Spinors are a mathematical abstraction related to vectors. This document describes the processes for defining what a spinor is.
The Kronecker delta \(\delta_{ii} = 1\), \(\delta_{ij} = 0\: i \neq j\).
The Levi-Civita symbol \(\epsilon_{ijk} = 0\: if\: i = j,\, or\, j = k,\, or\, k = i\).
\[\epsilon_{123} = \epsilon_{231} = \epsilon_{312} = 1\] \[\epsilon_{321} = \epsilon_{132} = \epsilon_{213} = -1\]
The mathematics of General Relativity is differential geometry. The mathematics of Quantum Theory is algebra using group theory.
A group is a collection of objects that share a symmetry called elements. If an element \(x\) is a member of group \(G\) is written \(x \in G\). Groups have composition operators, such as multiplication, that must obey rules for the collection to be a group. Groups are classified according the the type of symetry they represent.
An \(n x n\) square matrix \(A\) have an eigenvector \(\vec{v}\) and an eigenvalue \(\lambda\), that satisfy the eigenvector equation.
\[A\vec{v} = \lambda\vec{v}\]
The eigenvector equation can be rewritten in terms of the identity matrix \(I\).
\[(A - \lambda I)\vec{v} = 0\]
This can be solved using a determinant.
\[|A - \lambda I| = 0\]
If \(A\) is a square matrix then it makes sense to have powers of the matrix \(A^n\). Where \(A^0 = I\). It also makes sense to have a matrix exponential.
\[e^{A} = \sum_{n=0}^\infty\frac{A^n}{n!}\]
If the matrix is diagonal:
\[ A = \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} , A^2 = \begin{bmatrix} a^2 & 0 \\ 0 & b^2 \end{bmatrix} ... \]
\[e^{A} = \sum_{n=0}^\infty\frac{1}{n!} \begin{bmatrix} a^n & 0 \\ 0 & b^n \end{bmatrix} = \begin{bmatrix} e^a & 0 \\ 0 & e^b \end{bmatrix} \]
If the matrix is anti-diagonal and symmetrical:
\[ A = \begin{bmatrix} 0 & a \\ a & 0 \end{bmatrix} , A^2 = \begin{bmatrix} a^2 & 0 \\ 0 & a^2 \end{bmatrix} , A^3 = \begin{bmatrix} 0 & a^3 \\ a^3 & 0 \end{bmatrix} ... \]
\[e^{A} = \sum_{n=0}^\infty\frac{a^{2n}}{2n!}I + \frac{a^{2n}}{(2n+1)!}A = \begin{bmatrix} \cosh{a} & \sinh{a} \\ \sinh{a} & \cosh{a} \end{bmatrix} \]
If the matrix is anti-diagonal and anti-symmetrical:
\[ A = \begin{bmatrix} 0 & -a \\ a & 0 \end{bmatrix} , A^2 = \begin{bmatrix} -a^2 & 0 \\ 0 & -a^2 \end{bmatrix} , A^3 = \begin{bmatrix} 0 & a^3 \\ -a^3 & 0 \end{bmatrix} , A^4 = \begin{bmatrix} a^4 & 0 \\ 0 & a^4 \end{bmatrix} ... \]
\[e^{A} = \sum_{n=0}^\infty\frac{(-1)^na^{2n}}{2n!}I + \frac{(-1)^na^{2n}}{(2n+1)!}A = \begin{bmatrix} \cos{a} & -\sin{a} \\ \sin{a} & \cos{a} \end{bmatrix} \]
This will be needed later.
\[ A = \begin{bmatrix} 0 & ia \\ ia & 0 \end{bmatrix} , A^2 = \begin{bmatrix} -a^2 & 0 \\ 0 & -a^2 \end{bmatrix} , A^3 = \begin{bmatrix} 0 & -ia^3 \\ -ia^3 & 0 \end{bmatrix} , A^4 = \begin{bmatrix} a^4 & 0 \\ 0 & a^4 \end{bmatrix} ... \]
\[e^{A} = \sum_{n=0}^\infty\frac{(-1)^na^{2n}}{2n!}I + \frac{(-1)^na^{2n}}{(2n+1)!}A = \begin{bmatrix} \cos{a} & i\sin{a} \\ i\sin{a} & \cos{a} \end{bmatrix} \]
The matrix commutator \([X, Y]\) of matrices \(X\) and \(Y\) is \([X, Y] = XY - YX\).
