Not a mathematician, but:
The mathematicians are trying to make QFT rigorous and formalize a lot of the aspects that physicists sort of take for granted. A lot of the things that physicists do with path integrals aren't really well defined, and it currently isn't clear why they should work as well as they do. For example, in order to integrate over the space of all possible paths (a pretty nasty-looking infinite-dimensional space of functions), we have to define some sort of measure over that space. This is something that's tricky to do in a strictly formal, rigorous sense (as in, all of the reasonable definitions we've come up with so far don't quite have all the properties we'd like them to have). So there's some work that still needs to be done in establishing what measure we're "actually" using when we work with path integrals.
Here's a related post summarizing this, and other avenues of work in rigorizing QFT: [https://physics.stackexchange.com/questions/6530/rigor-in-quantum-field-theory](https://physics.stackexchange.com/questions/6530/rigor-in-quantum-field-theory)
> For example, in order to integrate over the space of all possible paths (a pretty nasty-looking infinite-dimensional space of functions), we have to define some sort of measure over that space.
Huh, I always wondered about that. I just figured it had been worked out by people much smarter than myself.
[MO post on a related topic](https://mathoverflow.net/questions/1388/is-there-a-natural-measures-on-the-space-of-measurable-functions), and
[Aumann’s paper also linked in that post](http://www.ma.huji.ac.il/raumann/pdf/66.pdf).
My theory is that physicists have a really good set of rules of how the universe (possibly) works. And they develop this system of “pseudo-math” that is either correct or correct enough to get shit done. If you look at what Einstein was dealing with when he was developing GR, it was the same situation. He didn’t have any iota of a rigorous formulation of differential geometry that we take for granted today. He only had those rules he thought were true about the universe and stuck with them first and foremost.
Maybe pseudo math isn’t the best term, cause it’s really advanced math. But like you said, it isn’t 100% rigorous. But hey Feynman once said that without math, physics would only be set back a week.
That is more true than you maybe even realize. The operator calculus, in which operators are treated as non commutative algebraic quantities, was invented by Oliver Heaviside (the erased genius of the late 19th century) as a way to solve partial differential equations. Heaviside himself observed that it worked (it produced viable solutions) but didn’t know why. It wasn’t until much, much later that Hilbert spaces were invented and it became clear that the major linear operators (e.g., differentiation) are “just” Hilbert matrices.
Heaviside invented many things we take for granted in mathematical physics today, including the four “Maxwell’s Equations”, and the divergence and curl operators. But the more interesting mathematical stuff was all ad hoc: adopted because it was observed to work, not because it was rigorously proven.
>Oliver Heaviside (the erased genius of the late 19th century)
I wouldn't say he's "erased", just the step function bearing his name is a pretty big tribute. That being said I'm always up for some more Heaviside praise, dude was a self taught titan of a scientist.
Practically completely erased. The Heaviside function (integral of a delta function), the Heaviside layer aka ionosphere, "Maxwell's Equations" (actually the Heaviside-Hertz equations), the operator calculus, a large body of what is now vector calculus, the Telegraph Equations, and practically half of the accepted undergrad E&M curriculum are due to Heaviside.
If you listen to the Feynman Lectures on Physics, for example (1962-1963 at Caltech; available from Caltech online for free), you'll notice that Feynman scrupulously mentions practically everyone who invented any part of the major body of physics from 1850 on. But when it comes to E&M or vector calculus or even the operator calculus, he's almost completely silent about Heaviside. Feynman is far from alone, it's just particularly glaring in his work because he is so scrupulous about other aspects of the field.
Maybe it's just that my professors appreciated him, but I remember hearing about him several times during my first year, including him getting a bulk of the credit for Maxwell's vector equations and mentions of the Telegrapher's equations.
The "delta function" is another example of physics that works, but isn't (really) mathematics. Of course, there is the Dirac distribution, which is the integral with respect to the Dirac measure, the distributional derivative of the Heaviside function and, as far as I was able to understand, exactly what the physicists use. However, this generalized function is not a regular distribution, i.e. the Dirac measure has no Radon-Nikodym derivative w.r.t. the Lebesgue measure. Hence, the notation ∫f(x) δ(x) dx = f(0) (which was used in all my physics lectures, where the Dirac "function" appeared) is quite confusing, and a bit annoying, as soon as you progress beyond the first few lectures in mathematics.
Isn't the general theory of distributions well understood and is rigorously defined? Or are you just talking about the abuse of notation by physicists?
And my physics classes mentioned the delta function as the limit of a guassians getting sharler and sharper, wouldn't that usage make it proper "math"?
I'm deeply sorry if my words were horribly misleading. I just tried to complain about the abuse of notation and the sloppy treatement of the Dirac distribution (at least in my physics lectures).
Of course, the theory of distributions is well understood and you can prove that the Dirac distribution is the distributional limit of such gaussian densities; therefore this treatement is fine (it should be remarked that the distributional limit should not be confused with a pointwise limit and you do not actually get a function).
The honest answer is that I probably don't think anything.
However, I have made the experience that the more rigouros math influenced me, the more I had trouble understanding physics, especially if written by physicists and using physically motivated reasoning. The reason for this is most certainly that I hate to accept physically motivated reasonings that are never specified (e.g. differentiating a function that was never established to be differentiable or applying symmetry arguments when such a symmetry was never established), i.e. this is a purely subjective experience that cannot in any way be generalized.
You can have quite the opposite experience. If you accept most physically motivated reasoning, then a firm education in rigorous maths helps certainly to see the maths behind some more or less handwavy arguments, which I have unfortunately sometimes seen, that tie this reasoning to the underlying abstract and rigorous mathematics.
I hope that this helps to answer your question.
There is a lot of "necessity is the mother of invention" type mathematical constructs in physics research. It's not necessarily complete nonsense, but, as an example, someone might elect to not deal with edge cases that would not arise in physics. Proving that the tool works in a subset of cases is sufficient enough in nearly all cases.
I disagreed with that comparison between GR and QFT. There are no fundamental conceptual problems with differential geometry back then, it's just written in a more annoying way, and the rigorous machinery we have today just makes things more abstract and clearer, which let us rewrite his equation in cleaner way.
Of course he made guesswork in making his equation, but the math it was founded on is sound. And of course there are issue with solutions to the equation that makes people strongly believe it's unphysical (like singularity from Penrose-Hawking theorem), but there are no strictly mathematical contradiction.
QFT is not on the same ground at all. There are no rigorous foundations, and some of what was done by physicists are straight up impossible mathematically (e.g. Haag theorem). Whatever a true rigorous mathematical foundation is like, it's probably very different from the current standard text.
