Spinors


 * This article is College level

Spinors

 * See also:

External link:An introduction to spinors

, and the action of an even-graded element on a spinor is given by ordinary quaternionic multiplication.

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Spinors in three dimensions

 * From Spinors in three dimensions

The association of a spinor with a 2×2 complex was formulated by Élie Cartan.

In detail,



Matrices of this form have the following properties, which relate them intrinsically to the geometry of 3-space:
 * det X = – (length x)2.
 * X 2 = (length x)2I, where I is the identity matrix.
 * $$\frac{1}{2}(XY+YX)=({\bold x}\cdot{\bold y})I$$
 * $$\frac{1}{2}(XY-YX)=iZ$$ where Z is the matrix associated to the cross product z = x &times; y.
 * If u is a unit vector, then −UXU is the matrix associated to the vector obtained from x by reflection in the plane orthogonal to u.
 * It is an elementary fact from that any rotation in 3-space factors as a composition of two reflections.  (Similarly, any orientation reversing orthogonal transformation is either a reflection or the product of three reflections.)  Thus if R is a rotation, decomposing as the reflection in the plane perpendicular to a unit vector u1 followed by the plane perpendicular to u2, then

Column vectors
Having effectively encoded all of the rotational linear geometry of 3-space into a set of complex 2&times;2 matrices, it is natural to ask what role, if any, the 2&times;1 matrices (i.e., the ) play. Provisionally, a spinor is a column vector
 * $$\xi=\left[\begin{matrix}\xi_1\\\xi_2\end{matrix}\right],$$ with complex entries ξ1 and ξ2.

The space of spinors is evidently acted upon by complex 2&times;2 matrices. Furthermore, the product of two reflections in a given pair of unit vectors defines a 2&times;2 matrix whose action on euclidean vectors is a rotation, so there is an action of rotations on spinors.

Often, the first example of spinors that a student of physics encounters are the 2&times;1 spinors used in Pauli's theory of electron spin. The are a vector of three 2&times;2 that are used as.

Example: u = (0.8, -0.6, 0) is a unit vector. Dotting this with the Pauli spin matrices gives the matrix:



S_u = (0.8,-0.6,0.0)\cdot \vec{\sigma}=0.8 \sigma_{1}-0.6\sigma_{2}+0.0\sigma_{3} = \begin{bmatrix} 0.0 & 0.8+0.6i \\ 0.8-0.6i & 0.0 \end{bmatrix} $$

The eigenvectors may be found by the usual methods of, but a convenient trick is to note that a Pauli spin matrix is an, that is, the squareof the above matrix is the identity matrix.

Thus a (matrix) solution to the eigenvector problem with eigenvalues of ±1 is simply 1 ± Su. That is,



S_u (1\pm S_u) = \pm 1 (1 \pm S_u) $$

One can then choose either of the columns of the eigenvector matrix as the vector solution, provided that the column chosen is not zero. Taking the first column of the above, eigenvector solutions for the two eigenvalues are:



\begin{bmatrix} 1.0+ (0.0)\\ 0.0 +(0.8-0.6i) \end{bmatrix}, \begin{bmatrix} 1.0- (0.0)\\ 0.0-(0.8-0.6i) \end{bmatrix} $$

The trick used to find the eigenvectors is related to the concept of, that is,  The same trick works in any, in particular the that are discussed below. These projection operators are also seen in theory where they are examples of pure density matrices.



While the two columns appear different, one can use a2 + b2 + c2 = 1 to show that they are multiples (possibly zero) of the same spinor.

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Column vectors

 * From Spinor:

"Quote from Elie Cartan: The Theory of Spinors, Hermann, Paris, 1966: "Spinors...provide a linear representation of the group of rotations in a space with any number $n$ of dimensions, each spinor having $2^\nu$ components where $n = 2\nu+1$ or $2\nu$." The star (*) refers to Cartan 1913."

(Note: $$\nu$$ is the number of an object can have in n dimensions.)

Although spinors can be defined purely as elements of a representation space of the spin group (or its Lie algebra of infinitesimal rotations), they are typically defined as elements of a vector space that carries a linear representation of the Clifford algebra. The Clifford algebra is an associative algebra that can be constructed from Euclidean space and its inner product in a basis independent way. and in applications the Clifford algebra is often the easiest to work with. After choosing an orthonormal basis of Euclidean space, a representation of the Clifford algebra is generated by gamma matrices, matrices that satisfy a set of canonical anti-commutation relations. The spinors are the column vectors on which these matrices act.

In three Euclidean dimensions, for instance, the Pauli spin matrices are a set of gamma matrices, and the two-component complex column vectors on which these matrices act are spinors. However, the particular matrix representation of the Clifford algebra, hence what precisely constitutes a "column vector" (or spinor), involves the choice of basis and gamma matrices in an essential way.

In three Euclidean dimensions, for instance, spinors can be constructed by making a choice of Pauli spin matrices corresponding to (angular momenta about) the three coordinate axes. These are 2×2 matrices with complex entries, and the two-component complex column vectors on which these matrices act by matrix multiplication are the spinors. In this case, the spin group is isomorphic to the group of 2×2 unitary matrices with determinant one, which naturally sits inside the matrix algebra. This group acts by conjugation on the real vector space spanned by the Pauli matrices themselves, realizing it as a group of rotations among them, but it also acts on the column vectors (that is, the spinors).

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Electron spin

 * From Spinor:

. This is true both in the relativistic and non-relativistic theory. The wavefunction of the non-relativistic electron has values in 2 component spinors transforming under three-dimensional infinitesimal rotations. The relativistic for the electron is an equation for 4 component spinors transforming under infinitesimal Lorentz transformations for which a substantially similar theory of spinors exists.

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