of the 2nd law of thermodynamics. So what happened here?

Well, in the real world, demons don't exist. And humans do not have the supernatural powers that demons have that would enable them to see individual gas particles moving super fast around in a box. But what if we set up a computer that could play the role of the demon? In principle, a computer could detect a gas particle much faster than a human, maybe even as fast as Maxwell's hypothetical demon. But if it does this, it has to either store the information about where each of the gas particles are, or temporarily watch each gas particle for a moment and then forget about it. If it stores this information, then it needs to fill up an exponentially large memory storage system. If it wants to keep the storage from getting out of hand, then it has to erase some of this information at some point... and erasure of information is an irreversible process. Because it is irreversible, it must dissipate heat. I mostly understand this part of Maxwell's demon. The other part I've always been a little bit fuzzy on though... what happens if the computer chooses to just store and store more and more information in its memory? Then, it will be filling up its memory with more and more information about the trajectories of the billions and billions of particles in the box. But does this in itself represent an increase in entropy? Or is it just the erasure of such information which increases entropy? It seems to me that storing anything at a memory location which could have previously taken multiple values but is then set to a single value represents a decrease in entropy. It would seem that storing it decreases entropy and then erasing it undoes that increasing it again. But I must be thinking about things a bit wrong there. I admit, this is where my understanding has always grown a bit fuzzy.

In the next part, I hope to actually get to wandering sets, and by extension, Boltzman's brain paradox, Poincare recurrence, and Liouville's Theorem. But maybe that's ambitious. To be continued...

There's a connected set of issues that rests at the very heart of physics, which I've always thought was not very clearly explained in any of my classes. I had a very vague understanding of it while I was an undergrad, and hoped that I'd be able to sharpen it up in graduate school. To some extent, I did, but I found that even after 6 years of graduate school in physics, these issues were still never quite cleared up in my classes, and I never had enough time while working on other things to read enough on my own about them to fill in all the gaps in my understanding. I have always had the impression that these issues *are* well understood, at least by somebody, but that not many physics professors do understand them fully and they tend to just avoid talking about them in actual classes, or talk about them in a very superficial way. I do think my adviser was among those who understood them, but I always felt a bit shy about wasting his time on questions that were purely for my own curiosity and unrelated to any research we were working on. Especially ones that would take a long time to sort out.

Recently, I've started thinking about them again, but through an unexpected route. A couple months ago, OkCupid added support for bitcoin, so as we got closer to the release of it, several of the people I work with would tend to get into idle conversations about bitcoin during the day. A few of them I got involved in, and in one of them we got on to the subject of bitcoin mining. One guy asked the group if it would be worth it to dedicate his server at home to bitcoin mining. This is where your computer works on various difficult mathematical tasks, like factoring large numbers, and helping to merge block chains into each other, and as a reward you collect newly minted bitcoins that are conjured into existence. One guy commented immediately that it would never be worth it with a standard PC, you'd have to buy very expensive specialized hardware to do it. The original asker of the question objected that he wasn't using his server for anything else anyway, but the cynic replied that with a standard PC, the cost of running the hardware in an increased electric bill was greater than the payoff. (An idle server draws less electricity than one engaged in difficult mathematical computation.) This got us onto a tangent, about whether you could cool the computer down to a point where it didn't cost as much to run because it wasn't dissipating as much heat. Naturally, we went from there to discussing why computing necessarily dissipates heat, and what the ultimate physical limits of that process is. He mentioned that only quantum computers were reversible and didn't dissipate heat. I was first made aware of this fact in 1998 during a gradate quantum computing class that I took as an undergrad at Georgia Tech, where we read some relevant papers on the subject. But while I have long had a superficial understanding for why this was true, and I can repeat the standard things that people say about it, I admitted to him that I never quite understood it fully. He then started talking rapidly about the many worlds interpretation of quantum mechanics, and how the so-called "measurement problem" is really just decoherence and thermodynamics. He said he thought it was fairly straightforward, and this is the point where I mentioned that I had a PhD in theoretical physics and had spent many years working on related topics, but felt that there was just always a missing piece for me conceptually surrounding entropy, Maxwell's demon, reversibility, and dissipation. He shrugged and said "well, it seems straightforward, but I guess if you've studied this more then there must be more to the story than I'm aware of." I told him I didn't want to get into discussing interpretations of quantum mechanics, because it was too complex a subject, and I probably couldn't say anything more on reversibility because I didn't know how exactly to articulate what was missing from my understanding of things.

