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what in the name of the world made string theorists think that the universe is made of strings? what is the basis for this assumption?

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mathman

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Yes but could you explain what led to the belief that the universe is made of strings?

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Strings were found handy because they get rid of infinities in our current theories, such as the singularity in general relativity. Also, one vibrational mode of a string was realized to be the graviton, so it was the first glimpse of hope to include gravity into the quantum theory.

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- removes UV divergences

- predicts gravity

- predicts gauge bosons

- predicts fermions

- it is more elegant than field theory because essentially only one kind of object (the superstring) is needed, in contrast with many kinds of particles/fields in the Standard Model based on field theory

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By the way, the case D=11 is the maximal number of dimensions in supersymmetric FIELD (not string) theories, while the relation of this with string theories is not yet completely understood.

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These super Virasoro algebras are central extensions of the contact superalgebras K(1|N), and as such appear to be exceptional algebraic objects. However, one thing that convinced me that string theory is wrong (apart from its disagreement with experiment) was my observation in http://www.arxiv.org/abs/physics/9710022 that these algebras are not as exceptional as people thought. The full superdiffeomorphism algebra vect(M|N) admits abelian Virasoro-like extensions for all M|N, and the restriction to K(1|N) is N-extended super-Virasoro. Hence, rather than being elements in the short list of central extensions, these algebras are undistinguished members of the infinite list of abelian, Virasoro-like extensions.

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The 2d worldsheet of strings is the only manifold which gives local conformal invariance. To copy and paste a post I made on another forum (and pulled mostly from GS&W - Superstrings) :

Strings were found handy because they get rid of infinities in our current theories, such as the singularity in general relativity. Also, one vibrational mode of a string was realized to be the graviton, so it was the first glimpse of hope to include gravity into the quantum theory.

In string theory the Lagrangian can take a form roughly (up to some factors etc) of

[tex]\mathcal{L} = \int d^{n+1}\sigma \sqrt{h}h^{\alpha \beta}(\sigma) g_{\mu\nu}(X) \partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}[/tex]

Not particularly pleasant, but it's very general and this form is used in standard quantum field theory where n=0, the objects are zero dimensional points with [tex]g_{\mu\nu} = \eta_{\mu\nu}[/tex]. This is just the mathematical extension to higher dimensions to see if anything is interesting. The integral is over n+1 dimensions because you're including time, so it's over an n+1 dimensional 'manifold' (just think surface or volume)

[tex]g_{\mu\nu}[/tex] is the space-time metric, the one which you'd see in general relativity. [tex]h_{\alpha \beta}[/tex] is the metric which describes the geometry of the n+1 dimensional manifold, basically the shape the quantum object 'sweeps out' as it moves through time and space, [tex]h^{\alpha \beta}[/tex] is it's inverse and [tex]\sqrt{h}[/tex] is the square root of the determinant of that metric. [tex]X^{\mu}[/tex] is the position function of the object. Just like [tex]\mathbf{x}=(x,y,z)[/tex] says where an object is in 3d space, [tex]X^{\mu}[/tex] does it in 10.

Local conformal invariance is sort of like a position dependent rescaling of the n+1 dimensional manifold (ie a rescaling of the surface where the rescaling changes with position) which leaves the integral unchanged. Specifically, it's the transformation [tex]h_{\alpha \beta} \to \Lambda(\sigma)h_{\alpha \beta} [/tex]. Crunching the numbers, this turns the integral into

[tex]\mathcal{L} = \int d^{n+1}\sigma \Lambda^{\frac{1}{2}(n+1)-1}\sqrt{h}h^{\alpha \beta}(\sigma) g_{\mu\nu}(X) \partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}[/tex]

But we want invariance, so [tex]\Lambda^{\frac{1}{2}(n+1)-1} = 1[/tex] so [tex]\frac{1}{2}(n+1)-1 = 0[/tex], so n=1. Therefore the

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But there's reason a theory should have local conformal invariance right?!The 2d worldsheet of strings is the only manifold which gives local conformal invariance.

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Of course. For example, this kills UV divergences in quantum loops. This is exactly why quantum gravity with strings works much better than that with point particles or fields.But there's reason a theory should have local conformal invariance right?!

