>be a math guy>decide to dabble in QFT>get told picrel is the most rigorous introduction to the subject>literally zero rigor>some math explanations are just outright wrong>some parts read like an undergrad end-of-the-semester math paper>horrendous notation, terminal case of subscriptivitus>something that would take a mathematician 1 page to explain takes up 5 pages of “derivations”What gives? This guy wrote an amazing book on GR and it wasn’t even his field.
>>16552272You dont want the most rigurous, you want something easy to get used to the ideas.Also, what is not rigurous in QFT? Explain a single non-rigurous thing in QFT
>>16552272>something that would take a mathematician 1 page to explain taAlso, dont be so proud of this shit bourbaki freak. We know you are oh so proud of making text as dense as possible and you fail at it because its a;ways riddled with mistakes. So much for rigour, you are not smarter for eschewing context and redundancy
>>16552401I didn’t say QFT isn’t rigorous. I said the book is trash.
>>16552414>I didn’t say QFT isn’t rigorous. IYou said QFT has zero rigor. QFT is as described in that book and others like it. If you think its just the book that is defective, name a single flaw of it
>>16552422Learn to read, retard.
>>16552272>>some math explanations are just outright wrong>>16552431Give me a single example
>>16552272>yet another mathematician filtered by qftPost some examples so that we can make fun of you for not understanding what you read.
>>16552452His proof that SL(2,C) is homeomorphic to R^3xS^3 claims that the polar decomposition of a matrix is Ue^H, where U is unitary and H is Hermitian. This is clearly wrong as exponentiating a Hermitian matrix gives you another unitary matrix. The actual polar composition is UH, where H has to be Hermitian AND positive semi-definite. The actual proof hinges on the latter fact if you actually do it correctly.
>>16552471How embarrassing for a mathematician to post this. e^iH is unitary. e^H is hermitian and positive definite.
>>16552492tell me what matrix I need to exponentiate to get the zero matrix
>>16552536You don't need the zero matrix when you're dealing with SL(2, C) because it only contains invertible matrices.
I remember some tale i heard when i was young that some mathematician went into an elementary school to watch lectures or to be a teacher, dont remember desu, and he was making mistakes with his long divisions or some such arithmetic operations kiddies study and he freaked out and thought he was a loser
>>16552547I literally don’t care. The polar decomposition theorem is valid for any matrix. This is the kind of “wrong” I’m talking about. Yeah, it technically works, but it’s blatantly wrong upon closer inspection. Another example: in the same chapter he mentions his “reasoning” as to why massless representations have to be restricted to the SO(2) little group as opposed to ISO(2).>the action of the little group on physical one-particle states is just a rotation around the momentum>a rotation by an angle 4pi around the momentum can be continuously deformed into no rotation at allNot only is this terribly put, but the reasoning is also entirely faulty. 4pi comes up in SU(2) vs SO(3) because the former is the double cover of the latter. But there is no covering group involved in either ISO(2) or SO(2). Both are not simply connected and have Z as the fundamental group. And 4pi CANNOT be continuously deformed into no rotation at all because those are in two separate homotopy classes of a circle.He is literally saying nonsense garbage as “justification” and I’m supposed to take this book seriously?
>>16552272jewish physics is all fake
>>16552613>but it’s blatantly wrong upon closer inspectionThere is absolutely nothing wrong with it you retard. Using the polar decomposition in the form he's mentioned is common and more sensible when you're proving topological properties of groups because you want to exclude non-invertible matrices. Stop doubling down on your retardation.>But there is no covering group involved in either ISO(2) or SO(2).I looked at his reasoning and nowhere does it involve anything about the representation restricted to the little group. He derives the fact about spins using the fact that the representation is defined on all of SL(2, C). If you just restrict yourself to representations of the little group like you're doing, then the massless particle would have different allowed values of helicities for each direction and there would be no relation between these helicities at all, a manifestly nonsensical result arising because you're making bizarre non-physical assumptions.
>>16552821>nowhere does it involve anything about the representation restricted to the little groupExcept the exact quote>a rotation around the momentumcease thy cope
>>16552846What quote? That's your quote and it's about how the little group is defined. Completely unrelated to Weinberg's derivation. This is the entire quote from that section"On the other hand, for zero mass the action of the little group on physical one-particle states is just a rotation around the momentum, and here there is no algebraic reason for a limitation to integer or half-integer helicity. There is, however, a topological reason: a rotation by an angle 2pi around the momentum can be continuously deformed into no rotation at all, sothe factor eхp(4 \pi \sigma) must be unity, and hence \sigma must be an integer or half-integer."I'm thinking you should go back to grade school and learn how to read.
