47.3 Weyl spinors

47.3 Weyl spinors

Concept links · terms present in this machine draft; source roles are unverified: eigenvalue · irreducible representation · charge operator · complexification

For the case $m=0$ of the Dirac equation, it turns out that there is an interesting operator acting on the space of solutions:

Definition (Chirality)

The operator

$${\gamma }_5=i{\gamma }_0{\gamma }_1{\gamma }_2{\gamma }_3$$

is called the chirality operator. It has eigenvalues $\pm 1$ and its eigenstates are said to have chirali $ty\pm 1$ . States with chiral $ity+1$ are called “right-handed”, those with chirality −1 are called $\cdot \cdot left-hande{d}^{\prime \prime }$

Note that the operator $J_W=-i{\gamma }_5={\gamma }_0{\gamma }_1{\gamma }_2{\gamma }_3$ satisfies $J_W^2=-1$ and provides a choice of complex structure on the space $V$ of real-valued solutions of the Dirac equation. We can complexify such solutions and write

$$\mathcal{V}\otimes \mathbf{C}={\mathcal{V}}_L\oplus {\mathcal{V}}_R\text{\hspace{1em}(47.15)}$$

where ${\gamma }_L$ is the $+i$ eigenspace of $J_W$ (the negative or left-handed chirality solutions), and ${\gamma }_R$ is the −$i$ eigenspace of $J_W$ (the positive or right-handed chirality solutions).

To work with $J_W$ eigenvectors, it is convenient to adopt a choice of $\gamma -$ -matrices in which ${\gamma }_5$ is diagonal. This cannot be done with real matrices, but requires complexification. One such choice was already described in 41.2, the chiral or Weyl representation. In this choice, the $\gamma$ matrices can be written in 2 by 2 block form as

$${\gamma }_0=-i\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\gamma }_1=-i\left(\begin{array}{cc}0& {\sigma }_1\\ -{\sigma }_1& 0\end{array}\right),{\gamma }_2=-i\left(\begin{array}{cc}0& {\sigma }_2\\ -{\sigma }_2& 0\end{array}\right),{\gamma }_3=-i\left(\begin{array}{cc}0& {\sigma }_3\\ -{\sigma }_3& 0\end{array}\right)$$

and the chirality operator is diagonal

$${\gamma }_5=\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)$$

We can thus write (complexified) solutions in terms of chiral eigenstates as

$$\psi =\left(\genfrac{}{}{0ex}{}{{\psi }_L}{{\psi }_R}\right)$$

where ${\psi }_L$ and ${\psi }_R$ are two-component wavefunctions, of left and right chirality respectively.

The Dirac equation 47.2 is then

$$-i\left(\begin{array}{cc}-im& \frac{\partial }{\partial t}+\mathbf{\sigma }\cdot \nabla \\ \frac{\partial }{\partial t}-\mathbf{\sigma }\cdot \nabla & -im\end{array}\right)\left(\genfrac{}{}{0ex}{}{{\psi }_L}{{\psi }_R}\right)=0$$

${}_{\mathrm{or}},$ in terms of two-component functions

$$\left(\frac{\partial }{\partial t}+\mathbf{\sigma }\cdot \nabla \right){\psi }_R=im{\psi }_L$$ $$\left(\frac{\partial }{\partial t}-\mathbf{\sigma }\cdot \nabla \right){\psi }_L=im{\psi }_R$$

When $m=0$ the equations decouple and one can consistently restrict attention to just right-handed or left-handed solutions, giving:

Definition (Weyl equations)

The Weyl wave equations for two-component spinors are

$$\left(\frac{\partial }{\partial t}+\sigma \cdot \nabla \right){\psi }_R=0\text{\hspace{1em}(47.16)}$$

$$\left(\frac{\partial }{\partial t}-\mathbf{\sigma }\cdot \nabla \right){\psi }_L=0\text{\hspace{1em}(47.17)}$$

Also in the massless case, the chirality operator satisfies

$$[{\gamma }_5,H_D]=0$$

$(H_D$ is the Dirac Hamiltonian 47.5) since for each $j$

$$[{\gamma }_5,{\gamma }_0{\gamma }_j]=0$$

This follows from the fact that commuting ${\gamma }_0$ through ${\gamma }_5$ gives three minus signs, commuting ${\gamma }_j$ through ${\gamma }_5$ gives another three. In this case chirality is a conserved quantity, and the complex structure $J_W=-i{\gamma }_5$ commutes with $H_D$ $J_W$ then takes positive energy solutions to positive energy solutions, negative energy to negative energy solutions, and thus commutes with the relativistic complex structure $J_r$

We now have two commuting complex structures $J_W$ and $J_r$ on $\nu ,$ and they can be simultaneously diagonalized (much like the situation in section 44.1.2). We get a decomposition

$${\mathcal{V}}_{J_r}^{+}={\mathcal{H}}_1={\mathcal{H}}_{1,L}\oplus {\mathcal{H}}_{1,R}$$

of the positive energy solutions into $+i({\mathcal{H}}_{1,L})$ and $-i({\mathcal{H}}_{1,R})$ eigenspaces of $J_W$ . Restricting to the solutions of equation 47.17 we get a decomposition

$${\mathcal{V}}_L={\mathcal{H}}_{1,L}\oplus {\overline{\mathcal{H}}}_{1,L}$$

into positive and negative energy left-handed solutions. We can then take Weyl spinor fields to be two-component objects

$${\Psi }_L(\mathbf{x}),{\overline{\Psi }}_L(\mathbf{x})$$

that are continuous basis elements of ${\mathcal{H}}_{1,L}$ and ${\overline{\mathcal{H}}}_{1,L}$ respectively. The action of the (double cover of the) Poincar´e group on space-time dependent Weyl fields will be given by

$${\Psi }_L(x)\to (a,\Lambda ){\Psi }_L(x)=S(\Lambda {)}^{-1}{\Psi }_L(\Lambda \cdot x+a)$$

where $\Lambda$ is an element of $Spin(3,1)$ and $S(\mathbf{\Lambda })\in SL(2,\mathbf{C})$ is the $\left(\frac{1}{2},0\right)$ representation (see chapter 41, where $S(\Lambda )=\Omega )$ . $\Lambda \cdot x$ is the $Spin(3,1)$ action on Minkowski space vectors.

