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wormholes, part IV: the Casimir Effect

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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.

wormholes, part III: negative energy

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So what is this thing called negative energy (also called "exotic matter", and could it exist somewhere, or if it doesn't exist naturally, is there a way we could somehow generate it?

The two main theories of fundamental physics today are General Relativity and Quantum Field Theory. General Relativity was developed as a way to understand the large scale structure of the universe (cosmology, astrophysics, etc), while quantum field theory was developed as a way to understand the small scale structure (quantum mechanics, subatomic particles, etc.) Putting the two together is still a work in progress and string theory so far seems to be the only promising candidate, but it is far from complete.

General Relativity by itself is usually referred to as a "classical" theory of physics, since it doesn't involve any quantum mechanics. But there has been a lot of work using a "semi-classical" theory called Quantum Field Theory in Curved Spacetime. This is basically quantum field theory but where the space the quantum fields live in is allowed to be slightly curved as opposed to perfectly flat. Because this doesn't work once the curvature becomes too strong, it's not a full theory of quantum gravity, and is only regarded as an approximation. But it has been good enough to get various interesting results (for example, the discovery of Hawking radiation).

In General Relativity by itself (usually referred to by string theorists as "classical GR"), there are a number of "energy conditions" which were conjectured early on, specifying what kinds of energy are allowed to exist. The main ones are the strong energy condition, the dominant energy condition, the weak energy condition, and the null energy condition. As I understand it, all of these are satisfied by classical physics. If there were no quantum mechanics or quantum field theory, then it would be easy to say that wormholes are impossible, since negative energy is not even a thing. But in quantum field theory, the situation is much more subtle. In Kip Thorne's 1989 paper he finds that a variant of the weak energy condition (AWEC = averaged weak energy condition) is the one which would need to be violated in order to construct his wormhole. I've seen more recent papers which focus more on ANEC (averaged null energy condition) though, so perhaps there have been wormhole geometries since discovered which only require violation of the null energy condition.

I'm not going to explain what the difference is between all of these different energy conditions. But I should explain the difference between the "averaged" conditions and the regular ("local") conditions. The weak energy condition says that the energy density measured by every ordinary observer at a particular location in space must be zero or positive. The surprising thing about quantum field theory is that this, as well as all of the other local conditions (local means at a particular point) are violated. In other words, in quantum field theory, negative energy is very much "a thing".

But hold your horses for a second there! Because the thing about quantum field theory is that, there are loads of different examples of weird things that can happen on short time scales and at short distances that cannot happen macroscopically. For example, virtual particles exist that travel faster than the speed of light, masses can be imaginary, and energy is not even strictly conserved (there are random fluxuations due to quantum uncertainty). There are particles and antiparticles being created out of the vacuum and then annihilated all the time (quantum foam). There are bizarre things called "ghosts" that can have negative probability (which I won't go into). But when you look at the macroscopic scale, none of these weird effects show up--through very delicate mathematics, they all cancel out and you end up having very normal looking physics on the large scale. It's like if you look at the individual behavior at the microscopic level, everything is doing something completely weird and bizarre. But if you take an average of what's happening, it all gets smoothed out and you have very solid reliable macroscopic properties: energy is conserved, probabilities are positive, everything moves at less than the speed of light, etc. These things have been proven and are well understood. So given everything I know about how quantum field theory works, my intuition would be that something similar happens for negative energy: it's the kind of thing that could happen momentarily on the microscopic scale, but would never be the kind of thing one would expect to see on the macroscopic scale. And that's the main reason I've always told people I don't think wormholes are possible, despite not having reviewed most of the relevant literature related to it until this month.

After reviewing the literature, I have seen that over the past 20 years, the case that negative energy cannot exist macroscopically in our universe has grown stronger. Since the mid 90's the focus has shifted from energy conditions to what are known as "quantum energy inequalities" or QEI's. I read a couple review papers on QEI's, and will try to summarize in my next part. The gist of it is that while negative energy can happen locally, there are limits which can be placed on how negative that energy can be. And the limits depend on what timescale you're looking at. If you want a very negative energy, you will only find that on a very short timescale. If you want only a little bit of negative energy, you might find it on a longer time scale. But once you get to timescales like a second or more, the amount of negative energy you can have at a point is indistinguishably different from zero. There is a related idea called "quantum interest". Quantum interest refers to the fact that: given any negative energy spike there will be some compensating positive energy spike in the near future to compensate it (and make it average out to zero). And the time you have to wait to have this "payback" in the energy balance is shorter the larger the initial spike.

