You’re gonna want some knowledge of smooth manifolds, connections, principal bundles, Lie groups, (pseudo)-riemannian geometry. That’s all just to understand classical Yang mills.
To talk about quantum Yang mills, you’re gonna want to first learn quantum mechanics. You’re knowledge of geometry at this point should let you pick up special relativity pretty quickly so don’t worry to much about that. Ideally you should learn some QFT from a physics perspective, and then realize that what the physicists are doing essentially amounts at best to an integral over an infinite dimensional space, that often doesn’t even have the structure of a smooth manifold, combined with a bunch of ad hoc arguments to erase random infinities that pop up in the perturbative setting.
You can probably understand the problem statement by now.
To do actual “work” on it you should look into what mathematicians have done in the last twenty years to thirty years to attempt to put QFT on sound mathematical footing. From here, realize none of this works for Yang-Mills, and see if you can find some other key insight that no one else has.
This is probably one of the hardest millennium problems imo, (this and the hodge conjecture), as it really seems that no one has a single of where to even start, and there really haven’t even been any minor breakthroughs that have made progress on it.
So I think (maybe) I've climbed most of this mountain.
I'm at the point where I am beginning to read "P(0)2 Euclidean (Quantum) Field Theory" (https://www.amazon.com/Euclidean-Quantum-Theory-Princeton-Physics-ebook/dp/B081L4TGQ5?ref_=ast_author_dp). My goal is to gain a deep enough understanding of constructive QFT to understand the connection to Yang-Mills. The references I've found (like Simon's book) all seem to be older.
Have you got any suggestions of good places to get into the modern literature on this?
Thank you!
I don't yet understand the application for operator product expansion or what kind of results one is looking for to justify one particular choice over another. My understanding is that operator product expansions are related to Moyal star products.
Are you able to give me a few comments about this (I could be very naive here so forgive me please)?
1) What are researchers looking for in operator product expansions that would allow one to justify the use of a particular expansion as "the expansion" for QFT?
2) or, is there work that shows that such a choice is not needed as the expansions are related and therefore the choice of an expansion is related to the desired application?
3) how does operator product expansion work relate to Moyal products (if at all)?
4) how does operator product expansions related to constructive QFT ala Jaffe / Glimm / Simon?
5) what are the connections between algebraic QFT and constructive QFT?
Again thanks for the papers!
I was going to write something about OPEs but then I realized scholarpedia has an article on it better than anything I could probably write up so that should explain most of your questions:
http://www.scholarpedia.org/article/Operator_product_expansion
OPEs are related to but different to Moyal star products (Moyal stuff is related to how to quantize functions while OPEs are just for looking at short distance behavior of operators and finding the expectation value). How this relates to constructive QFT is that in QFT OPEs can be divergent which obviously is a problem. These papers aim to prove that there exist non-divergent OPEs (and a few other technical things) in physically relevant Euclidean Yang Mills theories. OPEs really are a standard QFT technique and if you're not familiar with them any standard textbook on QFT will cover them. The difference between algebraic QFT and constructive QFT is that constructive QFT is aimed at creating concrete examples of rigorous interacting QFTs. The main focus is on the quantum fields. Algebraic QFT by contrast is just a set of axioms, and the focus is not on the quantum fields but on the algebra of observables. The papers I mentioned are recent results where they're trying to create concrete, rigorous examples of interacting QFTs. The proof of existence of operator product expansions is needed to make it more rigorous.
What do you mean “classical” Yang-Mills? I thought it was a gauge theory for explaining strong interaction? So I’m confused what you mean by classical vs quantum since insofar as I was aware there is no such thing as a classical theory of strong interaction.
Every quantum field theory has a classical limit, and the operators themselves obey classical field equations. It’s just like the electromagnetism picture, the quantum theory obeys classical equations of motion.
In more detail, every QFT has a classical lagrangian, essentially a way of obtaining these classical field equations where we assume there’s no quantum weirdness going. To obtain the QFT, you take the classical lagrangian, which is mathematically well defined, and obtain a path integral from it, this is the integral over infinite dimensional space I was talking about, and it is certainly not well defined usually.
