Application: Quantum mechanics on curved spaces - Lec 26 - Frederic Schuller

[@jmafoko]

heavy material handled by a great teacher. that motivation example is very important

[@majasaro]

when I have a better base in math I'll study this kind of things, thanks a lot to share your kwnoledge with us

[@mastershooter64]

it's been 4 years, do you have that base yet?

[@Smooth_Manifold]

Superb, as always.

[@samhuang2545]

I have no background in any physics, but it seems to me that those P_j, Q^j are local operators on the local sections of the bundles with some commutator relations and "formal" self-adjointness. More-precisely: P_j,Q^j are local operators, defined via covariant derivative along coordinate field frame adapted to a chart only?

[@wolphramjonny7751]

in what course did you ever heard of the Gelfand Triple? I could not follow anything, this lecture requires a triple PhD, in quantum gravity, knot theory, and set/category theory.

[@Anonymous-kj6cu]

LOL you got me cracking!

[@kapoioBCS]

But is the introduction of a metric well-defined for the Lebesgue integral formalism well defined? How we introduce this volume in the L^p Banach space since we already have a sesqui-linear inner product on L^2?

[@seanki98]

He starts with the inner product on L^2 you mentioned, and adjusts it so that it is consistent for general spaces.

[@tensorfeld295]

How can a C-Line-Bundle (E,pi_E,M,C) be an associated vectorbundle to the frame bundle (P,pi,M,GL(n,R)) with group GL(n,R)? If I try to construct the associated vectorbundle with (PxC)/GL(n,R) i need to act with GL(n,R) on C from the left ... how does this work?

[@tensorfeld295]

I think i was too restrictive with (PxC)/GL(n,R). I can construct the bundle wiht a representation rho: GL(n,R) -> GL(C). The equivalence relations are then (p,f)~(pg,rho(g^-1)f) with g in G, f in C, p in P.

[@pedidep]

I am wondering about a manifold with a boundary. The calculation will change a little bit.

[@TenzinLundrup]

First thank-you for the clear example of polar coordinates. One also has to be careful about boundary terms when integrating by parts in r, since psi and phi need not vanish at r = 0. OK now comes the differential geometry so I had better concentrate. Question after watching: What would have happened if we had defined the covariant derivative using the Levi-Civita connection. That is the first thing I would have tried. Also, it was unclear to me what the Lie group is that defines the associated fiber bundle in the example. Is the associated bundle C on which U(1) acts? If so, the importance of the existence of the group action was never mentioned.

[@tensorfeld295]

Is there a book or paper that covers the idea of elementary quantum physics on curved space? ^^

[@ULRecordings]

“Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics” from Robert M. Wald.

[@tensorfeld295]

@@ULRecordings Thank you! :D

[@hyperduality2838]

Changing the basis whilst keeping the vector constant is equivalent or dual to changing the vector whilst keeping the basis constant -- Duality. The covariant vector is dual -- vectors are dual to co vectors (forms). Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Sinh is dual to cosh -- hyperbolic functions are dual. Domains (the entire space, Groups) are dual to co domains (the base space, fields). Subgroups are dual to subfields -- the Galois correspondence. Certainty (predictability, syntropy) is dual to uncertainty (unpredictability, entropy) -- the Heisenberg certainty/uncertainty principle. Injective is dual to surjective synthesizes bijective or isomorphism. The force of gravity is dual -- changing the vector is dual to changing the basis. Potential energy is dual to kinetic energy -- gravitational energy is dual. "Always two there are" -- Yoda. Action is dual to reaction -- Sir Isaac Newton or the duality of force. Attraction (sympathy) is dual to repulsion (antipathy), push is dual to pull, stretch is dual to squeeze -- forces are dual. Apples fall to the ground because they are conserving duality. Yes is dual to no -- if you choose yes the no still exists -- duality (energy) is being conserved -- the 5th law of thermodynamics!

[@aldasorodaniel]

Muy bueno. Saludos

[@uniphyphysics6604]

sleepy camera man..