>>16552862>a rotation by an angle 2pi around the momentum*4 \pi. I made that mistake while formatting the copied quote.
>>16552862>what quote?top of p90>That's your quotehis quote directly from the book>Completely unrelated to Weinberg's derivationNo derivation involved. It’s not even supposed to be a derivation, retard. It’s supposed to be a “proof” that continuous spin representations cannot be realized in nature. And that “proof” is blatantly wrong.>There is, however, a topological reason: a rotation by an angle 2pi around the momentum can be continuously deformed into no rotation at allGo read what the fundamental group is, what the fundamental group of a circle is and why there is no continuous map from the trivial rotation to a 2pi SO(2) rotation, nor from the trivial rotation to a 4pi rotation, nor from a 2pi rotation to a 4pi rotation.
My goodness, this retard keeps doubling down on his retardation. >>16552877The fundamental group of SO(2) is irrelevant here you mouthbreather. This is because he's talking about the representation on all of SL(2, C) and not just the little group as I already mentioned before.
>>16552884It is literally not fucking irrelevant, you absolute mathlet. He is claiming you can deform a path spanning 4pi degrees on a circle into a point. This is literally wrong and that’s literally what a homotopy class is.>This is because he's talking about the representation on all of SL(2, C)No, you fucking retard. He fixes the momentum vector, so it’s not the whole group but only the little group. In non-niggerlicious language, he is talking about the stabilizer subgroup of SL(2,C) associated with its orbit in momentum space.
>>16552890>He is claiming you can deform a path spanning 4pi degrees on a circle into a point.IN FUCKING SO(3) YOU FUCKING RETARD. How could you not understand this when I've repeated it multiple times now? I should have already given up on you when you said e^H was a fucking unitary matrix.
>>16552901SO(3) IS THE WRONG SUBGROUP YOU ABSOLUTE MONGSO(3) IS THE LITTLE GROUP OF MASSIVE REPSHE IS TALKING ABOUT MASSLESS REPS FOR WHICH THE LITTLE GROUP IS ISO(2)MASSIVE REPS DONT EVEN ADMIT CONTINUOUS SPIN AT ALL BECAUSE SO(3) IS COMPACTholy fuck, you are one dumb stubborn fuck
Incredible. I give up trying to communicate the same fucking thing for the 4th time now to this fucking mouthbreathing braindead incel.
Let's recap the "communication" so far.>continuous deformations of loops have nothing to do with the fundamental group somehow>Weinberg is somehow still talking about SL(2,C) despite fixing the momentum and thus restricting the argument to ISO(2)>for some reason SO(3) is involved even though it's completely unrelated to the representations in question>no mention of the fact that Weinberg's non-argument works for both ISO(2) and SO(2) because it only uses the fundamental group, which is the same for both>no mention of the fact that the actual topological difference between these two is compactness, which Weinberg doesn't addressanon just needs to admit that he wants to suck Weinberg's decaying penis instead of thinking for himself.
>>16552536The exponential form of polar decomposition only applies to non-singular matrices. You can see stuff like this in Hall's book Lie Groups, Lie Algebras, and Representations chapter 2.
>arguing about mathematical schizo-babble that has nothing to do with realitywhere did physics go so wrong?
>>16553209>nothing to do with realityt. illiterate luddite
Reminder that if you believe in scientific realism, you literally believe that there is an immaterial symmetry group SU(3)×SU(2)×U(1)× P(1,3) living outside the universe and yet governing every interaction in the material universe without even being able to explain how immaterial rules act on matter.
>>16553319>dumb basement dwelling non-materialist/schizophrenic trying to be relevant by being retarded on an image board
>>16552422>You said QFT has zero rigor.No, he>>>>impliedthat the "book" in question has zero rigor. Are you retarded?
>>16552916Not really. The mathsfag is deranged.
>>16553445>ignores my counterarguments>makes blatant mistakes>doesn’t know basic definitions such as the fundamental group and the little group>confuses two completely separate representations and their little groups>REEEEEEE you’re deranged!That’s why nobody takes you guys seriously anymore. It used to be that physics and math went shoulder-to-shoulder. Wigner pioneered the method of induced representations. But today physicists scoff at mathematics because le physical intuition whatever the fuck that means.
>>16553585We simply don't care about any of that shit, lol. Anyway, a mathjanny will come and fix it, if it turns out to be important.
>>16553586Enjoy your echo chamber, my dude.
>>16553589Enjoy your sloppy seconds
>>16553592Sloppy seconds? Kek. You guys are just working on very specific aspects of mathematics pretending like it’s something else entirely. It’s like a child helping his mom cook but when she asks him to cut the veggies into regular-sized pieces, he throws a tantrum and says he’ll cut them according to his “intuition”.