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nrqed

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The 2d worldsheet of strings is the only manifold which gives local conformal invariance. To copy and paste a post I made on another forum (and pulled mostly from GS&W - Superstrings) :

In string theory the Lagrangian can take a form roughly (up to some factors etc) of

[tex]\mathcal{L} = \int d^{n+1}\sigma \sqrt{h}h^{\alpha \beta}(\sigma) g_{\mu\nu}(X) \partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}[/tex]

Not particularly pleasant, but it's very general and this form is used in standard quantum field theory where n=0, the objects are zero dimensional points with [tex]g_{\mu\nu} = \eta_{\mu\nu}[/tex]. This is just the mathematical extension to higher dimensions to see if anything is interesting. The integral is over n+1 dimensions because you're including time, so it's over an n+1 dimensional 'manifold' (just think surface or volume)

[tex]g_{\mu\nu}[/tex] is the space-time metric, the one which you'd see in general relativity. [tex]h_{\alpha \beta}[/tex] is the metric which describes the geometry of the n+1 dimensional manifold, basically the shape the quantum object 'sweeps out' as it moves through time and space, [tex]h^{\alpha \beta}[/tex] is it's inverse and [tex]\sqrt{h}[/tex] is the square root of the determinant of that metric. [tex]X^{\mu}[/tex] is the position function of the object. Just like [tex]\mathbf{x}=(x,y,z)[/tex] says where an object is in 3d space, [tex]X^{\mu}[/tex] does it in 10.

Local conformal invariance is sort of like a position dependent rescaling of the n+1 dimensional manifold (ie a rescaling of the surface where the rescaling changes with position) which leaves the integral unchanged. Specifically, it's the transformation [tex]h_{\alpha \beta} \to \Lambda(\sigma)h_{\alpha \beta} [/tex]. Crunching the numbers, this turns the integral into

[tex]\mathcal{L} = \int d^{n+1}\sigma \Lambda^{\frac{1}{2}(n+1)-1}\sqrt{h}h^{\alpha \beta}(\sigma) g_{\mu\nu}(X) \partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}[/tex]

But we want invariance, so [tex]\Lambda^{\frac{1}{2}(n+1)-1} = 1[/tex] so [tex]\frac{1}{2}(n+1)-1 = 0[/tex], so n=1. Therefore theONLYtheory which has local conformal invariance is that which works with 2 dimensional manifolds, one dimension of which is time, so the object is extended through 1 spacial dimension, a string.

Very interesting post.

But then the obvious question is: what about branes?? We hear that string theory contains higher-dimensional objects than strings and your point makes it clear that this will destroy conformal invariance. So why is the theory of branes still viable???

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Because branes are not FUNDAMENTAL objects, but emergent effective ones. As known even from QFT, an effective theory does not need to be renormalizable and/or infinities-free.Very interesting post.

But then the obvious question is: what about branes?? We hear that string theory contains higher-dimensional objects than strings and your point makes it clear that this will destroy conformal invariance. So why is the theory of branes still viable???

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nrqed

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Thank you very much for the reply, it is appreciated (since this is a technical question and string theory one on top of that, I was not very hopeful to get an answer on this board).Because branes are not FUNDAMENTAL objects, but emergent effective ones. As known even from QFT, an effective theory does not need to be renormalizable and/or infinities-free.

I thought that branes were as fundamental as strings in M theory. You are saying that they are effective degrees of freedom? An dthat strings are still the only fundamental degrees of freedom? I thought that branes were on par with strings, the main difference being that they are only "seen" in the nonperturbative regime (i.e. that perturbation theory only "reveals" the strings degrees of freedom). Am I completely off?

Let me ask a related question to make sure I get this right. Superstring theory was initially believed to be free of anomalies in 10 dimensions. That's the usual number quoted in early books (26 dimensions for the bosonic string and 10 for the superstring theory). But then wittens howed that actually, the correct number is 11 because the analysis that led to an answer of 10 dimensions was based on perturbative calculations and one had to use nonperturbative arguments to show that it was really 11 (which was neat since supergravity "lives" in 11 dimensions, if I recall correctly). If this picture is right, then why is the usual condition of conformal invariance now irrelevant ?

As I wrote this, I started to feel that the fact that the usual restriction due to conformal invariance does not apply anymore is an issue of perturbative vs nonperturbative analysis. Is that a possibility?

Thanks for your feedback.

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In M-theory, which is supposed to be an even more fundamental theory, strings are believed to be equally (non)fundamental as branes. The problem is that nobody actually knows what M-theory is.

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In M-theory, which is supposed to be an even more fundamental theory, strings are believed to be equally (non)fundamental as branes. The problem is that nobody actually knows what M-theory is.

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There is a good book by Michio Kaku "hyperspace" (Oxford Univ. press) that explains in a nice way the ideas in laymen terms.

If I remember well, there is a paragraph where he says, that a lot of people ask "why strings ? nature doesn't seem to have strings as a recurring theme, see the planets, the atom, etc, it's more like spheres isn't it...?" and he answers that strings are not less "natural", for example look at the molecule of ADN, it looks more like a string doesn't it...

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