A Penrose twistor theory correctly predicts 4 dimensions

[kodama]

string theory predicts dimensions but predicts 10 dimensions.

Penrose twistor theory correctly predicts 4 dimensions, doesn't this make it more successful?

 

[MathematicalPhysicist]

What other predicitions are made in twistor theory?

In the summer I began reading his treatise with Wolfgang Rindler (the green two volumes) there doesn't seem there something groundbreaking (but I only was reading through chapter one and haven't finished yet, I took hiatus from reading the book).

 

[haushofer]

Actually, string theory doesn't uniquely give the number of spacetime dimensions; it's also consistent in three spacetime dimensions, as far as i can tell.

 

[MathematicalPhysicist]

@haushofer doesn't string theory require extra dimensions?

 

[JorisL]

You can find out why string theories we use have extra dimensions in a day or two.(Depending on your level of physics education/knowledge)
I liked the treatment in Becker, Becker and Schwarz's book. But I think you can also find it in Zwiebach as well.

In short they are required to remove the central charge from the virasoro algebra that naturally pops up when you look at strings.

 

[kodama]

You can find out why string theories we use have extra dimensions in a day or two.(Depending on your level of physics education/knowledge)
I liked the treatment in Becker, Becker and Schwarz's book. But I think you can also find it in Zwiebach as well.

In short they are required to remove the central charge from the virasoro algebra that naturally pops up when you look at strings.

if there are no extra dimensions, doesn't this falsify string theory as a candidate fundamental theory of nature?

 

[haushofer]

@haushofer doesn't string theory require extra dimensions?

In lightcone gauge you need the Lorentz algebra still to be satisfied, which imposes constraints on the mass level parameter a and spacetime dimensions in order to avoid anomalies. But in 2+1 dimensions this anomaly is automatically avoided. I've never understood why people say string theory 'uniquely' predicts the number of spacetime dimensions: it doesn't. It does with the extra assumption that this number shouldn't be less than 4, but that's extra input.

 

[haushofer]

See e.g. http://arxiv.org/pdf/1106.1374.pdf

 

[MathematicalPhysicist]

In lightcone gauge you need the Lorentz algebra still to be satisfied, which imposes constraints on the mass level parameter a and spacetime dimensions in order to avoid anomalies. But in 2+1 dimensions this anomaly is automatically avoided. I've never understood why people say string theory 'uniquely' predicts the number of spacetime dimensions: it doesn't. It does with the extra assumption that this number shouldn't be less than 4, but that's extra input.

Ok, so the extra assumption that the number of dimensions should be greater than 4 is still an extra dimensions requirement of string theory. So is string theory also developped in 3+1 dimensions?

 

[haushofer]

I don't think so, because then the anomaly can't be avoided as far as i can tell. I'm not sure about e.g. 2+2 dimensions.

 

[JorisL]

if there are no extra dimensions, doesn't this falsify string theory as a candidate fundamental theory of nature?

If we can prove with 100% certainty that there are no small extra dimensions?

I suppose that would be pretty bad for string theory research. (understatement)

However it wouldn't invalidate all we did so far.
For example AdS/CFT has been succesfully used in analysing heavy-ion collisions.
Even though we believe our universe is de Sitter it improved our understanding of the measurements.

 

[kodama]

If we can prove with 100% certainty that there are no small extra dimensions?

I suppose that would be pretty bad for string theory research. (understatement)

However it wouldn't invalidate all we did so far.
For example AdS/CFT has been succesfully used in analysing heavy-ion collisions.
Even though we believe our universe is de Sitter it improved our understanding of the measurements.

there's this

Dark Energy, Inflation and Extra Dimensions
Paul J. Steinhardt, Daniel Wesley
(Submitted on 11 Nov 2008 (v1), last revised 7 Dec 2008 (this version, v2))
We consider how accelerated expansion, whether due to inflation or dark energy, imposes strong constraints on fundamental theories obtained by compactification from higher dimensions. For theories that obey the null energy condition (NEC), we find that inflationary cosmology is impossible for a wide range of compactifications; and a dark energy phase consistent with observations is only possible if both Newton's gravitational constant and the dark energy equation-of-state vary with time. If the theory violates the NEC, inflation and dark energy are only possible if the NEC-violating elements are inhomogeneously distributed in thecompact dimensions and vary with time in precise synchrony with the matter and energy density in the non-compact dimensions. Although our proofs are derived assuming general relativity applies in both four and higher dimensions and certain forms of metrics, we argue that similar constraints must apply for more general compactifications.
Comments: 26 pages, 1 figure. v2: reference added, typos corrected
Subjects: High Energy Physics - Theory (hep-th)
Journal reference: Phys.Rev.D79:104026,2009
DOI: http://arxiv.org/ct?url=http%3A%2F%2Fdx.doi.org%2F10%252E1103%2FPhysRevD%252E79%252E104026&v=5096d5ba
Report number: DAMTP-2008-104
Cite as: arXiv:0811.1614 [hep-th]

 

[MathematicalPhysicist]

@Demystifier I see you liked my post; from my memory I do remember that in the first volume of Rindler's and Penrose's book they call the following transformation $M(z) = (az+b)/(cz+d) \ z\in \mathbb{C}$ for $ad-bc=1$, "Spin Transformation", where if I recall correctly from books in pure maths and from courses from the mathematics department it's called "Moebius Transformation".