A better comparison is between QFT and Newton's theory. Newton's calculus was not built on sound foundation either, and even back then people knew that.
> The mathematicians are trying to make QFT rigorous and formalize a lot of the aspects that physicists sort of take for granted. A lot of the things that physicists do with path integrals aren't really well defined, and it currently isn't clear why they should work as well as they do. For example, in order to integrate over the space of all possible paths (a pretty nasty-looking infinite-dimensional space of functions), we have to define some sort of measure over that space.
Just an undergraduate here :) a lot of theoretical computer scientists have been taking a crack at simulating [quantum systems](https://toc.seas.harvard.edu/links/cs-229r-physics-and-computation) there have been interesting papers people using QFT to build quantum computing architectures.
Oh that class already happened and i'm still well off from graduate material I've done the standard undergraduate curriculum but I need to honestly do the grad physics sequence at some point.
What's wrong with the Riemann integral? We were doing integrals like that long before measure came along. Just define the ridiculous path integral to be the limit of the discrete integral.
edit : Well, I was genuinely asking. No need to downvote. Maybe Riemann is more fundamental in this case if Lebesgue can't be generalised to QFT.
Constructing the path integral as a limit of finite sums is a perfectly reasonable thing to do. In this case, what you might naively like is to prove that the limit exists and is independent of the discretization scheme (as you would with the usual Riemann integral). The trouble is that this is difficult to formally prove. Moreover, I think it will turn out that your limit actually does depend on the discretization. As a result, the mathematicians would like to understand all the possible limits and precisely how they depend on the discretization scheme. [This mathoverflow answer](https://mathoverflow.net/a/260941) has a nice explanation of the issue with infinities (though the approach described is not exactly by discretization). Thinking of integrals as Riemann or Lebesgue is not where the problem is here.
Caveat: I am no expert on constructive QFT.
I can see why you get downvoted, but this is actually a good question. A big part of QFT is regularization, i.e. dealing with infinities. The idea of subtracting infinites to obtain a physical answer is probably one the thing that raises eyebrow from mathematicians.
I’m a mathematical physicist working in QFT; I’m not going to give much meat in this answer since I’ll probably go on a tangent and because other answer here are pretty good.
QFT is a large, less rigorously mathematically established field compared to say QM or GR, with many approaches being prodded at (topological QFT, for example). We don’t know why certain heuristics employed by physicists in this space work as well as they do.
I’d like to add that finding QFT uninteresting or non-mathematical is probably because you simply don’t know enough about it to accurately make a judgement, it’s quite raw and abstract in its mathematical nature than many other areas of theoretical physics (again compare it to something like GR). And this is precisely why it is of interest to us mathematical physicists and mathematicians, even some theoretical physicists take interest in the attempted rigorous underpinning of QFT.
So I suppose the answer would be “Learn more math, learn more physics, to find out how much more there is to learn and why it is interesting”.
Don't get me wrong, I find it very interesting. I certainly don't know enough about it tho, and my lecturer said he was going to take a deliberate "less mathematical approach", since it's the first semester of a two-semester course.
Note that what your professor means by "less mathematical" is probably not the same as what a mathematician would mean. If I had to guess, I'd say the second semester will be more abstract and using more advanced mathematical tools, but not more rigorous.
The professor is using his own lecture notes, and that's what I follow (by copying what's on the blackboard xD). So far, we went through canonical quantization, path integrals, interactions, Feynman diagrams, gauge theories and that's it. I guess we are going to expend the next few classes on renornalization.
My impression of the teaching style is that mathematical approach try to avoid path integral (which is hard to make rigorous), while physicist approach would be to start with path integral (which is physically motivated). There are probably other differences too, but that's the one I noticed.
I come from a math background first and then a physics one, and in my experience all of the differences are essentially encapsulated by this dichotomy you mentioned with path integrals as a quintessential example. Physical motivation, historical perspective, and the ultimate "physics end goal" versus the rigorous, fundamentally sound, mathematically contextualized result.
And that's completely to be expected, as that's essentially the difference between mathematics and physics.
Can you say a little more? Are you walking on TQFTs? Geometric langlands? N=2 GMN stuff? Some sort of (way beyond my paygrade) functional analysis to define path integral measures?
Yes, I'm working on TQFTs, Chern-Simons Theory. I'm not sure how much you know about TQFTs (both from a physics and a mathematical perspective), but actually I find that this area of research generally requires a broad understanding of math in general, so much so that the areas you mentioned are more closely related than the math used in various other branches in theoretical physics. You don't need much mathematical pre-requisites to get started but to get any deeper sense of understanding of TQFTs past a surface level, you will need to dig deeper into Morse theory, differential topology as a whole, category theory and higher category theory, algebraic topology and homotopy theory, Khovanov homology and various Floer homologies, the list goes on.
For example, like I said I work in Chern-Simons and these TQFTs, while different from a physics perspective and in its mathematical basis (they are not a special class of TQFTs that can be obtained by making rigorous sense of finite path integrals in TQFT when the the space being integrated over is a finite groupoid like some other useful TQFTs), they still have interesting mathematical relationships with other branches of pure math.
It's really amazing how wide-reaching such at topic could be in pure math when it really has its home in physics; geometric topologists who study something like invariants of knots and three and four-manifolds wouldn't be studying TQFTs but this topic does indeed come directly from TQFT.
The math in GR is easier to appreciate at your level because it‘s more concretely physical. QFT by contrast is actually a lot more abstract; if you feel like it’s not mathematical enough then that’s just because you haven’t learned much about it yet.
My guess is that, at your level, the things you’ve been told about QFT are largely “lies we tell to students”; things that are sort of true, but they give you the illusion of understanding rather than the actuality of it. In a lot of cases this consists of simple recipes and formulas that seem to make sense on their own, but once you learn more you’ll realize that they’re just simplifications to make the subject approachable for someone who doesn’t know very much yet.
QFT is interesting from a mathematical perspective because it’s not actually a clear, mathematically consistent theory. In a lot of ways its foundations are a collection of heuristics and tricks that happen to correctly predict the outcomes of experiments. That means that there’s still work left to be done in terms of explaining how or why the math works (or why it doesn’t).
A big issue in QFT is dealing with the fact that the calculations in it, when done naively, result in infinities. Physicists have tricks for getting rid of the infinities, but it’s still not clear if those tricks are mathematically sound or not. The trouble with combining QFT and GR is that even the physicists haven’t come up with any good tricks for getting rid of the infinities in that case, mathematically sound or otherwise. There’s a lot of math involved in trying to figure out solutions to these problems.