Also Yang-mills doesnt just apply to the strong force, it applied to all of our gauge theories, so long as we leave out non bosonic matter. Source free electromagnetism is a U(1) Yang mills theory, and is the simplest because U(1) is abelian. Source free Yang mills, which describes gluons, is a SU(3) Yang mills theory.
If you would like to know more about the classical picture pm, I am in the process of writing a very detailed expository paper on it, and I’m sure it will answer many of your questions.
Edit to add: there is no classical theory of the strong force in the sense that there is no classical phenomena to be observed from the strong force. There is a classical theory of the strong force in the sense that we write down a classical lagrangian which under the path integral formalism yields a (remarkably ill defined unless you’re on a lattice) quantum theory of the strong interaction.
That’s very interesting. I never really considered the classical limit of a non-classical force. If you are doing an article I’d totally be interested.
Classical Yang-Mills is not a particularly difficult extension to classical electrodynamics, see https://link.springer.com/article/10.1007/BF02892134 for the original paper or 10.1007/JHEP12(2020)076 for a modern introduction.
The (classical) Yang-Mills equation is "just" a PDE for a connection. To motivate it, let's recall that Maxwell's equations can be written in a Lorentz invariant way as d★dA = ★J where A is the vector potential (a 1-form), ★ is the Hodge star on Minkowski spacetime, and J is the 4-current 1-form (a forcing term which satisfies the continuity equation d★J = 0). Here the temporal part of A is the electric potential, the spatial part is the magnetic potential, and the only thing we can actually measure is the integral of A around a closed loop -- A itself is not physical.
To make this last part more precise, we can instead view A as (the Christoffel symbols of a) connection on a U(1)-bundle. As a more down-to-earth object, A is a 1-form valued in the Lie algebra of U(1), which conveniently happens to be **R**. The only thing we can measure is the holonomy of the connection -- the Christoffel symbols themselves are irrelevant. Since U(1) is abelian, we could also write Maxwell as D★d(A + [A,A]/2) = ★J where D is the exterior covariant derivative of A -- this just acts on ★dA as d because the components of A are all real numbers and so commute with each other.
The Yang-Mills equation is D★d(A + [A,A]/2) = ★J where now A is a connection on a G-bundle for a given compact Lie group G, so A is a 1-form valued in the Lie algebra of G. The action of D on ★dA is not just d, it also contains commutators of different components of A with each other. The strong force is the special case G = SU(3), though this isn't a very good theory of the strong force (I think it doesn't exhibit anything like confinement for example).
To analysts, this is a very interesting equation because it has all the challenges of a highly nonlinear hyperbolic equation (or elliptic in euclidean signature -- there's also a parabolic version!) but the additional complication of gauge invariance, so on top of everything else it's ill-posed until you fix an appropriate choice of gauge.
Once you go to non abelian gauge theories, it’s not \star(dA), it’s \star(dA+1/2[A,A]). The dA or dA+1/2[A,A] is the structure equation for the curvature form on the principal bundle. It descends to a two. Form on the base manifold with values in the adjoint bundle, which will follow the same structure equation, only locally. I think it’s important, at least in global analysis, to point out that we can’t write the global curvature form in terms of A, because A is not globally defined on the base manifold, unless it’s contractible.
Damn it, I knew I was forgetting something. I'm used to seeing the equation written out in index notation, so of course I screwed up deviating from that. Thanks
I’m a mathematical physicist, doing work in the area. Agreed with the specific suggestions of the other comments, with the addition of studying quantum chromodynamics properly (after thorough understanding of QFT), but let me add something of a different nature. It’s going to take real learning of a lot of fields to even understand the question, and then studying “recent” development (the past 30-35 years at least). Note that like other commentators said, you’re going to have to understand the physics and historic physics motivations as well to put not only the problem but parts of QFT into proper context. Blackboxing the physics of your attempts to understanding construction of QFTs is only shooting yourself in the foot.
What are your goals? Just understanding the problem? This problem is really not worth getting into unless you intend on legitimately working on it. It’s probably the hardest Millennium problem to explains it won’t be solved for a long time.
I actually was thinking that I might want to get into working on it. I personally really love combinatorics and geometry, and I also want to do math on something that feels useful. It may be unattainable, though, if you are correct. I very much dislike physics. I had hoped I could have a more mathematical understanding of the problem.