I wonder why different names to the same thing, I gather mathematical physicists and pure mathematician don't share the same terminology.

 

[Demystifier]

@Demystifier I see you liked my post; from my memory I do remember that in the first volume of Rindler's and Penrose's book they call the following transformation $M(z) = (az+b)/(cz+d) \ z\in \mathbb{C}$ for $ad-bc=1$, "Spin Transformation", where if I recall correctly from books in pure maths and from courses from the mathematics department it's called "Moebius Transformation".

I wonder why different names to the same thing, I gather mathematical physicists and pure mathematician don't share the same terminology.

I don't know, perhaps only Penrose calls it spin transformation because he likes spinors so much because they are related to his twistors?

 

[weirdoguy]

One reservation I have (which applies to all current theories) is that they are strong on effect but weak on causes.

And it will never change because everytime you find a cause it raises a question about causes of that cause. It will never stop, so at one point you have to say "it is the way it is" and move on.

 

[MathematicalPhysicist]

I don't think so, because then the anomaly can't be avoided as far as i can tell. I'm not sure about e.g. 2+2 dimensions.

Is this anomaly can be refuted by experiments?

I mean there quite a lot of anomalies in nature, isn't "life" such an anomaly?

 

[haushofer]

"Anomaly" here means a gauge symmetry which is threatened to be broken by the quantization procedure. That's a problem because in the usual quantization procedure one does not want to change the amount of degrees of freedom. E.g., a classical massless vector field has two polarization states, which one wants to keep upon quantization. This means your quantization shouldn't break the U(1) gauge symmetry.

 

[no-ir]

"Anomaly" here means a gauge symmetry which is threatened to be broken by the quantization procedure. That's a problem because in the usual quantization procedure one does not want to change the amount of degrees of freedom. E.g., a classical massless vector field has two polarization states, which one wants to keep upon quantization. This means your quantization shouldn't break the U(1) gauge symmetry.

I know things are more complicated than this, but symmetry breaking is not something which is avoided at all costs in other parts of physics. Why would gauge anomalies be undesirable? Perhaps the broken-symmetry theory just is the true quantized theory, or perhaps you started from the wrong assumption of what to quantize to get the desired symmetries in the end, and that's all there is to it? I am sure I am missing something here, though.

 

[haushofer]

I know things are more complicated than this, but symmetry breaking is not something which is avoided at all costs in other parts of physics. Why would gauge anomalies be undesirable? Perhaps the broken-symmetry theory just is the true quantized theory, or perhaps you started from the wrong assumption of what to quantize to get the desired symmetries in the end, and that's all there is to it? I am sure I am missing something here, though.

Well, I see it like this: gauge symmetry is very useful to introduce coupling. But besides that, it's more of a redundancy.

E.g., take the photon. We measure it has two polarizations, and from QFT we think it is massless. So two on-shell degrees of freedom (dof) it is. Next we try to pack these into a representation of the Lorentz group. The smallest possibility is the real vector representation, but this gives us two degrees of freedom too much. This is where gauge symmetry kicks in: it enables us to write down the photon field and its dynamics in a manifestly Lorentz-covariant way.

From that point of view it would be weird (apart from being mathematically inconsistent!) if suddenly, upon quantization, we obtain extra dof's again because the gauge symmetry is lost.

The same goes for the world-sheet of a string. Conformal symmetry is used to rewrite the Nambu-Goto action into the Poyakov action by using Weyl rescalings. If these would be broken by gauge anomalies, that would mean the introduction of an extra dof.

 

[Urs Schreiber]

The following is well known but keeps being underappreciated:

1) Twistor theory exists not just in dimension 4, but also in dimensions 3 and 6 and to some extent in dimension 10.

Ingemar Bengtsson, Martin Cederwall,
"Particles, Twistors and the Division Algebras", Nucl.Phys. B302 (1988) 81-103
http://inspirehep.net/record/247269

Edward Witten,
"Twistor-like transform in ten dimensions",
Nuclear Physics,
Section B, Volume 266, Issue 2, p. 245-264. (1986) 10.
http://dx.doi.org/10.1016/0550-3213(86)90090-8

2) The reason "twistors work" is the fact that in these dimensions there is a magical coincidence by which a) Minkowski spacetime is identified with 2x2 hermitian matrices with entries in the complex numbers (for 4d) or real numbers (for 3d) or quaternions (for 6d), or octonions (for 10d): the generalized Pauli matrices. Moreover, in these dimensions the spin group happens to be isomorphic to the special linear group on two entries with coefficients in this number system.

For more on how this work see at "Twistor space" here

3) These algebraic facts that make twistors work in dimensions 3,4 6 and to some extent in dimension 10 are precisely the same algebraic facts that make the Green-Schwarz super-string work in these dimensions. See at division algebras and supersymmetry for more on this.So the mathematics that makes twistors work, is just the same mathematics that makes the Green-Schwarz superstring work. Both theories agree on which spacetime dimensions are possible.