Bundles
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Meanwhile, as for the Euler Class, we often meet it as a *top* class for the tangent bundle thereby prohibiting seeing it as a square root.
Additionally, Vilfredo Pareto's move towards ordinal utility can be seen as imparting a non-abelian bundle structure to welfare.
2010
Note to Geometers: A depiction of a fiber bundle is shared by both the US Senate seal and the Fascist Flag.
Odd, that.
GU: Don't conflate Spin 0 fields valued in the adjoint bundle / non-linear sigma models w/ higgs at LHC. Nature uses Spin 0 alternatively.
The definition of "tangent bundle" is a good example of how mathematical precision makes even the visual incomprehensible.
The definition of 'line bundle' is a good example of how mathematical precision makes even the incomprehensible physics 'anomaly' visual.
2018
@mkealz @jisheppard So far as we know, the universe we live in is generated by a principal fiber bundle. This is essentially the only visual mature example we have to show you one without equations. I didnât explain it. I pointed to it.
If someone I trusted said these words to me, Iâd invest in it.
2019
@GTAlien @SamHarrisOrg Love Toronto. Iâd do a gig there in a heartbeat if I could bundle it with a trip to @Perimeter and/or @FieldsInstitute to geek out.
My personal & overly condensed view of mathematics and physics in the 20th century would be summarized like this.
Mathematics began as a stool on the three legs of Algebra, Calculus, and Geometry where the last appeared to many to be the weakest leg. It turned out otherwise.
Repeatedly we find that any important problem from math or physics which we consider to be outside geometry/topology has a hidden geometrical nature to it. And there are only so many times you fall for that before you start to see geometry absolutely everywhere.
As for Weinberg, he is one of three people I can make the case for as our âGreatest Living Physicistâ. Iâve met him. But he still has big bets which are undecided (e.g. asymptotic safety). Witten is somehow even smarter but less accomplished in standard predictive theory. But...
I would say the one who awes me most is...CN Yang. I donât understand why I never hear his name as candidate. He has at least 3 of the greatest achievements: chirality for the weak force (w/ Lee), non-Abelian maxwell theory (w/ Mills), and the bundle revolution (w/ Simons/Wu).
2020
2021
2022
Huh. Letâs seeâŚ
Standard Model: Fiber Bundle
General Relativity: Fiber Bundle
Our universe: Derived from SM+GR
SoâŚuhâŚyeah. So far. Crazy right?
Weird flex, but it checked out.
In essence this is happening every time âyouâ move. When you see spectators doing âThe Waveâ the spectators are the medium. They donât move with the wave.
You are a wave. You excite a totally different portion of the medium wherever you go. That medium is called a vector bundle.
@CreatedInTheD The atom moves through space. But as a wave. If a wave moves through a small oil slick, the oil slick doesnât move with the wave. It briefly rises & falls in place when excited. The medium doesnât move. The thing that moves is the atom. The thing that stays is the Vector bundle.
A surprisingly deep simple question.
There appears to be a mysterious circle at every point in spacetime which physicists accept but cannot explain. And, every type of particle is endowed w/ a mysterious complementary âď¸. The spacetime âď¸ rotates the particleâs sympathetically.
The charge on the particle is the gearing ratio of the spacetime âď¸ with the particleâs âď¸. Itâs like a bicycle where the pedal gearâď¸ is the spacetime âď¸ and the particle âď¸ is the rear wheel âď¸. Positive charge is clockwise drive. Negative charge is counterclockwise.
An electrically neutral particle is like a particle not having a chain hooked up between the pedal and wheel. So a +2/3 Up Quark will be driven around 2 times clockwise for every three times an electron goes counter-clockwise with charge -1=-3/3.
That may sound weird. So be it.
@TEMguru That U(1) is the circle at every point in space time. Itâs minimal gauge coupling via a character is the chain between the gears. Câmon.
Uh. Thatâs *exactly* how itâs done. There is a principal U(1) (circle) bundle. But it isnât the U(1) that you refer to which is weak-hypercharge. And the analogy makes perfect sense based on internal quantum number
\chi_n:U(1) â> Aut(C)
before tensoring with the spinor bundles.
Let me just say that there is a community of academics who throw a lot of nasty anti-collegial scientific shade that just isnât scientifically accurate. Donât know what to do about that. These people try to cast a spell of Fear Uncertainty and Doubt.
I stand by what I say here.
@sluitel34 Let me help you then. You have a group:
G=SU(3) x SU(2) x U(1)
And a homomorphism:
rho: G â> U(16)
So
Spin(1,3) x G â> SL(2,C) x U(16)
represents on C^2 tensor C^16, and its conjugate, to give one generation of the Fermions (with Right handed neutrinos assumed). With me?