Just as in the Majorana case, parametrizing the space of solutions using fixed-time fields does not allow one to see the action of boosts on the fields. In addition, we know that relativistic field quantization requires use of the relativistic complex structure $J_r.$ , which is not simply expressed in terms of the fixed-time fields. To solve both problems we need to study the solutions in momentum space. To find solutions in momentum space we Fourier transform, using

$$\psi (t,\mathbf{x})=\frac{1}{(2\pi {)}^2}\int d^{4}p{e}^{i(-p_{0}t+\mathbf{p}\cdot \mathbf{x})}\tilde{\psi }(p_0,\mathbf{p})$$

and see that the Weyl equations are

$$(p_0-\sigma \cdot \mathbf{p}){\tilde{\psi }}_R=0$$ $$(p_0+\sigma \cdot \mathbf{p}){\tilde{\psi }}_L=0$$

Since

$$(p_0+\mathbf{\sigma }\cdot \mathbf{p})(p_0-\mathbf{\sigma }\cdot \mathbf{p})=p_0^2-(\mathbf{\sigma }\cdot \mathbf{p}{)}^2=p_0^2-|\mathbf{p}{|}^2$$

both ${\tilde{\psi }}_R$ and ${\tilde{\psi }}_L$ satisfy

$$(p_0^2-|\mathbf{p}{|}^2)\tilde{\psi }=0$$

so are functions with support on the positive $\left(p_0=|\mathbf{p}|\right)$ and negative $\left(p_0=-|\mathbf{p}|\right)$ energy null-cone. These are Fourier transforms of solutions to the massless Klein-Gordon equation

$$\left(-\frac{{\partial }^2}{\partial x_0^2}+\frac{{\partial }^2}{\partial x_1^2}+\frac{{\partial }^2}{\partial x_2^2}+\frac{{\partial }^2}{\partial x_3^2}\right)\psi =0$$

In the two-component formalism, one can define:

Definition (Helicity)

The operator

$$\frac{1}{2}\frac{\mathbf{\sigma }\cdot \mathbf{p}}{|\mathbf{p}|}\text{\hspace{1em}(47.18)}$$

on the space $of$ solutions to the Weyl equations is called the helicity operator. It has eigenvalues $\pm \frac{1}{2}$ , and its eigenstates are said to have helicity $\pm \frac{1}{2}$

The helicity operator is the component of the spin operator $\begin{array}{r}\mathbf{S}=\frac{1}{2}\sigma \end{array}$ along the direction of the momentum of a particle. Single-particle helicity eigenstates of eigenvalue $+\frac{1}{2}$ are said to have “right-handed helicity”, and described as having spin in the same direction as their momentum, those with helicity eigenvalue $-\frac{1}{2}$ are said to have “left-handed helici ${\mathrm{ty}}^{,5}$ and spin in the opposite direction to their momentum.

A continuous basis of solutions to the Weyl equation for ${\psi }_L$ is given by the wavefunctions

$$u_L(\mathbf{p})e^{i(-p_0{x}_0+\mathbf{p}\cdot \mathbf{x})}$$

where the polarization vector $u_L(\mathbf{p})\in {\mathbf{C}}^2$ satisfies

$$\sigma \cdot \mathbf{p}{u}_L(\mathbf{p})=-p_0{u}_L(\mathbf{p})$$

Note that the $u_L(\mathbf{p})$ are the same basis elements of a specific p-dependent $\mathbf{C}\subseteq \zeta$ ${\mathbf{C}}^2$ subspace first seen in the case of the Bloch sphere in section 7.5, and later in the case of solutions to the Pauli equation (the $u_{-}(\mathbf{p})$ of equation 34.8 is $u_L(\mathbf{p})$ for positive energy, the $u_{+}(\mathbf{p})$ of equation 34.7 is $u_L(\mathbf{p})$ for negative energy). The positive energy $\left(p_0=|\mathbf{p}|\right)$ solutions have negative helicity, while the negative energy $\left(p_0=-|\mathbf{p}|\right)$ solutions have positive helicity. After quantization, this wave equation leads to a quantum field theory describing massless left-handed helicity particles and right-handed helicity antiparticles. Unlike the case of the Majorana field, the theory of the Weyl field comes with a non-trivial internal symmetry, due to the action of the group $U(1)$ on solutions by multiplication by a phase, and this allows the introduction of a charge operator.

Recall that our general analysis of irreducible representations of the Poincar´e group in chapter 42 showed that we expected to find such representations by looking at functions on the positive and negative energy null-cones, with values in representations of $SO(2)$ , the group of rotations preserving the vector $p$. Acting on solutions to the Weyl equations, the generator of this group is given by the helicity operator (equation 47.18). The solution space to the Weyl equations provides the expected irreducible representations of helicity $\pm \frac{1}{2}$ and of either positive or negative energy.

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