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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 16:43, 24 July 2023
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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 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 | ; Infinitesimally | ||
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 acted upon by the spin group to be constructed, and the Clifford algebra of the first real vector space. To distinguish between the first vector space and the second, the elements of the former are simply referred to as vectors and the latter as spinors. This may be confusing because mathematically both are vector spaces whose elements are vectors, however physically the vectors of the first space have a more basic meaning as directions in physical coordinate 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> | :;1) | ||
::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>. The two-input bilinear form <math> q(\cdot ,\cdot ) </math> is (often) concretely the dot product, and the original quadratic form is recovered by plugging in the same vector twice <math> q(v,v) </math>. The intermediate mathematical structure which keeps track of the formal products of the vectors is known as the tensor algebra <math> T(V)</math> and notably does not depend on <math> q </math>. It helps to interpret the meaning of the unit <math> 1 </math> in the relation as another formally adjoined symbol. Then, the resulting Clifford algebra with the given modified multiplication rule/relation depending on <math> q </math> is denoted by <math> \mathcal{Cl}(V,q)=T(V)/(v\cdot v-q(v)1) </math> as the quotient of <math> T(V) </math> by the subspace of expressions which we want to evaluate to 0. | |||
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::The Spin group is then found within the Clifford algebra, | |||
== Notes == | == Notes == | ||