Spinor | Wikipedia audio article
Channel: wikipedia tts
Date: 25/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Spinor


00:03:37 1 Introduction
00:08:01 2 Mathematical definition
00:08:45 2.1 Overview
00:10:21 2.2 Clifford algebras
00:11:03 2.3 Spin groups
00:11:15 2.4 Terminology in physics
00:11:44 2.5 Spinors in representation theory
00:12:10 2.6 Attempts at intuitive understanding
00:13:31 3 History
00:15:27 4 Examples
00:17:22 4.1 Two dimensions
00:18:22 4.2 Three dimensions
00:20:04 5 Explicit constructions
00:21:55 5.1 Component spinors
00:22:50 5.2 Abstract spinors
00:23:11 5.3 Minimal ideals
00:23:19 5.4 Exterior algebra construction
00:27:17 5.5 Hermitian vector spaces and spinors
00:34:07 6 Clebsch–Gordan decomposition
00:34:34 6.1 Even dimensions
00:34:42 6.2 Odd dimensions
00:34:55 6.3 Consequences
00:35:08 7 Summary in low dimensions
00:42:07 8 See also
00:42:36 9 Notes
00:43:00 10 References
00:43:11 11 Further reading
00:43:31 2ημν1 (i.e. fix a convention for the gamma matrices). Then the assignment eμ → γμ extends uniquely to an algebra homomorphism Cℓ(V, g) → Mat(2k, C) by sending the monomial eμ1 … eμk in the Clifford algebra to the product γμ1 … γμk of matrices and extending linearly. The space Δ
00:45:24 Abstract spinors
00:47:11 0. If n
00:47:27 2k + 1 is odd, let W∗ be a maximal isotropic subspace with W ∩ W∗
00:47:58 Minimal ideals
00:49:01 0 (i.e., ω is nilpotent of order 2), and moreover, w′ω
00:49:22 2k, then the left ideal Δ
00:49:40 Cℓevenω and Δ−
00:50:43 0, (wi′)2
00:53:35 Exterior algebra construction
00:54:36 W ⊕ W′ where W′ is the chosen isotropic complement. Hence any v ∈ V decomposes uniquely as v
00:57:28 Hermitian vector spaces and spinors
00:58:52 Clebsch–Gordan decomposition
01:00:23 Even dimensions
01:05:57 Odd dimensions
01:07:26 Consequences
01:08:22 Summary in low dimensions
01:11:09 See also



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SUMMARY
=======
In geometry and physics, spinors  are elements of a (complex) vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a  slight (infinitesimal) rotation.   When a sequence of such small rotations is composed (integrated) to form an overall final rotation, however, the resulting spinor transformation depends on which sequence of small rotations was used: unlike  vectors and tensors,  a spinor transforms to its negative when the space is rotated through a complete turn from 0° to 360° (see picture). This property characterizes spinors.  It is also possible to associate a substantially similar notion of spinor to Minkowski space in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geometry by Élie Cartan  in 1913.  In the 1920s physicists  discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron and other subatomic particles.Spinors are characterized by the specific way in which they behave under rotations.  They change in different ways depending not just on the overall final rotation, but the details of how that rotation was achieved (by a continuous path in the rotation group).  There are two topologically distinguishable classes (homotopy classes) of paths through rotations that result in the same overall rotation, as famously illustrated by the belt trick puzzle (below).  These two inequivalent classes yield spinor transformations of opposite sign.  The spin group is the group of all rotations keeping track of the class. It doubly covers the rotation group, since each rotation can be obtained in two inequivalent ways as the endpoint of a path. The space of spinors by definition is equipped with a (complex) linear representation of the spin group, meaning that elements of the spin group act as linear transformations on the space of ...