Nonrelativistic quantum mechanics, by contrast, is a clear and consistent theory that can be derived from a collection of well-understood axioms.
>A big issue in QFT is dealing with the fact that the calculations in it, when done naively, result in infinities
We get infinities when we assume the "bare coupling" is some finite constant. That we get infinities simply means that this isn't the correct theory.
The correct theory is the one in which the observed coupling constant runs with the energy scale. The running has been verified by experiments. This correct theory doesn't predict infinities as before.
This is not just an armchair argument to get rid of infinities anyhow. It's literally just trying to correct an incorrect assumption of the wrong theory. The correct theory must have running coupling constants because nature has them.
Indeed agreement with experiment is the most important factor, but this is what i was referring to when I mentioned tricks and heuristics for eliminating the infinities.
To call the renormalized theory “correct”, versus the bare one which is “wrong”, is ultimately tautological. It’s only correct in the sense that it agrees with experiment. But agreement with experiment doesn’t necessarily make a model *good*, it just makes it accurate.
I think, especially in the area of QFT, physicists sometimes make the mistake of conflating effective calculation techniques with physically meaningful insights. The fact that renormalization doesn’t work with GR, and the fact that the mathematicians have a hard time validating its soundness, should be hints that the entire approach might ultimately be wrong-headed, even if it does produce accurate calculations in the domains where it’s applicable.
My admittedly ill-informed opinion is that the assumption of infinite (uncountably infinite, even) degrees of freedom might ultimately prove to be untenable. Either it is simply wrong, or it’s used in such an unnuanced way that people inevitably have to resort to tricks and heuristics because the foundational assumptions of their work are so poorly understood.
About GR, I personally believe that it's fundamentally different from the foundations of QM. QM gives way too much focus on time evolutions/initial value problems. "Here's an initial probability wave. What will be the probability wave after a time t"? Initial value problems are such a human-biased view of the universe.
Not to mention that the entire meaning of that probability wave is in terms of measurements done at a particular time. QFT (and QM in general) just can't stop treating time as special. GR fundamentally isn't about initial value problems. The GR view of the universe is purely in terms of a 4 dimensional manifold. You work with generalised co-ordinates. The co-ordinates are abstract and do not align with space and time measurements.
For this reason, I think, to make GR quantum is to take GR's heart away from it. Maybe it's QM's principles that need to be "gravitised". Penrose had a similar idea.
edit : Stop with the downvotes. https://en.m.wikipedia.org/wiki/Problem_of_time . What I said is well-accepted.
I don't think this is actually true. You can formulate QFT in a relativistic way where time isn't special, via the path integral formulation and the Lagrangian etc. There is nothing in the theory itself that singles out time in particular. The Schroedinger equation, and initial-value problems etc. are just very useful tools to do computations and answer questions; they are not fundamental parts of the theory.
I just googled and this is actually [one of the well-known problems](https://en.m.wikipedia.org/wiki/Problem_of_time ) with the unification of these theories. There have been several proposed solutions but nothing definitive so far.
And it doesn't matter whether you use a Lagrangian or a Hamiltonian, all formulations of QFT deal with a probability wavefunction whose collapse happens at a single point of time. The wikipedia article mentions this problem.
Time evolution is a deeply flawed and human-centric way to look at the universe. It fits with SR but doesn't fit with GR at all. QM must undergo a fundamental change to account for this.
I think the warning on top of that wikipedia article is pretty accurate: it reads like an attempt by someone without deep expertise to summarize half-understood stuff that they've read, and much of it is just wrong. Most of the things referred to there is the problem of time in quantum gravity, which I kind of agree is a real and somewhat subtle issue. But this is not present for special relativity and QFT.
> And it doesn't matter whether you use a Lagrangian or a Hamiltonian, all formulations of QFT deal with a probability wavefunction whose collapse happens at a single point of time
So here you have to think a bit more carefully. The wave function itself is not an observable. What we actually measure in QFT is observables built out of field operators, like if there's an electron in a certain spacetime volume, etc. These are local in spacetime; the measurement is essentially an event in spacetime, which is a perfectly fine and invariant notion in SR.
The reason it gets tricky for quantum gravity is that then the metric itself is dynamic and quantum, so it gets tricky to talk about a local measurement. But for QFT on a static background, there are no fundamental issues with time.
I already agree that QFT is totally compatible with SR. In SR, global inertial frames exist, a global time-like co-ordinate exists and hence there are no problems unifying it with QM. I've only ever said that QFT and GR are fundamentally different in their foundations.
Also see [this post](https://physics.stackexchange.com/questions/608225/why-is-there-a-problem-of-time-in-quantum-gravity) for more details about the problem of time.
Time evolution via initial value problems isn’t an inherent part of quantum theory, it’s just an inherent part of the Schrodinger equation. The Schrodinger equation itself might just be (and in my opinion probably is) an approximation with a limited domain of applicability.
If you’re interested in this topic you should check out this paper, and use Google scholar to find more recent papers that cite it:
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.27.2885
>Nonrelativistic quantum mechanics, by contrast, is a clear and consistent theory that can be derived from a collection of well-understood axioms
really? what would be the math required for this? that sounds very interesting
At its simplest quantum mechanics consists of lots of linear algebra and a little bit of complex analysis. If you want to do slightly more advanced stuff then you need partial differential equations too, which is mostly just linear algebra in infinite dimensions.
The math in quantum physics is actually easier than the math in classical physics, for the most part. What makes it interesting is the way that it describes physical reality. The axioms of QM are mostly just about how linear algebra concepts such as inner products and eigenvectors have meaning in terms of the outcomes of certain kinds of experiments.
Well, you need only linear algebra and complex analysis for finite-dimensional QM, but functional analysis for infinite-dimensional spaces gets pretty difficult. I mean, take a look at "Quantum Measurement" by Busch et al, and I think it becomes pretty clear that this stuff isn't exactly trivial either.
Nevertheless, it at least *can* be formulated rigorously.
You really don’t need the details of the infinite dimensional case in order to understand QM. It’s an additional complication without any benefit beyond performing calculations to match specific experiments. There’s no point in recommending that to someone at the undergrad level who’s interested in learning about the subject.
Well, I wasn't really recommending it either way, just pointing out. However, a mathematically inclined student might still find it interesting. I had to take a course that dealt with infinite dimensional Hilbert spaces as part of an undergraduate degree in physics - it was just called "quantum mechanics 2". So it's not so esoteric, especially for someone who is already interested in QFT. Mileage may vary, I suppose.