The matrix anti-commutator \(\{X, Y\}\) of matrices \(X\) and \(Y\) is \(\{X, Y\} = XY + YX\).
Consider rotations in the four dimensions \([ct, x, y, z]^T\). Rotations about \(x\), \(y\), and \(z\) are quite straightforward. When it comes to rotations about any other axis, it requires combining several rotations. The matrix terms can quickly become very complex. There is a better solution using an infinitesimal generator. This uses rotations of a very small value. Products of infinitesimals can be ignored as too small. An infinitesimal can be integrated to get the required result.
A rotation about \(x\) is defined by the matrix:
\[ R_x(\theta) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos\theta & \sin\theta \\ 0 & 0 & -\sin\theta & \cos\theta \end{bmatrix} \]
In this rotation replace \(\theta\) with the infinitesimal \(d\theta\). Then \(\cos d\theta = 1\) and \(\sin d\theta = d\theta\).
\[ R_x(d\theta) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & d\theta \\ 0 & 0 & -d\theta & 1 \end{bmatrix} = I + \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \end{bmatrix} d\theta \]
Now call the right hand matrix \(A_x\), so, \(R_x(d\theta) = I + A_xd\theta\).
Now, we define a vector \(u\) and its infinitesimal \(d\mathbf{u}\) caused by an infinitesimal rotation.
\[\mathbf{u} + d\mathbf{u} = R_x(d\theta)\mathbf{u} = (I + A_xd\theta)\mathbf{u}\]
\[d\mathbf{u} = A_xd\theta\mathbf{u}\]
\[\frac{d\mathbf{u}}{d\theta} = A_x\mathbf{u}\]
This differential equation can easily be integrated.
\[\mathbf{u} = \mathbf{u_0}e^{A_x\theta}\]
Where \(\mathbf{u_0}\) is a constant of integration.
As \(A_x\) is an anti-diagonal and anti-symmetrical matrix, we know that \(e^{A_x\theta} = R_x(\theta)\) and \(A_x\) is an infinitesimal generator.
We need to make a small change by defining a generator matrix \(J_x\) where \(A_x = iJ_x\). Then \(e^{iJ_x\theta} = R_x(\theta)\).
Applying the same process to the \(y\) and \(z\) rotations we have the following rotation and J matrices.
\[ R_x(\theta) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos\theta & \sin\theta \\ 0 & 0 & -\sin\theta & \cos\theta \end{bmatrix} \: J_x = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \end{bmatrix} \]
\[ R_y(\theta) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\theta & 0 & -\sin\theta \\ 0 & 0 & 1 & 0 \\ 0 & \sin\theta & 0 & \cos\theta \end{bmatrix} \: J_y = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & i \\ 0 & 0 & 0 & 0 \\ 0 & -i & 0 & 0 \end{bmatrix} \]
\[ R_z(\theta) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\theta & \sin\theta & 0 \\ 0 & -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \: J_z = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & -i & 0 \\ 0 & i & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \]
These \(J\) matrices satisfy the commutation \([J_i, J_j] = i\epsilon_{ijk}J_k\).
Now consider the Lorentz transformations. These are called boosts rather than rotations because they use hyperbolic functions rather than trigonometric functions. To simplify the equatiosn we define two variable \(\beta\) and \(\gamma\).
\[\beta = \frac{v}{c}\: \gamma = (1 - \beta^2)^{-\frac{1}{2}}\]
Then we can define the boost in the \(x\) direction.
\[ B_x = \begin{bmatrix} \gamma & \beta\gamma & 0 & 0 \\ \beta\gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \]
The determinant of the matrix is \(\gamma^2 - \beta^2\gamma^2 = 1\). So, we can set \(\cosh\phi = \gamma\) and \(\sinh\phi = \beta\gamma\).
\[ B_x = \begin{bmatrix} \cosh\phi & \sinh\phi & 0 & 0 \\ \sinh\phi& \cosh\phi & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \]
The rotational matrices were anti-symmetrical whereas the boosts are symmetrical. So, we can derive the infinitesimal boost matrices \(K_i\) in the same way that we derived the infinitesimal rotation matrices \(J_i\).