Every few years I pick up this topic, or one of a related set of topics that ties into it, and try to understand it, and I always make a bit more progress. I think the last bit of big progress I made was in reading Leonard Susskind's book on black holes and information, which helped me understand both black hole complimentarity and more about the density matrix and how entropy and information work in quantum mechanics. But this conversation renewed my interest again, so for the past month or so my mind has occasionally drifted back to it, and a couple days ago I managed to stumble onto the Wikipedia page for "Wandering Sets" which I think is a huge part of the missing piece for me. For some reason, wandering sets were never mentioned in any of my undergrad or graduate classes, and yet they seem absolutely crucial to understanding what the word "dissipation" means. Until yesterday, I had honestly never even heard of them. It's no wonder that I went through undergrad and graduate school always feeling frustrated when professors would use the word "dissipation" without giving any definition for it and just assuming that we all knew what it meant. Unfortunately, I feel like the Wikipedia page is poorly written and there are two facts they mention which seem obviously contradictory to me.

I will explain in part 2 what I've learned so far, and how wandering sets are related to the ergodic theorem, Liouville's theorem, and pretty much every foundational area of physics. And then I'll go into what I still don't understand about it--perhaps after this I need to find the right book that covers this stuff. It seems very weird to me that they always gloss over it in physics classes.

In parts 2 and 3, I discussed one example of a quantum anomaly, the chiral anomaly generated by the electroweak sphaleron that could explain the baryon asymmetry of the universe (how there got to be more matter than antimatter).

Now let's take a look at another kind of quantum anomaly: the conformal anomaly. Personally, I consider the conformal anomaly to be the most interesting kind of anomaly of them all (although perhaps if I'd stayed in physics longer I would have discovered even more interesting ones, you never know).

What is the conformal anomaly?

The conformal anomaly is an anomaly that shows up in a lot of different quantum field theories, in one way or another. It has to do with a particular kind of symmetry called conformal symmetry. Said in one sentence, conformal symmetry is the symmetry group of transformations in a vector space that preserves angles, but not scale. This would include any kind of rigid rotations or scaling, but not stretching or twisting. It would also include more complicated transformation although I'm not sure how to describe them (maybe I will link to a picture in the next post). For example, in ordinary 3-dimensional space if you had a hologram of the Wizard of Oz's head, floating in space... a conformal transformation on the hologram would be to make the head look bigger or smaller, or rotate it around, perhaps upside down or sideways. An example of something that would *not* be a conformal transformation would be squishing the head in such a way that the face looked wider than it usually does, or taller than it usually does. In other words, if it looks distorted in a way that changes the "aspect ratio", then you've gone outside of the conformal symmetry group. If you stick to only rotations and scaling (or other things that preserve all angles within the hologram), then you're still in the conformal symmetry group. Of course, the Wizard of Oz's head is an example of something that does *not* have conformal symmetry. It does not have conformal symmetry because if you do a conformal transformation on it, it looks different (like, rotated or scaled).

