Application: Spin structures - lec 27 - Frederic Schuller

[@jackozeehakkjuz]

1:13:00 what an insightful shot. It is a shame the lecture had to finish so abruptly. On the other hand, we have the oportunity to write the GR operators all by ourselves. Thank you a lot for all the effort.

[@tanchienhao]

the best video ive found on mathematical treatment of spin! btw i covered Spin(3) group in one of my videos if you want to see it in detailed steps but this lecture covered pretty much everything on Spin(3) anyway.

[@synaestheziac]

36:49 I love how he calls attention to the fact that he had to go and memorize the sigma matrices after the earlier lecture where he needed a student to tell him what they were

[@yatmingcheung2772]

2:48 There may be a typo in the definition of SO(n). A reflection can also preserve the standard inner product but a reflection does not belong to SO(n) and belongs to O(n).

[@wolfart3984]

Thank you I wish see more and more about your lectures

[@nmdwolf]

At around 22:00, isn't it (a bit) more formal when you express the group elements by matrix exponentials (which you can) and then differentiate the equation U^dagger U = 1? And again for the trace, by working with the formula that the determinant of the matrix exponential is equal to the exponential of the trace we don't have to use such an approximation.

[@hr9653]

@19:00 I think we can make some sense because it's a curve going through identity and epsilon is the curve parameter.

[@millerfour2071]

6:00, 18:48, 29:00, 36:20, 1:35:36

[@kapoioBCS]

In 1:27:30 the manifold must also me oriented

[@hyperduality2838]

Double cover = dual cover! Antipodal points identify for the rotation group SO(3) -- north poles are dual to south poles on a sphere. Perpendicularity, orthogonality = duality in mathematics! "Perpendicularity in hyperbolic geometry is measured in terms of duality" -- universal hyperbolic geometry, Professor Norman J. Wildberger, mathematician. 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! Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation. The left handed spinor/mobius loop is dual to the right handed spinor/mobius loop synthesizes the Klein bottle -- topology. Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung. Thesis is dual to antithesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic. Null vectors (light rays, lines) are dual to null points (fermions, spinors) -- twistor space. Points are dual to lines -- the principle of duality in geometry.

[@hyperduality2838]

Twistor space is an equivalent or dual description of space/time. Space is dual to time -- Einstein.

[@mathgeek43]

At 59:51, it is claimed that the definition rho(U) given is an element of GL(R,3). However, I thought that the tensor product produced a tensor which should output a real number. Therefore, shouldn't the definition given be the elements of the rho(U) matrix or rho(U)^i _j ? If I am wrong, then could someone explain how this definition outputs a vector?

[@quaereverum3871]

In my understanding: epsilon^i (x) e_j is going to be a map from the tensor product of "the dual space of R^3" with "R^3", to R (in this case). You can think of this intuitively as a column vector of row vectors, for example. Each of the three entries of the column vector, is a row vector with 3 elements. This then becomes a matrix, hence an element of GL(3,R). This intuitive notion is just to get the message across, you should probably not think of it that way generally speaking. It maps to R because it takes as input a row vector and a column vector. As you probably know, a row vector times a matrix times a column vector produces a scalar. One thing that should make it clear immediately is the relation dim(V(x)W)=dim V x dim W. So the dimension of our tensor product should be 3x3.

[@synaestheziac]

23:05 “dirty as hell”

[@postbodzapism]

No sound

[@JackSarfatti]

Excellent lectura!