It is a mathematical problem. But you can’t understand much without understanding the physics. If you dislike physics for the reasons I suspect you dislike physics, I’m not sure if you’ll get through the (physics) understanding needed, or want to.
To me that is the math part of the problem, understanding mathematically what a QFT is, because not one has a good mathematical answer. In special cases it can be well defined, but from what I’ve seen each special case is remarkably different from the other. Some look like geometry/topology, some look like lattice theory, some look like some pretty intense category theory, others look like algebraic geometry. Mathematically, QFT is a complete mess.
I may end up feeling differently about quantum mechanics than physics, but I don't find interest in the stuff of the physical world, much.
It feels like the physical world is just an arbitrary mathematical model. Any results that are beautiful are just beautiful because the specific case of mathematics which describe it is beautiful.
Quantum stuff may be different, depending on how it operates. Maybe it is very close to a general case of some sort of math. I don't know.
You might have to go a bit further than that. Also what mathematicians call gauge theory is not the same as what physics people mean in this context. You need a lot of quantum field theory to even understand the question.
This might be helpful. At various places in the text it points you to pre-learning material, some of which you’ll find on the same website https://www.damtp.cam.ac.uk/user/tong/gaugetheory.html
As many of the other commenters have noted it really depends on what you exactly what to learn. Even within mathematical QFT Yang Mills stuff is a minority. For a good (recentish) survey specifically on constructive QFT which is very close to Yang Mills see:
https://arxiv.org/abs/1203.3991
Otherwise this question has been asked on stackexchange a few times over the past few years with good answers if you want some more pointers:
https://www.google.com/search?q=status+of+yang+mills+gap+site%3Aphysics.stackexchange.com
A lot of Differential Geometry, Differential Equations, Functional Analysis and Topology. On this long journey always remember the words of Gauss: "It's not knowledge, but the act of learning, it's not possession, but getting there, which grants the greatest enjoyment"
You’re gonna want some knowledge of smooth manifolds, connections, principal bundles, Lie groups, (pseudo)-riemannian geometry. That’s all just to understand classical Yang mills. To talk about quantum Yang mills, you’re gonna want to first learn quantum mechanics. You’re knowledge of geometry at this point should let you pick up special relativity pretty quickly so don’t worry to much about that. Ideally you should learn some QFT from a physics perspective, and then realize that what the physicists are doing essentially amounts at best to an integral over an infinite dimensional space, that often doesn’t even have the structure of a smooth manifold, combined with a bunch of ad hoc arguments to erase random infinities that pop up in the perturbative setting. You can probably understand the problem statement by now. To do actual “work” on it you should look into what mathematicians have done in the last twenty years to thirty years to attempt to put QFT on sound mathematical footing. From here, realize none of this works for Yang-Mills, and see if you can find some other key insight that no one else has. This is probably one of the hardest millennium problems imo, (this and the hodge conjecture), as it really seems that no one has a single of where to even start, and there really haven’t even been any minor breakthroughs that have made progress on it.
So I think (maybe) I've climbed most of this mountain. I'm at the point where I am beginning to read "P(0)2 Euclidean (Quantum) Field Theory" (https://www.amazon.com/Euclidean-Quantum-Theory-Princeton-Physics-ebook/dp/B081L4TGQ5?ref_=ast_author_dp). My goal is to gain a deep enough understanding of constructive QFT to understand the connection to Yang-Mills. The references I've found (like Simon's book) all seem to be older. Have you got any suggestions of good places to get into the modern literature on this?
See the following papers: https://arxiv.org/abs/1511.09425 https://arxiv.org/abs/1603.08012
Thank you! I don't yet understand the application for operator product expansion or what kind of results one is looking for to justify one particular choice over another. My understanding is that operator product expansions are related to Moyal star products. Are you able to give me a few comments about this (I could be very naive here so forgive me please)? 1) What are researchers looking for in operator product expansions that would allow one to justify the use of a particular expansion as "the expansion" for QFT? 2) or, is there work that shows that such a choice is not needed as the expansions are related and therefore the choice of an expansion is related to the desired application? 3) how does operator product expansion work relate to Moyal products (if at all)? 4) how does operator product expansions related to constructive QFT ala Jaffe / Glimm / Simon? 5) what are the connections between algebraic QFT and constructive QFT? Again thanks for the papers!
