Editing 20: Sir Roger Penrose - Plotting the Twist of Einstein’s Legacy
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[https://en.wikipedia.org/wiki/Roger_Penrose Sir Roger Penrose] is arguably the most important living descendant of [https://en.wikipedia.org/wiki/Albert_Einstein Albert Einstein's] school of geometric physics. In this episode of [[The Portal Podcast|The Portal]], we avoid the usual questions put to Roger about quantum foundations and quantum consciousness. Instead we go back to ask about the current status of his thinking on what would have been called “Unified Field Theory” before it fell out of fashion a couple of generations ago. In particular, Roger is the dean of one of the only rival schools of thought to have survived the “String Theory wars” of the 1980s-2000s. We discuss his view of this [https://en.wikipedia.org/wiki/Twistor_theory Twistor Theory] and its prospects for unification. Instead of spoon feeding the audience, however, the material is presented as it might occur between colleagues in neighboring fields so that the Portal audience might glimpse something closer to scientific communication rather than made for TV performance pedagogy. We hope you enjoy our conversation with Professor Penrose. | [https://en.wikipedia.org/wiki/Roger_Penrose Sir Roger Penrose] is arguably the most important living descendant of [https://en.wikipedia.org/wiki/Albert_Einstein Albert Einstein's] school of geometric physics. In this episode of [[The Portal Podcast|The Portal]], we avoid the usual questions put to Roger about quantum foundations and quantum consciousness. Instead we go back to ask about the current status of his thinking on what would have been called “Unified Field Theory” before it fell out of fashion a couple of generations ago. In particular, Roger is the dean of one of the only rival schools of thought to have survived the “String Theory wars” of the 1980s-2000s. We discuss his view of this [https://en.wikipedia.org/wiki/Twistor_theory Twistor Theory] and its prospects for unification. Instead of spoon feeding the audience, however, the material is presented as it might occur between colleagues in neighboring fields so that the Portal audience might glimpse something closer to scientific communication rather than made for TV performance pedagogy. We hope you enjoy our conversation with Professor Penrose. | ||
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[[File:ThePortal-Ep20 RogerPenrose-EricWeinstein.png|600px|thumb|Eric Weinstein (right) talking with Sir Roger Penrose (left) on episode 20 of The Portal Podcast]] | [[File:ThePortal-Ep20 RogerPenrose-EricWeinstein.png|600px|thumb|Eric Weinstein (right) talking with Sir Roger Penrose (left) on episode 20 of The Portal Podcast]] | ||
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:;Summary | :;Summary | ||
::The following diagram summarizes the relationship with the structures so far: | ::The following diagram summarizes the relationship with the structures so far: | ||
[[File:Spinor_construction.png|frameless| | [[File:Spinor_construction.png|frameless|]] | ||
</div> | </div> | ||
::The hook arrows denote the constructed embeddings, the double arrow gives the spin double cover of the rotations, and the downwards vertical arrows are the representations of the two groups acting on their respective vector spaces. The lack of an arrow between the two vector spaces indicates no direct relationship between vectors and spinors, only between their groups. There is however a bilinear relationship, meaning a map which is quadratic in the components of spinors to obtain components of vectors, justifying not only that spin transformations are square roots of rotations but spinors as square roots of vectors. When n is even, the spinor space will decompose into two vector spaces: <math> \ | ::The hook arrows denote the constructed embeddings, the double arrow gives the spin double cover of the rotations, and the downwards vertical arrows are the representations of the two groups acting on their respective vector spaces. The lack of an arrow between the two vector spaces indicates no direct relationship between vectors and spinors, only between their groups. There is however a bilinear relationship, meaning a map which is quadratic in the components of spinors to obtain components of vectors, justifying not only that spin transformations are square roots of rotations but spinors as square roots of vectors. When n is even, the spinor space will decompose into two vector spaces: <math> \delta = S^+ \oplus S^- </math>. In a chosen spin-basis these bilinear maps will come from evaluating products | ||
of spinors from the respective subspaces under the gamma matrices as quadratic forms. | |||
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In the cases of Euclidean and indefinite signatures of quadratic forms, a fixed orthonormal of <math> V </math> can be identified with the identity rotation such that all other bases/frame are related to it by rotations and thus identified with those rotations. Similarly for the spinors, there are "spin frames" which when choosing one to correspond to the identity, biject with the whole spin group. This theory of group representations on vector spaces and spinors in particular were first realized by mathematicians, in particular [https://www.google.com/books/edition/The_Theory_of_Spinors/f-_DAgAAQBAJ?hl=en&gbpv=1&printsec=frontcover Cartan] for spinors, in the study of abstract symmetries and their realizations on geometric objects. However, the next step was taken by physicists. | |||
</div> | </div> | ||
;Finitely | ;Finitely | ||