Take rod of metal with temperature distribution in equilibrium. Place lots of thermometers along the rod, they should all measure the same temperature. But, each thermometer has a different measuring units, they have a different and local gauge. So, if you plot the number each thermometer measures in a graph, it would NOT be a straight line. Someone who did not know they were all measuring in different local gauges, different local units, would not know the rod was in equilibrium temperature. They would think the temperature of the rod is different, wherever each thermometer is at. A bad derivative such as dT/dX would not be zero because the numbers are changing (only because we placed Thermometers with different units). This derivative is NOT good because we know the temperature is constant throughout the rod, so our derivative should keep track of the real, physical temperature. So we introduce into the derivative, a term which keeps track of each thermometers unit (how each of them gauges temperature). This new derivative will now be completely independent of whatever units you pick for any thermometer you place on that rod, the derivative will tell you the TRUE temperature distribution of the rod, no matter whether each thermometer is using individually different units. This is one way I like to think of gauge theory. Notice how symetry in this system, with this new derivative is now local. Meaning you can take one individual thermometer, change it's units, but your theory will not change based on this local unit transformation. This is local gauge invariance = system with local symmetry.
That's a pretty good analogy, but definitely not complete. In gague theories the rotor is non-vanishing, so you can't really think about thermometers with some absolute calibration. Maldacena gave a wonderful monetary analogy that I believe is superior. If you have time check out his talk: th-cam.com/video/OQF7kkWjVWM/w-d-xo.html
Exactly. Poetry is an art form of communication. There is no perfect poetry but if there was it would be understood by everyone. Also every language would have a different "perfect poem"
My favourite way of thinking about Gauge Symmetry is Fiber Bundles formalism. Local symmetry is then just invariance of laws of physics under reparametrisation of fibres. It's just symmetry that says that "coordinates are not physical" just as in General Relativity for example. Coordinates are irrelevant only underlining geometric structure matters.
Different people have different ways to grasp a concept. Some may find a rigorous analytical formalism more on-point, but in the end a visualisation always helps us appreciate what the theory truly stands for
Topologists love to draw pictures but the subject has to be dense so that visual ideas can be formalized and generalized to apply to as many situations as possible in the same way. If we only work with pictures then we have to treat every object like a torus, sphere, Möbius strip, etc. individually; topology gives us a single language for studying all shapes (and more exotic structures) at once
Cao For other fundamental interactions, which, it is believed, can be described as gauge interactions, we find that the theoretical structures of the corresponding theories are exactly parallel to that of gravity. There are internal symmetry spaces: phase space, for electromagnetism, which looks like a circle.
Lorentz symmetry is not usually what one thinks of when talking about gauge symmetries, at least not in the standard model of particle physics. In the standard model, the fundamental symmetries are spacetime symmetry (that is the Lorentz symmetry discussed here, which is a non-compact group) which acts on space and time globally (ignoring gravity), and then there is gauge symmetry (compact Lie groups/algebras) which act locally on the equations that describe the dynamics of all the known particles (the Lagrangian). Because the gauge symmetry is a symmetry of the equations, there isn't an easy way to understand what the gauge symmetry physically represents, although attempts have been made for example, as the curvature from higher dimensional compactified spaces ala Kaluza and Klein etc. As for a simple example of how to conceptualize gauge covariant derivatives more generally, I think Ashley Cotrell's example in these comments nails it. For completeness, I should mention that there is a formulation of general relativity using vielbiens where one can actually think of the Lorentz symmetry as a gauge.
Atiyah: As far as gravity is concerned, Einstein’s General Relativity is a beautiful and complete theory. But as Einstein realized it has to be extended to account for other physical forces, the most notable being electromagnetism. It is perhaps no accident that the first and most significant step in this direction was taken by a mathematician - Hermann Weyl. He showed that, by adding a fifth dimension, electromagnetism could also be interpreted as curvature. His idea was that the size of a particle could alter as it passed through an electromagnetic field. In analogy with railways it was called a gauge theory, and this name has stuck through subsequent evolutions of the theory. Unfortunately for Weyl, Einstein immediately objected on physical grounds that this would have meant different atoms of, say hydrogen, would have different sizes depending on their past history, in contradiction with observation. Given this devastating critique, it is remarkable but fortunate that Weyl’s paper was still published, with Einstein’s objection as an appendix. Clearly the beauty of the idea attracted the editor, despite the fatal flaw. In fact, beauty often wins such contests, because with the advent of quantum mechanics, with its complex wave functions, it was pointed out by Kaluza and Klein that Weyl’s gauge theory could be salvaged if one interpreted the variable as a phase rather than a length. A pure phase shift by itself is not physically observable and so Weyl’s theory avoids the Einstein objection.
This was very interesting. Visualisation is definitely an easy way into a subject for most people (including myself). Once you've been doing something for while it becomes more and more abstract and subconscious so you can't explain in a way that makes sense to noobies. Personally I agree with Lex that building bridges is one of the most interesting things you can do.
I love the fact he brings up Lorentz. Pull up any video about 'Einstein' on TH-cam and chances are they'll examine concepts that were originally conceived by Lorentz rather than Einstein
Tbh mathematicians like Lorentz, Riemann and Hilbert have done more to GR than Einstein. Einstein just had the spot of piecing it all together and giving it a physical explanation.
I feel that the term "gauge" (for one reason or another) has the unfortunate quality of obscuring an idea that most people have already come across in an undergraduate linear algebra course. Geometric objects, whether physical or abstract, are often poked and prodded with respect to a coordinate system, i.e. with respect to a basis of some vector space. Simply put, a gauge is just a choice of basis. The problem is that our choice of basis is arbitrary, e.g. think of two different observers looking at the same vector from different angles. We say that an object or quantity is gauge invariant (or admits a gauge symmetry) if its existence is independent of the gauge. Intuitively, a vector (and thus positions, velocities, etc.) should be an example of such an object since bases are typically introduced after the definition of a vector space! The remaining details (e.g. thinking of the collection of transformations between gauges as a group, generalizing to manifolds and their tangent bundles, etc.) are just technical bells and whistles that allow us to speak more generally about gauge invariant objects on arbitrary geometric spaces (such as the sphere or a blob), but they shouldn't distract you from the essential notion of an object existing independently of how I choose to describe it.
