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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 16:18, 24 July 2023
, 24 July 2023→Spinors
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Spinors have two main instantiations: the infinitesimal quantity usually in finite dimensions as the value of a vector field at a point, or as the vector field taken over a finite region of space(time). | Spinors have two main instantiations: the infinitesimal quantity usually in finite dimensions as the value of a vector field at a point, or as the vector field taken over a finite region of space(time). | ||
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For the infinitesimal quantity in finite dimensions, spinors are constructed via an algebraic means known as representation theory. The key fact that supports the construction is that the Lie groups of rotations <math> SO(n,\mathbb{R}) </math> acting on linear n-dimensional space admit double coverings <math> Spin(n) </math> such that two elements of the spin group correspond to a single rotation. | For the infinitesimal quantity in finite dimensions, spinors are constructed via an algebraic means known as representation theory. The key fact that supports the construction is that the special orthogonal Lie groups of rotations <math> SO(n,\mathbb{R}) </math> acting on linear n-dimensional space admit double coverings <math> Spin(n) </math> such that two elements of the spin group correspond to a single rotation. Five algebraic structures comprise the story here: an n-dimensional real vector space, the n-choose-2 dimensional rotation group acting on it, the spin group, another vector space whose elements are the spinors to be constructed, and the Clifford algebra of the first real vector space. | ||
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The Clifford algebra is constructed directly from the first vector space, by formally/symbolically multiplying vectors and simplifying these expressions according to the rule: <math> v\cdot v = -q(v)1 </math> where q is the length/quadratic form of the vector that the rotations must preserve, or utilizing the polarization identities of quadratic forms: <math> 2q(v,w)=q(v+w)-q(v)-q(w)\rightarrow v\cdot w + w\cdot v = -2q(v,w) </math> | |||
== Notes == | == Notes == | ||