>>16553595Delusional. All interesting mathematics has its origins in physics. Physics is literally the cum that physics chads shoot into fertile mathbrains, where it blooms into gay mathematical flowers.
>>16553597*its origins in pre 1950 physicsThe only things physics has contributed since are some very particular aspects of differential geometry like spinor bundles. And the way physicists approach those is the sloppiest shit ever.
>>16553600That's why I said "interesting" math. Doing a bunch of shit with no connection to reality is only interesting in the deranged minds of mathsissies.
>>16553605>t. works on N=8 supergravity in 10 dimensions
>>16553614Bro I'm not a string theorist. They are related to mathsissies like you.
>>16553621Ok, let me tone it down a bit>mashes random groups together with SU(3)xSU(2)xU(1) with no rhyme or reason because le dark matter o algoso heckin’ realistic!
>>16552272QFT is an inherently unrigorous theory. It's just a set of mathematical tricks to produce calculations that agree with experiment.
>>16552272"Rigor" is a meme. It usually obscures the underlying idea if you try to pin down all the pathological things that could occur at the very outset. Unlike what some people suggest, QFT is not just some ad hoc collection of recipes but you have to take certain things for granted like the validity of the path integral.If you want to talk about Weinberg's book in particular, the reason some people say it is rigorous is because he has a very good 2nd chapter on representations of the Poincare group, which a lot of other textbooks don't cover, and he has a lot of (often unnecessary) technical detail on the canonical approach to QFT. This technical detail is not the same thing as rigor, but I think that is why the book got that reputation.Weinberg isn't the best book on QFT even for a mathematician. The canonical approach is rather dated, and pretty much everything is based on the path integral nowadays. The route to QFT through the Euclidean path integral and analogies to statistical mechanics is an easier way to understand in my opinion.
>>16553641can your math make better predictions?
>>16552613>4pi comes up in SU(2) vs SO(3) because the former is the double cover of the latter. But there is no covering group involved in either ISO(2) or SO(2).Weinberg was talking about the full rotation group SO(3). The state is in a representation of the little group (which here involves a single generator J_3), but it is also in a representation of the full Poincare group, and the little group is just a subgroup of that. Can you take a non-half integer representation of the little group SO(2) and extend it to the full rotation group? No, for topological reasons, as Weinberg points out.
>>16553959>like the validity of the path integralThere is a literal no-go theorem for interaction path integrals in mathematics which states that there is no notion of Lebesgue measure on infinite-dimensional Banach separable spaces. There is an analogous no-go theorem in axiomatic QFT called the Haag theorem which states the same but in the language of creation/annihilation operators of Hamiltonian mechanics.
>>16553984No, how many times do I have to repeat this? This paragraph is taken from a chapter on massless representations. SO(3) has nothing to do with massless representations. Their little group is ISO(2).>Can you take a non-half integer representation of the little group SO(2)That’s not the full little group. The full little group is ISO(2) = R^2 /rtimes SO(2). The R^2 bit is what leads to continuous representations because that group is non-compact. Muh loops has nothing to do with continuous vs discrete because that’s an entirely different topological property of local compactness vs compactness respectively.
>>16553985I don't know about your no-go theorem for path integrals, but I have looked at Haag's theorem before and it is not saying the same thing. Haag's theorem is ultimately something rather trivial. When you turn on an interaction in QFT the energy density of the new interacting vacuum is non-zero. If you have an unbounded volume (as is required by the Wightman axioms) the total energy of the vacuum diverges, so the vacuum of the interacting theory is not in the domain of the Hamiltonian of the free theory. This is simply cured by taking everything in a finite volume and setting the volume to infinity at the end, and it is not really a problem for QFT.
>>16553992No you are misunderstanding. It's okay, I do it some times too. Back up and take a deep breath.Take a rotation by 4pi using only the generator J_3. This is in the little group of the massless particle. There is no reason you can't have a representation of the little group with arbitrary spin right? Okay good, let's put it in one.But wait the state is also in a representation of the full Poincare group, not just the little group. What happens if I deform the 4\pi rotation within the Poincare group to the identity (which is sensible, and allowed)? Do you get the point now?
>>16554002https://en.m.wikipedia.org/wiki/Infinite-dimensional_Lebesgue_measure>inb4 wikipediareferences are given>This is simply cured by taking everything in a finite volume and setting the volume to infinity at the endThat’s a pathological approach that breaks translation invariance and then magically reintroduces it through some ill-defined limit. Whenever you take the “continuum limit” in physics, what you guys are actually doing is making vast leaps of faith about properties of locally compact groups based on properties of compact groups. This is “okayish” until it stops making sense, the canonical example being path integrals.