This isn't right, it's not a "choice of coordinates" thing, it's a "symmetries" thing. The appropriate language to describe it by the actions of Lie groups on a manifold. "Gauge groups" are the structure groups of principal bundles. It's complicated because it is, and for no other reason. It's an elegant and beautiful theory, but it's a graduate level+ thing that I'm sure very few people have successfully rigorously self-taught to the point where they can solve exercises successfully. TLDR: bundles have coordinates (but not a basis, they aren't vector bundles), and the nature of the bundle is independent of choice of coordinates, but that choice is not what a "gauge" is.
We were forcing thermal phenomena in the last Hilbert confined entropy regulated gauge group conformations that Alan Guth said forget it, it's like trying to not go back at near the speed of light which at least sees our mass in the effort
Dirac: Perhaps I should say at this point, to give you some physical insight into all this, that we shall later be applying this theory to electrodynamics and we shall then find that the quality of being physical is just the quality of being gauge-invariant.
Wow, this was a great discussion. Both people were saying new things and adding something to the space, which is different from most postcasts i sometimes see. Would like to hear more from blue shirt guy.
I’m a little sad this didn’t help me find clarity about how gauge symmetry really works. The best I’ve got is that there’s a sense in which a circle, which stretches out in the complex plane (I think?), can be mapped onto space and becomes a rigorous way to describe electromagnetic fields… Hope one of these physicists who keep coming by Lex’s studio finds a way to describe this simply soon. Eric Weinstein specializes in rendering statements of obscure genius with few footholds for people who aren’t specialists… That’s not fair… I just have no idea what he’s talking about as soon as he says “gauge theory.”
Shortest answer I can give is, consider normal transformations, like rotation or translation. This just changes the global way you look at something. Now imagine this would be possible locally. If you try this, something goes wrong, a certain quantity (the action) is changed under these local transformations, and physicists agree it shouldn't (for certain transformations, not translation and rotation, but for example phase change, a quantum mechanical thing). To fix this, we change our notion of change, the derivative, to the covariant derivative. Covariant in this context just means that it has a term that counteracts the transformation. With this notion of the covariant derivative (which generalizes to connections on principal bundles but forget that unless you're in for a deep dive), we get certain relations between different fields (which represent particles) by looking at how they can change while keeping this action constant under these transformations, these interactions can for instance be between photons and electrons. But more generally, with more complicated gauge groups (or Lie groups acting on the principal bundles from earlier), aka local transformations, we get more exotic particles/forces. At this point in time, we've made a theory that combines a whole lot of different symmetries (U(1), SU(2), SU(3)) together into a model that describes the interaction of all the forces/particles we know, except gravity. This exception is the root of many fancy new physics theories, like supersymmetry and string theory. But so far, we can only combine all the other forces in what's called the standard model, which is a gauge theory. Hope this helps :)
Neural networks are beautiful when represented. I think graphical representations are extremely powerful, for example in quantum computing. If we didn't have those diagrams of wires and boxes, I'm not sure we would all be able to understand a quantum computer program.
I’m with you on this, Lex❣️ I think it would be much more exciting & engaging to use the visual representations of these abstractions in conjunction with the mathematical formulae & algorithms, especially for visual people like myself, who use TH-cam specifically for the added information & understanding that video, as a visual medium, provides‼️☑️🎯🙏🏻🤴🏻🗽👼🏻
The problem is that no matter how much someone is a visual person (I am and used to try an approach math that way, I still do sometimes out of habit) the visualizations simply can't represent the real marvel of these structures, or even what's actually going on and when you really dive into things, that becomes apparent and they stop being interesting very quickly. Math is the way it is for a reason and it isn't (always) a matter of preference -- the notation is just a language that compresses concepts into efficiently communicable convention. That hurdle is what everyone spends time talking about because it's the first one and most people fail to traverse it, but the difficult part is actually in the concepts themselves and they're difficult for the same reason they can't be visualized. The only way to really understand them deeply enough to understand their beauty is to get into the weeds and interact with the abstractions themselves. There's more than one way to skin that cat, but the unfortunate reality is that no matter which way you choose, it's eventually going to require a knife if you want to do it properly not waste a whole lot of time accomplishing very little.
I‘m half way down the video and no word of Noether? Explaining gauge theories without any mentioning of charge, topology or Noether is weird …. (I’m a physicist). But yes Lex is right, visualizing stuff is really important. The physicist in the video is a typical „algebraic-conceptional“ thinker: he understands concepts well, but visualizing is not his mode of thinking. This is one of the biggest divider in phycisitsts, there are visualizers, intuitionists, and conceptional thinkers (or any combination of the above). It does make a difference to visualize, many peoples mode of thinking is that; in fact it’s one of the best skills as a physicist, many physicists struggling with high level abstract/dense math can mitigate many shortcomings by visualizing. There is a lack of important visualizations, but the reason is quite straight forward: mathematicians often do not care about stuff like this. They want to prove, they want to relate abstract structures to others and generalize it to any usecases. Physicists are different, using their lacking knowledge and intuition to find one detailed, explicit solution to a hard problem. This is the exact reason why mathematical physicists make good theory confirmers, but really bad physicists in general. Some things get lost in expressive rigorous mathematics. It’s even worse for mathematicians, they are really bad physicists (I’m generalizing a lot, but you get my point). Physics is some kind of game, between lack of knowledge, intuition and useful handwavy-ness. When mathematicians say: Physics is just bad/not rigorous math, it’s a really dumb statement, misunderstanding the multiplicity of information that needs to be condensed. Condensed matter physics is the perfect example, you can never describe a material fully. It’s not possible. But with the right set of ideas, stuff can be broken down to smaller pieces and with good approximations, models can be constructed. By time, the approximations ALWAYS break down (due to new measurements or general disagreement with certain parameters from the get go), and the job then becomes to find new approximations, which will always lead to new dynamics and in the best case becomes the old model again when tuning a set of parameters to extreme values (this is called an „effective“ theory, and object being effectively described by something easier, ignoring complex stuff, i.e. „dismissing friction“ would be an effective dynamic, in the real model, one needs to take friction into account). In a more general sense, physics is like the interplay between „distance“ (from the truth) and „precision. The more distant you are (more „effectively“ describe your thing), there less precise you are but you get an nice general understanding, i.e. take newton mechanics, but the more precise you get, the harder your problems become but also the closer it is to the truth. So in a sense, even better approximation gets us closer to the truth but it’s harder to do.
