Penrose twistor theory accurately predicts 4 dimensions, contrasting with string theory, which suggests up to 10 dimensions. While string theory can be consistent in fewer dimensions, it requires additional assumptions, such as the minimum number of dimensions being greater than 4. The discussion highlights that anomalies in string theory, particularly in lightcone gauge, necessitate extra dimensions to avoid inconsistencies. Furthermore, twistor theory is applicable not only in 4 dimensions but also in 3, 6, and 10 dimensions, sharing mathematical foundations with the Green-Schwarz superstring theory.
PREREQUISITESThe discussion is beneficial for theoretical physicists, mathematicians interested in quantum field theory, and researchers exploring the implications of dimensionality in fundamental theories of nature.
JorisL said: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.
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.MathematicalPhysicist said:@haushofer doesn't string theory require extra dimensions?
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 said: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.
kodama said:if there are no extra dimensions, doesn't this falsify string theory as a candidate fundamental theory of nature?
JorisL said: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.
I don't know, perhaps only Penrose calls it spin transformation because he likes spinors so much because they are related to his twistors?MathematicalPhysicist said:@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.
SafetyNow said:One reservation I have (which applies to all current theories) is that they are strong on effect but weak on causes.
Is this anomaly can be refuted by experiments?haushofer said: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.
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 said:"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.
Well, I see it like this: gauge symmetry is very useful to introduce coupling. But besides that, it's more of a redundancy.no-ir said: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.