The association of a spinor with a 2×2 complex traceless Hermitian matrix was formulated by Élie Cartan.
It is an elementary fact from linear algebra that any rotation in 3-space factors as a composition of two reflections.
Thus if R is a rotation which decomposes as the reflection in the plane perpendicular to a unit vector
To motivate this introduction of spinors, suppose that X is a matrix representing a vector x in complex 3-space.
Suppose further that x is isotropic: i.e., Then since the determinant of X is zero there is a proportionality between its rows or columns.
Thus a spinor may be viewed as an isotropic vector, along with a choice of sign.
Note that because of the logarithmic branching, it is impossible to choose a sign consistently so that (3) varies continuously along a full rotation among the coordinates x.
In spite of this ambiguity of the representation of a rotation on a spinor, the rotations do act unambiguously by a fractional linear transformation on the ratio ξ1:ξ2 since one choice of sign in the solution (3) forces the choice of the second sign.
In particular, the space of spinors is a projective representation of the orthogonal group.
As a consequence of this point of view, spinors may be regarded as a kind of "square root" of isotropic vectors.
When the space is real, however, spinors possess some additional structure which in turn facilitates a complete description of the representation of the rotation group.
Suppose, for simplicity, that the inner product on 3-space has positive-definite signature: With this convention, real vectors correspond to Hermitian matrices.
Furthermore, real rotations preserving the form (4) correspond (in the double-valued sense) to unitary matrices of determinant one.
In modern terms, this presents the special unitary group SU(2) as a double cover of SO(3).
As a consequence, the spinor Hermitian product is preserved by all rotations, and therefore is canonical.
If, however, the signature of the inner product on 3-space is indefinite (i.e., non-degenerate, but also not positive definite), then the foregoing analysis must be adjusted to reflect this.
Suppose then that the length form on 3-space is given by: Then the construction of spinors of the preceding sections proceeds, but with
Instead, the anti-Hermitian form defines the appropriate notion of inner product for spinors in this metric signature.
This form is invariant under transformations in the connected component of the identity of O(2,1).
The fact that this is a quartic invariant, rather than quadratic, has an important consequence.
If one confines attention to the group of special orthogonal transformations, then it is possible unambiguously to take the square root of this form and obtain an identification of spinors with their duals.
The differences between these two signatures can be codified by the notion of a reality structure on the space of spinors.
The eigenvectors of that spin matrix are the spinors for spin-1/2 oriented in the direction given by the vector.
Thus a (matrix) solution to the eigenvector problem with eigenvalues of ±1 is simply 1 ± Su.
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.
In atomic physics and quantum mechanics, the property of spin plays a major role.
, of a particle corresponds to the sum of the orbital angular momentum (i.e., there only integers are allowed) and the intrinsic part, the spin.