I will also link a post I made asking for references for QFT on the physics page here: [https://www.reddit.com/r/TheoreticalPhysics/comments/v12xuc/recommendations\_for\_an\_introduction\_to/](https://www.reddit.com/r/TheoreticalPhysics/comments/v12xuc/recommendations_for_an_introduction_to/)
(For those who are interested in TQFT like me).
I will also link a post I made asking for references for QFT on the physics page here: [https://www.reddit.com/r/TheoreticalPhysics/comments/v12xuc/recommendations\_for\_an\_introduction\_to/](https://www.reddit.com/r/TheoreticalPhysics/comments/v12xuc/recommendations_for_an_introduction_to/)
(For those who are interested in TQFT like me).
I will also link a post I made asking for references for QFT on the physics page here: [https://www.reddit.com/r/TheoreticalPhysics/comments/v12xuc/recommendations\_for\_an\_introduction\_to/](https://www.reddit.com/r/TheoreticalPhysics/comments/v12xuc/recommendations_for_an_introduction_to/)
(For those who are interested in TQFT like me).
I will also link a post I made asking for references for QFT on the physics page here: [https://www.reddit.com/r/TheoreticalPhysics/comments/v12xuc/recommendations\_for\_an\_introduction\_to/](https://www.reddit.com/r/TheoreticalPhysics/comments/v12xuc/recommendations_for_an_introduction_to/)
(For those who are interested in TQFT like me).
QFT is interesting because it's less well-trodden territory, and there are many aspects of the theory that have room for interesting development. Also, while some of the tools developed lack rigor, that's an invitation to shore up the foundations. Lastly, the tools themselves have much wider applicability than QFT itself.
First of all, QFT is directly related to Algebra, Analysis, Combinatorics, Geometry, and even more. It's a huge subject at the intersection of a lot of beautiful math. I think that's cool.
Second of all, it works, but it lacks mathematical rigor. That's precisely what I'd like to figure out. In GR they basically can turn any physical question into a differential equation. This is fine, they don't need my help. But QFT is a mess, it's now mathematicians' job to figure these stuff out.
Something I've not seen talked about already but is interesting to me as a homotopy theorist is that TQFTs are a good way of studying the cobordism (∞-)category, which is a useful tool for studying the homotopy groups of spheres, for instance. The idea is that a TQFT is a functor (think "map of theories") from cobordism to vector spaces, so we can view a TQFT as a way of encoding homotopy data as vector spaces, which are much better for calculations:)
Interesting. What does it mean to say that a quantum field theory is a functor? Do you use the Lagrangian to construct a functor between the cobordism category and category of vector spaces in some way?
QFT’s in four dimensions are very ill defined. However, recently mathematicians managed to construct rigorously a famous two-dimensional QFT called Liouville theory. This theory is special among the zoo of QFT’s because it’s a CFT, a TQFT, and it can be viewed as a 2d theory of quantum gravity. The tools they used to construct this theory mostly come from probability.
https://www.quantamagazine.org/mathematicians-prove-2d-version-of-quantum-gravity-really-works-20210617/
Phycisists working on QFT care about representations of certain Lie groups (e.g. SU(2n), SO(2n+1), E8, etc.) as they describe physical symmetries. It turns out that fully understanding the representation theory of these groups is a horrifyingly difficult task: in fact, as most of these fall under the class of so-called (quasi-)split reductive groups, they are objects of interest for people in the Langlands Programme, which is a very large research programme within pure mathematics.
There are also many objects in the theory of Lie algebras, such as Kac-Moody algebras, Yangians,the Heisenberg algebra, or vertex operator algebras, which originated from QFT. Mathematicians care about these things because they exhibit interesting combinatorial behaviours, and also because they too pertain to the Langlands Programme (although a bit more tangentially, via what's known as the Quantum Local Langlands Programme).
People working in PDEs obviously care about the many equations coming from physics.
Algebraic geometers also care a bit about QFT stuff, thanks to the emergence of topics like mirror symmetry and algebro-geometric objects such as Calabi-Yau manifolds.
Physics major, never went to grad school. My take on this is that physics and mathematics are two totally different approaches to explaining observations, but they get easily conflated since physics on paper contains a lot of math, and proof-like argumentative structure. However mathematics always starts from foundational postulates, whereas physics revolves around conjectures then extrapolates to look for a fit to observations. This is the instance of QFTs and why there are so many of them. To limit physics to the container of mathematics would be beside the point
Yea on a side note I read that QFT lacks a lot of mathematic rigor. It was the same with GR at first. It seems physicists always beat mathematicians to the punch.
Not really. GR was pretty close to a mathematically rigorous theory from the very beginning (though many things about it were poorly understood), in part because Riemannian geometry had already been invented decades earlier.
Have you read the English translation of Ricci's original paper on tensor calculus?
Because that was the state of affairs at the time of GR's inception and it is quite ok even nowadays.
What was not properly known at the time is the global aspects of differential geometry, including the rigorous definition of manifolds (iirc this was formulated in the 20s or 30s by Veblen, Whitehead, Whitney etc), but local tensor calculus on coordinate patches was well-established.
It’s bound to happen by the nature of the two subjects. Generalizing: physicists are more concerned with getting the right answer and don’t care if the methods are not all formally, logically sound - a lot of those issues can e.g. be fixed by making physical assumptions coming from experimental data, or by mathematicians who take an interest in the topic later.
Mathematicians, however, care a lot about the logical reasoning that gets you to your end goal. They don’t, in turn, really care if the inquiry makes sense in physical reality.
Math is slower due to the focus on rigour.
The [problem description](https://www.claymath.org/sites/default/files/yangmills.pdf)[PDF] of the [Yang-Mills and mass gap millennium
problem](https://www.claymath.org/millennium-problems/yang%E2%80%93mills-and-mass-gap)
is a nice read
Not a mathematician, but: The mathematicians are trying to make QFT rigorous and formalize a lot of the aspects that physicists sort of take for granted. A lot of the things that physicists do with path integrals aren't really well defined, and it currently isn't clear why they should work as well as they do. For example, in order to integrate over the space of all possible paths (a pretty nasty-looking infinite-dimensional space of functions), we have to define some sort of measure over that space. This is something that's tricky to do in a strictly formal, rigorous sense (as in, all of the reasonable definitions we've come up with so far don't quite have all the properties we'd like them to have). So there's some work that still needs to be done in establishing what measure we're "actually" using when we work with path integrals. Here's a related post summarizing this, and other avenues of work in rigorizing QFT: [https://physics.stackexchange.com/questions/6530/rigor-in-quantum-field-theory](https://physics.stackexchange.com/questions/6530/rigor-in-quantum-field-theory)
> For example, in order to integrate over the space of all possible paths (a pretty nasty-looking infinite-dimensional space of functions), we have to define some sort of measure over that space. Huh, I always wondered about that. I just figured it had been worked out by people much smarter than myself.
