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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 =)

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