[@mirkostappert7467]

Has anyone found the lecture notes, he references at 9:20

[@letmeoffendyou]

www.reddit.com/r/math/comments/77zdq3/lecture_notes_for_frederic_schullers_lectures_on/

[@alanrapoport6784]

Those are not the Statistical Mechanics ones. Does anyone know where to find any material (notes, videos, whatever) about other courses of Prof. Schuller besides this andd the QM one?

[@TernaryM01]

@@alanrapoport6784 Winter School on Gravity and Light

[@95riedl]

@@alanrapoport6784 Theoretical Mechanics: www.video.uni-erlangen.de/course/id/272 But it is in german

[@mrnarason]

@@95riedl it's also on youtube, use autotranslate from german to english kzbin.info/www/bejne/hJe5iX1sqdFgmqc

[@discoverrealityclover9620]

My question is...{x,y,..} and {r, theta} represent the same coordinate system, in physics; if we want to test whether the theory is general enough must we not appeal to a completely different coordinate system {X,Y,..} with its own radial system {R, THETA}.. Since we cannot use x and r at the same time in the same local space. I have more pedantry...but lm tired.

[@seanki98]

I think the polar coordinates happen to just be a useful proxy to motivate everything. He checks towards the end of the lecture that this formalism works for any general coordinate systems, on even curved spaces.

[@chymoney1]

I have no idea what he’s talking about way too much abstract math lingo

[@noname6878]

Did you watch the rest of the lectures? The point is that he applies what he did the entire semester.

[@alechewitt2347]

yeah, this isn't the best lecture for a physics major

[@mrnarason]

@@alechewitt2347 This is what german undergrads learn, get on their level

[@alechewitt2347]

@@mrnarason it has nothing to do with level, it has to do with notation, his notation is different, I see this notation in my math classes, not my physics classes.

[@rounak5106]

@@mrnarason Can you suggest more such (mathematically) rigorous lectures in (theoretical) physics (gauge theories , differential forms in physics, exterior algebra and others) with english as language of instruction?

[@Igdrazil]