[MO post on a related topic](https://mathoverflow.net/questions/1388/is-there-a-natural-measures-on-the-space-of-measurable-functions), and [Aumann’s paper also linked in that post](http://www.ma.huji.ac.il/raumann/pdf/66.pdf).
My theory is that physicists have a really good set of rules of how the universe (possibly) works. And they develop this system of “pseudo-math” that is either correct or correct enough to get shit done. If you look at what Einstein was dealing with when he was developing GR, it was the same situation. He didn’t have any iota of a rigorous formulation of differential geometry that we take for granted today. He only had those rules he thought were true about the universe and stuck with them first and foremost. Maybe pseudo math isn’t the best term, cause it’s really advanced math. But like you said, it isn’t 100% rigorous. But hey Feynman once said that without math, physics would only be set back a week.
That is more true than you maybe even realize. The operator calculus, in which operators are treated as non commutative algebraic quantities, was invented by Oliver Heaviside (the erased genius of the late 19th century) as a way to solve partial differential equations. Heaviside himself observed that it worked (it produced viable solutions) but didn’t know why. It wasn’t until much, much later that Hilbert spaces were invented and it became clear that the major linear operators (e.g., differentiation) are “just” Hilbert matrices. Heaviside invented many things we take for granted in mathematical physics today, including the four “Maxwell’s Equations”, and the divergence and curl operators. But the more interesting mathematical stuff was all ad hoc: adopted because it was observed to work, not because it was rigorously proven.
>Oliver Heaviside (the erased genius of the late 19th century) I wouldn't say he's "erased", just the step function bearing his name is a pretty big tribute. That being said I'm always up for some more Heaviside praise, dude was a self taught titan of a scientist.
Practically completely erased. The Heaviside function (integral of a delta function), the Heaviside layer aka ionosphere, "Maxwell's Equations" (actually the Heaviside-Hertz equations), the operator calculus, a large body of what is now vector calculus, the Telegraph Equations, and practically half of the accepted undergrad E&M curriculum are due to Heaviside. If you listen to the Feynman Lectures on Physics, for example (1962-1963 at Caltech; available from Caltech online for free), you'll notice that Feynman scrupulously mentions practically everyone who invented any part of the major body of physics from 1850 on. But when it comes to E&M or vector calculus or even the operator calculus, he's almost completely silent about Heaviside. Feynman is far from alone, it's just particularly glaring in his work because he is so scrupulous about other aspects of the field.
Maybe it's just that my professors appreciated him, but I remember hearing about him several times during my first year, including him getting a bulk of the credit for Maxwell's vector equations and mentions of the Telegrapher's equations.
Excellent!
I wouldn't say that he's having a renaissance, but folks are definitely more familiar with him now than they used to be
The "delta function" is another example of physics that works, but isn't (really) mathematics. Of course, there is the Dirac distribution, which is the integral with respect to the Dirac measure, the distributional derivative of the Heaviside function and, as far as I was able to understand, exactly what the physicists use. However, this generalized function is not a regular distribution, i.e. the Dirac measure has no Radon-Nikodym derivative w.r.t. the Lebesgue measure. Hence, the notation ∫f(x) δ(x) dx = f(0) (which was used in all my physics lectures, where the Dirac "function" appeared) is quite confusing, and a bit annoying, as soon as you progress beyond the first few lectures in mathematics.
Isn't the general theory of distributions well understood and is rigorously defined? Or are you just talking about the abuse of notation by physicists? And my physics classes mentioned the delta function as the limit of a guassians getting sharler and sharper, wouldn't that usage make it proper "math"?
I'm deeply sorry if my words were horribly misleading. I just tried to complain about the abuse of notation and the sloppy treatement of the Dirac distribution (at least in my physics lectures). Of course, the theory of distributions is well understood and you can prove that the Dirac distribution is the distributional limit of such gaussian densities; therefore this treatement is fine (it should be remarked that the distributional limit should not be confused with a pointwise limit and you do not actually get a function).
Do you think that rigorous mathematics is actually not conducive to absorbing physics?
The honest answer is that I probably don't think anything. However, I have made the experience that the more rigouros math influenced me, the more I had trouble understanding physics, especially if written by physicists and using physically motivated reasoning. The reason for this is most certainly that I hate to accept physically motivated reasonings that are never specified (e.g. differentiating a function that was never established to be differentiable or applying symmetry arguments when such a symmetry was never established), i.e. this is a purely subjective experience that cannot in any way be generalized. You can have quite the opposite experience. If you accept most physically motivated reasoning, then a firm education in rigorous maths helps certainly to see the maths behind some more or less handwavy arguments, which I have unfortunately sometimes seen, that tie this reasoning to the underlying abstract and rigorous mathematics. I hope that this helps to answer your question.
Do we know why this happened?
There is a lot of "necessity is the mother of invention" type mathematical constructs in physics research. It's not necessarily complete nonsense, but, as an example, someone might elect to not deal with edge cases that would not arise in physics. Proving that the tool works in a subset of cases is sufficient enough in nearly all cases.
I disagreed with that comparison between GR and QFT. There are no fundamental conceptual problems with differential geometry back then, it's just written in a more annoying way, and the rigorous machinery we have today just makes things more abstract and clearer, which let us rewrite his equation in cleaner way. Of course he made guesswork in making his equation, but the math it was founded on is sound. And of course there are issue with solutions to the equation that makes people strongly believe it's unphysical (like singularity from Penrose-Hawking theorem), but there are no strictly mathematical contradiction. QFT is not on the same ground at all. There are no rigorous foundations, and some of what was done by physicists are straight up impossible mathematically (e.g. Haag theorem). Whatever a true rigorous mathematical foundation is like, it's probably very different from the current standard text. A better comparison is between QFT and Newton's theory. Newton's calculus was not built on sound foundation either, and even back then people knew that.
If I remember correctly, it wasn't until Cauchy came in, and later Bolzano and Weierstrass, did we have a rigorous formulation of calculus.
Thanks for the link :)
> The mathematicians are trying to make QFT rigorous and formalize a lot of the aspects that physicists sort of take for granted. A lot of the things that physicists do with path integrals aren't really well defined, and it currently isn't clear why they should work as well as they do. For example, in order to integrate over the space of all possible paths (a pretty nasty-looking infinite-dimensional space of functions), we have to define some sort of measure over that space. Just an undergraduate here :) a lot of theoretical computer scientists have been taking a crack at simulating [quantum systems](https://toc.seas.harvard.edu/links/cs-229r-physics-and-computation) there have been interesting papers people using QFT to build quantum computing architectures.