Gotta run for now, but I still have more to summarize on this. To be continued in part IV!

wormholes, part II: Kip Thorne

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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.

wormholes, part I

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I saw Christopher Nolan's latest film Interstellar a couple weeks ago on opening night. I hadn't even heard about it before, my company just decided to buy us all tickets to see it in IMAX, one of the nice things they do for employees once a month or so. I loved it and it left a big impression on me, and I intend to go back and see it again as soon as I get a chance. (Don't worry, I'm not going to spoil anything about the plot here--the intention of this series of posts is not to talk about the movie but about wormholes.)

I mostly loved it for the story, but also that you see concepts from theoretical physics like time dilation, black holes, and wormholes being applied in a relatively mainstream hollywood movie. Not all of it was very accurate scientifically, but it was still really exciting to see these things show up on the big screen in such a prominent spectacular way.

Lots of people were asking me about the time dilation effects after the movie, and it took me a few days to remember exactly how everything works and do enough back of the envelope calculations, but I eventually came to the conclusion that there is no realistic way in which the effects portrayed could have worked the way they did in the movie. But something similar could have maybe possibly happened if things had been a bit more complicated (for instance if there were two black holes nearby instead of just one). Maybe Kip Thorne (who consulted on the movie) suggested something like this but they decided they didn't care enough about the details to bother getting everything exactly right.

Regarding the wormhole itself, I feel like a bit of a hypocrite for getting excited about it. I've always been annoyed at how big the disconnect is between what the sci-fi community thinks is plausible and what is actually plausible according to our latest and most up to date scientific knowledge. As I've always told people who ask me, traversable wormholes are most likely completely impossible, not something that any civilization no matter how advanced could create. But they show up in sci fi all the time as if all we need to do is gain enough technology and then we can figure out how to build one. Worse, the depictions of them in sci fi are usually nothing like a real wormhole would look like, in the unlikely case it turned out somehow they were possible. At least, in Interstellar, they got this right--a real wormhole would look like a sphere, not like a hoop as I've seen in most sci fi.

After I watched the film, I started thinking about wormholes and the different conversations I've had with people, many of whom tend to be very enthusiastic about the possibility of using them for interstellar travel some day. I always have tried to emphasize that it's very unlikely that they are possible at all, even in principle. But I realized that the truth is--I have never looked into the science behind them deeply enough to know exactly what the reasons for this are, and what possible loopholes there could be that might allow some advanced civilization to build one. So for the past couple weeks I did my due diligence and looked through the current scientific literature to find out what the present state of knowledge is. What is the most solid argument against them being possible--are they almost certainly impossible, or just probably impossible, or is it that we really just don't know whether they are impossible? I think my answer to this is about the same as when I started looking through the literature--they're either very likely impossible or almost certainly impossible depending on who you believe, but as of yet nobody has succeeded in coming up with an absolutely rock solid proof that they are impossible. However, I now know much more of the details than I did a few weeks ago, so I'd like to share them and let you be the judge!
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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!Collapse )

gymnastics

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I started doing gymnastics in late-2009, after switching to that from a brief flirtation with parkour. The problem is, I moved so much between 2009 and 2013 (from California to Illinois in 2010, then to Pennsylvania in 2012, then to Queens, New York 6 months later, and finally to New Jersey in mid-2013. Every time I moved I had to find a new gym and figure out the schedule and how to get there.

In Illinois, I was pretty sure the closest gym was nearly 3 hours away in Chicago, so I tried to drive up there at least one Saturday per month to practice, but it was pretty impractical. Eventually I found one closer but it was almost time to move again by that point. So for the first 3.5 years it was nearly impossible for me to get consistent practice in. On top of that, the gyms I went to were all "open gyms", with only very minimal guidance from instructors so I was basically just teaching myself.

Well I'm pleased to announce that now that I've settled down, the past year (starting in mid-2013) I've actually been going pretty regularly, almost every week, and I'm finally starting to make some progress. Even more important than the regularity is that the gym I go to now has actual instructors that teach an actual adult class. I've realized now that my form was so bad I had very little chance of doing anything non-trivial until I got the proper training and feedback I needed to improve my body position.

There's a beginner class, an intermediate class, and an advanced class, and I've just recently decided to start attending the intermediate class rather than the beginner, even though I'm still kind of on the border. I'm generally one of the best when I go to the beginner, but one of the worst when I go to the intermediate--I get different things out of each when I go.

Also, now that I own a house with an actual lawn, I can practice in the spring and summer in the backyard on the weekends. It's more difficult because the grass isn't as springy as a gymnastics floor, but I've found that if I can do something on the grass then it means I have *really* got it down.

