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String theorists say that there are many more dimensions out there, but they are too small to be detected.

  1. However, I do not understand why there are ten dimensions and not just any other number?

  2. Also, if all the other dimensions are so coiled up in such a tiny space, how do we distinguish one dimension from the other?

  3. If so, how do we define dimension?

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(1) It's all in the mathematics. (2) Can you distinguish your everyday 3 dimensions? Nope. So then there's no problem if the curled up ones are indistinguishable (not saying they are, though). [Atleast, I think it's this.] –  Manishearth Jul 12 '12 at 12:27
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Lenny Susskind has once shown a simplified mathematical argument that one needs 26 dimensions to allow for the tachionic ground state of bosonic string theory. The number of dimensions can then reduced to 9 + 1 dimensions turning to superstrings. –  Dilaton Jul 12 '12 at 13:53
    
What dimensions are in general, Prof Strassler nicely explains in a series of articles starting with this one. As the later articles in the series explain, the (large) extra dimensions could in principle have been detected by the discovery of Kaluza-Klein particles at the LHC for example. –  Dilaton Jul 12 '12 at 13:53
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One would need to analyze a whole spectum of such particles to experimentaly determin their shape as is explained here. From a theoretical point of view, the shape of the extra dimensions is discribed by moduli fields. Darn, now this has become too long for one comment :-P –  Dilaton Jul 12 '12 at 13:54
    
Duplicates: TOo many to list, just refer to: meta.physics.stackexchange.com/q/4653 @RodyOldenhuis: Uh... That doesn't give explanation... . . . –  Dimensio1n0 Jul 30 '13 at 15:39

3 Answers 3

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One can posit mathematical string theories in any dimensions of any kind.

However, I do not understand why there are ten dimensions and not just any other number?

The specific dimensions arise from the requirements of the known physics encapsulated in the Standard Model and other data coming from particle physics, plus the requirement of General Relativity and its quantization. The Special Unitary groups whose representations accommodate the SM need at least these dimensions. There are models with more dimensions than this.

Also, if all the other dimensions are so coiled up in such a tiny space, how do we distinguish one dimension from the other?

We cannot move into the coiled ones, only in x.y,z. We do not need to distinguish them, as we do not distinguish the molecules in the air. The predictions from this type of theory on the behavior of particles is the only way of checking for their existence: consistency of theory with data.

If so, how do we define dimension?

A space variable ( centimeters) or time one ( seconds) that is continuous and maps the real numbers, each dimension at 90degrees to the rest, an extension of how we define normal x,y,z.That some are curled should not bother one. The coordinates over the earth are curled over the sphere's surface, for example, though the 90degree does not hold there. It would hold on the surface of a cylinder , z from -infinity to infinity, x from 0 to 2*r*pi.

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This answer is not accurate, you cannot formulate any sort of string theory in 60 dimensions, if you have too many spatial dimensions, there are too many degrees of freedom on the horizon. Regarding spherical coordinates, these are orthogonal. –  Ron Maimon Jul 13 '12 at 1:45
    
@RonMaimon Sure, they will not make physics sense but the mathematics will be there,no? As for spherical coordinates on the sphere surface, the angles are not 90degrees. think of the poles. en.wikipedia.org/wiki/Spherical_trigonometry . I will make it clear I mean the surface coordinates. Thanks. –  anna v Jul 13 '12 at 4:35
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I see what you mean. But if you have ghosts in the theory, and divergent loop integrals, what does the mathematics mean? I agree that in all dimensions less than 10 and in all dimensions less than 26 you can formulate a fermionic/bosonic string theory (if you use a Polyakov noncritical string or linear dilaton), but in general I don't like to say this for dimensions higher than 26, because ghosts are not the same kind of problem qualitatively. BTW, only the poles are bad in spherical coordinates. –  Ron Maimon Jul 13 '12 at 4:52

(1) String Theory is a very mathematical theory based on some natural assumptions, and this ends up relating Quantum Mechanics and General Relativity, as we want. Some of the equations in String Theory, however, have a proportionality constant $c$ in it, called the central charge. And when we manipulate these equations and set them equal to each other, we see that they ONLY make sense if $c=26$. This $c$ is the dimension of space that String Theory is a priori defined over, so now we see that we need 26 dimensions to not have absurdities... BUT that only made use of the bosonic particles in the world -- we forgot about fermions!! This is where Supersymmetry comes into play, and it throws in the fermions, and the equations are perturbed and leads to a new dimension of 10 for everything to make sense.

(2) Just because we can't see it, doesn't mean it's not there... we can't see atoms with the eye, but we can use tools to see them... same thing happens here, our current technology can't see them, but we hope to change this in the future. EVEN BETTER though, is that the formula for gravitational force should actually be different because of these extra small dimensions -- thus we plan to figure these extra dimensions out by testing the gravitational force at small distances and seeing a perturbation to the standard inverse-square law of Newton. These extra dimensions are what is supposed to make gravity so weak compared to the other forces of nature.

(3) a dimension is just a coordinate axis... so time is a dimension too. And just like your clock, this axis can repeat itself and not stretch to infinity.