So if the Wizard of Oz's head does not have conformal symmetry, then what *would* have conformal symmetry? Well, the rotation part is easy. In order to be symmetric under rotations, you'd have to have something like a sphere, where it looks the same no matter how you rotate it. Although a sphere itself won't work, because a sphere is not scale invariant--it has a particular radius. If you expand the sphere or shrink the sphere, you can tell that it's different. It has grown or it has gotten smaller. In other words, the sphere has a particular size, a "characteristic length" associated with it. For something to be truly conformally symmetric, it would need to have no size or all sizes at once. A simple example in ordinary 3d space would be a point--a second example would be the entire space. In either case, no matter how much you scale it up or down, it still looks the same. The size of the point is 0, even if you multiply 0 by 100. The size of the entire space is infinite, and even if you multiply it by 100 it's still infinite. I'm not sure there are other examples in 3d space, but quantum field theories happen in a much more complicated space than 3 dimensional space. One important thing with quantum field theories is that usually they are mathematically defined in such a way that there are a lot of "non-physical" properties of them, as well as some "physical" properties. And in order to obey conformal symmetry, you don't have to worry about the non-physical aspects, just the physical ones. To use a fairly loose analogy, if you hypothetically performed a rotation on a being that had a soul, and the actual material flesh and bones were symmetric (came back to the same position after the rotation) but you could tell that the soul had been rotated, then that being would still be considered symmetric for "all practical purposes" since it only differs by something that's non-physical and hence non-measurable. In physics, non-physical things are treated as an issue of "useful redundancy" in a theory. You deliberately include things in your metaphysical framework that are extra, beyond the measurable parts of the theory, because it makes the math easier or more elegant, but then you ignore them in making physical predictions. Often, there can be multiple ways of defining a theory where the non-physical elements are different, but the physical ones are identical. These are treated, essentially, as part of the subjective "language" that you're using to describe the world rather than part of the actual physical objective world itself... as are all metaphysical things. You can have equivalent models of reality even if you start from different frameworks or perspectives--that's a really important thing in physics actually, that shows up all over the place, but especially in relativity and quantum mechanics. These are often called "dualities" (for instance, wave-particle duality).

The rotational aspect of conformal symmetry is not the interesting part, it's more the scaling that's interesting. If something remains the same when you expand or shrink it, it's called "scale invariant". All conformal quantum field theories (also called "conformal field theories") are scale invariant. If a quantum field theory isn't scale invariant, then it doesn't have conformal symmetry and is therefore not considered a conformal field theory.

Now, to move beyond the more visualizable case of the hologram in 3d space, what does conformal symmetry mean in an actual quantum field theory, which is a mathematical structure that requires infinite-dimensional space to define? Well, you can sort of picture a quantum field theory itself as being "the laws of physics" for a particular universe. If those laws have particular constants in them that define a particular length scale, then just like the sphere that has a particular radius, they would not be scale-invariant and therefore have no conformal symmetry. One way to look at it is to ask the question of "what would happen if I were to double the size of everything in the universe... would the laws of physics change?" Although there is a bit more to it than that, because in quantum field theory, length is something that is connected to energy (and therefore mass), which is connected to momentum, which is connected to time, which is then connected back to space. So if you double all the lengths, you also have to halve all the energies and momenta, and double all the time intervals. All of these 4 types of things (space, time, energy, and momentum) can be measured in the same "natural units" instead of the more widely known "engineering units" (kilograms, meters, seconds, etc.) where they look like completely different types of things. Natural units in physics, are units where you consider the speed of light to be equal to 1, and Planck's constant (the fundamental constant of quantum mechanics, that separates the weird microscopic quantum world from the more normal macroscopic classical world) also equal to 1. (Technically, it is really Dirac's constant that is set to 1, but since everyone calls it Planck's constant colloquially I will too.)

Units are a pretty fundamentally important thing in physics. If I had to pick a "second most important thing in physics" besides symmetry, I would probably pick units. And they probably seem pretty boring if you have only experienced them at the high school or undergrad level. But later on, they actually become more and more interesting as you learn to do "dimensional analysis" and other neat tricks, that actually expose deep symmetries and properties of the theories or systems you're considering. One way of saying why units are so important, is that they are related to what I was saying earlier, about the difference between the metaphysical frameworks we use (the "language" that reality is constructed out of) and reality itself. Units are symbols that label things, they are the way in which things are measured, so they are connected to the whole idea of measurement and observation, and the fact that if you measure something you have to have some kind of measuring stick to measure it by. When people talk about the "role of the observer" in quantum mechanics, and say that the "measurement paradox" is what makes quantum mechanics so deep and interesting... I would agree, although I would also add that units are deep and interesting for exactly the same reason. And this is what the term "dimensional transmutation" has to do with.