Recently I took some videos of me doing a few things. The ones that came out the best were my front-to-back cartwheel and front handspring. However, when I was editing the videos I was struck by an interesting thought about the physics of gymnastics. When you run the video in reverse, the cartwheel changes chirality from right-handed to left-handed. But when you run the handspring in reverse, it changes from a front handspring to a back handspring. (By the way, the term "front-to-back" cartwheel just means that you start facing forward and end up facing backwards, but in between while you're rotating the body twists around--as oppose to a regular cartwheel where you just start out facing to the side and don't twist as you rotate. Either one demonstrates the change in chirality, but the front-to-back version is a bit more fun to do and watch.)

Anyway, there are some really neat parallels between this and particle physics which I hadn't fully realized before. So I want to post the videos and then explain how gymnastics can be used to illustrate the concept of "discrete spacetime symmetries" in particle physics. I have it in a couple formats, including animated gif, but the gif version loads so slowly in a web browser that I need to cut down the size or something before posting it--or maybe I'll give up on that and just upload a video to Youtube instead. Still figuring out what the best way is to do it--but soon!

wandering sets part 10: mystery solved!

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When I wrote part 9 I was thinking I was pretty confused and I was not sure I had really made any progress on answering the basic question that I had set out to answer with this series. But I'm pleased to announce that within 24 hours after writing that one, I started thinking and piece by piece, it just all came together. I think I have pretty much solved the mystery, aside from some minor threads that might still need to be wrapped up. (Just didn't get a chance to write it down until today.) That was way faster than I'd imagined it would take.

The main thing I wasn't seeing is how mixing (whether the ordinary process of two gasses mixing in a box, or the more esoteric quantum measurement process) relates to wandering sets. And the lynchpin that was missing, that holds everything together, and explains how mixing relates to wandering sets, is "what is the identity of the attractor?"

I realized that if I could pinpoint what the attractor was in the case of mixing, then I would see why mixing is a wandering set (and hence, a dissipative process). Soon after I asked myself that question, the answer became pretty obvious. The attractor in the case of mixing--and indeed, in any case where you're transitioning from a non-equilibrium state to thermodynamic equilibrium--is the macrostate with maximal entropy. In other words, the macrostate that corresponds to "thermodynamic equilibrium".

I think the reason I wasn't seeing this is because I was thinking too much about the microstates. But from the point of view of a microscopic description of physics, any closed system is always conservative--all of the physics is completely reversible. You can only have dissipation in two ways. One is fairly trivial and uninteresting, and that's if the system is open and energy is being sucked out of it. Sucking out energy from a system reduces its state space, so from within that open system, ignoring the outside, you start in any corner of a higher dimensional space and then you get pulled into an attractor that represents the states which have lower total energy. If energy keeps getting sucked out, it will eventually all leave and you'll just be left in the ground state (which would in that case be the attractor).

But there's a much more interesting kind of dissipation, and that's when you course grain a system. If you don't care about some of the details of the microscopic state, but you only care about the big picture, then you can use an approximate description of the physics, you can just keep track of the macrostate. And that's where the concept of entropy comes into play, and that's when even closed systems can involve dissipation. There's no energy escaping anywhere, but if you start in a state that's not in thermodynamic equilibrium, such as two gasses that aren't mixed at all, or that are only halfway mixed, or only partially mixed anywhere in between... from the point of view of the macrostate space, you'll gradually get attracted towards the state of maximal entropy. So it's the macrostate phase space that is where the wandering sets comes in, in this case. Not the microstates! The physics of the evolution of the macrostate involves a dissipative action, meaning it contains wandering sets; and it is an irreversible process because you don't have the microstate information that would be required in order to know how to reverse the process.

So how does this work in the case of a quantum measurement? It's really the same thing, just another kind of mixing process. Let's say you have a quantum system that is just a single spin (a "qubit") interacting with a huge array of spins comprising the "environment". Before this spin interacts, it's in a superposition of spin-up and spin-down. It is in a pure state, similar to the state where two gasses are separated by a partition. Then you pull out the partition (in the quantum case, you allow the qubit to interact with its environment, suddenly becoming entangled with all of the other spins). In either case, this opens up a much larger space, increasing the dimensionality of the microstate space. Now in order to describe the qubit, you need a giant matrix of correlations between it and all of the other spins. As with the mixing case I described earlier, you could use a giant multidimensional Rubik's cube to do this. The only difference is that classically, each dimension would be a single bit "1" or "0", while this quantum mechanical mixing process involves a continuous space of phases (sort of ironic that quantization in this case makes something discrete into something continuous). If this is confusing, just remember that a qubit can be in any superposition of 1 and 0, and therefore it takes more information to describe it's state than a classical bit requires.

