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8:27 AM
hi
 
 
2 hours later…
10:25 AM
S matrix should have been difficult to define in QED because it has massless particles like photons, but most QFT books avoid this and find out cross sections and what not...
Where can I find a discussion of why this works?
 
@Sanjana why would it have been difficult to define?
 
When we define S matrix we evolve some initial state to final state and these states are required to be free particle states; but massless particles provide long range interactions. So we cannot define free particle states
Probably in better words..
 
10:40 AM
@Sanjana really interesting
 
I was reading Collin's book on Regge theory and found that they have gotten rid of EM and Newtonian Gravitation precisely because of this reason, while defining the S matrix
I think this manifests itself in the form that one should not use Born approximation for calculating the Rutherford cross section (although it works out to give the correct result)
:65702175 Hmm. That's the adiabatic hypothesis. Idk if that holds for QED or other theories with massless dof
 
oh
sorry i will re-write the comment : in books, we switch off the interactions. also, the Green's function decays exponentially even for the massless theory
nvm i think it does not decay exponentially for massless theories
 
Hmm
S matrix theory seems wonderful. Too bad it didn't'work out. Fortunately Nima is giving new life to it in recent times...
 
@Sanjana The discussion of why this works is mainly under the heading of IR divergence. A better way to handle this is via the infraparticle, but then you have to give up the standard Dirac equation form of the electron with the self-energy correction. This is not an easy replacement because things like the LSZ reduction currently assumes that as a basic ingredient. In the end, we really really really don't care about the ultra-soft photons that nobody can actually measure in experiment.
 
i think it decays according to $\frac{e^{-mr}}{r}$, so it decays decay according to 1/r for m=0
so the interaction r stronger for massless but they still die at infinity
 
10:52 AM
@RyderRude Coulomb interaction dies at infinity but not fast enough.
 
@Sanjana I mean, this is a pretty silly objection because the definition of "asymptotic free states" as it is usually done in intro QFT doesn't work anyway if you're being serious about it (Haag's theorem!)
 
@naturallyInconsistent Yeah. Although one needs to include these soft photons into the theoretical calculation to cancel the IR divergences...
 
@Sanjana im not familiar with this. is there a bound on the decay rate for us to b able to define asymptotic particles
 
@naturallyInconsistent The paper in the screenshot above is authored by Schwartz, who discuss something of this sort in his book...But he glosses over the S matrix thing
 
10:53 AM
both these things ultimately lead to divergences you need to cancel by renormalization/regularization ad hoc later, but the massless problem isn't really worse than the overall renormalization problem
 
@ACuriousMind Ohhh
 
@ACuriousMind but if you accept the Causal Perturbation Theory scheme, then it does the UV divergence properly, and so it kinda makes more sense to deal with the IR divergence properly, so that we don't have any ad hoc regularisation at all.
 
@naturallyInconsistent Sanjana was talking about "most QFT books", not CPT :P
 
I know...~
 
Now what's that :o
 
11:01 AM
@Sanjana somebody found a scheme that delicately handles the "zero distance" UV divergence in usual QFT, chopping things up in a way that respects the causality conditions rather than the brute force way that is usually done, and by defining everything only ever in terms of the renormalised quantities, it avoids having to deal with bare quantities at all.
 
@RyderRude Usually we define the asymptotic states to be such that they "look like" (superposition of) plane waves. But if you plug in the Coulomb potential for example in time independent Schrodinger equation and explicitly look at the asymptotics, they don't even "look like" plane waves; there is some extra log term in the exponential
@naturallyInconsistent wtf! And why isn't this popular...or is it???
Let me go check it
 
@Sanjana ooh
 
Thanks for bringing this uppp
@RyderRude E.g. Look at Tong's notes on "Applications of QM", section 10.4
 
@Sanjana thanks
 
@Sanjana we gain the lack of counterterms and such, but we lose in a much longer exposition, and a much more complicated perturbation expansion. Also, very few people have heard of it.
 
11:11 AM
it says it works for 1/r^n but fails for 1/r
so we never obtain free-like behavior when massless interaction r allowed
 
@Sanjana it's not popular because it's a lot of mathematical technicalities to arrive at the same results :P
people make a big show of not liking the "infinities" in QFT but it turns out they dislike formal math even more :P
2
as far as I know, rigorous scattering theory for systems without mass gap - which is what you're looking for when you want a proper S-matrix for soft photons - is still a somewhat open problem (while for systems with mass gap this has long been known in the form of Haag-Ruelle theory, another mathematically rigorous version of a part of QFT that you won't find in the average QFT book)
 
12:02 PM
@ACuriousMind ei, is infraparticle not rigorous?
 
12:40 PM
@naturallyInconsistent I don't think the scattering theory for infraparticles is fully developed, see e.g. the intro of arxiv.org/abs/2105.05723
 
12:51 PM
dam holding back infinite tears bursts
 
@RyderRude I ain't watching a tutorial for that
 
@Mr.Feynman and on miao miao side, it is more like miao miao should be authoring that tutorial
 
I was implying that :P
 
1:06 PM
miao miao was replying in a way that could mean both ways miehehehe
 
@Mr.Feynman wud only take a day for the checklist
 
 
4 hours later…
6:03 PM
Question. Youtube has lots of videos of chaotic motion of double pendulum. But I'm looking for a specific kind that I don't see anywhere. I'd like a pendulum with two sections constrained to a plane, with the first section pivoting around a fixed point and the second section pivoting around a point of the second (so far there are lots of videos), but such that the two pivot points and the center of mass of the first section aren't collinear. What keyword should I search for that describes this?
(This would be in a vertical plane so gravity affects the system.)
 
6:49 PM
@Sanjana assuming Haag aka formal nonsense: "The practical resolution to this impasse is to ignore Haag’s theorem, ignore that charged particles cannot be isolated and other assumptions, and simply use the LSZ reduction theorem as if it were true, computing S-matrix elements with Feynman diagrams." You start getting into this waffle about "The problem can be traced to the impossibility of isolating single-particle states at asymptotic times" and on and on
 
 
5 hours later…
11:21 PM
I think I had never the brain of a mathematician. I was only curious, how things are. What is proven from what, that had always low prio for me. For me, world of math is a discovery of an existing world and not creating one.
I think, sphalerons are construction of the human mind, to bridge over the terrible gap between GUT and LHC energies. It does not mean that they don't exist, although I doubt that they could be stable.
 

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