\[ B_x = \begin{bmatrix} \cosh\phi & \sinh\phi & 0 & 0 \\ \sinh\phi & \cosh\phi & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \: K_x = \begin{bmatrix} 0 & -i & 0 & 0 \\ -i & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \]
\[ B_y = \begin{bmatrix} \cosh\phi & 0 & \sinh\phi & 0 \\ 0 & 1 & 0 & 0 \\ \sinh\phi & 0 & \cosh\phi & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \: K_y = \begin{bmatrix} 0 & 0 & -i & 0 \\ 0 & 0 & 0 & 0 \\ -i & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \]
\[ B_z = \begin{bmatrix} \cosh\phi & 0 & 0 & \sinh\phi \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \sinh\phi & 0 & 0 & \cosh\phi \end{bmatrix} \: K_z = \begin{bmatrix} 0 & 0 & 0 & -i \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ -i & 0 & 0 & 0 \end{bmatrix} \]
As \(K_x\) is an anti-diagonal and symmetrical matrix, we know that \(e^{iK_x\theta} = B_x(\theta)\) and \(K_x\) is an infinitesimal generator.
Note that there is a relationship between the \(K\) and \(J\) matrices.
\[[K_i, K_j] = -i\epsilon_{ijk}J_k\]
A Hermitian matrix is a square complex matrix that is its own conjugate transpose.
\[a_{ij} = \bar{a}_{ji}\]
Wolfgang Pauli defined a set of three complex 2 x 2 matrices. Any Hermitian 2x2 matrix can be written as a combination of the Pauli Matrices with real coefficients. The identity matrix I is sometimes referred to as the zeroth Pauli matrix \(\sigma_0\). They form a quaternion.
\[ \sigma_0 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \quad \sigma_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \quad \sigma_2 = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} \quad \sigma_3 = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \]
The three matrices (excluding $_0) can be compacted into a single expression using the Kronecker delta.
\[ \sigma_j = \begin{bmatrix} \delta_{j3} & \delta_{j1} - i\delta_{j2} \\ \delta_{j1} + i\delta_{j2} & -\delta_{j3} \end{bmatrix} \]
The matrices are involutory, meaning that each matrix is its own inverse.
\[ \sigma_1^2 = \sigma_2^2 = \sigma_3^2 = i\sigma_1 \sigma_2 \sigma_3 = I \]
The matrices also commute and anti-commute.
\[[\sigma_i, \sigma_j] = 2i\epsilon_{ijk}\sigma_k\]
\[\{\sigma_i, \sigma_j\} = 2\delta_{ij}\sigma_0\]
Consider a matrix \(U\) of two complex numbers \(a\) and \(b\). The complex conjugates are \(a^*\) and \(b^*\).
\[ U = \begin{bmatrix} a & b \\ -b^* & a^* \end{bmatrix} \]
If two such matrices are multiplied they form a 4-dimensional vector space or a quaternion. This is 4-dimensional as there are only two independent entries in each matrix, but each entry is from the 2-dimensional set of complex numbers! This is called the doubling procedure.
\[ \begin{bmatrix} a & b \\ -b^* & a^* \end{bmatrix} \begin{bmatrix} c & d \\ -d^* & c^* \end{bmatrix} = \begin{bmatrix} ac - bd^* & ad + bc^* \\ -b^*c - a^*d^* & -b^*d + a^*c^* \end{bmatrix} \]
Each of the matrices only has two variables. Only the first column is required as the second column can easily be generated from the first.
\[\vec{u} = \begin{bmatrix} a \\ -b^* \end{bmatrix} \]
This is a spinor. It is actually a form of rotation.
We have seen the complex matrix \(U\) with elements \(a\) and \(b\) and their complex conjugates.
\[ U = \begin{bmatrix} a & b \\ -b^* & a^* \end{bmatrix} \]
Its conjugate transpose \(U^*\) or \(U^\dagger\), which if it is unitary is also its inverse \(U^{-1}\).
\[ U^\dagger = \begin{bmatrix} a^* & -b \\ b^* & a \end{bmatrix} \]
Unitary requires that \(aa^* + bb^* = 1\).
The conjugate transpose is also called the Hermitian Conjugate, or the Hermitian Adjoint.
\[U^\dagger U = UU^\dagger = I\]
We are going to use the Pauli matrices to transform cartesian spacetime coordinates. The spacetime vector is.
\[ x = \begin{bmatrix} ct \\ x \\ y \\ z \end{bmatrix} \]
Now, transform \(x\) using the Pauli matrices.
\[H = x^\mu\sigma_\mu = ct\sigma_0 + x\sigma_1 + y\sigma_2 + z\sigma_2\]
\[ H = \begin{bmatrix} ct + z & x - iy \\ x + iy & ct - z \end{bmatrix} \]
The determinant is the invariant \(c^2t^2 - x^2 - y^2 - z^2\). We need to find a unitary matrix \(U\) that transforms \(H\) to \(H' = UHU^\dagger\).