@sluitel34 Now the U(1) âď¸ of the original description lives inside the SU(2) x U(1) via bundle reduction or symmetry breaking as you see fit. The gearing ratio I mentioned is simply the integer indexing all irreducible representations of U(1) which are all 1-dimensional characters. Clear?
@sluitel34 Every U(1) character can be visualized as two circular gears connected by a chain with some integer ratio of the circumferences. Negative integer representations are ones with the chain having a half twist. The trivial representation has no chain at all.
Hope that helps.
@sluitel34 @FrankWilczek Not true at all. @FrankWilczek correctly points out that there is something super compelling about SO(10) Grand Unified Theory. Both space time and internal representations are spinorial if this is true.
I just donât know from what position youâre speaking so authoritatively.
@sluitel34 @FrankWilczek This should be in any book that discusses the standard model via groups, representations, bundles, etc.
@WKCosmo @PasseVivant Itâs a decent first answer for dynamics as in Hamiltonian systems. But there are a lot of places where symmetries intrude where that simple answer seems less convincing. Principal bundle structure groups for example. Or discrete symmetries. Etc. Etc.
@WKCosmo @PasseVivant Uh, no. Is âStructure group of a principal bundleâ or âDiscrete groupâ buzzwords to you? That doesnât sound like a physicist to me.
Sorry. Iâll move on. I thought this was a Professional conversation. Be well. Bye.
According to physics, youâre a wave. A conscious wave.
As a conscious wave, you were curious as a child. The most natural question for a conscious wave is probably âIf Iâm but a conscious wave, in what medium am I an excitation?â
Yet most waves never ask this question.
Why? đ
The short answer is âYou appear to be a wave in a structure called a Fiber Bundle.â of which many have never heard.
I talk about Fiber Bundles a lot because they appear to underlie all of existence, and am thus very confused by physicists who donât discuss them. Itâs so odd.
For years this has been the leading image of a fiber bundle on Google Image search. This I take as proof that the human race is slightly insane: Our leading image of the underlying medium of existence itself looks to me like a bandaid/plaster that has been ripped off a hairy arm.
We created this picture so that you would have a picture of what a âFiber Bundle with Gauge Potentialâ actually is. So that everyone could see in what type of structure they actually vibrate.
So far as I know, this is the only animation of its kind:
Would love to get back to explaining things about the true wonder of our existence.
If you are fascinated by Entanglement, Quantum Weirdness, Relativity Theory, The Multiverse, String Theory etc, most of you would be better served studying fiber bundles:
@McLuhanStates @LueElizondo There is a lot of loose talk about dimensionality. Keep in mind that I have zero direct evidence of the phenomena. So this is wildly premature.
My interest here is that GU replaces one manifold with two in a bundle structure and adds BOTH temporal and spatial dimensions.
Q5: So letâs see. Inflation is a field like temperature. But a field in a fiber bundle over âž-dimensional path spaces of loops of preferences/prices valued in non-commuting groups leading to non linearities not addressed by economists? What about actual geography!â
A5: Fair. đ
2023
2024
Buckaroo?
Ok. ButâŚ.Also a laypersonâs term for flatland viewing an ambient space for an embedded/immersed sub-manifold. Also for a Kaluza Klein theory or a general fiber bundle projection. Or for worm holes and non trivial topology. Etc
In other words, it means almost nothing.
I'm confused. This lecture doesn't negate the geometric foundations of GR. Einstein differentiates between how gravity and electromagnetism relate to the structure of space, all the while pointing to his ultimate goal of unification. As for the rest of the original article linked, I'm unsure how the quotes from Einstein support the author's title. GR is indeed a geometric theory; however, Einstein's viewpoint was that its geometric nature doesn't singularly distinguish it from the broader domain of physics, where geometry has always played a fundamental role. If anything, Einstein is saying not to confuse the map with the territory.
He is correctly anticipating the Simons-Yang discovery of the âWu Yang dictionaryâ.
Maxwell became Yang Mills Yang Mills became Simons Yang. Simons Yang became the Wu Yang Dictionary. Wu Yang was (except for one entry) was Ehressmann fiber bundle geometry.
Think of metric geometry, fiber geometry and symplectic geometry as the geometry of symmetric metric 2-tensors, fiber bundle connections and anti-symmetric 2 tensors respectively.
[Note for Curt: This is the whole point of Geometric Unity. They are three geometries. Which are all one geometry, and that is only possible in the rarest of circumstances. Which we are in oddly.
Metric Geometry: General Relativity GR Fiber Geometry: Standard Model SM Symplectic Geometry: Hamiltonian Quantization of the SM. ]