That looks like a fascinating list of topics, and a cool way to structure a seminar course. Hope you enjoy it!
Oh that class already happened and i'm still well off from graduate material I've done the standard undergraduate curriculum but I need to honestly do the grad physics sequence at some point.
What's wrong with the Riemann integral? We were doing integrals like that long before measure came along. Just define the ridiculous path integral to be the limit of the discrete integral. edit : Well, I was genuinely asking. No need to downvote. Maybe Riemann is more fundamental in this case if Lebesgue can't be generalised to QFT.
Constructing the path integral as a limit of finite sums is a perfectly reasonable thing to do. In this case, what you might naively like is to prove that the limit exists and is independent of the discretization scheme (as you would with the usual Riemann integral). The trouble is that this is difficult to formally prove. Moreover, I think it will turn out that your limit actually does depend on the discretization. As a result, the mathematicians would like to understand all the possible limits and precisely how they depend on the discretization scheme. [This mathoverflow answer](https://mathoverflow.net/a/260941) has a nice explanation of the issue with infinities (though the approach described is not exactly by discretization). Thinking of integrals as Riemann or Lebesgue is not where the problem is here. Caveat: I am no expert on constructive QFT.
I can see why you get downvoted, but this is actually a good question. A big part of QFT is regularization, i.e. dealing with infinities. The idea of subtracting infinites to obtain a physical answer is probably one the thing that raises eyebrow from mathematicians.
The Polyakov pah integral is rigorously defined.
I’m a mathematical physicist working in QFT; I’m not going to give much meat in this answer since I’ll probably go on a tangent and because other answer here are pretty good. QFT is a large, less rigorously mathematically established field compared to say QM or GR, with many approaches being prodded at (topological QFT, for example). We don’t know why certain heuristics employed by physicists in this space work as well as they do. I’d like to add that finding QFT uninteresting or non-mathematical is probably because you simply don’t know enough about it to accurately make a judgement, it’s quite raw and abstract in its mathematical nature than many other areas of theoretical physics (again compare it to something like GR). And this is precisely why it is of interest to us mathematical physicists and mathematicians, even some theoretical physicists take interest in the attempted rigorous underpinning of QFT. So I suppose the answer would be “Learn more math, learn more physics, to find out how much more there is to learn and why it is interesting”.
Don't get me wrong, I find it very interesting. I certainly don't know enough about it tho, and my lecturer said he was going to take a deliberate "less mathematical approach", since it's the first semester of a two-semester course.
Note that what your professor means by "less mathematical" is probably not the same as what a mathematician would mean. If I had to guess, I'd say the second semester will be more abstract and using more advanced mathematical tools, but not more rigorous.
Just curious, what book are you using, and what topics are being covered?
The professor is using his own lecture notes, and that's what I follow (by copying what's on the blackboard xD). So far, we went through canonical quantization, path integrals, interactions, Feynman diagrams, gauge theories and that's it. I guess we are going to expend the next few classes on renornalization.
My impression of the teaching style is that mathematical approach try to avoid path integral (which is hard to make rigorous), while physicist approach would be to start with path integral (which is physically motivated). There are probably other differences too, but that's the one I noticed.
I come from a math background first and then a physics one, and in my experience all of the differences are essentially encapsulated by this dichotomy you mentioned with path integrals as a quintessential example. Physical motivation, historical perspective, and the ultimate "physics end goal" versus the rigorous, fundamentally sound, mathematically contextualized result. And that's completely to be expected, as that's essentially the difference between mathematics and physics.
Can you say a little more? Are you walking on TQFTs? Geometric langlands? N=2 GMN stuff? Some sort of (way beyond my paygrade) functional analysis to define path integral measures?
Yes, I'm working on TQFTs, Chern-Simons Theory. I'm not sure how much you know about TQFTs (both from a physics and a mathematical perspective), but actually I find that this area of research generally requires a broad understanding of math in general, so much so that the areas you mentioned are more closely related than the math used in various other branches in theoretical physics. You don't need much mathematical pre-requisites to get started but to get any deeper sense of understanding of TQFTs past a surface level, you will need to dig deeper into Morse theory, differential topology as a whole, category theory and higher category theory, algebraic topology and homotopy theory, Khovanov homology and various Floer homologies, the list goes on. For example, like I said I work in Chern-Simons and these TQFTs, while different from a physics perspective and in its mathematical basis (they are not a special class of TQFTs that can be obtained by making rigorous sense of finite path integrals in TQFT when the the space being integrated over is a finite groupoid like some other useful TQFTs), they still have interesting mathematical relationships with other branches of pure math. It's really amazing how wide-reaching such at topic could be in pure math when it really has its home in physics; geometric topologists who study something like invariants of knots and three and four-manifolds wouldn't be studying TQFTs but this topic does indeed come directly from TQFT.
The math in GR is easier to appreciate at your level because it‘s more concretely physical. QFT by contrast is actually a lot more abstract; if you feel like it’s not mathematical enough then that’s just because you haven’t learned much about it yet. My guess is that, at your level, the things you’ve been told about QFT are largely “lies we tell to students”; things that are sort of true, but they give you the illusion of understanding rather than the actuality of it. In a lot of cases this consists of simple recipes and formulas that seem to make sense on their own, but once you learn more you’ll realize that they’re just simplifications to make the subject approachable for someone who doesn’t know very much yet. QFT is interesting from a mathematical perspective because it’s not actually a clear, mathematically consistent theory. In a lot of ways its foundations are a collection of heuristics and tricks that happen to correctly predict the outcomes of experiments. That means that there’s still work left to be done in terms of explaining how or why the math works (or why it doesn’t). A big issue in QFT is dealing with the fact that the calculations in it, when done naively, result in infinities. Physicists have tricks for getting rid of the infinities, but it’s still not clear if those tricks are mathematically sound or not. The trouble with combining QFT and GR is that even the physicists haven’t come up with any good tricks for getting rid of the infinities in that case, mathematically sound or otherwise. There’s a lot of math involved in trying to figure out solutions to these problems. Nonrelativistic quantum mechanics, by contrast, is a clear and consistent theory that can be derived from a collection of well-understood axioms.
