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 traceless 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
\vec{x} → X =\left(\begin{matrix}x3&x1-ix2\\x1+ix2&-x3\end{matrix}\right).
In physics, this is often written as a dot product
X\equiv{\vec\sigma} ⋅ {\vecx}
{\vec\sigma}\equiv(\sigma1,\sigma2,\sigma3)
\detX=-|{\vecx}|2
\det
X2=|{\vecx}|2I
1 | |
2 |
(XY+YX)=({\vecx} ⋅ {\vecy})I
1 | |
2 |
(XY-YX)=iZ
{\vecz}={\vecx} x {\vecy}
{\vecu}
-UXU
{\vecx}
{\vecu}
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
{\vecu}1
{\vecu}2
U2U1XU1U2
{\vecx}
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
\xi=\left[\begin{matrix}\xi1\\\xi2\end{matrix}\right],
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. Clearly, if
X\mapstoRXR-1
{\vecx} ⋅ {\vec\sigma}.
Michael Stone and Paul Goldbar, in Mathematics for Physics, contest this, saying, "The spin representations were discovered by ´Elie Cartan in 1913, some years before they were needed in physics."
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.,
2=0. | |
{x} ⋅ {x}=x | |
3 |
X=2\left[\begin{matrix}\xi1\\\xi2\end{matrix}\right]\left[\begin{matrix}-\xi2&\xi1\end{matrix}\right].
}
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
C=\left(\begin{matrix}0&1\\-1&0\end{matrix}\right),
A fortiori, if the roles of ξ and x are now reversed, the form Q(ξ) = 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(μ, ξ). This bilinear form then transforms tensorially under a reflection or a rotation.
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 (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
x2
i
x2
(x1,x2,x3)
\left(\begin{matrix}x3&x1-x2\\x1+x2&-x3\end{matrix}\right)
\langle\mu|\xi\rangle=\bar{\mu}1\xi2-\bar{\mu}2\xi1
In either case, the quartic form
\langle\mu|\xi\rangle2=\hbox{length}\left(Q(\bar{\mu},\xi)\right)2
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
\xi*=K\bar{\xi}.
The particular form of the inner product on vectors (e.g., or) determines a reality structure (up to a factor of -1) by requiring
\bar{X}=KXK
See also: quaternions and spatial rotation.
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 matricesthat are used as spin operators.
Given a unit vector in 3 dimensions, for example (a, b, c), one takes adot product with the Pauli spin matrices to obtain a spin matrix forspin in the direction of the unit vector.
The eigenvectors of that spin matrix are the spinors forspin-1/2 oriented in the direction given by the vector.
Example: u = (0.8, -0.6, 0) is a unit vector. Dotting this with the Paulispin matrices gives the matrix:
Su=(0.8,-0.6,0.0) ⋅ \vec{\sigma}=0.8\sigma1-0.6\sigma2+0.0\sigma3=\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 linear algebra, but a convenient trickis 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 (matrix) solution to the eigenvector problem with eigenvalues of±1 is simply 1 ± Su. That is,
Su(1\pmSu)=\pm1(1\pmSu)
One can then choose either of the columns of the eigenvector matrix as the vector solution, provided that the column chosenis 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 ofideals, that is, the matrix eigenvectors (1 ± Su)/2 are projection operators or idempotents and therefore each generates anideal in the Pauli algebra. The same trickworks in any Clifford algebra, in particularthe Dirac algebra that is discussed below. These projectionoperators are also seen in density matrix theorywhere they are examples of pure density matrices.
More generally, the projection operator for spin in the (a, b, c) directionis given by
1 | |
2 |
\begin{bmatrix}1+c&a-ib\\a+ib&1-c\end{bmatrix}
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(r), one has with spin: ψ = ψ(r, σ), where σ takes the following discrete set of values:
\sigma=-S ⋅ \hbar,-(S-1) ⋅ \hbar,...,+(S-1) ⋅ \hbar,+S ⋅ \hbar
\vec{J}
M2(C) ⊗ R3
⋅ :R3 x M2(C) ⊗ R\toM2(C)