Having gone over most of what conformal symmetry is, and begun to introduce the interesting aspects of units in physics (I have a lot more to say about them though), I will break here and then in part 5 we should be ready to jump right into dimensional transmutation itself, which is what happens when you have conformal symmetry broken by a quantum anomaly. Dimensional transmutation is the core of what I wanted to talk about in this series of posts, although as usual... it has taken me much longer than I had expected to introduce the subject =)

Incidentally, some of what I'm going to talk about here is the expanded version of one of the short background sections of my dissertation, section 1.5 (pages 14-19). In fact, I thought of writing this series of posts as I was writing that section, it just took me a while (over a year) to make it. If you want you can read that part here: http://physics.ucsc.edu/~jeff/dissertation.pdf. It's not very long, and some of it is about electric-magnetic duality and other things specific to the model I was working on, which I won't discuss here. But it's dense so you probably want to have some kind of physics background if you attempt reading it, preferably a particle physics background. Also, even with a background what I'm going to write here will have more depth and hopefully be more insightful, I'm mainly just linking to it for completeness and to indicate how this stuff fits into the research I published in grad school.

Speaking of my dissertation, I posted it about a year ago when it was finished, and encouraged people to find the "Easter Egg" I put in the front material. Nobody took me up on that, so I figure at this point I should just give it away. If you want to see the Easter Egg, click on the above link and read the last line of page xii!

So--What is a quantum anomaly? This seems like the best place to start.

Everything in particle physics is based on symmetry. If there's one principle that is widely acknowledged to be behind every other principle or "law" of physics in the universe, it is undoubtedly symmetry. Like many scientific and engineering fields, the particle physics community has its own professional magazine, to help keep members of the community informed about important advances going on in the field and current events. Appropriately, the name of this magazine is "Symmetry Magazine" (http://www.symmetrymagazine.org/). In physics, a symmetry is defined as any group of transformations you can perform on something that leaves important physical properties unchanged.

But there are a lots of different types of symmetries. And of all the different types of symmetry, there are several broad categories. One distinction is between exact symmetries and approximate symmetries. This distinction is pretty obvious--for an exact symmetry things are perfectly symmetric, while for an approximate symmetry they are not quite perfectly symmetric. Another distinction is between local (also known as "gauge") symmetries and global symmetries. A third distinction is between spacetime symmetries and internal symmetries. A forth distinction is between unbroken symmetries and spontaneously broken symmetries (also called "hidden symmetries"). And a fifth distinction is between non-anomalous symmetries and anomalous symmetries. Each of these categories has all kinds of examples within it. Explaining what all of these are would take us off track, but I mention them just to put the idea of anomalous symmetries in context--this isn't the only distinction you can make, and you can have just about all combinations of these general categories of symmetries show up in physical theories. But the anomalous/non-anomalous is nevertheless an important one.

The anomalous/non-anomalous distinction has to do with the way in which quantum theories are constructed out of classical theories. Actually, many of today's leading theorists would object to the way I said "constructed" here because really things should be viewed the other way around. In practice, what we do to define a quantum field theory is to start with a classical field theory and then "quantize" it by promoting all physical observables from regular variables into non-commuting measurement operators. However, in principle the world was not "constructed" in this way, that's just the way that humans have of understanding the world. We like to think classically, so we start with something classical and build something quantum out of it. But the world is not classical, it's quantum. So really the right way to think of it is that some quantum theories have "classical limits" where you imagine Planck's constant becoming zero and all commutation relations between different measurement operators vanishing. Some quantum theories even have more than one classical limit. An anomalous symmetry is when the classical limit of the theory is symmetric but the full quantum theory is not symmetric.