A mobius strip is a representation of a one sided object but a truly one sided object is impossible to go around. It's kind of a weird thought but I wonder if something can be so small that it only has one side and is impossible to go around.
I'm not certain by any means, but I dont think it's a matter of size. For an object to be one sided my guess is that it would have to have one side that existed in the physical realm but while the other "side" was immaterial. Picture walking up to the front of a prop house made for a motion picture of some sort. You look at it, touch it, notice its colors, textures, scents, and then you walk around to the other side, but as soon as you get to an angle perpendicular to the face of the house it vanishes because it'd have no thickness. You go completely to the back, turn so that you're facing the direction you previously confirmed the house to be in, and you'd see right through where it was. You'd pass your hand through where you thought it should be and feel nothing. Finally you'd walk straight through where the house must have been without any resistance, make a 180 degree turn, and see the front of the house staring back at you again.
I'm horrible at mathematics and algebra but I'm very visual and got an A+ in geometry in one of the hardest private schools in the US in which i got a 2.0 GPA. Then I got into psychedelics heavily and saw sacred geometry and all these higher dimensional shapes and now I'm falling in love with physics. Now I actually understand the hopf fibration, hilbert curves, other space filling curves, platonic solids and finding passion for physics and math by going the opposite of the traditional trajectory. It's all linked and it's crazy how psychedelics can actually show us visual learners the inherent higher dimensions of physical reality. What does that say about psychedelics specifically dmt? Cause I've seen hopf vibration and hilbrrt space geometry long before I ever saw or knew of those and I've seen sierpinski triangles on lsd long before ever seeing as much as a picture of all this?
The Hidden variables of Pandora's box: you might not want to find out Stephen Hawking's joke about black holes not being sexual has mined this anti-logical and yet even more intelligent truth
I was taught that gauge theory has to do with the invariance of a Lagrangian action with respect to a local change of coordinates or other parameters in the Lagrangian. I'm not hearing the same simplicity expressed in this talk.
Naked singularly can surely be reached by trying to strip your Tableau of factors through complexity, complexity simplifying, frozen phenomena becoming redressed in a new bizarre orange lattice lab synthesized at Southern Oregon University 🎓
People get confused by gauge theory because they are presented the idea as if it is a single construct with one specific, right way of cracking the code to understanding a new field of math on its own. In effect, gauge theory is a way of thinking about mathematics as it applies to dynamical systems as a whole. Gauge theory is made up by the combination of most fields of math appearing in physics as an attempt to see the relationship they all share. This allows us to simplify things and filter out a lot of noise. In understanding it this way, it's not just an application of finding symmetries as they correlate with conservation (applying Noether's theorem already gets you there)... it's also two other fundamental things: 1. It tells us that it doesn't matter how we write out our physics. If two ways of describing a system work with equal accuracy, then those two ways of describing the system are equivalent. It means that if a hypothetical alien species that doesn't understand our construct of linguistics start deriving math, we can predict where our predictions will match no matter what happens due to the inherent ratios and constants we measure in nature. We will be forced to come to a similar conclusion to be accurate, even if the interpretation or method of getting there is entirely foreign. It gets us as close as possible to "why" these systems work when the most we can approximate is "how" they work. 2. It thus allows for a more rigorous way of developing a philosophy of physics for which all the underlying mathematical fields and axioms play a role in coming together to yield a result that can be explained with various, equivalent analogies. The primary fields of math you will find in a focused study of gauge theory are typically Lie algebras (and their groups), differential geometry, spinor analysis, and (increasingly more common) geometric/Clifford algebras. It gets even wilder when you begin to see gauge theory through the lens of vertex algebras and W-algebras (which allow us to jump around between quantum and classical field theories on the fly utilizing structures that coexist in those theories). These things make up a vast sum of the math in physics when you think of what composes them. The issue in all this? Two authors writing a book on mathematical gauge theory are unlikely to focus on remotely the same topics or goals, and will likely end up writing two books that feature entirely different content. Neither of those authors are then likely to even use the word "gauge theory" in any mathematical sense beyond the cover that they use to catch your attention. Reading gauge theory books for me has been closer to the experience of reading more rigorous physics texts where you can't help but take a deep dive into how that specific author engages with mathematics as a whole. It's like reading a philosophy of physics book, except the author isn't afraid to throw in the most advanced tools we know of in mathematics across every page alongside proofs because they are often self-aware that these are not trivial topics. Reading gauge theory textbooks has been a very different experience from many others, as you can at least find a dozen or more calculus textbooks that cover more or less the same material with a similar approach besides a few quirks. You gain a deeper appreciation for mathematical physics when you realize the reason why gauge theory can't be sufficiently summarized by reading a text by one or two authors on it (despite aspects of it clearly existing for over 100 years now). It's a journey of discovering how you recognize mathematical patterns and use them to find connections across the field.
@@freniisammii It provides a unified foundation for zeros, complex numbers, quaternions, 3D rotation, linear and bilinear algebra, exterior forms algebra, >3D spaces, spacetime algebras, lie algebras, quantum logic lattice, differential geometry, conformal and projective spaces, group representations, and combinatoric hypergraphs.
@@RecursionIs am a professional mathematician. Few concepts are as well loved by crackpots as "geometric algebra". Geometric algebra is not even close to being a framework for understanding all of "differential geometry". It's just linear algebra/module theory in a different language, nothing new is added, and there is not really a compelling reason to use this different language when the standard abstract language contains the same information, and is understood by everyone. For gauge theory, you need smooth manifolds, Riemannian manifolds and the Covariant derivative, fiber bundles, more particularly principal bundles, which means you need lie groups and group actions. Bunches of graduate level math. So no, geometric algebra cannot describe guage theory. It is a re-description of tensor products and the algebras that result from it. You need much more than linear algebra for gauge theory.
Lex, your search for clarity will require you to thoroughly understand the MATH!!! Math is the source of the reality that is being exposed, particularly in the quantum world. Utilizing one abstraction to describe another abstraction will lead you to ZERO.
