The undergrad physics curriculum includes the standard college sequence of calculus and differential equations. Many physics students take more math than this, or today, probably some computer science. We took a "physics" course called "theoretical physics" that filled some gaps in the math sequence, most notably multivariate calculus in cylindrical and spherical coordinates, path integrals, and some stuff like that. Specific upper level physics courses also introduced math topics as needed.
Graduate coursework re-iterated the same "theoretical physics" course, but it was called "mathematical methods." We used a textbook by Arfken, that covered the same multivariate calculus plus vector and tensor math. Finding out what the current textbook du jour is for this course, and perusing the table of contents, would give you a pretty good idea of what to expect.
I didn't go into scattering theory as a specialty, but we covered it in some of the other courses, and I remember doing difficult integrals and differential equations in spherical polar coordinates. The coursework quickly runs you out of problems that can be solved in closed form. I'd be shocked if the real work isn't done today with mostly numerical techniques.
You need to work up to (and through) Quantum Field Theory to understand this.
Thankfully, you don't need to be as good or as thorough as Gerard 't Hooft as you learn all those things!
QFT itself has many prerequisites; you need to be well versed in special relativity, electromagnetism, classical field theory and quantum mechanics to get a hold on it. Each of these areas have their own mathematical prerequisites, which include algebra and calculus, multivariable calculus, pdes, complex analysis and fourier analysis.
General relativity and string theory are also comparative to QFT in scope, prerequisite knowledge and utility in other areas of mathematical physics. You would need to learn these if you wanted to start reading a bit more broadly in scattering amplitudes or in another part of math phys, but you can go quite a long way in scattering amplitudes without too much of them.
As someone who has taught myself significantly more than formal education could, I know how valuable it is to have this sort of knowledge trail accessible. Much appreciated.
Do you have any guesses for what might be the connection between these two different scattering amplitudes in seemingly disconnected physical phenomenon?
Dixon et al. find a duality linking form factors and scattering amplitudes. Scattering amplitudes are written as a 'loop expansion', which you could think of as a bit like a Taylor expansion, and I think a remarkable feature of their result is that it appears to hold for all orders of this expansion. I've seen some results relating form factors and scattering amplitudes at only the first order of this expansion before.
Their result is based on a mathematical observation that the functions that the scattering amplitudes are built from at each loop order (called polylogarithms) have a natural operation on them called the 'antipodal map', and in applying this operation to every polylogarithm in the amplitude they recover the form factor. It's intuitive to me that if the mathematical building blocks of your amplitude behave nicely under some operation, you might expect the amplitude to have nice properties under that operation also.
So it's clear mathematically why this new relationship exists, and I guess the question is, 'what does it mean physically?'. Dixon et al. comment that they don't know, and I don't really have any thoughts on this either. What I can say is that, over the last 20 or 30 years there have been many different discoveries of duality between seemingly disparate areas of mathematical physics. The first and most famous was the AdS-CFT relationship which relates gravity to gauge theory, and there is lots of work in amplitudes currently on a different relationship which writes gravitational physics as the square of physics in gauge theory. There are many others.
So, my experience working in the field has taught me that there are many unexpected dualities and relationships between different physical theories, and that nature is more deeply connected at a fundamental level than is indicated by the initial (apparently disparate) mathematical formulations of those theories.
This appears to be a development in a toy model. It might or might not apply to the standard model.
Still, at least it isn't a condensed matter experiment being misconstrued as a particle physics breakthrough.
Hey don't throw shade on quasiparticles, they're people too.
As far as I can tell, protons and neutrons are quark-gluon quasiparticles, but we’re happy to call studying protons and neutrons “particle physics”.
> complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum
That is, this description fits protons and neutrons relative to quarks and gluons: very cold quark-gluon regions behave as if protons and neutrons exist.
Protons and neutrons, on the other hand, exist in free space. You can accelerate protons and fire them at the moon (or whatever). You can't do that with phonons or electron holes. They are composite particles (not fundamental) but they aren't quasiparticles.
(quarks, weirdly, can't exist in free space. If you try to pull a proton apart you will have to put so much energy into it that you generate more quarks which then immediately couple up with the proton's quarks, possibly in a novel combination producing new composite particles. So you have to study composite particles to study quarks)
I’m also unsure that’s true, even ignoring that issue:
I can shoot an electron hole by moving the chunk of metal it’s situated in, the same way I can shoot a proton by shooting the constituent quarks and gluons.
