Spinors in three dimensions


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.

Formulation

The association of a spinor with a 2×2 complex Hermitian matrix was formulated by Élie Cartan.
In detail, given a vector x = of real 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:
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 play. Provisionally, a spinor is a column vector
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.
There were some precursors to Cartan's work with 2×2 complex matrices: Wolfgang Pauli had used these matrices so intensively that elements of a certain basis of a four-dimensional subspace are called Pauli matrices σi, so that the Hermitian matrix is written as a Pauli vector In the mid 19th century the algebraic operations of this algebra of four complex dimensions were studied as biquaternions.
As per the book, by Michael Stone and Paul Goldbar, Mathematics for Physics, "The spin representations were discovered by ´Elie Cartan in 1913, some years before they were needed in physics.", thus contradicting the above statement on the precursor to Cartan's work as being done by Pauli.

Isotropic vectors

Spinors can be constructed directly from isotropic vectors in 3-space without using the quaternionic construction. 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 the matrix may be written as an outer product of two complex 2-vectors:
This factorization yields an overdetermined system of equations in the coordinates of the vector x:
subject to the constraint
This system admits the solutions
Either choice of sign solves the system. 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 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 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. Specifically, introducing the matrix
the system is equivalent to solving X = 2 ξ tξ C for the undetermined spinor ξ.
A fortiori, if the roles of ξ and x are now reversed, the form Q = x defines, for each spinor ξ, a vector x quadratically in the components of ξ. If this quadratic form is polarized, it determines a bilinear vector-valued form on spinors Q. This bilinear form then transforms tensorially under a reflection or a rotation.

Reality

The above considerations apply equally well whether the original euclidean space under consideration is real or complex. 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 correspond to unitary matrices of determinant one. In modern terms, this presents the special unitary group SU as a double cover of SO. 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, 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 x2 replacing i x2 in all the formulas. With this new convention, the matrix associated to a real vector is itself real:
The form is no longer invariant under a real rotation, since the group stabilizing is now a Lorentz group O. 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.
In either case, the quartic form
is fully invariant under O, where Q is the vector-valued bilinear form described in the previous section. 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. In the language of representation theory, this implies that there is only one irreducible spin representation of SO up to isomorphism. If, however, reversals are also allowed, then it is no longer possible to identify spinors with their duals owing to a change of sign on the application of a reflection. Thus there are two irreducible spin representations of O, sometimes called the pin representations.

Reality structures

The differences between these two signatures can be codified by the notion of a reality structure on the space of spinors. Informally, this is a prescription for taking a complex conjugate of a spinor, but in such a way that this may not correspond to the usual conjugate per the components of a spinor. Specifically, a reality structure is specified by a Hermitian 2 × 2 matrix K whose product with itself is the identity matrix: K2 = Id. The conjugate of a spinor with respect to a reality structure K is defined by
The particular form of the inner product on vectors or ) determines a reality structure by requiring
Thus K = i C is the reality structure in Euclidean signature, and K = Id is that for signature. With a reality structure in hand, one has the following results:

Spinors of the Pauli spin matrices

Often, the first example of spinors that a student of physics
encounters are the 2×1 spinors used in Pauli's theory of electron spin.
The Pauli matrices are a vector of three 2×2 matrices
that are used as spin operators.
Given a unit vector in 3 dimensions, for example, one takes a
dot product with the Pauli spin matrices to obtain a spin matrix for
spin in the direction of the unit vector.
The eigenvectors of that spin matrix are the spinors for
spin-1/2 oriented in the direction given by the vector.
Example: u = is a unit vector. Dotting this with the Pauli
spin matrices gives the matrix:
The eigenvectors may be found by the usual methods of
linear algebra, but a convenient trick
is to note that a Pauli spin matrix is an involutory matrix, that is, the square of the above matrix is the identity matrix.
Thus a solution to the eigenvector problem with eigenvalues of
±1 is simply 1 ± Su. That is,
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:
The trick used to find the eigenvectors is related to the concept of
ideals, that is, the matrix eigenvectors /2 are projection operators or idempotents and therefore each generates an
ideal in the Pauli algebra. The same trick
works in any Clifford algebra, in particular
the Dirac algebra that is discussed below. These projection
operators are also seen in density matrix theory
where they are examples of pure density matrices.
More generally, the projection operator for spin in the direction
is given by
and any non zero column can be taken as the projection operator. While the
two columns appear different, one can use a2 + b2 + c2 = 1 to show that they are multiples of the same spinor.

General remarks

In atomic physics and quantum mechanics, the property of spin plays a major role. In addition to their other properties all particles possess a non-classical property, i.e., which has no correspondence at all in conventional physics, namely the spin, which is a kind of intrinsic angular momentum. In the position representation, instead of a wavefunction without spin, ψ = ψ, one has with spin: ψ = ψ, where σ takes the following discrete set of values:
The total angular momentum operator,, of a particle corresponds to the sum of the orbital angular momentum and the intrinsic part, the spin. One distinguishes bosons and fermions.