Tell me more ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.
up vote 10 down vote favorite
3

One way that I've seen to sort-of motivate string theory is to 'generalize' the relativistic point particle action, resulting in the Nambu-Goto action. However, once you see how to make this 'generalization', it becomes obvious how to write down the action, not just for a string, but for manifolds of higher dimension as well. In fact, Becker-Becker-Schwarz (the main source I happen to be learning from) actually do this. But (as far as I have read), they merely write down the action and do nothing further with it.

My question is: what happens when we proceed along the same lines as string theory, but when replacing a string with a 2-manifold, the simplest example of which would be the 2-sphere, a "shell/membrane"? Assuming 3 spatial dimensions, this is the highest dimensional manifold we can consider (because we don't allow for non-compact manifolds). Furthermore, there is only one compact manifold of dimension 1; however, there are infinitely many compact manifolds of dimension 2, which could potentially make the theory much richer (and probably much more difficult). For example, in string theory, we are stuck with $S^1$ (if you insist upon having no boundaries), but if you allow for 2 dimensions, we could consider the sphere, the torus, etc.

Because it seems that this path is not ever presented (in fact, I've never even heard of it), I would presume that something goes wrong. So then, what exactly does go wrong?

share|improve this question
May be connected with the fact you wouldn't have all the richness of the CFT you get on a 2d worldsheet. 2d is a special case where the conformal group is infinite dimensional? – twistor59 Feb 28 at 11:22
1  
They had started talking of membranes instead of strings at some point, I have this picture of a drum surface vibrating :). The wiki article seems to say M-Theory is not really mebranes: en.wikipedia.org/wiki/M-theory – anna v Feb 28 at 11:31
@twistor59: I'm not a real string theorist, but I don't really see where in ST the Virasoro algebra instead of the conformal algebra is crucial. Of course, with the normal conformal group, there is much less control and we might not be able to do as many things explicitly. – Vibert Feb 28 at 12:41
3  
Because T-duality transforms (among other things) the von Neuman boundary condition of a string that is not stuck on something to the Dirichlet boundery condition which means that there has to be something the string can stick on, D-branes which can be higher dimensional have to be there too. That's the way a have Lenny Susskind seen introducing D-branes. – Dilaton Feb 28 at 13:31
1  
The shells are correctly called "branes". They actually do appear in string theory but their dynamics is generally determined either by strings attached to them or more general, matrix-like or noncommutative, maths. A direct analogy of the strings to higher-dimensional objects leads to a world volume theory inside the membranes or branes in which the internal gravity doesn't decouple, so at least, one runs into the same problems with defective naively quantized gravity, but now in the world volume. In world sheets, all the physical polarizations of 2D gravity decouple. – Luboš Motl Mar 1 at 13:25

1 Answer

up vote 11 down vote

A classical$^1$ theory of (relativistic) $p$-dimensional membranes exists for any non-negative integer $p$. Such classical membrane-like objects appears in the full non-perturbative formulation of string theory.

The problem arises if one tries to quantize a membrane (in a first quantized sense) using standard perturbative quantization methods, which have been successfully applied to $0$-dimensional point-particles and $1$-dimensional strings. This program has so far failed for membranes with internal dimension $p\geq 2$. See e.g. Ref. 1 and Ref. 2. Intuitively, the problem is that membranes can grow spikes/tubes that don't cost energy.

Nicolai et. al. have later given a second-quantized interpretation of membranes inside M(atrix) theory. See e.g. Ref. 3.

References:

  1. B. de Wit, J. Hoppe, and H. Nicolai. On the Quantum Mechanics of Supermembranes, Nucl. Phys. B305 (1988) 545.

  2. B. de Wit, W. Lüscher, and H. Nicolai, The supermembrane is unstable, Nucl. Phys. B320 (1989) 135.

  3. H. Nicolai and R. Helling, Supermembranes and M(atrix) Theory, Lectures at Trieste Spring School (1998). (Hat tip: alexarvanitakis.)

--

$^1$ Here the word classical means $\hbar=0$.

share|improve this answer
1  
This is outdated; see e.g arXiv:hep-th/9809103v1 "SUPERMEMBRANES AND M(ATRIX) THEORY" by Hermann Nicolai and Robert Helling. The point is that membrane theory may be regarded as a "second-quantized" theory already. – alexarvanitakis Feb 28 at 20:12
@alexarvanitakis: Thanks. I will update the answer accordingly. – Qmechanic♦ Feb 28 at 20:22
I should also mention that it seems that standard quantization techniques do not really work, and that people like to think of the quantized supermembrane as the gauge theory of the group SU($\infty$). This is explained in some detail in the paper I linked to above. – alexarvanitakis Feb 28 at 20:25

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.