With GR, time, space and light frequency can expand together or compress together. Gravitational time dilation matches a side view of the "gravity well," while the idea of space compressing in front of a moving mass matches a top view best, light from the top of the tower is bluer, local light is redder than that. It works best by taking non-local light-frequencies at face value locally, as if born as seen and unchanged in transit, locally. It's "hole"-friendly and pro-spaghettification. The opposing view, a mass-compression-centric view, flips that idea around by supposing all shift happens in transit. Your feet age faster than your head if standing. The compressive view appears more sophisticated, as in representing gravity waves, the compression intensity is most easily visualized as maximized in front of a moving mass due to some finite speed of gravity and light.
Matter apparently should show temperature dependence tending toward omni-retro-reflectivity in gravitational energy exchanges, this is the simplest path to understanding cold conglomerate asteroids Ryugu and Bennu as well as warmer "Dark Matter"-effects including slow-growing lateral disk-to-disk axial spin couplings carried in DM filament effects. Gravitational quanta should be able to convey ultra-slow oscillation carrying rotational inertia, this is apparently the simplest path to understanding the ultra-regular spherical voids honeycombing the DM structure throughout the latest DM-effect map slices seen close to CMB scale. A way for spinning gravity flow to be concentrated more finely than light quanta seems to be needed to explain wavefunction stretching and noisy light conduction in entanglement. Notwithstanding wormhole fans, there's no point in getting lost in math and stuck in extradimensional space-time box thinking if one can describe a quantum of gravitational energy flow correctly. Seems gravity should be quantized with composite energy dipoles, where a classical graviton is equivalent to a negative energy effect monopole, the composite graviton has positive energy partially coating negative energy effects and a resolving limit with the negative energy close to Planck scale. The negative component of a self-compressing composite dipole can subtract from interaction-size and energy normally associated with positive-only energy quanta. A suitable model for a cold proton could rely on three perpendicularly-intersecting flat disks. Disk surfaces could somehow be partly reflective to gravitational quanta. Extra dimensions might make sense for explaining the insides of particles, in the severely-limited ways of expressing those insides after collision, but they should always be recognized as an admission of limited dynamical insight.
hello Lex. two words. Nassim Haramein. get him on the podcast & dramatically simplify this shit. The complexity is a product of tenured intellectuals who are no longer capable of thinking simplicity
This is completely nonsense. I'm a professional research mathematician. Gauge theory is not simple. Period. Whatever it is you're listening to is not correct, or complete, or both. There is a zero percent chance that you can listen to something "simple" and be able to correctly solve exercises in the field of gauge theory. It takes years of training. This notion that anyone talking about complexity is "exaggerating" is so American..it's annoying. Learning gauge theory is the *beginning* of doing research, so a young PhD *begins* there. Lord only knows what you would say about the complexity of their original research...
Please can we all work on using filler words like like like ya know like ya know These guys are clearly super smart let’s speak as if that’s the case! Just pause if you need to! Pass it on
Said a lot of things, but at the same time said nothing. If you dont understand some topic in Physics or Math or cant visualize it, it is your problem, things in science are always defined in the best and simplest way they can be defined.
@@nickhowatson4745 I think physics assume too much that's all. I think they are so excited to get to the next level of understanding they are willing to compromise the foundations to do so. We really don't know what the foundation is yet.
@@timmothyburke scientific fields which enjoy the existence of experimentally observable outcomes can afford quite a lot of assumption before meaningful progress is compromised -- mathematical/theoretical physics can and does take liberties beyond what is experimentally viable, but that's why theoretical is in the name
It's not idiotic at all. If I stand in Chicago, then F = ma, and if I stand in Hong Kong, then F = ma. We have no reason to believe otherwise as all evidence supports this. I do agree, though, that not *_all_* assumed symmetries are actually symmetries. In fact, this was a problem during the 50's where up to that point it was believed that parity was a fundamental symmetry, and in most things it is a symmetry, but certain interactions were discovered to violate parity, proving it wasn't a symmetry. It could be that some of these symmetries aren't actually symmetries at all, but it's perfectly reasonable to assume they are when all current evidence supports that they are. It's not idiotic to assume something that has overwhelming supporting evidence for, but it *_would_* be idiotic to assume something that had overwhelming contradictory evidence disproving it.
The more math I learn the less I want a visualization of a mathematical concept. The visualization befouls and trivializes and cheapens an otherwise beautiful and generalized idea for the sake of vulgar communication.
some people, probably the most intellectually gifted, can figure out solutions to problems simply through showing visualization. Like einstein visualized the clock slowing down.
People say this as though he didn't spend decades working through the rigorous nuts and bolts to get to a point where he could express those visualizations (which really only scratch the surface of what's actually going on) Some of those visualizations are extremely useful in gaining a superficial understanding of something, but the real beauty in the structures themselves will almost always lie much deeper than anything a visualization can represent
![[TH] 2024 PMGC League | Survival Stage วันที่ 1 | PUBG MOBILE Global Championship](http://i.ytimg.com/vi/FVFT-PFwU80/mqdefault.jpg)
![[MV] A leap of faith...ความเชื่อใจครั้งสุดท้าย (From "Petrichor the series")](http://i.ytimg.com/vi/U1piZH2CNXA/mqdefault.jpg)
Take rod of metal with temperature distribution in equilibrium. Place lots of thermometers along the rod, they should all measure the same temperature. But, each thermometer has a different measuring units, they have a different and local gauge. So, if you plot the number each thermometer measures in a graph, it would NOT be a straight line. Someone who did not know they were all measuring in different local gauges, different local units, would not know the rod was in equilibrium temperature. They would think the temperature of the rod is different, wherever each thermometer is at. A bad derivative such as dT/dX would not be zero because the numbers are changing (only because we placed Thermometers with different units). This derivative is NOT good because we know the temperature is constant throughout the rod, so our derivative should keep track of the real, physical temperature. So we introduce into the derivative, a term which keeps track of each thermometers unit (how each of them gauges temperature). This new derivative will now be completely independent of whatever units you pick for any thermometer you place on that rod, the derivative will tell you the TRUE temperature distribution of the rod, no matter whether each thermometer is using individually different units.
This is one way I like to think of gauge theory. Notice how symetry in this system, with this new derivative is now local. Meaning you can take one individual thermometer, change it's units, but your theory will not change based on this local unit transformation. This is local gauge invariance = system with local symmetry.
you know what you can do with your rod, what a convoluted pretentious load. of bollocks
@@outsidethepyramid nah, you're just too stupid. It's okay though. Not your fault.