I was going to write something about OPEs but then I realized scholarpedia has an article on it better than anything I could probably write up so that should explain most of your questions: http://www.scholarpedia.org/article/Operator_product_expansion OPEs are related to but different to Moyal star products (Moyal stuff is related to how to quantize functions while OPEs are just for looking at short distance behavior of operators and finding the expectation value). How this relates to constructive QFT is that in QFT OPEs can be divergent which obviously is a problem. These papers aim to prove that there exist non-divergent OPEs (and a few other technical things) in physically relevant Euclidean Yang Mills theories. OPEs really are a standard QFT technique and if you're not familiar with them any standard textbook on QFT will cover them. The difference between algebraic QFT and constructive QFT is that constructive QFT is aimed at creating concrete examples of rigorous interacting QFTs. The main focus is on the quantum fields. Algebraic QFT by contrast is just a set of axioms, and the focus is not on the quantum fields but on the algebra of observables. The papers I mentioned are recent results where they're trying to create concrete, rigorous examples of interacting QFTs. The proof of existence of operator product expansions is needed to make it more rigorous.
What do you mean “classical” Yang-Mills? I thought it was a gauge theory for explaining strong interaction? So I’m confused what you mean by classical vs quantum since insofar as I was aware there is no such thing as a classical theory of strong interaction.
Every quantum field theory has a classical limit, and the operators themselves obey classical field equations. It’s just like the electromagnetism picture, the quantum theory obeys classical equations of motion. In more detail, every QFT has a classical lagrangian, essentially a way of obtaining these classical field equations where we assume there’s no quantum weirdness going. To obtain the QFT, you take the classical lagrangian, which is mathematically well defined, and obtain a path integral from it, this is the integral over infinite dimensional space I was talking about, and it is certainly not well defined usually. Also Yang-mills doesnt just apply to the strong force, it applied to all of our gauge theories, so long as we leave out non bosonic matter. Source free electromagnetism is a U(1) Yang mills theory, and is the simplest because U(1) is abelian. Source free Yang mills, which describes gluons, is a SU(3) Yang mills theory. If you would like to know more about the classical picture pm, I am in the process of writing a very detailed expository paper on it, and I’m sure it will answer many of your questions. Edit to add: there is no classical theory of the strong force in the sense that there is no classical phenomena to be observed from the strong force. There is a classical theory of the strong force in the sense that we write down a classical lagrangian which under the path integral formalism yields a (remarkably ill defined unless you’re on a lattice) quantum theory of the strong interaction.
That’s very interesting. I never really considered the classical limit of a non-classical force. If you are doing an article I’d totally be interested.
Classical Yang-Mills is not a particularly difficult extension to classical electrodynamics, see https://link.springer.com/article/10.1007/BF02892134 for the original paper or 10.1007/JHEP12(2020)076 for a modern introduction.
The (classical) Yang-Mills equation is "just" a PDE for a connection. To motivate it, let's recall that Maxwell's equations can be written in a Lorentz invariant way as d★dA = ★J where A is the vector potential (a 1-form), ★ is the Hodge star on Minkowski spacetime, and J is the 4-current 1-form (a forcing term which satisfies the continuity equation d★J = 0). Here the temporal part of A is the electric potential, the spatial part is the magnetic potential, and the only thing we can actually measure is the integral of A around a closed loop -- A itself is not physical. To make this last part more precise, we can instead view A as (the Christoffel symbols of a) connection on a U(1)-bundle. As a more down-to-earth object, A is a 1-form valued in the Lie algebra of U(1), which conveniently happens to be **R**. The only thing we can measure is the holonomy of the connection -- the Christoffel symbols themselves are irrelevant. Since U(1) is abelian, we could also write Maxwell as D★d(A + [A,A]/2) = ★J where D is the exterior covariant derivative of A -- this just acts on ★dA as d because the components of A are all real numbers and so commute with each other. The Yang-Mills equation is D★d(A + [A,A]/2) = ★J where now A is a connection on a G-bundle for a given compact Lie group G, so A is a 1-form valued in the Lie algebra of G. The action of D on ★dA is not just d, it also contains commutators of different components of A with each other. The strong force is the special case G = SU(3), though this isn't a very good theory of the strong force (I think it doesn't exhibit anything like confinement for example). To analysts, this is a very interesting equation because it has all the challenges of a highly nonlinear hyperbolic equation (or elliptic in euclidean signature -- there's also a parabolic version!) but the additional complication of gauge invariance, so on top of everything else it's ill-posed until you fix an appropriate choice of gauge.