But after the interaction, we just want to know what state the qubit is in--we don't really care about all of these extra correlations with the environment, and they are random anyway. They are the equivalent of thermal noise, non-useful energy. So therefore, we shift from our fine grained description to a more course grained one. We define the macrostate as just the state of the single qubit, but averaged over all of the possibilities for the environmental spins. Each one involves a sum over its up and its down state. And if we sum over all of those different spins, that's accomplished by taking the trace of the density matrix, which I mentioned in part 9. Tracing over the density matrix is how you course grain the system, averaging over the effects of the environment. As with the classical mixing case, putting this qubit in contact with the environment suddenly puts it in a non-equilibrium state. But if you let it settle down for a while, it will quickly reach equilibrium. And the equilibrium state, the one with the highest entropy, is one where all of the phases introduced are essentially random, ie there are no special extra correlations between them. So the microstate space is a lot larger, but there is one macrostate that the whole system is attracted to. And in that macrostate, when you trace over the spins in the environment, you wind up with a single unique state for the qubit that was measured. And that state is a "mixed state", it's no longer a coherent superposition between "0" and "1" but a classical probability distribution between "0" and "1". The off diagonal elements of the density matrix have gone to zero. So while the microstate space has increased in dimensionality, the macrostate space has actually *decreased*! This is why I was running into so much confusion. There's both an increase in dimensionality AND a decrease in dimensionality, it just depends on whether you're asking about the space of microstates or the space of macrostates.

Mystery solved!

I'm very pleased with this. While I sort of got the idea a long time ago listening to Nima Arkani-Hamed's lecture on this, and I got an even better idea from reading Leonard Susskind's book, it really is all clear to me now. And I have to thank wandering sets for this insight (although in hindsight, I should have been able to figure it out without that).

I would like to say "The End" here, but I must admit there is one thread from the beginning--Maxwell's Demon--which I never actually wrapped up. I suspect that my confusion there, about why erasure of information corresponds to entropy increase, and exactly how it corresponds, is directly related to my confusion between macrostate and microstate spaces. So I will write a tenative "The End" here, but may add some remarks about that in another post if I think of anything more interesting to say. Hope you enjoyed reading this series as much as I enjoyed writing it!

The End

wandering sets part 9: still confused

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I have to admit, in trying to tie all of this together, I have realized that there still seems to be something big that I don't understand about the whole thing. And there is at least one minor mistake I should correct in part 8. So from here on out, we're treading on thin ice, I'm doing something more akin to explaining what I don't understand rather than describing a solution.

It seemed that if I could understand wandering sets, then all of the pieces would fit together. And it still seems that way, although the big thing I still don't get about wandering sets is how they related to mixing. And that seems crucial.

The minor mistake I should correct in part 8 is my proposed example of a completely dissipative action. I said you could take the entire space minus the attractor as your initial starting set, and then watch it evolve into the attractor. But this wouldn't work because the initial set would include points that are in the neighborhood of the attractor. However, a minor modification of this works--you would just need to start with a set that excludes not only the attractor but also the neighborhood around it.

In thinking about this minor problem, however, I realized there are also some more subtle problems with how I presented things. First, I may have overstated the importance of dimensionality. In order to have a completely dissipative action, you could really just use any space which has an attractor that is some subset of that space, where it attracts any points outside of it into the attractor basin. The subset wouldn't necessarily have to have a lower dimension--my intuition is that in thermodynamics that would be the usual case, although I must admit that I'm not sure and I don't want to leave out any possibilities.

This leads to a more general point here that the real issue with irreversibility need not be stated in terms of dimension going up or down--a process is irreversible any time there is a 1-to-many mapping or a many-to-1 mapping. So a much simpler way of putting the higher/lower dimensionality confusion on my part is that I often am not sure whether irreversible processes are supposed to time evolve things from 1-to-many or from many-to-1. Going from a higher to lower dimensional space is one type of many-to-1 mapping, and going from lower to higher is one type of 1-to-many mapping. But these are not the only types, just types that arise as typical cases in thermodynamics, because of the large number of independent degrees of freedom involved in macroscopic systems.

Then there's the issue of mixing. I still haven't figured out how mixing relates to wandering sets at all. Mixing very clearly seems like an irreversible process of the 1-to-many variety. But the wandering sets wiki page seems to be describing something of the many-to-1 variety. However, they say at the top of the page that wandering sets describe mixing! I still have no idea how this could be the case. But now let's move on to quantum mechanics...