@@outsidethepyramid i am bollox
Great expansion, thanks! 😁
That's a pretty good analogy, but definitely not complete. In gague theories the rotor is non-vanishing, so you can't really think about thermometers with some absolute calibration. Maldacena gave a wonderful monetary analogy that I believe is superior. If you have time check out his talk:
th-cam.com/video/OQF7kkWjVWM/w-d-xo.html
"A poet knows he has achieved perfection not when there is nothing left to add, but when there is nothing left to take away." -Leonardo da Vinci
Exactly. Poetry is an art form of communication. There is no perfect poetry but if there was it would be understood by everyone. Also every language would have a different "perfect poem"
He didn't say that
@@turtle926 This guy did: en.wikipedia.org/wiki/Antoine_de_Saint-Exup%C3%A9ry
My favourite way of thinking about Gauge Symmetry is Fiber Bundles formalism. Local symmetry is then just invariance of laws of physics under reparametrisation of fibres. It's just symmetry that says that "coordinates are not physical" just as in General Relativity for example. Coordinates are irrelevant only underlining geometric structure matters.
So then what is a broken gauge symmetry?
Different people have different ways to grasp a concept. Some may find a rigorous analytical formalism more on-point, but in the end a visualisation always helps us appreciate what the theory truly stands for
Topologists love to draw pictures but the subject has to be dense so that visual ideas can be formalized and generalized to apply to as many situations as possible in the same way. If we only work with pictures then we have to treat every object like a torus, sphere, Möbius strip, etc. individually; topology gives us a single language for studying all shapes (and more exotic structures) at once
Could not agree more Lex about disciplines being in love w visually demonstrating their mental landscape.
LOL WHY HATING ON TOPOLOGY?????????!!!!! it's gr8
Cao
For other fundamental interactions, which, it is believed, can be described as gauge interactions,
we find that the theoretical structures of the corresponding theories are exactly parallel to that
of gravity. There are internal symmetry spaces: phase space, for electromagnetism, which looks
like a circle.
Lorentz symmetry is not usually what one thinks of when talking about gauge symmetries, at least not in the standard model of particle physics. In the standard model, the fundamental symmetries are spacetime symmetry (that is the Lorentz symmetry discussed here, which is a non-compact group) which acts on space and time globally (ignoring gravity), and then there is gauge symmetry (compact Lie groups/algebras) which act locally on the equations that describe the dynamics of all the known particles (the Lagrangian). Because the gauge symmetry is a symmetry of the equations, there isn't an easy way to understand what the gauge symmetry physically represents, although attempts have been made for example, as the curvature from higher dimensional compactified spaces ala Kaluza and Klein etc. As for a simple example of how to conceptualize gauge covariant derivatives more generally, I think Ashley Cotrell's example in these comments nails it. For completeness, I should mention that there is a formulation of general relativity using vielbiens where one can actually think of the Lorentz symmetry as a gauge.
''Prose and poetry'' what a wonderful example!
Atiyah:
As far as gravity is concerned, Einstein’s General Relativity is a beautiful and complete theory.
But as Einstein realized it has to be extended to account for other physical forces, the most
notable being electromagnetism. It is perhaps no accident that the first and most significant
step in this direction was taken by a mathematician - Hermann Weyl. He showed that, by
adding a fifth dimension, electromagnetism could also be interpreted as curvature.
His idea was that the size of a particle could alter as it passed through an electromagnetic
field. In analogy with railways it was called a gauge theory, and this name has stuck through
subsequent evolutions of the theory.
Unfortunately for Weyl, Einstein immediately objected on physical grounds that this would
have meant different atoms of, say hydrogen, would have different sizes depending on their
past history, in contradiction with observation. Given this devastating critique, it is remarkable
but fortunate that Weyl’s paper was still published, with Einstein’s objection as an appendix.
Clearly the beauty of the idea attracted the editor, despite the fatal flaw. In fact, beauty often
wins such contests, because with the advent of quantum mechanics, with its complex wave
functions, it was pointed out by Kaluza and Klein that Weyl’s gauge theory could be salvaged
if one interpreted the variable as a phase rather than a length. A pure phase shift by itself is
not physically observable and so Weyl’s theory avoids the Einstein objection.
This was very interesting. Visualisation is definitely an easy way into a subject for most people (including myself). Once you've been doing something for while it becomes more and more abstract and subconscious so you can't explain in a way that makes sense to noobies. Personally I agree with Lex that building bridges is one of the most interesting things you can do.
Imagine the Mandelbrot Set forming infinitely inside a horn torus. I love that kind of stuff
It just means you could stand to get in touch with things and exercise explaining things like people were 5.
A periodic verbalable audit, so to speak.
One source of "visualization" of abstract mathematical ideas, especially those related to topology, is "Mathematical Impressions" by A. T. Fomenko.
I love the fact he brings up Lorentz. Pull up any video about 'Einstein' on TH-cam and chances are they'll examine concepts that were originally conceived by Lorentz rather than Einstein
Tbh mathematicians like Lorentz, Riemann and Hilbert have done more to GR than Einstein. Einstein just had the spot of piecing it all together and giving it a physical explanation.
He did not explain gauge theory
I feel that the term "gauge" (for one reason or another) has the unfortunate quality of obscuring an idea that most people have already come across in an undergraduate linear algebra course. Geometric objects, whether physical or abstract, are often poked and prodded with respect to a coordinate system, i.e. with respect to a basis of some vector space. Simply put, a gauge is just a choice of basis. The problem is that our choice of basis is arbitrary, e.g. think of two different observers looking at the same vector from different angles. We say that an object or quantity is gauge invariant (or admits a gauge symmetry) if its existence is independent of the gauge. Intuitively, a vector (and thus positions, velocities, etc.) should be an example of such an object since bases are typically introduced after the definition of a vector space! The remaining details (e.g. thinking of the collection of transformations between gauges as a group, generalizing to manifolds and their tangent bundles, etc.) are just technical bells and whistles that allow us to speak more generally about gauge invariant objects on arbitrary geometric spaces (such as the sphere or a blob), but they shouldn't distract you from the essential notion of an object existing independently of how I choose to describe it.
This isn't right, it's not a "choice of coordinates" thing, it's a "symmetries" thing. The appropriate language to describe it by the actions of Lie groups on a manifold. "Gauge groups" are the structure groups of principal bundles. It's complicated because it is, and for no other reason. It's an elegant and beautiful theory, but it's a graduate level+ thing that I'm sure very few people have successfully rigorously self-taught to the point where they can solve exercises successfully.
