I continue to read Sheldon Katz’s book “Enumerative Geometry and String Theory”. Here’s question I posted on sci.physics. If you have any answer, comment or ANYTHING, please (rather: I BEG YOU, CRYING ON THE FLOOR!!!) share them. It’s really hard to find someone who is mathematician and who’s interested enough in physics to answer even such simple questions… In my department there is only one more guy I didn’t yet ask (about what follows or some other physical questions) , but I don’t trust him – he was a physicist in his youth.
!Clarification update! I didn’t mean that if you’re physicist then I don’t want to see your comments – I do even more! It’s just that I have generally bad experience with talking about physics with physicists on my university. Especially with this guy I mentioned above, which was particularly dissapointing, ’cause he’s sort of both mathematician and physicist…
So here goes my post:
Yo,
I’m reading Sheldon Katz’s book “Enumerative geometry and string theory”, chapter 11. I know nothing about QFT or physics so please use polite language (for example, D-brane, string, SUSY are very impolite (unless [explained or given with reference] for dummies) while connection, Sobolev space, Poincare duality etc are honey for my ears).
SK says that we’ll be doing this QFT on (n+1) or (n+0) dimensional X, called spacetime; and that we’ll be interested in fields on X of mainly two types:
1) sections of vector bundles
2) maps from X to another manifold Y
First question: what physical meaning have fields of second type? I hope that fields of first type are old-school fields, like 4-form coming from Maxwell equations, etc. – am I right?
Second “question”: well, actually I know that unfortunately I’m not right, because SK gives following weird example: (0+1) dimensional X and fields are real-valued functions. He introduced Action Functional S(x(t)) = “here stands precisely the standard integral used for deriving Newton’s laws of motions)”. Why it’s weird for me? Because X is called “space-time” and here it isn’t space-time. Rather, on X there are fields which can be interpreted as functions from time to “real” spacetime. So it’s weird – is it normal? Should I abandon this nice idea that space-time in QFT roughly corresponds to real space-time?
Please, be polite. Thanks!
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[ sirix ]————sirix-at-univ-szczecin-pl————-
https://sirix.wordpress.com (my maths&physics-oriented blog)
Just noticed that I again used this stupid comma before “that”. Fixed it, as well as some other typos.
If I learn something nontrivial, I’ll report here.
Fightin' the resistance of matter 
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March 24, 2007 at 8:29 am
John Armstrong
For the first question, the best example I can give is the string worldsheet. Think of a little (closed) string flying through spacetime. It traces out a tube in spacetime as it moves. Instead of thinking of a loop in space changing position as time goes by, think of a cylinder mapped into spacetime.
The second question is best answered by a simpler version of the first. Don’t think of a particle moving through space. Think of a curve in spacetime — the worldline.
It really broke my brain the first time I ran into this, but on my third pass through Green, Schwartz, and Witten’s book it hit me: spacetime coordinates are fields on a 1-dimensional “spacetime” parametrized by the proper time of the particle. Similarly for strings, spacetime coordinates are fields on the 2-dimensonal “spacetime” parametrized by the position along the string and the proper time of the point along that string.
Hope this helps.
March 25, 2007 at 1:08 am
sirix
I get the concept. However, this is weird way of looking at a particle (as a field on a timeline). Is it possible to say in few words why looking this way on things is fruitful?
Also, do physicists consider fields on “real” spacetime (I mean spacetime in which we live, spacetime as it is in general relativity) which take values in some strange manifolds?
(I know that, apart from vector bundles, they’re interested also in principal bundles; I don’t know why, unfortunately. I hear that section of a principle bundle is what physicists call “gauge”, but I don’t get it at all)
I lent a book “Quantum Fields and Strings: A Course Mathematicians”, by many people, including Pierre Deligne (THE Pierre Deligne ;-). It covers one year of efforts of Princeton mathematicians to understand modern ideas of physics. It starts nice (say, “mathematically”), but it’s two volumes, ~700 pages each…
March 25, 2007 at 1:51 am
John Armstrong
I’ve got the Deligne et. al. book. I still haven’t gotten it all down. What you really have to realize about it is that it’s more like a collection of books than any one book, and it’s all aimed at people past their Ph.D.s in the field already. Not saying you can’t figure your way through it, but it’s really hard.