>>16554016>This is in the little group of the massless particleMy God, what prevents you from opening the book and literally seeing that it’s ISO(2) and not SO(2)? There are THREE generators of ISO(3), those being J3, P1 and P2. And I don’t give the slightest shit about the full Poincare group here because Weinberg is exclusively talking about the little group in his non-argument.
>>16554026*ISO(2)I’m so fuming at this retardation or Jewish tricks or whatever it is that you’re doing that I can’t even type properly.
>>16554017>This is “okayish” until it stops making senseThis is a healthy approach to working on new things. You don't need to get everything right in the beginning. The culture of rigor in mathematics is sick. The theoretical physicist point of view is to not fix it until it is broken. And probably the cases where it is broken aren't relevant for physics. (And sometimes they are, but that's okay we'll figure it out when it's necessary.)
>>16554026ISO(2) or SO(2) doesn't matter, both are subgroups of the Poincare group and the argument about deforming to the identity is taking place within the full group (or specifically the full rotation subgroup).
>>16554031Path integrals are 70 years old by now. And it’s not healthy when someone literally proves a theorem saying you can’t do that. Imagine someone continuing to search for trivial non-zeroes off the critical line after the Riemann hypothesis gets confirmed. This is insanity.>>16554035THESE ARE TWO DIFFERENT GROUPS YOU FUCKING MORON. OF COURSE IT FUCKING MATTER. ONE IS NON-COMPACT AND THE OTHER IS COMPACT. NON-COMPACT GROUPS DONT ADMIT DISCRETE REPRESENTATIONS, WHICH IS WHAT CONTINUOUS SPIN IS ALL ABOUT. Holy fuck, genuinely kys.
>>16554041*discrete unitary representationsbut this should be fucking obvious because that’s the only relevant type of representations in QM
>>16554041Calm down. I have Weinberg's book open in front me. I understand what you are saying but you are not understanding what I am saying, perhaps because you are angry. I'm not the other guy you were arguing with. Just read my posts again.
>>16554049I understand what you’re saying. You are repeating the exact same mistake over and over again and I’m trying my best to tell you why you’re wrong but you just wouldn’t listen. The momentum vector is already fixed. If it’s fixed, the entire group cannot be considered because that would “unfix” the momentum vector (move it from its orbit to another orbit of the action of SL(2,C) on R^1,3). So considering the whole group for a particular class of representations is wrong. It just plainly is. You keep acting like a stubborn oaf because you are refusing to understand this simple fact.https://en.m.wikipedia.org/wiki/Group_action#Orbit-stabilizer_theorem
>>16554059It's not a mistake, the argument Weinberg mentions involves changing the momentum vector. We are already done with the little group at this point. The little group was used to get equation (2.5.42) which tells how a given vector transforms under *arbitrary* Lorentz matrices \Lambda. Up to this point it looks like \sigma could be an arbitrary real number. Up to this point I hope we agree.Now Weinberg mentions an argument for why it is half integer. He doesn't really go into detail on the top of page 90, but you have to read between the lines. Before page 90 he had section 2.7 on projective representations, and that is what this is really what this is about.Do a rotation by 4\pi with J_3. This is equal to the identity within the Poincare group, but if \sigma is not half integer (2.5.42) tells you there is some overall phase left over after rotating 4\pi. But that is still okay, maybe we can either redefine things to eliminate the phase, or maybe we are dealing with an intrinsically projective representation. But Weinberg's Appendix B deals with intrinsically projective representations and the claim is that if the group is simply connected (and there are no central charges) then you can redefine things to get rid of the phase. SO(3) is not simply connected (which is why a 2\pi rotation can introduce a phase for fermions) but SU(2) is, and since we are looking at a 4\pi rotation it is fair to look at the covering group.If you still don't get it or think something is wrong after that, I give up.
>>16554131>you have to read between the linesI wanted to make a detailed response, but then I noticed this phrase. Nah, I’m not going to read between the lines. What a Jewish thing to say. Either make your point clear or fuck off, Moshe.
>>16554143I made my point in the paragraph below. An attitude like that where you always think you are the smartest person in the room is why you won't be able to progress in either math or physics. Remember this is an "introduction" to QFT, You should feel embarassed getting stuck on some little issue and throwing the whole thing away.
>>16554150Yeah, don't care. I don't take anyone who says>actually bro that's not what he said. He actually said something else. You have to read between the lines.seriously. Doesn't matter if it's an introductory textbook or a paper only 3 people in the world can understand.
>>16552431You're talking to an H1B who worships Einstein--reading comprehension isn't their strong point.