\[ \begin{bmatrix} ct' + z' & x' -iy' \\ x' + iy' & ct' - z' \end{bmatrix} = \begin{bmatrix} a & b \\ -b^* & a^* \end{bmatrix} \begin{bmatrix} ct + z & x - iy \\ x + iy & ct - z \end{bmatrix} \begin{bmatrix} a^* & -b \\ b^* & a \end{bmatrix} \]
Mutiply the matrices.
\[ \begin{bmatrix} ct' + z' & x' -iy' \\ x' + iy' & ct' - z' \end{bmatrix} = \begin{bmatrix} cat + az + bx + iby & ax - iay + bct - bz \\ -cb^*t - b^*z + a^*x + ia^*y & -b^*x + ib^*y + ca^*t - a^*z \end{bmatrix} \begin{bmatrix} a^* & -b \\ b^* & a \end{bmatrix} \]
Expanding given \(aa^* + bb^* = 1\).
\[ct' + z' = caa^*t + aa^*z + a^*bx + ia^*by + ab^*x - iab^*y + bb^*ct - bb^*z = ct + x(a^*b + ab^*) + iy(a^*b - ab^*) + z(aa^* - bb^*)\] \[ct' - z' = cb^*bt + b^*bz -a^*bx -ia^*by - ab^*x + iab^*y +ca^*at - a^*az = ct - x(a^*b + ab^*) - iy(a^*b - ab^*) - z(a^*a - b^*b)\] \[x' - iy' = -cabt - abz - b^2x - ib^2y + a^2x - ia^2y + cabt - abz = x(aa - bb) - iy(aa + bb) - 2zab\] \[x' + iy' = -ca^*b^*t - a^*b^*z + a^*a^*x + ia^*a^*y - b^*b^*x + ib^*b^*y + ca^*b^*t - a^*b^*z = x(a^*a^* - b^*b^*) + iy(a^*a^* + b^*b^*) - 2za^*b^*\]
Adding and subtracting pairs of terms and divide by \(2\).
\[ct' = ct\] \[x' = \frac{1}{2}x(aa + a^*a^* - bb - b^*b^*) - \frac{1}{2}iy(aa - a^*a^* + bb - b^*b^*) - z(ab + a^*b^*)\] \[iy' = -\frac{1}{2}x(aa - a^*a^* + bb - b^*b^*) + \frac{1}{2}iy(aa + a^*a^* + bb + b^*b^*) + z(ab - a^*b^*)\] \[z' = x(a^*b + ab^*) + iy(a^*b - ab^*) + z(aa^* - bb^*)\]
There is no transformation in the time dimension. This is unfortunate as the process won’t work for Lorentz transforms.
if \(a = \cos\frac{\theta}{2}\) and \(b = i\sin\frac{\theta}{2}\) then \(a^* = \cos\frac{\theta}{2}\) and \(b^* = -i\sin\frac{\theta}{2}\).
\[t' = t\] \[x' = x\] \[y' = y\cos\theta + z\sin\theta\] \[z' = -y\sin\theta + z\cos\theta\]
This describes a rotation about the \(x\) axis.
\[U_1 = \begin{bmatrix} \cos \frac{\theta}{2} & i\sin \frac{\theta}{2} \\ i\sin \frac{\theta}{2} & \cos \frac{\theta}{2} \end{bmatrix} = e^{\frac{1}{2}i\sigma_1\theta} \]
if \(a = \cos\frac{\theta}{2}\) and \(b = \sin\frac{\theta}{2}\) then \(a^* = \cos\frac{\theta}{2}\) and \(b^* = \sin\frac{\theta}{2}\).
\[t' = t\] \[x' = x\cos\theta - z\sin\theta\] \[y' = y\] \[z' = x\sin\theta + z\cos\theta\]
This describes a rotation about the \(y\) axis.
\[U_2 = \begin{bmatrix} \cos \frac{\theta}{2} & \sin \frac{\theta}{2} \\ -\sin \frac{\theta}{2} & \cos \frac{\theta}{2} \end{bmatrix} = e^{\frac{1}{2}i\sigma_2\theta} \]
If \(a = e^{\frac{i\theta}{2}}\) and \(b = 0\) then \(a^* = e^{-\frac{i\theta}{2}}\) and \(b^* = 0\).
