Bundles
This article is a stub. You can help us by editing this page and expanding it.2009
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
Melanie, youâre one of my favorite stable wave collections co-propagating along the base-space of this twisted chiral Spinor bundle we call reality.
It seems a crime to waste our time discussing âMany-Worldsâ or âString Theoryâ over the geometric beauty of our existence. Thanks!
Gauge Symmetry is essentially the study of horizontal cross-sections to those circles pictured in the GIF under *variable* amounts of rotation of the circles themselves.
Donât know why no one seems to say things like that...but thatâs what it is.
As for the âdefinitionâ given...
1st: The GIF pictured is a bundle, but NOT a vector bundle. It is called a Principal Bundle. If you want a vector bundle think MĂśbius band.
2nd: The horizontal cross section I mentioned are used to create the differential operators they mention.
3rd: The âfunctionsâ which get differentiated by the operators are called âSectionsâ. They are not pictured here.
Hope this helps. But you are looking at an actual gauge theoretic structure. This is the real thing and not an analogy. Thatâs why I use it to explain this all. đ
@katoi In fact it is. This bundle pictured is the 720 degree double cover of the 360 degree regular rotation bundle.
This is the âSpin double cover of the orthonormal frame bundle of the sphere.â
If you will.
@Chrisfalchen That concept of a bundle structure is our most fundamental picture of reality.
@natanlidukhover Circles are 1-dimensional manifolds depicted in 2-dimensional planes. Mathematicians count dimensions differently.
@NoGodOnlyReason Because it is likely the most familiar of any designs that one can put on a round sphere. Itâs merely an aid to show that there is a regular two dimensional sphere in this three dimensional mix.
2021
The Nakamoto Collective is almost the only forward looking thing I can think of. 11 years ago, I was unable to get our Prime Broker to take seriously that a small Hedge Fund wanted to speculate on some new concept. It was so cumbersome that I gave up and wrote an essay instead...
Bitcoin at the time felt totally sketchy as a financial instrument as it was tied to contraband. But I didnât see it as money. If I did, I would be unimaginably wealthy if I didnât lose it all to digital theft, accidental loss or spending it . But I am an idiot in these matters.
The reason I was interested in it was more complex. If Bitcoin was digital gold, and gold was a quantum mechanical wave, then some group had created a:
1) Novel
2) Locally enforced
3) Digital
4) Conservation law
Called the blockchain. And money was but one thing it could be.
Can you imagine. Some group was creating as-if physics inside the network. Bitcoins to me were âwavesâ propagating not in vector bundles, but on networked computers as substrate.
This was genius. I reasoned at the time that it didnât make sense to me as a medium of exchange.
And I hated the blockchain. What the Satoshi collective had done was genius. But there should be no ledger. Gold, as a wave, doesnât tell you where it has been. So instead I dreamed of meeting the Satoshis and getting rid of that damn implementation by using digital bundles.
So, Satoshis if youâre out there, you havenât needed fame or been eager to cash in. That is likely because you get where this is going. Please find me or someone who can explain how AU works as a wave in a bundle. Letâs build a new digital physics around local conservation laws.
I wrote this 11 years ago. I always thought youâd read it & come find me. You werenât rich then. You were either a government project, a collective or a lone genius. But you inspired me like little in our time. This was my attempt to get you to reach out:
Thank you. For everything. And congratulations. Not on your wealth, but for giving us all the means of escape. For creating something truly new. And for having it up for for so long and proving the naysayers wrong. I have no words.
What you created wasnât money but hope. đ
No. Your job is to liberate physics.
Mine, to liberate you. Presumably some of you understand the peril weâre in. We canât stay here.
I didnât mention Geometric Unity to the đ from the early-mid 1980s until 2013. Letâs see if the Satoshi collective can go that long. #TimeToGo
I have contributed nothing to the vision of distributed computing. I donât talk much about the essay even.
Iâm not a Bitcoiner. Keep me apart from that discussion and I will always support you all.
People confuse the new as-if physics for money. So I stayed out to do my vision.
I would however come to any credible meeting about freeing Satoshiâs genius from the loss of anonymity to the ledger that is the blockchain.
Bitcoiners are the logical saviors of physics. And post-Einsteinian space travel and local digital conservation laws are out best hopes.
But Bitcoiners must stop fetishizing the BTC/USD exchange rate and wealth. Set your sights higher. Become the numeraire. Time to fund the worthy outside your communities. There arenât many. Become our adults.
And also look for an update from me around April Fools day. Hopefully.
Itâs time. Put distributed computing on a bundle. Stop worshiping the initial blockchain innovation. Time to move to full digital physics with space-time replaced by the networked computers in distributed computing, and by the Observerse in actual physics. We canât stay here. đ
Iâd propose total reassessment of the National Physics program.
Much greater autonomy for theorists.
*Much* higher salaries.
Much greater *diversity* of approaches.
More high precision work.
Fewer graduate programs.
Physics = economic/security priority.