In quantum mechanics, one can think of the measurement process in terms of a quantum Hilbert space (sort of the analog of state space in classical mechanics) where different subspaces (called "superselection sectors") "decohere" from each other upon measurement. That is, they split off from each other, leading to the Many Worlds terminology of one world splitting into many. Thinking about it this way, one would immediately guess that the quantum measurement process therefore is a 1-to-many process. 1 initial world splits into many different worlds. However, if you think of it more in terms of a "collapse" of a wavefunction, you start out with many possibilities before a measurement, and they all collapse into 1 after the measurement. So thinking about it that way, you might think that quantum physics involves the many-to-1 type of irreversibility. But which is it? Well, this part I understand, mostly... and the answer is that it's both.

The 1-to-many and many-to-1 perspectives can be synthesized by looking at quantum mechanics in terms of what's called the "density matrix". Indeed, you need the density matrix formulation in order to really see how the quantum version Lioville's theorem works. In the density matrix formulation of QM, instead of tracking the state of the system using a wavefunction--which is a vector whose components can represent all of the different positions of a particle (or field, or string) in a superposition--you use a matrix, which is sort of like the 2 dimensional version of a vector. By using a density matrix instead of just a vector to keep track of the state of the system, you can distinguish between two kinds of states--pure states and mixed states. A pure state is a coherent quantum superposition of many different possibilities. Whereas a mixed state is more like a classical probability distribution over many different pure states. A measurement process in the density matrix formalism, then, is described by a mixing process that evolves a pure state into a mixed state. This happens due to entanglement between the original coherent state of the system and the environment. When a pure state becomes entangled in a random way with a large number of degrees of freedom, this is called "decoherence". What was originally a coherent state (nice and pure, all the same phases), is now a mixed state (decoherent, lots of random phases, too difficult to disentangle from the environment).

What happens is that you originally represent the system plus the environment by a single large density matrix. And then, once system becomes entangled with environment, the matrix decomposes into the different superselection sectors. These are different sub matrices, each of which represents a different pure state. The entire matrix is then seen as a classical distribution over the various pure states. As I began writing this, I was going to say that because it was a mixing process, it went from 1-to-many. But now that I think of it, because the off-diagonal elements between the different sectors end up being zero after the measurement, the final space is actually smaller than the initial space. And I think that's even before you decide to ignore all but one of the sectors (which is where the "collapse" part comes in, in collapse based interpretations). From what I recall, the off-diagonal elements wind up being exactly zero--or so close to zero that you could never tell the difference--because you assume the way in which the environment gets entangled is random. As long as each phase is random (or more specifically--as long as they are uncorrelated with each other), when you sum over a whole lot of them at once, they add up to zero--although I'd have to look this up to remember the details of how that works.

I was originally going to say that mixed states are more general and involve more possibilities than pure states, so therefore evolving from a pure state to a mixed state goes from 1-to-many, and then when you choose to ignore all but one of the final sectors, you go back from many-to-1, both of these being irreversible processes. However, as I write it out, I remember 2 things. The first is what I mentioned above--even before you pick one sector out you've already gone from many-to-1! Then you go from many-to-1 again if you were to throw away the other sectors. And the second thing I remember is that, mathematically pure states never really do evolve into mixed states. As long as you are applying the standard unitary time evolution operator, a pure state always evolves into another pure state and entropy always remains constant. However, if there is an obvious place where you can split system from environment, it's tradition to "trace over the degrees of freedom of the environment" at the moment of measurement. And it's this act of tracing that actually takes things from pure to mixed, and from many to 1. I think you can prove that from a point of view of inside the system, whether you trace over the degrees of freedom in the environment or not is irrelevant. You'll wind up with the same physics either way, the same predictions for all future properties of the system. It's just a way of simplifying the calculation. But when you do get this kind of massive random entanglement, you wind up with a situation where tracing can be used to simplify the description of the system from that point on. You're basically going form a fine grained approximation of the system+environment to a more course grained approximation. So it's no wonder that this involves a change in entropy. Although whether entropy goes up or down in the system or in the environment+system, before or after the tracing, or before or after you decide to consider only one superselection sector--I'll have to think about and answer in the next part.

This is getting into the issues I thought I sorted out from reading Leonard Susskind's book. But I see that after a few years away from it, I'm already having trouble remembering exactly how it works again. I will think about this some more and pick this up again in part 10. Till next time...
orangegray
I think I did a decent job in part 7 of getting across the main paradox in physics that has confused me over the years. And there's a very similar paradox that I found on the wandering sets Wikipedia page (http://en.wikipedia.org/wiki/Wandering_set), so now seems like a good time to return to that.