TLDR: bundles have coordinates (but not a basis, they aren't vector bundles), and the nature of the bundle is independent of choice of coordinates, but that choice is not what a "gauge" is.
We were forcing thermal phenomena in the last Hilbert confined entropy regulated gauge group conformations that Alan Guth said forget it, it's like trying to not go back at near the speed of light which at least sees our mass in the effort
Dirac:
Perhaps I should say at this point, to give you some physical insight into all this, that we shall later
be applying this theory to electrodynamics and we shall then find that the quality of being physical is just the
quality of being gauge-invariant.
Wow, this was a great discussion. Both people were saying new things and adding something to the space, which is different from most postcasts i sometimes see. Would like to hear more from blue shirt guy.
Its like they just refuse to explain the force at a distance part, and just settle at the explaining the conservation part.
I’m a little sad this didn’t help me find clarity about how gauge symmetry really works. The best I’ve got is that there’s a sense in which a circle, which stretches out in the complex plane (I think?), can be mapped onto space and becomes a rigorous way to describe electromagnetic fields… Hope one of these physicists who keep coming by Lex’s studio finds a way to describe this simply soon. Eric Weinstein specializes in rendering statements of obscure genius with few footholds for people who aren’t specialists… That’s not fair… I just have no idea what he’s talking about as soon as he says “gauge theory.”
His math site is so helpful, but when talks he obfuscates on purpose. Weird
Shortest answer I can give is, consider normal transformations, like rotation or translation. This just changes the global way you look at something. Now imagine this would be possible locally. If you try this, something goes wrong, a certain quantity (the action) is changed under these local transformations, and physicists agree it shouldn't (for certain transformations, not translation and rotation, but for example phase change, a quantum mechanical thing). To fix this, we change our notion of change, the derivative, to the covariant derivative. Covariant in this context just means that it has a term that counteracts the transformation. With this notion of the covariant derivative (which generalizes to connections on principal bundles but forget that unless you're in for a deep dive), we get certain relations between different fields (which represent particles) by looking at how they can change while keeping this action constant under these transformations, these interactions can for instance be between photons and electrons. But more generally, with more complicated gauge groups (or Lie groups acting on the principal bundles from earlier), aka local transformations, we get more exotic particles/forces. At this point in time, we've made a theory that combines a whole lot of different symmetries (U(1), SU(2), SU(3)) together into a model that describes the interaction of all the forces/particles we know, except gravity. This exception is the root of many fancy new physics theories, like supersymmetry and string theory. But so far, we can only combine all the other forces in what's called the standard model, which is a gauge theory. Hope this helps :)
Distribution of mass and energy within particles. The coupling of thermodynamics and mass.
Prior to Kepler and Newton, planet orbits were thought to be perfect circles, because symmetry.
Bret Victors work is all about visualizing code/math
Thank you for your video.
Neural networks are beautiful when represented. I think graphical representations are extremely powerful, for example in quantum computing. If we didn't have those diagrams of wires and boxes, I'm not sure we would all be able to understand a quantum computer program.
Hidden awareness: "... about rulers being deformed (laughs) ..."
"and isn't it ironic" - alanis morriset
I’m with you on this, Lex❣️ I think it would be much more exciting & engaging to use the visual representations of these abstractions in conjunction with the mathematical formulae & algorithms, especially for visual people like myself, who use TH-cam specifically for the added information & understanding that video, as a visual medium, provides‼️☑️🎯🙏🏻🤴🏻🗽👼🏻
The problem is that no matter how much someone is a visual person (I am and used to try an approach math that way, I still do sometimes out of habit) the visualizations simply can't represent the real marvel of these structures, or even what's actually going on and when you really dive into things, that becomes apparent and they stop being interesting very quickly.
Math is the way it is for a reason and it isn't (always) a matter of preference -- the notation is just a language that compresses concepts into efficiently communicable convention. That hurdle is what everyone spends time talking about because it's the first one and most people fail to traverse it, but the difficult part is actually in the concepts themselves and they're difficult for the same reason they can't be visualized. The only way to really understand them deeply enough to understand their beauty is to get into the weeds and interact with the abstractions themselves.
There's more than one way to skin that cat, but the unfortunate reality is that no matter which way you choose, it's eventually going to require a knife if you want to do it properly not waste a whole lot of time accomplishing very little.
I‘m half way down the video and no word of Noether? Explaining gauge theories without any mentioning of charge, topology or Noether is weird …. (I’m a physicist).
But yes Lex is right, visualizing stuff is really important. The physicist in the video is a typical „algebraic-conceptional“ thinker: he understands concepts well, but visualizing is not his mode of thinking. This is one of the biggest divider in phycisitsts, there are visualizers, intuitionists, and conceptional thinkers (or any combination of the above). It does make a difference to visualize, many peoples mode of thinking is that; in fact it’s one of the best skills as a physicist, many physicists struggling with high level abstract/dense math can mitigate many shortcomings by visualizing.
There is a lack of important visualizations, but the reason is quite straight forward: mathematicians often do not care about stuff like this. They want to prove, they want to relate abstract structures to others and generalize it to any usecases. Physicists are different, using their lacking knowledge and intuition to find one detailed, explicit solution to a hard problem. This is the exact reason why mathematical physicists make good theory confirmers, but really bad physicists in general. Some things get lost in expressive rigorous mathematics. It’s even worse for mathematicians, they are really bad physicists (I’m generalizing a lot, but you get my point). Physics is some kind of game, between lack of knowledge, intuition and useful handwavy-ness.
When mathematicians say: Physics is just bad/not rigorous math, it’s a really dumb statement, misunderstanding the multiplicity of information that needs to be condensed. Condensed matter physics is the perfect example, you can never describe a material fully. It’s not possible. But with the right set of ideas, stuff can be broken down to smaller pieces and with good approximations, models can be constructed. By time, the approximations ALWAYS break down (due to new measurements or general disagreement with certain parameters from the get go), and the job then becomes to find new approximations, which will always lead to new dynamics and in the best case becomes the old model again when tuning a set of parameters to extreme values (this is called an „effective“ theory, and object being effectively described by something easier, ignoring complex stuff, i.e. „dismissing friction“ would be an effective dynamic, in the real model, one needs to take friction into account).