Admit String Thy failure.
We need to hire people who will upset the living hell out of the people doing the hiring.
We need to put fundamental physics theory in receivership. No theory lead advance in fundamental physics for almost 50 years, yet no soul searching about who lost physics?? Are we kidding?
Itâs time to stop listening to the same voices as if they hadnât failed. This is a national priority, not a cult of personality for a STEM generation that had their time..and then ate their own young across every field. Is no one following what we did to destroy our own capacity?
Â
Â
Or should we do yet more 2D Yang Mills on irrelevant groups in non physical signatures? Squarks/Sleptons? Ha!
Letâs say it clearly as everyone young is terrified to say it: the baby boomer theorists were successful as geometers while avoiding actual physics over entire careers.
By mumbling âQuantum Gravityâ every 2 minutes as a mantra and recasting actual High Energy Physics as âPhenomenologyâ they mis-educated an entire generation to think âtoy physicsâ was real physics. Itâs unbelievable.
Toy physics is real geometry & topology. But it ainât physics.
Real physics:
A) Works with dimension 4.
B) Works with SU(3)xSU(2)xU(1).
C) Uses observed quantum numbers.
D) Accepts Lorentzian Signature.
E) Focuses on 3 Generation.
There is *Nothing* wrong with toy models now and then. But we are talking *entire careers* playing with toys.
We tell people who are basically mathematicians that theyâre physicists.
Well, they arenât. Physicists say things about the world. And those things *need* to be potentially wrong to qualify as physics.
We have a culture of people who canât *afford* error. So they just do math.
Also, to be a fundamental physicist you really should be telling us what we now have wrong. Every advance partially recovers the one before it but also invalidates it, telling us where to look for error. Weâve made hidden assumptions so you have to tell your elders they goofed.
Well, young people canât say that to elders who hold their academic lives in the palms of their hands. Thatâs why young/iconoclast physicists need FU salaries.
Elder âYou should work on AdS/CFT or âBH informationâ if you want to get a job.â
Young Colleague: âHow does NO sound?â
When I say âThere are only two true generations of Fermions.â Iâm potentially wrong.
When I listed quantum numbers of the remaining particles, Iâm potentially wrong.
As when I claimed Pati-Salam is a maximal compact subgroup of the normal bundle of metrics.
Thatâs not a bug.
Yet you canât do this in academic depts.
Moral: we destroyed our ability to self-police. Peer review wonât work. We need to go back to doing physics. Whatâs holding us back may not be physics but the political economy of academic labor, citation, reputation & attribution.đ
One last thought. If there arenât very compelling UAF revelations coming our way, Iâd redirect our interest in aliens towards terrestrial physics done by humans. If there were such revelations, then Iâd *still* look to physics before tech, as đ˝ *still* implies new physics to me.
I feel string theorists know all the things wrong with this statement. Yet you & your community remain silent.
Letâs try it differently: âIf Einstein had never been born, Differential Geometry & Variational Calculus would have found General Relativity anyway.â
How am I wrong?
If Einstein had never been born ... string theory would have found general relativity anyway. The lowest vibrations of the string contain spin-two massless particles (the graviton) which in turn can be used to generate the entire theory of general relativity.
âIf Maxwell and Yang had never been born, Bundle Geometry & Variational Calculus would have found Yang-Mills anyway. If Bohr and Planck had never been born Symplectic Geometry of line bundles would have found quantum theory anyway.â
Again: am I wrong?
Just to finish up for completeness:
âIf Dirac had never been born, Index Theory & Bordism would have found Quantum Field Theory anyway as an enhanced extraordinary cohomology theory.â
Iâm sorry, but all my statements are as or more accurate than what you tweeted.
Why do string theorists pretending to do physics get to BS everyone actually trying to do physics.
We have worked out a world where string theorists and their supporters attack everyone else but say much more outrageous bullshit to the public than any other group by far.
Should we discuss? Perhaps I misunderstood you @michiokaku. But, if so, you are welcome to educate me on my show. But I feel you are *incredibly* aggressive against all non string theorists and you are not comparably challenged by all who know better for reasons I canât fathom.
The proposal to extend the Cost-Of-Living framework to changing ordinal preferences using differential geometry is now up at @uchicago here, as our inflation numbers soar. Thanks to my mathematical colleague @edfrenkel for going over the draft Tues night: https://economics.uchicago.edu/sites/economics.uchicago.edu/files/Welfare_Chicago_Draft.pdf
The paper gives our proposed construction of the solution to the long standing changing preference problem in intertemporal market welfare theory, discussed in the opening quote, via a welfare map through parallel translation in a natural âž-dimensional principal fiber bundle.
Also: thanks to Australian mathematician @Tim_Melon for going over the draft, and offering several ideas for exposition, as well as catching some omissions and symbolic errors in his reading. Much obliged.
Likewise to @BrookeDallas who improved the draft in a careful reading.
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
Â
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.
Â
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.
Â
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. ]