Domino Valdano

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.

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.]