In a more general sense, physics is like the interplay between „distance“ (from the truth) and „precision. The more distant you are (more „effectively“ describe your thing), there less precise you are but you get an nice general understanding, i.e. take newton mechanics, but the more precise you get, the harder your problems become but also the closer it is to the truth. So in a sense, even better approximation gets us closer to the truth but it’s harder to do.
Wonderful Gauge theory
A mobius strip is a representation of a one sided object but a truly one sided object is impossible to go around. It's kind of a weird thought but I wonder if something can be so small that it only has one side and is impossible to go around.
I'm not certain by any means, but I dont think it's a matter of size. For an object to be one sided my guess is that it would have to have one side that existed in the physical realm but while the other "side" was immaterial. Picture walking up to the front of a prop house made for a motion picture of some sort. You look at it, touch it, notice its colors, textures, scents, and then you walk around to the other side, but as soon as you get to an angle perpendicular to the face of the house it vanishes because it'd have no thickness. You go completely to the back, turn so that you're facing the direction you previously confirmed the house to be in, and you'd see right through where it was. You'd pass your hand through where you thought it should be and feel nothing. Finally you'd walk straight through where the house must have been without any resistance, make a 180 degree turn, and see the front of the house staring back at you again.
I'm horrible at mathematics and algebra but I'm very visual and got an A+ in geometry in one of the hardest private schools in the US in which i got a 2.0 GPA. Then I got into psychedelics heavily and saw sacred geometry and all these higher dimensional shapes and now I'm falling in love with physics. Now I actually understand the hopf fibration, hilbert curves, other space filling curves, platonic solids and finding passion for physics and math by going the opposite of the traditional trajectory.
It's all linked and it's crazy how psychedelics can actually show us visual learners the inherent higher dimensions of physical reality. What does that say about psychedelics specifically dmt? Cause I've seen hopf vibration and hilbrrt space geometry long before I ever saw or knew of those and I've seen sierpinski triangles on lsd long before ever seeing as much as a picture of all this?
Is that what those [heiro]glyphs mean?
The Hidden variables of Pandora's box: you might not want to find out Stephen Hawking's joke about black holes not being sexual has mined this anti-logical and yet even more intelligent truth
I think we are working on behalf of the university and the universe to add and subtract
Seems like gauge theory is just a way to measure a system taking into account all the variables that can change that measurement
Robert Ghrist, Elementary Applied Topology
I was taught that gauge theory has to do with the invariance of a Lagrangian action with respect to a local change of coordinates or other parameters in the Lagrangian. I'm not hearing the same simplicity expressed in this talk.
Well that’s because they don’t want you to know about it man.
Naked singularly can surely be reached by trying to strip your Tableau of factors through complexity, complexity simplifying, frozen phenomena becoming redressed in a new bizarre orange lattice lab synthesized at Southern Oregon University 🎓
Locked versus unlocked symmetry
I want your attempt at a theory of every thing explained In paradox only
People get confused by gauge theory because they are presented the idea as if it is a single construct with one specific, right way of cracking the code to understanding a new field of math on its own. In effect, gauge theory is a way of thinking about mathematics as it applies to dynamical systems as a whole.
Gauge theory is made up by the combination of most fields of math appearing in physics as an attempt to see the relationship they all share. This allows us to simplify things and filter out a lot of noise. In understanding it this way, it's not just an application of finding symmetries as they correlate with conservation (applying Noether's theorem already gets you there)... it's also two other fundamental things:
1. It tells us that it doesn't matter how we write out our physics. If two ways of describing a system work with equal accuracy, then those two ways of describing the system are equivalent. It means that if a hypothetical alien species that doesn't understand our construct of linguistics start deriving math, we can predict where our predictions will match no matter what happens due to the inherent ratios and constants we measure in nature. We will be forced to come to a similar conclusion to be accurate, even if the interpretation or method of getting there is entirely foreign. It gets us as close as possible to "why" these systems work when the most we can approximate is "how" they work.
2. It thus allows for a more rigorous way of developing a philosophy of physics for which all the underlying mathematical fields and axioms play a role in coming together to yield a result that can be explained with various, equivalent analogies.
The primary fields of math you will find in a focused study of gauge theory are typically Lie algebras (and their groups), differential geometry, spinor analysis, and (increasingly more common) geometric/Clifford algebras. It gets even wilder when you begin to see gauge theory through the lens of vertex algebras and W-algebras (which allow us to jump around between quantum and classical field theories on the fly utilizing structures that coexist in those theories). These things make up a vast sum of the math in physics when you think of what composes them.
The issue in all this? Two authors writing a book on mathematical gauge theory are unlikely to focus on remotely the same topics or goals, and will likely end up writing two books that feature entirely different content. Neither of those authors are then likely to even use the word "gauge theory" in any mathematical sense beyond the cover that they use to catch your attention. Reading gauge theory books for me has been closer to the experience of reading more rigorous physics texts where you can't help but take a deep dive into how that specific author engages with mathematics as a whole. It's like reading a philosophy of physics book, except the author isn't afraid to throw in the most advanced tools we know of in mathematics across every page alongside proofs because they are often self-aware that these are not trivial topics.
Reading gauge theory textbooks has been a very different experience from many others, as you can at least find a dozen or more calculus textbooks that cover more or less the same material with a similar approach besides a few quirks. You gain a deeper appreciation for mathematical physics when you realize the reason why gauge theory can't be sufficiently summarized by reading a text by one or two authors on it (despite aspects of it clearly existing for over 100 years now). It's a journey of discovering how you recognize mathematical patterns and use them to find connections across the field.
So I didn't understand gauge theory any better.
This clip is dumb.
I love how this missing visual mathematics that Lex describes is literally geometric algebra.
I've never personally seen how geometric algebra helped visualise these advanced forms of mathematics. Do you have any examples of this.
@@freniisammii It provides a unified foundation for zeros, complex numbers, quaternions, 3D rotation, linear and bilinear algebra, exterior forms algebra, >3D spaces, spacetime algebras, lie algebras, quantum logic lattice, differential geometry, conformal and projective spaces, group representations, and combinatoric hypergraphs.
@@RecursionIs am a professional mathematician. Few concepts are as well loved by crackpots as "geometric algebra".