As for gauge fields, that I can explain, but you’d actually do better to pick up Gauge Fields, Knots, and Gravity, by John Baez and Javier Munian.
March 25, 2007 at 11:00 pm
sirix
It would be pretty hard for me to obtain this book, since the only copy in Warsaw (Poland?) is in institute of physics – I think they wouldn’t lend me a book, only allow to read it in a library.
So, I encourage You strongly to share your knowledge :-)
March 31, 2007 at 8:34 am
theoreticalminimum
sirix,
The “raw” material for the 2-volume “Quantum Fields and Strings: A Course Mathematicians” can be accessed free from here – it is likely you already know this.
Besides, I can provide you with a scan-copy of “Gauge Fields, Knots, and Gravity” if you need it ;-]. You will need to have a djvu reader like Lizardtech installed to be able to open it.
April 1, 2007 at 1:00 pm
sirix
Well, I’d be delighted if You’d send it to me :-) It’d indeed save me much trouble. My e-mail is at the end of “About” page.
April 3, 2007 at 11:17 am
sirix
theoreticalminimum: Big thanks!
June 19, 2007 at 4:40 am
A.J. Tolland
Hi Sirix,
Maybe I can help a bit:
Fields of the first type are the sorts of fields that appear in the Standard Model: electron fields, quark fields, Higgs fields, and the like; the corresponding classical fields are all sections of various vector bundles on spacetime. The classical electron field, for example, is a section of a bundle of odd-parity Dirac spinors.
Because spacetime is usually assumed to be flat Lorentz space, most particle theorists also think of gauge fields as being fields of the first type. Gauge fields are really connections, but the only bundle on flat
is the trivial bundle, and so we can identify gauge fields with Lie algebra valued 1-forms, which are again bundle sections. (If you’re wondering how physicists talk about non-trivial topology on spacetime…well, what’s happening here is that they’re only interested in functions which have certain reasonably growth conditions at infinity, which is thought of as the 3-sphere.)
Likewise, the gravitational field, which is a metric, is a type 1 field: it’s a section of the second symmetric power of the cotangent bundle of spacetime. No complete quantum theory here yet, though, so maybe not a great example.
Fields of the second kind are crucial in string theory, for the reason that John Armstrong explained. Hopefully, he won’t mind if I repackage some of his explanation.
This point is maybe best understood by analogy:
In perturbative quantum field theory, we almost forget that we’re studying quantum fields; they only appear in the Higgs expectation values. Instead, we think almost entirely in terms of particles. You can think of these particles by just drawing the worldline the particle traces out as it moves through space and time.
What this means is that the motion of a particle in spacetime is almost entirely described by a field theory on a one-dimensional spacetime, the particle’s worldline. So particle motion is mostly described by a 1d quantum field theory.
I say “almost” because sometimes particles collide, worldlines meeting at a vertex. You can’t handle this with just 1d QFT; some additional rules are needed. But perturbative QFT is basically just 1d QFT describing some free particles moving in spacetime, together with some extra rules specifying what happens when particles collide.
So, the benefit is this: You can discuss most of particle physics using only quantum mechanics and some additional rules. Unless we’re interested in strong-coupling effects like instantons or confinement, we can think entirely in terms of particles, and basically forget the fields.
And perturbative string theory is the same thing, except that 1) you describe strings moving in space by replacing the 1d QFT with a 2d QFT of strings valued in spacetime, and 2) you don’t need any additional rules telling you how two strings interact when they meet. It’s basically just Feynman rules for strings. Unfortunately, we don’t know how what the analogue of QFT from which these stringy Feynman rules are derived.
QFTs of the second kind are crucial in string theory, but don’t appear too often in good old experimentally-tested physics. They’re sometimes used to approximate the physics of quarks and gluons, but that’s the only use I know of off the top of my head.