To start with one example which has some pretty sweeping consequences, let's take baryogenesis. The "baryon asymmetry" of the universe (also called "matter-antimatter asymmetry") refers to the puzzling fact that there is more matter in the universe than antimatter... and yet the underlying laws of physics look like they are symmetric with respect to matter and anti-matter--whatever can happen to matter can happen to anti-matter, and vice versa. If you started out with only pure energy (no matter) existing at the big bang, as the standard cosmological picture requires, and then evolved things forward in time as the universe cooled and expanded... you would have to end up with equal amounts of matter and antimatter, as long as the laws of physics really did have this symmetry built in. There has been a slight asymmetry observed at particle accelerators, but not nearly enough to account for the overall imbalance between matter and antimatter. (And incidentally, it's a good thing we have this imbalance, because without it life could not exist because the matter and antimatter would just annihilate with each other. But that doesn't help explain the mechanism for the origin of the imbalance, what is known as "baryogenesis".) What does provide at least some of the explanation however, is that if you look closer at the laws of physics governing matter and antimatter, you find a quantum anomaly.

This anomaly is known as the "electroweak sphaleron". It is one kind of quantum anomaly that fits into a category of anomalies called "instantons". These types of anomalies have to do with the fact that in quantum field theory, you can have multiple vacua within the same universe that have effects on each other. ("Vacua" is the plural of "vacuum"). This surely sounds like crazy talk to anyone outside of the field, however the word "vacuum" in quantum field theory just means "ground state". The vacuum in quantum field theory is the state when all of the quantum fields are in their lowest energy state. That doesn't mean they are at zero energy, due to the zero point energy, but it means it's lower than any other locally accessible state. So if there are multiple "lowest energy states" that the fields could get into that each have the same energy, then these are the "vacua states" of the theory. They are also called "degenerate vacua" (degenerate meaning "having the same energy"). In addition to that possibility, sometimes you can have a "false vacuum" or a "metastable vacuum" appear. If you notice I said that a vacuum had to be the lowest energy state of any other locally accessible states. By local I mean that you could smoothly transition from one state to the other in a classical way. But quantum mechanics allows for more "non-local" effects like quantum tunneling. If something would be the lowest energy state if there were no quantum tunneling allowed, then it is called a "false vacuum" or a "metastable vacuum". Sometimes the rate at which it can quantum tunnel into the "true vacuum" (the global energy minimum rather than just a local minimum) is so small that the false vacuum could exist for billions of years and you'd never know it was really the false vacuum. This is one possibility for the way our universe could end... we could be living in a false vacuum and suddenly tunnel into a true vacuum that has a greater degree of symmetry but no life allowed in it. Locally, a bubble would spontaneously form somewhere in space, and then expand until it filled the entire universe. However, given that this hasn't happened for the many billions of years the universe has been around and stable, I wouldn't bet on it happening any time soon, even if we are living in a false vaccuum.

So instantons are a form of quantum tunneling that happens between two different vacua states of a quantum field theory, sometimes between different degenerate vacua or other times from false vacuum to true vacuum. So the fact that there are multiple vacua states in the Standard Model allows for this instanton effect called the electroweak sphaleron to occur. The reason why you can't transition from one of those states to another classically is because they are in different "topological sectors" of field configuration space. Topology is the branch of mathematics that deals with deforming shapes into other shapes, and deciding which shapes can be smoothly deformed into each other and which cannot (because they would require ripping or tearing the shape apart). The fields in one topological sector cannot be smoothly (locally) deformed into a configuration that's in another topological sector--it would be like trying to untie a knot without letting go of either end of a string. Miraculously however, quantum tunneling allows you to effectively untie a knot without ever letting go of either end of the string. This is the instanton process and it results in a quantum anomaly--a symmetry that is there classically but violated quantum mechanically. Since all laws of physics are in the form of symmetries, a quantum anomaly is essentially a law of physics that is valid classically but can be broken (slightly) quantum mechanically. If you like, perhaps it could be described as sort of like "bending" the laws of physics every once in a while. More precisely, it's just that what is possible versus impossible is a bit more permissive in quantum mechanics than in classical mechanics because there are fewer symmetries forbidding things from happening.