They define a wandering point as a point which has a neighborhood in phase space which, after some time in the future, never gets back to where it intersects itself again. Similarly, they define a wandering set as a set whose points never intersect each other again after a certain time in the future. One seemingly minor caveat, which may be important, is that the intersection doesn't have to be exactly zero (no points in common), just so long as it has measure zero in the entire space. Measure is sort of like volume (but more mathematically rigorous). So for example, the set of points in a 2D plane has zero volume in a 3D space. So if two 3D objects intersect only in a 2D plane, it doesn't count as a true intersection, since the volume of that intersection is zero. Same goes for higher dimensional spaces, except the definition of volume is different.

Going hand in hand with their definition of a wandering set is their definition of a "dissipative action". The action is the specific rules that time-evolve the system from the past into the future. It's defined as dissipative if and only if there is some wandering set in the space under that action. If there is no wandering set in the space, then it's a conservative action.

But now for the contradiction I thought I saw on the page (after re-reading it many more times, I've figured out why it's not actually a contradiction). They define one more thing, a "completely dissipative action", whose definition seemed to me at first to be completely incompatible with their definition of a dissipative action. They define a completely dissipative action as an action that time-evolves a wandering set of positive measure into the future in such a way that the path it sweeps out through the space (it's "orbit") ends up taking up the entire space--or more precisely, the measure of its orbit is the same as the entire space. The reason this seemed to be a contradiction to me is that I was picturing a case such as mixing, where you start out with an initial condition that takes up some limited subspace of the entire phase space (like one cube of the Rubik's cube we talked about earlier), and then after you time-evolve it forward it ends up expanding to fill the whole space. But if it expands to fill the whole space, then it can't be a wandering set, because the intersection between the whole space and the original set is non-zero (it's the original set)!

So how does one resolve this paradox? Well, the main mistake I was making was confusing the final image of the set after it gets time-evolved with its orbit. The final image is where it is in a single snapshot in time, while the orbit is like the image you get when you leave a camera lens open for a long time (called "time exposure" in photography, I believe). Basically, it's the union of all of the different images it takes up as it progresses in time, not just a single snapshot. If it were just the final image, then their definition of a completely dissipative action is indeed contradictory, and can't coexist with their definition of a wandering set.

Ok, so it's the orbit, not the final image. Even then, it's a bit hard to imagine a scenario that would be "completely dissipative". The reason I was hoping it would be easier to imagine this scenario is because I was hoping that maybe the simplest kind of dissipation would be complete dissipation. And maybe understanding that would be a big step in the process towards understanding any kind of dissipation. In order to imagine what kind of a scenario would work, we need to find a case where the original set never wanders back to it's starting point but ends up sweeping out a path that fills all of space. To do that, it's best to think about what the reason would be that a set might never wander back to its original starting point. In most normal situations in physics, if you've got things moving around according to nice simple laws of physics, and you didn't start at any special starting point, you'd expect the motion to fill the whole space and eventually wander back an infinite number of times. The only case where it wouldn't get back, is if somehow it gets trapped in some subspace. This could be a single point that it ends up approaching, or a line, or a plane, or even a circle for instance. For example, if you had a planet that goes near a solar system and gets sucked into the orbit of that solar system, it would end up getting trapped in an ellipse, and never continue its nice straight motion, never getting back to the original starting point, even if the galaxy were inside a giant box. There's a name for this kind of occurrence in physics, it's called an "attractor". Basically, it seems that in order to have dissipation in the sense describe on the wandering set Wikipedia page, you would need some form of attractor. The ellipse I described would be a regular attractor, but in chaos theory you also have weirder more fractal patterns called "strange attractors". Chaos theory (also known as non-linear dynamics) seems intimately connected with the topic of dissipation, as many of the dissipative systems I mentioned in the beginning (such as hurricanes) are chaotic systems. I wasn't kidding when I said the question I'm wondering about here involves "all areas of physics"! :-)

So now unlike the mixing case, where you're going from a small dimensionality to a higher dimensionality, we've got the opposite happening. You start the motion in a space of higher dimensionality, and then get trapped in an attractor of lower dimensionality. Because no matter where you start in the higher dimensional space, you tend to end up in the lower dimensional attractor, you've got a many-to-1 mapping from initial to final states. In other words, you've got irreversibility and hence dissipation! And this higher to lower dimensional transition seems much more similar to the collapse of the wavefunction in quantum mechanics, which also goes from higher to lower. As opposed to the mixing case which seems to go from lower to higher. So this is not just an issue of quantum mechanics working one way, and thermodynamics working the other--now we have the same paradox appearing solely within regular classical thermodynamics, (which hearkens back to my earlier point that this issue has been around longer than quantum mechanics).