The very word "curved space" is highly questionable, potentialy problematic, and may be fallacious. Indeed, where on earth does a coordonate system, polar for instance, have anything to do with a presupposed physical space"? Even a simple sheet of "flat paper" is no way responsable nor pertubed in any way, by the fantasy of a mathematician, that choose to draw on it eighther cartesian or polar. And it gets even worse in full generality, since the so called "space-time world wide sheet of paper" doesn't exists in the very sens that IT IS NOT AN OBSERVABLE, but only so far, a MATHEMATICAL SIMPLIFICATORY MODEL on which particular physical models hapens to be built with SOME accuracy and heuristics. More over hell of much more care must be given to immersions and embedding concerns that may trigger whether the mathematical structures of manifold and fiber bundles are sufficiant or even consistant with obvious Phenomenology. Obviously either in Bohm Aharonov situation or "black halls" or "dark matter" cosmology, ORBIFOLDS, instead of manifolds, are standing up, with hell of a change since, at minima, the very heart of the TOPOLOGY, is drastically changed. A simple HALL can makes hell of a difference in Physics. It is not even clear at all that the "CONTINUUM" paradigm, based on the "real numbers" fairy tail, holds in the heart of actual Physics. Not talking about the very serious problem of chimeric INFINITY, that has such a love/hate recurrent afair in the very heart of Physics (and not less in "pure mathematics"). Where in rigor can one just throw away BORDER CONDITIONS in the integrals, under the false excuse that it is "far away" or "rapidly decreasing" "wave functions"... Is this a serious way to build theoretical Physics, by presuposing such very strong arbitrary asumptions, even after at least Mach warnings about GLOBAL INFLUENCE ON ILLUSIORY "LOCAL", or more recent ENTANGLEMENT concerns. Even a simple "light ray" crosses the cosmos, in a NULL PROPER TIME "JOURNEY", meaning, strictly speaking, IT DOESN'T TRAVELS, whatever STORY any exterior observator may say on what HE SEES about this very mysterious NON TRAVELING "TRAVELER" of "light". Has it even TRAVELED,... before "crashing" into the eye of the observator. What the hell was doing his "wave function" "before" the "crash"? Does "before" even means anything précise? And if yes, what? For whome, if any questionable uniqueness could be brought to this question? Thus is this "time" and "space" question even a singled valued "function"? Why not a multivalued one? And even if it hapens to be single valued, what the hell should be its input variable? Time and Space would be functions of what underlying variables? All these questions mean to open a very serious destruction/construction self critic of our naïve representations that leads us, even in 2020, to keep falling in the platonic cave allegory. Five genious at least, Kant, Poincaré, Granger, Souriau, Connes, have shown in many ways that even our "space-time" representation is highly ptoblematic. Kant, Poincaré, Granger have constantly shown why "space-time" canot be "physicalised" since it is NOT an OBSERVABLE ! Poincaré showed that 4-space-time is just a MATHEMATICAL commodity drastically simplificatory compare to the actual topology of the world that is at minima more alike an organic neuronic meta worldwide network, in which our own psycho neuronic invisible roots may well plug deap in and even be fusionaly part of it. Give mathematical genius to a farmer or à gardener, and he will entirely destroy our modern psycho rigid formal way of looking at Physics. And perhaps worse with an artist, or even a true polytechnical genius. So forget about fallacious "curved space" misleading rhetoric, or get hell of rigor on such sliping soap arbitrary INTERPRETATION, seeking seriously if it is more than an hallucinatory fairy tail mantra. The only thing we interact with is SPECTRAL. There is neighther "particles" nor "waves", nor "quanta". We only see SPECTRA, full stop! So we aught to build a rafined SPECTRAL GEOMETRY (alike Alain Connes) that can reproduce usual illusory phenomenology that we think we observe through the naïve inadequat mathematical layer glasses that we built historically. Vectors genius & "lies" have served hell alot but is extremely arbitrary, limited and misleading. Quaternions was a less clumsy tool, but still a neolithic hamer even against areas measures. Linear algebra improved hell of a lot until differential forms braught the right synthetic matured tool. But now that Structures have emerged since Lagrange, Galois, Lie and Poincaré, its no more for a while a question of lack of proper powerfull mathematical tools, but a philosophical question of epistemology : what model under what pertinent representation Nature is best described. Talking à priori about "curved space" without deeply questioning what the hell are de TALKING about is not serious. It's not Mathematics that is here in question, but pure thaughts about REPRESENTATIONS , pure consciousness of the meaning of fundamental CONCEPTS, and pure RIGOR. One shall never again separate PHYSICS, MATHEMATICS AND PHILOSOPHY. The first is about WHAT, the second about HOW, and the third about WHO. Forget one of the TRINITY, and you run in disaster !

[@rektator]

The curvature of a manifold is independent of the choice of local representation. So from the get go what you say makes no sense.

[@Igdrazil]

@@rektator Pathetic! To start with, where have you seen me talking about the "curvature of a manifold". The point of discussion is "SPACE", physical "space". So you didn't understood a word of my point, though explicite enough and detailed. You just examplify swamp mind of confusion and fog of naïve rhetoric. Alike those who slip on black bananas repeating like perroquets, that correlation coefficient equal to one (between two statistical variables) means "100% correlation". Nothing is more wrong, it says nothing more than the unitarity of statistical verctor in play in 2n dimension, full stop! The rest is fantasia... But not only do you fool yourself in false arguments, moreover you even fake playing the "clever" guy, with chieldish dose of arrogance, what a shame!

[@Igdrazil]

@@SirTravelMuffin Well you know, It's ok if you don't understand anything in Science. But it seems you'r even worse at psychology, or even humor. Perhaps would you discover some talent in shuting up?

[@mastershooter64]

@@SirTravelMuffin i dont know about that but he definitely won in being an ignorant a-hole