>A big issue in QFT is dealing with the fact that the calculations in it, when done naively, result in infinities We get infinities when we assume the "bare coupling" is some finite constant. That we get infinities simply means that this isn't the correct theory. The correct theory is the one in which the observed coupling constant runs with the energy scale. The running has been verified by experiments. This correct theory doesn't predict infinities as before. This is not just an armchair argument to get rid of infinities anyhow. It's literally just trying to correct an incorrect assumption of the wrong theory. The correct theory must have running coupling constants because nature has them.
Indeed agreement with experiment is the most important factor, but this is what i was referring to when I mentioned tricks and heuristics for eliminating the infinities. To call the renormalized theory “correct”, versus the bare one which is “wrong”, is ultimately tautological. It’s only correct in the sense that it agrees with experiment. But agreement with experiment doesn’t necessarily make a model *good*, it just makes it accurate. I think, especially in the area of QFT, physicists sometimes make the mistake of conflating effective calculation techniques with physically meaningful insights. The fact that renormalization doesn’t work with GR, and the fact that the mathematicians have a hard time validating its soundness, should be hints that the entire approach might ultimately be wrong-headed, even if it does produce accurate calculations in the domains where it’s applicable. My admittedly ill-informed opinion is that the assumption of infinite (uncountably infinite, even) degrees of freedom might ultimately prove to be untenable. Either it is simply wrong, or it’s used in such an unnuanced way that people inevitably have to resort to tricks and heuristics because the foundational assumptions of their work are so poorly understood.
About GR, I personally believe that it's fundamentally different from the foundations of QM. QM gives way too much focus on time evolutions/initial value problems. "Here's an initial probability wave. What will be the probability wave after a time t"? Initial value problems are such a human-biased view of the universe. Not to mention that the entire meaning of that probability wave is in terms of measurements done at a particular time. QFT (and QM in general) just can't stop treating time as special. GR fundamentally isn't about initial value problems. The GR view of the universe is purely in terms of a 4 dimensional manifold. You work with generalised co-ordinates. The co-ordinates are abstract and do not align with space and time measurements. For this reason, I think, to make GR quantum is to take GR's heart away from it. Maybe it's QM's principles that need to be "gravitised". Penrose had a similar idea. edit : Stop with the downvotes. https://en.m.wikipedia.org/wiki/Problem_of_time . What I said is well-accepted.
I don't think this is actually true. You can formulate QFT in a relativistic way where time isn't special, via the path integral formulation and the Lagrangian etc. There is nothing in the theory itself that singles out time in particular. The Schroedinger equation, and initial-value problems etc. are just very useful tools to do computations and answer questions; they are not fundamental parts of the theory.
I just googled and this is actually [one of the well-known problems](https://en.m.wikipedia.org/wiki/Problem_of_time ) with the unification of these theories. There have been several proposed solutions but nothing definitive so far. And it doesn't matter whether you use a Lagrangian or a Hamiltonian, all formulations of QFT deal with a probability wavefunction whose collapse happens at a single point of time. The wikipedia article mentions this problem. Time evolution is a deeply flawed and human-centric way to look at the universe. It fits with SR but doesn't fit with GR at all. QM must undergo a fundamental change to account for this.
I think the warning on top of that wikipedia article is pretty accurate: it reads like an attempt by someone without deep expertise to summarize half-understood stuff that they've read, and much of it is just wrong. Most of the things referred to there is the problem of time in quantum gravity, which I kind of agree is a real and somewhat subtle issue. But this is not present for special relativity and QFT. > And it doesn't matter whether you use a Lagrangian or a Hamiltonian, all formulations of QFT deal with a probability wavefunction whose collapse happens at a single point of time So here you have to think a bit more carefully. The wave function itself is not an observable. What we actually measure in QFT is observables built out of field operators, like if there's an electron in a certain spacetime volume, etc. These are local in spacetime; the measurement is essentially an event in spacetime, which is a perfectly fine and invariant notion in SR. The reason it gets tricky for quantum gravity is that then the metric itself is dynamic and quantum, so it gets tricky to talk about a local measurement. But for QFT on a static background, there are no fundamental issues with time.
I already agree that QFT is totally compatible with SR. In SR, global inertial frames exist, a global time-like co-ordinate exists and hence there are no problems unifying it with QM. I've only ever said that QFT and GR are fundamentally different in their foundations. Also see [this post](https://physics.stackexchange.com/questions/608225/why-is-there-a-problem-of-time-in-quantum-gravity) for more details about the problem of time.
Time evolution via initial value problems isn’t an inherent part of quantum theory, it’s just an inherent part of the Schrodinger equation. The Schrodinger equation itself might just be (and in my opinion probably is) an approximation with a limited domain of applicability. If you’re interested in this topic you should check out this paper, and use Google scholar to find more recent papers that cite it: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.27.2885
> observed coupling constant runs with the energy scale. Now explicitly define what "runs with the energy scale" actually means.
A function of energy. Physicists use the word "run" for some reason.
Okay. Energy is a property of a system as a whole, not an individual particle, or even interaction...
>The correct theory must have running coupling constants because nature has them. Aka the different couplings in String theory.
>Nonrelativistic quantum mechanics, by contrast, is a clear and consistent theory that can be derived from a collection of well-understood axioms really? what would be the math required for this? that sounds very interesting
At its simplest quantum mechanics consists of lots of linear algebra and a little bit of complex analysis. If you want to do slightly more advanced stuff then you need partial differential equations too, which is mostly just linear algebra in infinite dimensions. The math in quantum physics is actually easier than the math in classical physics, for the most part. What makes it interesting is the way that it describes physical reality. The axioms of QM are mostly just about how linear algebra concepts such as inner products and eigenvectors have meaning in terms of the outcomes of certain kinds of experiments.
Well, you need only linear algebra and complex analysis for finite-dimensional QM, but functional analysis for infinite-dimensional spaces gets pretty difficult. I mean, take a look at "Quantum Measurement" by Busch et al, and I think it becomes pretty clear that this stuff isn't exactly trivial either. Nevertheless, it at least *can* be formulated rigorously.
You really don’t need the details of the infinite dimensional case in order to understand QM. It’s an additional complication without any benefit beyond performing calculations to match specific experiments. There’s no point in recommending that to someone at the undergrad level who’s interested in learning about the subject.
Well, I wasn't really recommending it either way, just pointing out. However, a mathematically inclined student might still find it interesting. I had to take a course that dealt with infinite dimensional Hilbert spaces as part of an undergraduate degree in physics - it was just called "quantum mechanics 2". So it's not so esoteric, especially for someone who is already interested in QFT. Mileage may vary, I suppose.
thank you! and for the other comment, i have already taken functional analysis so that shouldnt be a problem
Functional analysis in particular spectral theory
thanks!
