# Question about the foundations of string theory

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
At the present time string theory is a theoretical model that has not been verified or disproven by any experiment. It is simply made up, but a lot of physicists believe that eventually it will be proven out by experiment.

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

If you don't want to think of elementary particles as 0-dimensional "dots", you have to go with strings or with some other more fundamental structure of nature, e.g. branes or spinfoams.
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.

Demystifier
Gold Member
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?
- 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

What about the eleven dimensions? Other theories had 26? don't these numbers seem arbitrary? Why not 17?Why not 37(heh ,thirty seven)?

Demystifier
Gold Member
It turns out that quantum anomalies cancel out only for D=10 in the case of superstrings and only for D=26 in the case of (unrealistic) bosonic strings. There is nothing arbitrary about this. You may also think about that as a weakness of the superstring theory, and this weakness is certainly not the only one. But your original question was about the advantages, not about the weaknesses, am I right?

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.

One can also consider extended world-sheet susy, as is done in the book by Green-Schwarz-Witten in section 4.5. However, such theories are dismissed, because the N = 2 theory has critical dimension D = 2, and the N = 4 theory has critical dimension D = -2. Only the theories with N = 0 (D = 26) and N = 1 (D = 10) are phenomenologically viable, i.e. have D >= 4.

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.

If you don't want to think of elementary particles as 0-dimensional "dots", you have to go with strings or with some other more fundamental structure of nature, e.g. branes or spinfoams.
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.
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

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

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 $$g_{\mu\nu} = \eta_{\mu\nu}$$. 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)

$$g_{\mu\nu}$$ is the space-time metric, the one which you'd see in general relativity. $$h_{\alpha \beta}$$ 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, $$h^{\alpha \beta}$$ is it's inverse and $$\sqrt{h}$$ is the square root of the determinant of that metric. $$X^{\mu}$$ is the position function of the object. Just like $$\mathbf{x}=(x,y,z)$$ says where an object is in 3d space, $$X^{\mu}$$ 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 $$h_{\alpha \beta} \to \Lambda(\sigma)h_{\alpha \beta}$$. Crunching the numbers, this turns the integral into

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

But we want invariance, so $$\Lambda^{\frac{1}{2}(n+1)-1} = 1$$ so $$\frac{1}{2}(n+1)-1 = 0$$, so n=1. Therefore the ONLY theory 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.

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The 2d worldsheet of strings is the only manifold which gives local conformal invariance.

But there's reason a theory should have local conformal invariance right?!

Demystifier
Gold Member
But there's reason a theory should have local conformal invariance right?!
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.

nrqed
Homework Helper
Gold Member
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

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

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 $$g_{\mu\nu} = \eta_{\mu\nu}$$. 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)

$$g_{\mu\nu}$$ is the space-time metric, the one which you'd see in general relativity. $$h_{\alpha \beta}$$ 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, $$h^{\alpha \beta}$$ is it's inverse and $$\sqrt{h}$$ is the square root of the determinant of that metric. $$X^{\mu}$$ is the position function of the object. Just like $$\mathbf{x}=(x,y,z)$$ says where an object is in 3d space, $$X^{\mu}$$ 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 $$h_{\alpha \beta} \to \Lambda(\sigma)h_{\alpha \beta}$$. Crunching the numbers, this turns the integral into

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

But we want invariance, so $$\Lambda^{\frac{1}{2}(n+1)-1} = 1$$ so $$\frac{1}{2}(n+1)-1 = 0$$, so n=1. Therefore the ONLY theory 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???

Demystifier
Gold Member
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???
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.

nrqed
Homework Helper
Gold Member
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.

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).

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?

Demystifier
Gold Member
In string theory, strings are fundamental while branes and fields are not.

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.

Demystifier