For this case of moving from somewhere in the bulk space into a smaller attractor, the definition of a "completely dissipative action" makes more sense. If you pick as your starting set, the entire space except the attractor, and then all of those points move into the attractor, you have exactly satisfied the definition. The orbit includes both everything outside of the attractor (which is what you started with) as well as everything in the attractor (what you end up with, or near enough to count as the same measure). So the orbit does indeed take up the entire space. But the set is still wandering, since there is no intersection between itself and the final attractor. Presumably, an action that is only partially dissipative (as opposed to completely dissipative) would include an attractor which captures starting points in some of the rest of the space, but not all.

We're getting closer and closer, so hopefully in the next part I'll be able to resolve this higher/lower dimensionality paradox in both thermodynamics and quantum mechanics (or if not both, at least one of them).
orangegray
In parts 2 through 5 I explained a bit about how wandering sets and thermodynamics works. And in part 6 I explained a bit about how quantum mechanics works. Now we can begin to bridge the gap and see how the two different angles from which I've been approaching this question intersect.

One of the biggest confusions I've had in trying to piece this together over the years is in mixing up whether the process of dissipation involves a transition from a higher dimensional space to a lower, or from a lower to a higher. I think it is both depending on how you look at it, but you have to keep straight what space you're talking about and what you mean.

If you look at it from the point of view of quantum mechanics, dissipation comes from the measurement process which involves projection matrices (or projection "operators" more generally) which take many possibilities and collapse them down to one. It's common to hear people use the word "reduction" in phrases like "the reduction of the state vector" to mean measurement in quantum mechanics. And measurement is the only time something irreversible happens, the rest of the laws of quantum mechanics are entirely reversible. So you would think intuitively, that a reduction or a collapse involves going from a higher dimensional space to a lower dimensional space. That's what a projection is mathematically. For example, when you walk in the sun outside and it's not directly overhead, you are followed around by a shadow. Your shadow is a 2-dimensional projection of your 3-dimensional self on the ground. A shadow is one of the simplest kinds of projections, but mathematically a projection refers to anything that reduces a higher dimensional object to a lower dimensional image. That's what the measurement operators used in quantum mechanics do, but because they are acting within the quantum Hilbert space, they project a space of ridiculously large dimensionality down to one of slightly lower dimensionality (but usually, still infinite).

On the other hand, if you look at it from the point of view of thermodynamics, dissipation happens only when entropy increases. The microscopic laws of physics, even in classical mechanics, are completely reversible and non-dissipative. The only irreversibility that comes into play is when the available phase space of a system increases. Let's walk through a concrete example of a mixing process step by step and see why it is irreversible and why it increases entropy.

First, imagine that you just have 3 classical particles in a box. They just bounce around in the box according to Newton's laws of physics. They move in straight lines unless they bounce off of a wall, in which case their angle of refraction equals their angle of incidence, just as a billiard ball bounces off of the wall of a pool table. It's easy to see that these laws are reversible, and that if you applied them backwards, you'd see basically the same thing happening, it's just that all 3 particles would be moving backward along their original paths instead of forward. Nothing weird or spooky or irreversible about it. But now let's conceptually divide that box into a left side and a right side, and keep track of which side each of the 3 particles are in. If the microstate of this system is the exact positions of all 3 particles plus the exact direction that each of them is moving in, then let's call the "macrostate" a single number between 0 and 3 that equals how many particles are on the right side of the box. To get at this number, we can construct a simplified microstate phase space which is a list of 3 booleans specifying which side of the box they are on. For example, if particles A and B are on the left side of the box, and particle C is on the right side, our list would be (left,left,right). If they were all on the right side, it would be (right,right,right). The macrostate can be deduced from the microstate (by summing up the number of right's in our list), but the reverse is not true as some of the macrostates correspond to more than one microstate. For example, the macrostate "2" could either be (left,right,right) or (right,right,left).

The full microstate phase space is what we talked about earlier--it's an 18-dimensional space, 3 times 3 coordinates for position, and 3 times 3 coordinates for momentum. But in order to understand maxing, we only really have to visualize a simplified microstate phase space based on our list of 3 right/left booleans--in order to do so, you need to picture something that looks like a mini Rubik's cube. A regular Rubik's cube consists of 3x3x3 = 27 cubes (if you include the center cube, which doesn't actually have any colors painted on it, and you can't actually see). But they also sell "mini" Rubik's cubes that are only 2x2x2 = 8 cubes. They are much easier to solve, but not completely trivial if I recall. Each of the 8 cubes in a mini Rubik's cube corresponds to one of the 8 microstates of our system: (left, left, left), (left, left, right), (left,right,left), ... etc., ... (right,right,right). Each of the particles can be in one of 2 possible states, but there are 3 particles so the space is 3-dimenionsal. But because we're only concerned with left-versus-right the space is discrete rather than continuous.