A baryon is a proton or a neutron (or any other combination of 3 quarks, although those are the only two stable combinations). The Standard Model has a different topological sector for each baryon number (baryon number is the total number of protons and neutrons in the universe, minus the number of anti-protons and anti-neutrons since they are essentially "negative baryons", the opposite of baryons). So there is one vacuum where the total number of baryons is 0 (perfect symmetry between matter and anti-matter)--presumably this is how the universe started out. But there is another vacuum where the total number is 1, and another where it is 2 (and also ones where it is -1 or -2). In our universe, the baryon number is about 10^80, so we are in a topological sector that is pretty far from the symmetric sector that things started in. How did we get here? Well, possibly through the electroweak sphaleron process. However, in order to fully understand why we would have tunneled all the way in this direction, rather than the opposite direction or just done a random walk through the different sectors, you also need something that tilts the energy levels of the different vacua states so that the ones on the matter side have lower energy (and are therefore favored) while the ones on the antimatter side have higher energy and are therefore "false vacua". One of the papers I published in grad school was a paper on how this might be done.

Well, I think I will break here until part 3. But to give a sneak preview of where we're headed after this. The quantum anomaly in baryon number helps explain how we could have more matter than antimatter in the universe. However, there is another question (which at first glance might seem related but is not, really) of where most of the baryonic mass in the universe comes from. (By baryonic I mean, as opposed to dark matter which makes up most of the mass in the universe.) This is one misconception that I think a lot of non-physicists have. They think that matter has to have mass, or that the two are even synonymous or something. Neither matter nor anti-matter has to weigh anything or have any inertia... for a long time it was believed that neutrinos, for instance, were massless. This was a part of the Standard Model and has only been recently revised within the past decade or two. It turns out that they do have a very tiny mass (at least 2 of the 3 flavors do, we don't know for sure whether the 3rd flavor does) but there is nothing in the laws of physics themselves that say that matter has to have mass. The way in which the quarks and leptons get mass is through their interactions with the Higgs boson. The quarks make up protons and neutrons. However, if you add up the masses of the 3 quarks in a single proton or neutron, you don't get anywhere near the total mass of the proton or neutron. Most of the mass of protons and neutrons is not due to their constituent quarks, only a tiny fraction of it is. Where does the rest of this mass come from? It turns out, it comes from a quantum anomaly. The conformal anomaly is the relevant anomaly here (yes, the same anomaly responsible for the 10-dimensional requirement of string theory), and the surprising result of it in this case is that protons and neutrons get most of their mass from dimensional transmutation! I'll begin explaining that in part 3.

[Update: wrote this out this morning and then was thinking about a few of the things I wrote more today, and wondering whether I have remembered everything right. The main two things I'm not sure of is whether sphalerons are considered a type of instanton or just similar to instantons--a typical instanton is at a maximum of the potential energy of a field configuration, while a sphaleron is at a saddle point. Whether they are just similar or a sphaleron is considered a type of instanton I'm not entirely sure, although everything I've said should be true of both of them I believe. The second thing is, I'm not sure my description of each vacuum having a different baryon number is quite right. I may be mixing up some things from quantum chromodynamics, relating instanton winding number to baryon number, with the case of the electroweak sphaleron. I seem to recall other ways of looking at it where each vacuum has a slightly different definition of baryon number and therefore there is some overlap between different rungs on the energy spectrum, and a possibility for transition between them. I will think about these issues more and clarify and/or correct any details I messed up in my next post if need be.]