3:29 and it was Einstein's genius to realize 3:31 well they're both right. Maxwell's right 3:34 and Galileo's right. what can what gives 3:37 here and he said well maybe it's the way 3:40 we measure space and time. maybe space 3:43 and time are personal things and they 3:45 depend upon your motion and in order to 3:48 get a measurement so each person's space 3:50 is in some and time in some sense unique 3:52 to them and that was the The Genesis of 3:55 special relativityNo, that was not the genesis of special relativity. That describes what Lorentz published in 1895, and Einstein's paper was not until 1905. Lorentz used Maxwell's equations to show how space and time can change to make observations independent of velocity of the frame, such as with the Michelson-Morley experiment.
Perhaps Krauss would object that Lorentz did not say that the theory is about the way we measure space and time. But Einstein did not either. That was done by Poincare and Minkowski.
there's an 4:21 absolute in the sense that that if you 4:24 think of the world as 4:26 four-dimensional time being an extra 4:28 dimension 4:30 then when I'm moving with respect to you 4:32 what I'm really kind of doing is 4:34 rotating in this four-dimensional space 4:36 so my space is your time and your time 4:38 is my space a little bit and when those 4:40 get mixed up you explain the wonderful 4:43 results of Einstein and so we now say 4:46 that we live in a four-dimensional 4:47 melski spaceThat idea was Poincare's in 1905, and built on by Minkowski in 1907. Einstein did not have anything to do with it.
If instead of living in 6:34 a four-dimensional world we live in a 6:35 five-dimensional world and somehow 6:37 electromagnetism is related to the 6:38 curvature of that extra Dimension that 6:40 you can't perceive well two people uh 6:44 kuta a mathematician and Klein a 6:46 physicist independently in in 1919 to 6:50 1926 came up with the same idea and 6:52 showed that this idea actually worked 6:54 mathematically if you assumed we live in 6:57 a five-dimensional universe and this 6:58 extra Dimension was invisible I and in 7:00 fact curled up on a very small 7:02 scale it was Klein who wanted it curved 7:04 up in a very small scale by the way and 7:07 I don't know if you can figure out why 7:08 kuta didn't didn't care he was a 7:10 mathematician why did he care the clein 7:12 wanted to say if there's an extra 7:13 Dimension if you don't see it there has 7:15 to be a reason and if it's curled up on 7:17 a very small scale then you can't 7:18 measure it in in in in experiments and 7:20 we can talk about that but in any case 7:22 if there was that extra Dimension and 7:24 and you could discuss a curvature in 7:26 that extra Dimension that you couldn't 7:28 directly see it's Remnant in the 7:30 four-dimensional its projection on the 7:32 four-dimensional universe that we can 7:33 see would give the equations of 7:35 electromagnetism it was an remarkable 7:38 idea turned out to be wrong because it 7:41 it also gave a little change to gravity 7:43 which we don't see and it got left asideIt was only wrong because Kaluza and Klein botched it up. Hermann Weyl was ahead of them with a similar idea in 1918, and that idea was essentially the modern gauge theory of electromagnetism. It can be viewed as a fifth dimension of spacetime, and it is not wrong.
He then rambles about string thoery having 22 extra dimensions. That theory really is wrong, or as Peter Woit would say, not even wrong.
You could say that the Standard Model has a group structure U(1)xSU(2)xSU(3) with 12 extra dimensions. We do not see them as spatial dimensions, as they have symmetries such that we only see the curvature effects. In that sense, we do have extra dimensions that are mostly hidden because of symmetries.
Here is Krauss giving a similar explanation of extra dimensions. He describes the extra dimensions as something that theorists have liked since 1919, but which have always failed. Maybe new accelerator experiments will detect extra dimensions.
It baffles me that he can say all this without mentioning that our very best modern Physics theory, the Standard Model, is a theory of extra dimensions. He talks about extra dimensions that are too small or too big or obscured for some other reason. In the Standard Model, the extra dimensions are obscured by gauge symmetries.
Yang and Mills extended Heisenberg's isospin symmetry (SU(2)) to a local symmetry by introducing massless vector bosons to mediate interactions between particles. Their approach was inspired by quantum electrodynamics (QED), where U(1) gauge invariance leads to the electromagnetic field. The original formulation of gauge symmetries by Yang and Mills did not explicitly rely on the geometric concepts of connections, bundles, or curvature. Instead, it was based on extending global symmetries to local symmetries through algebraic and operational principles. It relied on the algebraic structure of Lie groups and their associated Lie algebras. Curvature is just an abstract interpretation. The fundamental fields of the Standard Model (fermion fields, gauge boson fields, etc.) are defined on regular 3+1 dimensional spacetime. The "extra dimensions" or "curvature" in gauge theories are properties of the mathematical description, not of the physical fields themselves.
ReplyDeleteYes, Yang and Mills did not explicitly cite the geometric concepts, but they were being somewhat dishonest. They had learned about gauge theories, and I think they had been told the geometric interpretation.
ReplyDeleteI don't know what you mean by "properties of the mathematical description". The curvature is the physical field. There is no difference. You can argue about whether the fields are real, but that has been accepted since Maxwell. The curvature is equally real.
Also, the extra dimensions of gauge theories are just as real. You seem to be saying that they are mathematical and not physical. They are just as physical as Maxwell's equations.
The Standard Model is formulated on flat spacetime (Minkowski space), and its fields (fermions, gauge bosons, Higgs) are defined in this context. The curvature in gauge theories refers to the field strength of gauge fields, not the curvature of spacetime. The "extra dimensions" in gauge theories are internal dimensions associated with the gauge symmetry group, not physical spacetime dimensions. The physical predictions of the theory (e.g., scattering amplitudes, particle masses, forces) are derived from the behavior of these fields and their interactions, without any need to invoke additional geometric structures as "real." The differential equations themselves are self-contained and do not rely on geometric interpretations to make physical predictions.
DeleteAnd just to show how mathematicians wasted our time, Deser elegantly and naturally derived GR from flatspace spin-2 fields by self-interaction in a few lines back in the 70s. The entire problem with present field theory isn't unification or "quantization" but renormalization because the Wightman axioms stupidly assume operator linearity, which any undergrad engineering student would die laughing seeing it applied to the signal analysis of experiments.
You can ignore geometric interpretations in all of Physics if you want to, but I do not recommend it. The geometry makes everything easier and more intuitive.
DeleteThe extra dimensions in any theory are not physical spacetime dimensions, as spacetime is four dimensions. The curvature of a gauge theory is as real as any other physical field, and comes from those extra dimensions.
You are ignoring geometry. You are obsessing over a META-geometry on top of interacting fields. It's abstract and far from intuitive and misleads about the ontology of the theory. It's like saying math is really about sets. Or is it about homotopy types? Or is it about categories? Math is about tautologies we invent and isn't based on other clunky metaphors. It's all redundant metaphysics.
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