Now imagine that instead of 3 particles, we had an entire mole of particles--that is, Avagadro's number of particles, 6.02x10^23 particles. Any quantity of gas that you could fit in an actual box and hold in your hand would realistically have to have at least this number of particles, and probably a lot more! So what happens to our space of states? Now, instead of being 3 dimensional it is 6.02x10^23 dimensional--quite a bit larger. But still each dimension can only have 2 possible states. Our 3-particle system had only 2^3 = 8 microstates. But this system has an unimaginably large number of microstates, 2^(6.02x10^23) states! Equilibrium for this system means that you have allowed the particle to bounce around long enough that they are nice and randomly mixed (roughly equal numbers on the left and the right). The entropy of any system is simply the natural log of the number of accessible microstates it has. If the system is in equilibrium, then nearly all states are accessible so the entropy is log(2^(6.02x10^23)) = 4x10^23. A very small number of those states correspond to most of the particles being on the left, or most being on the right, but these are such a negligible fraction out of the number above, that it doesn't change the answer.

But what if we started the system in a state where all of the particles just happened to be on one side of the box? In other words, in the state (left,left,left,left,...,left,left) where there are 6.02x10^23 left's? This is a very special initial condition, similar to the special state the universe started in which I mentioned earlier. Because there is only 1 microstate where all of the particles are on the left, this state is extremely unlikely compared to the state where roughly equal numbers are on the left and on the right. The entropy of this state is just log(1) = 0. The true entropy of a gas like this is of course more than 0, but for our purposes here, we only care about the entropy associated with its mixing equally on either side of the box. The rest of the entropy is locked up in the much larger microstate phase space mentioned earlier, before we simplified it down to only the part that cares about which half of the box the particles are in.

The main point of all of this, which I'm getting to is... if all of the particles start out on one side of the box, and then later they are allowed to fill the whole box, you've drastically increased the entropy, because there are a lot more possible places for the particles to be. A slightly more complicated version of what we just went through is if you had two different kinds of particles, let's call them blue and red. Imagine all of the red particles started on the left side, and all of the blue particles started on the right side, perhaps because there is initially a divider in between. Then when you lift the divider, the two of them mix with each other, and after it reaches equilibrium, roughly equal numbers of red and blue will be on both sides. This is what is meant by "mixing" in thermodynamics. There are many many more ways in which they could be mixed than the one way in which they could all be on their own sides, so there is a lot more entropy in the mixed state at the end than there was in the separated state in the beginning. Unlike the version of this where only 3 particles were involved, this version is irreversible in the sense that: it's extremely unlikely, and pretty much inconceivable, that you would ever see a mixed box of these particles naturally and spontaneously sort themselves out to all blue on one side and all red on the other, whereas you would find nothing surprising whatsoever if initially unmixed red and blue gases gradually mixed with each other and wound up in a perfectly homogeneously purple mixture at the end.

In this example, the available state space before the particles are allowed to mix involves only 1 state. Afterwards, it involves something with a size on the order of 2 raised to the power of Avogadro's number of states. So the entropy should increase by something on the order of Avogadro's number. It seems like what has happened is that the state space at the beginning was very small and low dimensional, and then at the end it is very large with high dimensionality. Naively, this appears to be exactly the opposite of what happens during a measurement in quantum mechanics. But somehow, that's not the case--what's going on really is exactly the same. I think what's happening is that we're just confusing two different spaces here. And it's a confusion that I've often made in thinking about this. Where we'll have to go from here to resolve this paradox is to discuss open vs closed systems, and to explore a little bit the many worlds interpretation of quantum mechanics, and what happens to entropy in different parts of the multiverse as different branches of the universal wave function evolve forward in time. I'll leave you with one final piece of the paradox which seems directly related to this high-low dimensionality confusion: if the total entropy remains exactly the same in all branches of the multiverse, then you would think that every time a quantum measurement is performed and different branches split off from each other, the entropy would get divided among them and hence be less and less in each branch over time. (Because surely, the dimensionality of the accessible states in one branch is less than the dimensionality of the accessible states in the combined trunk before the split?) And yet, exactly the opposite happens--while the total entropy remains exactly constant, the entropy in each single branch increases more and more every time there is a split!

To be continued...

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