\[t' = t\] \[x' = x\cos\theta + y\sin\theta\] \[y' = -x\sin\theta + y \cos\theta\] \[z' = z\]
This describes a rotation about the \(z\) axis.
\[U_3 = \begin{bmatrix} e^{\frac{\theta}{2}} & 0 \\ 0 & e^{-\frac{\theta}{2}} \end{bmatrix} = e^{\frac{1}{2}i\sigma_3\theta} \]
We now have the unitary transformation matrices.
\[U_k = e^{\frac{1}{2}i\sigma_k\theta}\]
Note the significance of the half angle. In vector space a rotation of \(2\pi\) returns a vector to its original direction. In this space, a rotation of \(4\pi\) is required to return the vector to its original direction. It is like a vector on a Möbius strip; it has to traverse the strip twice to get to its origial position.
The rotation \(J\) matrices satisfy a commutation.
\[[J_i, J_j] = i\epsilon_{ijk}J_k\]
The Lorentz boosts and rotation matrices have a relationship between the \(K\) and \(J\) matrices.
\[[K_i, K_j] = -i\epsilon_{ijk}J_k\]
These two relationships have a different right hand side. Multiply the \(K\) matrices by \(\pm i\).
\[[\pm iK_i, \pm iK_j] = i\epsilon_{ijk}J_k\]
This means that two boosts about different axes becomes a rotation about the third.
Now define two generators \(A\) and \(B\).
\[A_i = \frac{J_i + iK_i}{2};\: B_i = \frac{J_i - iK_i}{2}\]
These have commutations.
\[[A_i, A_j] = i\epsilon_{ijk}A_k\] \[[B_i, B_j] = i\epsilon_{ijk}B_k\] \[[A_i, B_j] = 0\]
All of these have an algebra \(SO(4)\).
The exisitng generators are \(SO(4)\). The covering group \(SL(2)\) using Pauli Matrices can cover the \(SO(4)\) group as they both have two elements. First define an algebra.
\[\bigg(\frac{\sigma_1}{2}, \frac{\sigma_2}{2}, \frac{\sigma_3}{2}, \frac{i\sigma_1}{2}, \frac{i\sigma_2}{2}, \frac{i\sigma_3}{2}, \bigg) \equiv (j_1, j_2, j_3, k_1, k_2, k_3)\]
The Pauli matrices have the relations:
\[\bigg[\frac{1}{2}\sigma_i, \frac{1}{2}\sigma_j\bigg] = \frac{1}{2}i\epsilon_{ijk}\sigma_k\]
Hence:
\[[j_i, j_j] = i\epsilon_{ijk}j_k,\: [j_i, k_j] = i\epsilon_{ijk}k_k,\:[k_i, k_j] = i\epsilon_{ijk}j_k\]
These are the same algebras as for the \(J\) and \(K\) generators. This means that there is a mapping between \(J_i\) and \(j_i\) and between \(K_i\) and \(k_i\).
A spinor is often defined, somewhat cryptically, as a two-component, vector-like quantities with special transformation properties. We need to clarify what “vector-like” and “special transformation properties” actually mean.
The term “vector like” simply means that it takes the form of a two-vector.
\[ \mathbf{\xi} = \begin{bmatrix} \xi_1 \\ \xi_2 \end{bmatrix} \]
A two-component vector can be transformed using one or more 2 x 2 matrices. Transformations are often used to change the vector into another vector using scaling, rotation, and translation. Another, and in this case more useful, transformation is from one coordinate system into another \(\mathbf{\xi \rightarrow \xi'}\).
We have derived a number of algebras that commutate in the same way and that there are mappings between them. One such algebra combines rotations and boosts in the form \(e^{J_i \pm iK_i}\). This algebra can now be mapped to Pauli matrices. We have \(J_k \rightarrow j_k \rightarrow \frac{1}{2}i\sigma_k\theta\) and \(iK_k \rightarrow ik_k \rightarrow -\frac{1}{2}\sigma_k\phi\).
We now have two unitary 2x2 transformations for spinor rotations and boosts.
The right handed spinor \(U = \varphi_R\).
\[\varphi_R = e^{\frac{1}{2}i\sigma_k\theta-\frac{1}{2}\sigma_k\phi}\]
The left handed spinor \(U = \varphi_L\).
\[\varphi_L = e^{\frac{1}{2}i\sigma_k\theta+\frac{1}{2}\sigma_k\phi}\]
These are also known as Weyl spinors.