/sci/ - Science & Math

>>11426147
It should already be obvious to you. If it's not, your IQ is clearly under 140 and you should quit attempting to understand that which you can't.
t. someone who read and understood that paper in 7th grade

>>11426179
fuck anon, holy shit, time to go into engineering i guess :-(

>>11426205
Yep. Smart people don't have to work to understand things.

>>11426220
could you give me the babby-tier translation of what it says, high-IQ senpai?

>>11426147
both zwiebach and schwarz give an explicit reasoning/derivation of d=26, so i am not sure what you are looking for anyway

>>11426232
zwiebach says something along the lines of "we will not attempt to do the calculation here" and GSW uses 26 dimensions as an assumption stated in chapter 1 and they only get around to "justifying" it later in a few different ways, none of which seem to me to be very convincing relative to what polchinski says. i have both textbooks so if you want to point me to the smoking-gun page where they show a real proof, please do

how would one even conclude the necessity of more dimensions for some equation to work? where do dimensions become an argument or output of the function? or do they use other weird tricks

>>11426255
zwiebach says "we will not attempt the calculation" about the calculation of a specific commutator. There isnt anything special going on there, it is just algebra. I havent read Polchinski, what does he say?

>>11426283
polchinski says something like this, which i am very unsatisfied with.... pic related

i just don't like how he alludes to analytical continuation to solve divergent sums. i figure if "string theory is a finite theory" then this kind of slight-of-hand should be avoidable

>>11426298
ah yes, both zwiebach and schwarz use the same argument. I also dont know how to make it rigorous and would be interested to see proper reasoning instead, but i don think you will find that in the original paper. sorry i couldnt help you

>>11426309
thanks for your feedback. this is a real mystery to me. i can't find anything where somebody honestly derives that bosonic string theory must be 26 dimensional.... all i can find is quickly glossing it over and references to divergent sums. i would really like to see something that is more physically reasonable or at least mathematically rigorous

bumpzinga

>>11426309
hey anon, i dug through the literature and i eventually found a really nice paper from 1973 that does a derivation that i am happy with, using some ken wilson style renormalization arguments to justify it, pic related

>>11426147
In essence, it's the minimum number of spatial dimensions for the compatibility of spatial Lorentz symmetry and internal conformal symmetry. If you construct a [math]classical[/math] string theory with the Nambu-Goto action, canonical quantization takes coefficients of the canonical fields (as symplectic coordinates on the worldsheet) to conformal generators, which has Virasoro relation [math][L_n,L_m] = 2nL_0 + \frac{d-2}{12}n(n^2-1)[/math]. Indeed, if you interpret, as in a conventional CFT, the central charge as counting the number of chiral zero modes, then it stands to reason that string theory, as a background spacetime theory, have chiral zero modes associated with the number of available spacetime dimensions you can move in.
However, canonical quantization does [math]not[/math] preserve the emergent Lorentz invarance generated by the canonical fields. A computation of [math][M_i,M_j][/math] yields a term [math]\frac{d-2}{24}[/math] aside from the desired classical result, [math]precisely[/math] due to the existence of the central charge. This is the reason why we need [math]d = 26[/math] for bosonic strings.
When you add anticommutative fermions, however, you can "shrink" the conformal algebra by grading it against the fermion counting operator [math](-1)^F[/math], which increases your central charge to [math]\frac{d-2}{8}[/math].

>>11429681
sure, everything you said makes sense and follows logically (to the degree i am capable of judging) aside from your numerics. the question is really "where do your factors of 1/12 and 1/24 come from?" (i guess in your last one you have 8 and that too is mysterious.)

so do you have an explanation for those numerical factors that doesn't rely on appealing to analytic continuation for divergent sums? this gets to the heart of why you, yukari-poster, are not a good poster here. you make these obscure arguments without actually backing it up with anything. i already posted a nice intelligible paper which answers my own question, >>11429581
and your much-more-obscure mouth-flapping falls far short of being nearly as helpful to physically-minded people

>>11429708
Those numerical factors come from the choice of the Verma module of the Virasoro labeled by the central charge [math]c[/math]. You can compute it by computing the first few terms in the OPE of the energy-stress tensor, which goes back to the fact I've iterated that [math]c[/math] counts chiral gapless states in the theory. Check out Streater and Velo's monographs on gauge theory, an article therein contains a pretty thorough derivation of the above; for elementary CFT, which it seems that you need, check out di Francesco.
Sorry I couldn't handhold you in a 4chan post.

>>11429717
>. Check out Streater and Velo's monographs on gauge theory, an article therein contains a pretty thorough derivation of the above; for elementary CFT, which it seems that you need, check out di Francesco.
i actually already posted the exact 5 pages of the 5-page article which answers my question, so i think i'm happy with that resolution. but thanks for directing me to some unnamed books by authors without any page numbers or chapter numbers or anything. that's super helpful. i mean, far be it from ebin yukari-poster from actually writing a derivation on 4chan or actually screenshotting it from his ebin obscure textbooks