Spinors in Three Dimensions - Formulation

Formulation

This algebra admits a convenient description, due to William Rowan Hamilton, by means of quaternions. In detail, given a vector x = (x₁, x₂, x₃) of real (or complex) numbers, one can associate the matrix of complex numbers:

Matrices of this form have the following properties, which relate them intrinsically to the geometry of 3-space:

• det X = - (length x)2.
• X2 = (length x)2I, where I is the identity matrix.
• where Z is the matrix associated to the cross product z = x × 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 linear algebra 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 the matrix U₂U₁XU₁U₂ represents the rotation of the vector x through R.

Having effectively encoded all of 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. Furthermore, the product of two reflections in a given 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. Clearly, if X → RXR-1 is a representation of a rotation, then replacing R by -R will yield the same rotation. In fact, one can easily show that this is the only ambiguity which arises. Thus the action of a rotation on a spinor is always double-valued.