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20: Sir Roger Penrose - Plotting the Twist of Einstein’s Legacy: Difference between revisions
20: Sir Roger Penrose - Plotting the Twist of Einstein’s Legacy (edit)
Revision as of 23:00, 24 July 2023
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[[File:Spinor_construction.png|frameless|]] | [[File:Spinor_construction.png|frameless|]] | ||
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::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 | ::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. 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. | ||
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;Finitely | |||
Spinor-valued functions or sections of spinor bundles are the primary objects of particle physics, however they can be realized in non-quantum ways as a model for the electromagnetic field per Penrose's books. Analyzing the analogue of the vectors in this scenario, they represent tangents through curves at a particular point in space/on a manifold. The quadratic form then becomes a field of its own, operating on the tangent spaces of each point of the manifold independently but in a smoothly varying way. Arc length of curves can be calculated by integrating the norm of tangent vectors to the curve along it. In general relativity, this is the metric that is a dynamical variable as the solution to Einstein's equations. In order to follow the curvature of the manifold, vector fields are only locally vector valued functions, requiring transitions between regions as in the tribar example previously. Similarly, because the spin and orthogonal groups are so strongly coupled, we can apply the same transitions to spinor-valued functions. The physical reason to do this is that coordinate changes which affect vectors, thus also affect spinor fields in the same way. | |||
[[File:Penroseblackhole.jpg|thumb|Because the Minkowski inner product can be 0, the tangent vectors for which it is form a cone which Penrose depicts as smoothly varying with spacetime]] | |||
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The metric directly gives a way to differentiate vector fields, or finitely comparing the values at different points via parallel translation along geodesics (curves with minimal length given an initial point and velocity). Using this, a derivative operator can be given for spinor fields. It is usually written in coordinates with the gamma matrices: | |||
<math> Ds(x)=\sum_{\mu=1}^n\gamma_{\mu}\nabla_{e_{\mu}}s(x) </math> where the <math> \nabla_{e_{\mu}} </math> are the metric-given derivatives in the direction of an element of an orthonormal basis vector at x. Their difference from the coordinate partial derivatives helps to quantify the curvature. These orthonormal bases also vary with x, making a field of frames which like before can locally be identified with an <math> O(n,\mathbb{R}) </math>-valued function and globally (if it exists) defines an orientation of the manifold. | |||
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Dirac first wrote down the operator in flat space with partial derivatives instead of covariant derivatives, trying to find a first-order operator and an equation: | |||
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<math> (iD-m)s(x)=0 </math> | |||
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whose solutions are also solutions to the second order Klein-Gordon equation | |||
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<math> (D^2+m^2)s(x)=(\Delta+m^2)s(x)=0 </math> | |||
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But it was Atiyah who actually named the operator, and utilized its geometric significance. On a curved manifold it does not square to the Laplacian, but differs by the scalar curvature: | |||
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<math> D^2s(x)=\Delta s(x)+\frac{1}{4}Sc s(x) </math> This is known as the Lichnerowicz formula. | |||
== Notes == | == Notes == | ||