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

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

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.

[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.]

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:
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.]

## Comments

vaelynphispoonlessAlthough the page seems to contradict itself somewhat. Early on they mention the Pontryagin index as a topological invariant in Yang Mills theories. But later on in the section on "various numbers of dimensions" they say that for SU(N) groups it's the 2nd Chern class that serves as the topological invariant, while for SO(N) groups it's the Pontryagin index.

The only reconciliation I can think of for that is that, from what I understand, the SU(2) subgroup is really the only thing that matters for all of the SU(N) groups when it comes to instantons, at least the normal kind. But SU(2) is isomorphic to SO(3) so maybe you can use the Pontryagin index for that reason. But then in that case, I don't know why the 2nd Chern class would be interesting.

vaelynphiI need more sleep!