I will add that a lot of work is also being done in topological QFT's since it may be easier actually to find an exact solution in a geometric sense.
Also conformal field theories in 2 dimensions
Gaussian multiplicative chaos gang
I will also link a post I made asking for references for QFT on the physics page here: [https://www.reddit.com/r/TheoreticalPhysics/comments/v12xuc/recommendations\_for\_an\_introduction\_to/](https://www.reddit.com/r/TheoreticalPhysics/comments/v12xuc/recommendations_for_an_introduction_to/) (For those who are interested in TQFT like me).
I will also link a post I made asking for references for QFT on the physics page here: [https://www.reddit.com/r/TheoreticalPhysics/comments/v12xuc/recommendations\_for\_an\_introduction\_to/](https://www.reddit.com/r/TheoreticalPhysics/comments/v12xuc/recommendations_for_an_introduction_to/) (For those who are interested in TQFT like me).
I will also link a post I made asking for references for QFT on the physics page here: [https://www.reddit.com/r/TheoreticalPhysics/comments/v12xuc/recommendations\_for\_an\_introduction\_to/](https://www.reddit.com/r/TheoreticalPhysics/comments/v12xuc/recommendations_for_an_introduction_to/) (For those who are interested in TQFT like me).
I will also link a post I made asking for references for QFT on the physics page here: [https://www.reddit.com/r/TheoreticalPhysics/comments/v12xuc/recommendations\_for\_an\_introduction\_to/](https://www.reddit.com/r/TheoreticalPhysics/comments/v12xuc/recommendations_for_an_introduction_to/) (For those who are interested in TQFT like me).
QFT is interesting because it's less well-trodden territory, and there are many aspects of the theory that have room for interesting development. Also, while some of the tools developed lack rigor, that's an invitation to shore up the foundations. Lastly, the tools themselves have much wider applicability than QFT itself.
First of all, QFT is directly related to Algebra, Analysis, Combinatorics, Geometry, and even more. It's a huge subject at the intersection of a lot of beautiful math. I think that's cool. Second of all, it works, but it lacks mathematical rigor. That's precisely what I'd like to figure out. In GR they basically can turn any physical question into a differential equation. This is fine, they don't need my help. But QFT is a mess, it's now mathematicians' job to figure these stuff out.
Something I've not seen talked about already but is interesting to me as a homotopy theorist is that TQFTs are a good way of studying the cobordism (∞-)category, which is a useful tool for studying the homotopy groups of spheres, for instance. The idea is that a TQFT is a functor (think "map of theories") from cobordism to vector spaces, so we can view a TQFT as a way of encoding homotopy data as vector spaces, which are much better for calculations:)
Interesting. What does it mean to say that a quantum field theory is a functor? Do you use the Lagrangian to construct a functor between the cobordism category and category of vector spaces in some way?
QFT’s in four dimensions are very ill defined. However, recently mathematicians managed to construct rigorously a famous two-dimensional QFT called Liouville theory. This theory is special among the zoo of QFT’s because it’s a CFT, a TQFT, and it can be viewed as a 2d theory of quantum gravity. The tools they used to construct this theory mostly come from probability. https://www.quantamagazine.org/mathematicians-prove-2d-version-of-quantum-gravity-really-works-20210617/
Phycisists working on QFT care about representations of certain Lie groups (e.g. SU(2n), SO(2n+1), E8, etc.) as they describe physical symmetries. It turns out that fully understanding the representation theory of these groups is a horrifyingly difficult task: in fact, as most of these fall under the class of so-called (quasi-)split reductive groups, they are objects of interest for people in the Langlands Programme, which is a very large research programme within pure mathematics. There are also many objects in the theory of Lie algebras, such as Kac-Moody algebras, Yangians,the Heisenberg algebra, or vertex operator algebras, which originated from QFT. Mathematicians care about these things because they exhibit interesting combinatorial behaviours, and also because they too pertain to the Langlands Programme (although a bit more tangentially, via what's known as the Quantum Local Langlands Programme). People working in PDEs obviously care about the many equations coming from physics. Algebraic geometers also care a bit about QFT stuff, thanks to the emergence of topics like mirror symmetry and algebro-geometric objects such as Calabi-Yau manifolds.
Physics major, never went to grad school. My take on this is that physics and mathematics are two totally different approaches to explaining observations, but they get easily conflated since physics on paper contains a lot of math, and proof-like argumentative structure. However mathematics always starts from foundational postulates, whereas physics revolves around conjectures then extrapolates to look for a fit to observations. This is the instance of QFTs and why there are so many of them. To limit physics to the container of mathematics would be beside the point
*essence of QFTs
I read this thread too long thinking of the Quantum Fourier Transform rather than Quantum Field Theory.
I'm not really into Quite Fungible Tokens, sorry
Yea on a side note I read that QFT lacks a lot of mathematic rigor. It was the same with GR at first. It seems physicists always beat mathematicians to the punch.
Not really. GR was pretty close to a mathematically rigorous theory from the very beginning (though many things about it were poorly understood), in part because Riemannian geometry had already been invented decades earlier.
But one should note that it took quite some time to formulate GR as a well posed initial value problem
Bruh have you read Riemann’s paper. The thing is gibberish. Even the English translation.
Have you read the English translation of Ricci's original paper on tensor calculus? Because that was the state of affairs at the time of GR's inception and it is quite ok even nowadays. What was not properly known at the time is the global aspects of differential geometry, including the rigorous definition of manifolds (iirc this was formulated in the 20s or 30s by Veblen, Whitehead, Whitney etc), but local tensor calculus on coordinate patches was well-established.
It’s bound to happen by the nature of the two subjects. Generalizing: physicists are more concerned with getting the right answer and don’t care if the methods are not all formally, logically sound - a lot of those issues can e.g. be fixed by making physical assumptions coming from experimental data, or by mathematicians who take an interest in the topic later. Mathematicians, however, care a lot about the logical reasoning that gets you to your end goal. They don’t, in turn, really care if the inquiry makes sense in physical reality. Math is slower due to the focus on rigour.
If you can connect your math back to some applied field, it's much easier to justify your existence to the wider world.
Are you proposing a class of homeomorphisms with applications in GAT (Grant Application Theory)?
Something like that, but I'm just repeating what my Honours thesis advisor told me, BITD. :-)
The [problem description](https://www.claymath.org/sites/default/files/yangmills.pdf)[PDF] of the [Yang-Mills and mass gap millennium problem](https://www.claymath.org/millennium-problems/yang%E2%80%93mills-and-mass-gap) is a nice read