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I don't know how to make sense of compactified time, especially in string theory. Even if you compactify euclidean time, thermal string theory is hard to make sense of too, because gravity doesn't allow thermal ensembles of infinite extent. Regarding the central charge argument, it's fine, but it doesn't require 10 dimensions per-se, just an equivalent central charge, so you can have a non-geometric compactification. –  Ron Maimon Jul 13 '12 at 1:42
    
+1 about the mention of different quantisations (Canonical, path integral, light cone gauge, etc.) having to be equivalent. –  Dimensio1n0 Jun 18 '13 at 3:36

For bosonic string theory... see this

For superstrings...

In the Ramond sector . . . .

\begin{array}{l}0 = {{\hat G}_0}\left| \psi \right\rangle \\{\rm{ }} = \sum\limits_{n = - \infty }^\infty {{{\hat \alpha }_{ - n}}\cdot\;\;{{\hat d}_n}} \left| \psi \right\rangle {\rm{ }}\\{\rm{ }} = \left( {{{\hat \alpha }_0}\cdot\,{{\hat d}_0} + \sum\limits_{n = 1}^\infty {\left( {{{\hat \alpha }_{ - n}}\cdot\;\;{{\hat d}_n} + {{\hat d}_{ - n}}\cdot\;\;{{\hat \alpha }_n}} \right)} } \right)\left| \psi \right\rangle {\rm{ }}{\kern 1pt} \,\\{\rm{ }} = \left( {\left( {\frac{1}{2}{\ell _P}{p^\mu }} \right)\,\cdot\,\left( {\frac{1}{{\sqrt 2 }}{\gamma ^\mu }} \right) + \sum\limits_{n = 1}^\infty {\left( {{{\hat \alpha }_{ - n}}\cdot\;\;{{\hat d}_n} + {{\hat d}_{ - n}}\cdot\;\;{{\hat \alpha }_n}} \right)} } \right)\left| \psi \right\rangle \\{\rm{ }} = \left( {\frac{1}{{2\sqrt 2 }}{\ell _P}{\gamma ^\mu }{p_\mu } + \sum\limits_{n = 1}^\infty {\left( {{{\hat \alpha }_{ - n}}\cdot\;\;{{\hat d}_n} + {{\hat d}_{ - n}}\cdot\;\;{{\hat \alpha }_n}} \right)} } \right)\left| \psi \right\rangle \left( {} \right)\\{\rm{ }} = \left( {\frac{1}{{2\sqrt 2 }}{\ell _P}{\gamma ^\mu }{p_\mu } + \sum\limits_{n = 1}^\infty {\left( {{{\hat \alpha }_{ - n}}\cdot\;\;{{\hat d}_n} + {{\hat d}_{ - n}}\cdot\;\;{{\hat \alpha }_n}} \right)} } \right)\left| \psi \right\rangle \\\left( {\frac{1}{{2\sqrt 2 }}{\ell _P}{\gamma ^\mu }{p_\mu } + \sum\limits_{n = 1}^\infty {\left( {{{\hat \alpha }_{ - n}}\cdot\;\;{{\hat d}_n} + {{\hat d}_{ - n}}\cdot\;\;{{\hat \alpha }_n}} \right)} } \right)\left| \psi \right\rangle = 0\\\left( {{\gamma ^\mu }{p_\mu } + \frac{{2\sqrt 2 }}{{{\ell _P}}}\sum\limits_{n = 1}^\infty {\left( {{{\hat \alpha }_{ - n}}\cdot\;\;{{\hat d}_n} + {{\hat d}_{ - n}}\cdot\;\;{{\hat \alpha }_n}} \right)} } \right)\left| \psi \right\rangle = 0\end{array}

This is the Dirac-Ramond Equation.

In the Ramond sector (still)...

$${\hat L_0}\left| \psi \right\rangle = \hat G_0^2\left| \psi \right\rangle $$=

$${\hat L_0}\left| \psi \right\rangle = \hat G_0^2\left| \psi \right\rangle $$

$$a = 0$$

Now, consider some Level 1 Neveu-Schwarz Spurious State Vector $\left| \varphi \right\rangle = {\hat G_{ - 1/2}}\left| \chi \right\rangle $

$$0 = {\hat G_{1/2}}\left| \chi \right\rangle = {\hat G_{3/2}}\left| \chi \right\rangle = \left( {{{\hat L}_0} - a + \frac{1}{2}} \right)\left| \chi \right\rangle $$

So, $a = \frac{1}{2}$ in the Neveu - Schwarz sector.

Now, we consider a Ramond Spurious State Vector $\left| \varphi \right\rangle = {\hat G_0}{\hat G_{ - 1}}\left| \chi \right\rangle $ ; where ${\hat F_1}\left| \chi \right\rangle = \left( {{{\hat L}_0} + 1} \right)\left| \chi \right\rangle = 0$

$$0 = {\hat L_1}\left| \psi \right\rangle = \left( {\frac{{{{\hat G}_1}}}{2} + {{\hat G}_0}{{\hat L}_1}} \right){\hat G_{ - 1}}\left| \chi \right\rangle = \frac{{D - 10}}{4}\left| \chi \right\rangle $$

Thus, $D=10$.

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@Unknown commenter: Yes, I'll remove the [ ] s. –  Dimensio1n0 Sep 14 '13 at 14:19

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