You can’t separate an electron hole from the metal, but you also can’t separate the proton from quarks and gluons.
I was talking about protons and neutrons.
What distinguishes a proton, which gains its mass as a hole in the quark-gluon fuzz permeating the universe, from an electron hole in metal?
I think that gets you something:
You can understand protons and neutrons as holes in the “crystallized” space that came out of the universe cooling to its current state — that the fuzz of quarks and gluons is something like a “metal” in which protons and neutrons are “holes”.
You can certainly formulate the math such that hadrons, or any other particle for that matter, behave like holes in a substrate, indeed this is the basis of quantum field theory. But unlike quasiparticles where that substrate is something in the universe like a block of metal, for regular particles the substrate is the universe.
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> You can certainly formulate the math such that hadrons, or any other particle for that matter, behave like holes in a substrate, indeed this is the basis of quantum field theory.
Okay — I’m with you here.
> But unlike quasiparticles where that substrate is something in the universe like a block of metal, for regular particles the substrate is the universe.
Okay — but so what?
Why is it different that it’s within a metal than the stuff the universe cooled into? In both cases, we’re discussing an emergent object with new quantum numbers in some substrate:
- electron holes are a hole in the fuzz of electrons in metals
- protons are a hole in the background fuzz of quarks and gluons in cold aether
> It is mathematically useful to think of space as being full of virtual particles including quarks, gluons, and electrons, but there isn't actually a quark-gluon "fuzz" permeating the universe.
I’m not sure I follow: proton mass is related to flux tubes suppressing that fuzz in the area of the proton.
Empty space has energy, which manifests as that particle fuzz.
Because that's the point of classification - we want to distinguish between case A and case B. Everything in existence is an emergent property of the universe, but you can't really have any meaningful conversation if the only noun in your vocabulary is "stuff."
> protons are a hole in the background fuzz of quarks and gluons in cold aether
No, there is no cold aether, and protons are not holes in anything. They are composite particles. To the degree that their constituent parts can be described as excitations of quantum fields, so can they, but they are no less real.
> I’m not sure I follow: proton mass is related to flux tubes suppressing that fuzz in the area of the proton. Empty space has energy, which manifests as that particle fuzz.
Proton mass comes from the confinement of its constituent parts. Only about 9% of the proton's mass is a result of the quarks it contains interacting with the quark field, which again is not literally a "fuzz" of discrete particles. Flux tubes are just shapes of magnetic fields. Protons are not just their mass, they have many properties.
Right — but why?
What is useful about that classification if the two types of particles emerge the same way, as holes within a substrate?
What do we gain in “composite” vs “quasi”?
> No, there is no cold aether, and protons are not holes in anything.
There is — we can measure vacuum energy as existing. We can also measure spacetime as existing, directly. We know there’s an aether.
> Proton mass comes from the confinement of its constituent parts. […] Flux tubes are just shapes of magnetic fields.
Flux tubes are the region where the connection between quarks suppresses the fuzz from vacuum energy. That’s where the non-quark mass comes from. (As you call it, confinement.)
If we accept this reasoning to imply that the universe is a giant computer, we only have the unknowable left: was it purposefully (intelligently) designed or did it spontaneously arise? As far as I can tell, the probability of either outcome is equal according to our limited experiences, and we will never have access to a definitive answer one way or the other.
For any of you paper-writers out there, some advice: the abstract shouldn't contain undefined acronyms, and should be readable by anyone who you would expect to be at a conference where you present your paper. Exchanging a bit of the technical correctness for readability is the entire point of the abstract, and the source paper's abstract could really use some work.
As an experimental physicist, everything but the MHV (maximally helicity violating) was readable/comprehensible to me. Theorists in the field will read that as easily as breathing.
Yes, I guess this is pretty much the case here. There's no much way to simplify this
Conciseness is important in abstracts and it beats general simplifications.
I'm not in particle physics, but have read textbooks and papers in many areas of theoretical physics, and I understand enough of this abstract to know what it's going to do.
If MHV is an acronym you think should be expanded, it is not. There's well over 500,000 hits on google for MHV physics, and only 2900 hits for the expanded term. It is a widely used term in the field.