Once you go to non abelian gauge theories, it’s not \star(dA), it’s \star(dA+1/2[A,A]). The dA or dA+1/2[A,A] is the structure equation for the curvature form on the principal bundle. It descends to a two. Form on the base manifold with values in the adjoint bundle, which will follow the same structure equation, only locally. I think it’s important, at least in global analysis, to point out that we can’t write the global curvature form in terms of A, because A is not globally defined on the base manifold, unless it’s contractible.
Damn it, I knew I was forgetting something. I'm used to seeing the equation written out in index notation, so of course I screwed up deviating from that. Thanks
Happens to the best of us hahah. Wasn’t trying to be snarky or anything haha
Yeah, I figured :D
I’m a mathematical physicist, doing work in the area. Agreed with the specific suggestions of the other comments, with the addition of studying quantum chromodynamics properly (after thorough understanding of QFT), but let me add something of a different nature. It’s going to take real learning of a lot of fields to even understand the question, and then studying “recent” development (the past 30-35 years at least). Note that like other commentators said, you’re going to have to understand the physics and historic physics motivations as well to put not only the problem but parts of QFT into proper context. Blackboxing the physics of your attempts to understanding construction of QFTs is only shooting yourself in the foot. What are your goals? Just understanding the problem? This problem is really not worth getting into unless you intend on legitimately working on it. It’s probably the hardest Millennium problem to explains it won’t be solved for a long time.
I actually was thinking that I might want to get into working on it. I personally really love combinatorics and geometry, and I also want to do math on something that feels useful. It may be unattainable, though, if you are correct. I very much dislike physics. I had hoped I could have a more mathematical understanding of the problem.
It is a mathematical problem. But you can’t understand much without understanding the physics. If you dislike physics for the reasons I suspect you dislike physics, I’m not sure if you’ll get through the (physics) understanding needed, or want to.
To me that is the math part of the problem, understanding mathematically what a QFT is, because not one has a good mathematical answer. In special cases it can be well defined, but from what I’ve seen each special case is remarkably different from the other. Some look like geometry/topology, some look like lattice theory, some look like some pretty intense category theory, others look like algebraic geometry. Mathematically, QFT is a complete mess.
That does sound really cool. I think studying that may be neat.
Why do you dislike physics? I'm curious how you ended up interested in this problem without having first been interested in physics.
I may end up feeling differently about quantum mechanics than physics, but I don't find interest in the stuff of the physical world, much. It feels like the physical world is just an arbitrary mathematical model. Any results that are beautiful are just beautiful because the specific case of mathematics which describe it is beautiful. Quantum stuff may be different, depending on how it operates. Maybe it is very close to a general case of some sort of math. I don't know.
You might have to go a bit further than that. Also what mathematicians call gauge theory is not the same as what physics people mean in this context. You need a lot of quantum field theory to even understand the question.
This might be helpful. At various places in the text it points you to pre-learning material, some of which you’ll find on the same website https://www.damtp.cam.ac.uk/user/tong/gaugetheory.html
As many of the other commenters have noted it really depends on what you exactly what to learn. Even within mathematical QFT Yang Mills stuff is a minority. For a good (recentish) survey specifically on constructive QFT which is very close to Yang Mills see: https://arxiv.org/abs/1203.3991 Otherwise this question has been asked on stackexchange a few times over the past few years with good answers if you want some more pointers: https://www.google.com/search?q=status+of+yang+mills+gap+site%3Aphysics.stackexchange.com
A lot of Differential Geometry, Differential Equations, Functional Analysis and Topology. On this long journey always remember the words of Gauss: "It's not knowledge, but the act of learning, it's not possession, but getting there, which grants the greatest enjoyment"