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Concept
Spinors in three dimensions
Résumé
In mathematics, the spinor concept as specialised to three dimensions can be treated by means of the traditional notions of dot product and cross product. This is part of the detailed algebraic discussion of the rotation group SO(3).
The association of a spinor with a 2×2 complex Hermitian matrix was formulated by Élie Cartan.
In detail, given a vector x = (x1, x2, x3) of real (or complex) numbers, one can associate the complex matrix
In physics, this is often written as a dot product , where is the vector form of Pauli matrices. Matrices of this form have the following properties, which relate them intrinsically to the geometry of 3-space:
where denotes the determinant.
where I is the identity matrix.
where Z is the matrix associated to the cross product .
If is a unit vector, then is the matrix associated with the vector that results from reflecting in the plane orthogonal to .
The last property can be used to simplify rotational operations. It is an elementary fact from linear algebra that any rotation in 3-space factors as a composition of two reflections. (More generally, any orientation-reversing orthogonal transformation is either a reflection or the product of three reflections.) Thus if R is a rotation which decomposes as the reflection in the plane perpendicular to a unit vector followed by the reflection in the plane perpendicular to , then the matrix represents the rotation of the vector through R.
Having effectively encoded all the rotational linear geometry of 3-space into a set of complex 2×2 matrices, it is natural to ask what role, if any, the 2×1 matrices (i.e., the column vectors) play. Provisionally, a spinor is a column vector
with complex entries ξ1 and ξ2.
The space of spinors is evidently acted upon by complex 2×2 matrices. As shown above, the product of two reflections in a pair of unit vectors defines a 2×2 matrix whose action on euclidean vectors is a rotation. So there is an action of rotations on spinors. However, there is one important caveat: the factorization of a rotation is not unique.
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