Geometric algebra is not even close to being a framework for understanding all of "differential geometry". It's just linear algebra/module theory in a different language, nothing new is added, and there is not really a compelling reason to use this different language when the standard abstract language contains the same information, and is understood by everyone.
For gauge theory, you need smooth manifolds, Riemannian manifolds and the Covariant derivative, fiber bundles, more particularly principal bundles, which means you need lie groups and group actions. Bunches of graduate level math. So no, geometric algebra cannot describe guage theory. It is a re-description of tensor products and the algebras that result from it. You need much more than linear algebra for gauge theory.
Sam gambhir was a prume example
Lex, your search for clarity will require you to thoroughly understand the MATH!!! Math is the source of the reality that is being exposed, particularly in the quantum world. Utilizing one abstraction to describe another abstraction will lead you to ZERO.
With GR, time, space and light frequency can expand together or compress together. Gravitational time dilation matches a side view of the "gravity well," while the idea of space compressing in front of a moving mass matches a top view best, light from the top of the tower is bluer, local light is redder than that. It works best by taking non-local light-frequencies at face value locally, as if born as seen and unchanged in transit, locally. It's "hole"-friendly and pro-spaghettification. The opposing view, a mass-compression-centric view, flips that idea around by supposing all shift happens in transit. Your feet age faster than your head if standing. The compressive view appears more sophisticated, as in representing gravity waves, the compression intensity is most easily visualized as maximized in front of a moving mass due to some finite speed of gravity and light.
No explanation. No visualization.
Matter apparently should show temperature dependence tending toward omni-retro-reflectivity in gravitational energy exchanges, this is the simplest path to understanding cold conglomerate asteroids Ryugu and Bennu as well as warmer "Dark Matter"-effects including slow-growing lateral disk-to-disk axial spin couplings carried in DM filament effects. Gravitational quanta should be able to convey ultra-slow oscillation carrying rotational inertia, this is apparently the simplest path to understanding the ultra-regular spherical voids honeycombing the DM structure throughout the latest DM-effect map slices seen close to CMB scale. A way for spinning gravity flow to be concentrated more finely than light quanta seems to be needed to explain wavefunction stretching and noisy light conduction in entanglement.
Notwithstanding wormhole fans, there's no point in getting lost in math and stuck in extradimensional space-time box thinking if one can describe a quantum of gravitational energy flow correctly. Seems gravity should be quantized with composite energy dipoles, where a classical graviton is equivalent to a negative energy effect monopole, the composite graviton has positive energy partially coating negative energy effects and a resolving limit with the negative energy close to Planck scale. The negative component of a self-compressing composite dipole can subtract from interaction-size and energy normally associated with positive-only energy quanta.
A suitable model for a cold proton could rely on three perpendicularly-intersecting flat disks. Disk surfaces could somehow be partly reflective to gravitational quanta. Extra dimensions might make sense for explaining the insides of particles, in the severely-limited ways of expressing those insides after collision, but they should always be recognized as an admission of limited dynamical insight.
Math =/= and "cool kids"
Is the proper word for duplicating the same physics throughout Spacetime "translation" or transmission? The latter, to me, seems more accurate.
hello Lex. two words. Nassim Haramein. get him on the podcast & dramatically simplify this shit. The complexity is a product of tenured intellectuals who are no longer capable of thinking simplicity
This is completely nonsense. I'm a professional research mathematician. Gauge theory is not simple. Period. Whatever it is you're listening to is not correct, or complete, or both. There is a zero percent chance that you can listen to something "simple" and be able to correctly solve exercises in the field of gauge theory. It takes years of training.
This notion that anyone talking about complexity is "exaggerating" is so American..it's annoying. Learning gauge theory is the *beginning* of doing research, so a young PhD *begins* there. Lord only knows what you would say about the complexity of their original research...
One sign that an interview is not going well is when the interviewer has to speak as much as, or more than, the interviewee.
A podcast and an interview are two different things
it’s a conversation
Please can we all work on using filler words like like like ya know like ya know
These guys are clearly super smart let’s speak as if that’s the case! Just pause if you need to! Pass it on
Said a lot of things, but at the same time said nothing. If you dont understand some topic in Physics or Math or cant visualize it, it is your problem, things in science are always defined in the best and simplest way they can be defined.
Lex is so shockingly ignorant and proud to be so. It's embarrassing.
Assuming symmetry in translation through space is idiotic.
why? its a self evident fundamental property. The Symmetry of Translation through spacetime gives rise to Newtons 1st, 2nd and 3rd Laws Of Motion.
@@nickhowatson4745 I think physics assume too much that's all. I think they are so excited to get to the next level of understanding they are willing to compromise the foundations to do so. We really don't know what the foundation is yet.
@@nickhowatson4745 For example, try questioning the "2nd Law" of thermodynamics, you will be treated like an idiot.
@@timmothyburke scientific fields which enjoy the existence of experimentally observable outcomes can afford quite a lot of assumption before meaningful progress is compromised -- mathematical/theoretical physics can and does take liberties beyond what is experimentally viable, but that's why theoretical is in the name
It's not idiotic at all. If I stand in Chicago, then F = ma, and if I stand in Hong Kong, then F = ma. We have no reason to believe otherwise as all evidence supports this.
I do agree, though, that not *_all_* assumed symmetries are actually symmetries. In fact, this was a problem during the 50's where up to that point it was believed that parity was a fundamental symmetry, and in most things it is a symmetry, but certain interactions were discovered to violate parity, proving it wasn't a symmetry. It could be that some of these symmetries aren't actually symmetries at all, but it's perfectly reasonable to assume they are when all current evidence supports that they are. It's not idiotic to assume something that has overwhelming supporting evidence for, but it *_would_* be idiotic to assume something that had overwhelming contradictory evidence disproving it.
The more math I learn the less I want a visualization of a mathematical concept. The visualization befouls and trivializes and cheapens an otherwise beautiful and generalized idea for the sake of vulgar communication.
some people, probably the most intellectually gifted, can figure out solutions to problems simply through showing visualization. Like einstein visualized the clock slowing down.
People say this as though he didn't spend decades working through the rigorous nuts and bolts to get to a point where he could express those visualizations (which really only scratch the surface of what's actually going on)
Some of those visualizations are extremely useful in gaining a superficial understanding of something, but the real beauty in the structures themselves will almost always lie much deeper than anything a visualization can represent