Dirac Spinors and Spin Groups

Previously, we introduced Weyl spinors, which are two-component complex vectors that represent the state of a spin-1/2 particle. Crucially, Weyl spinors are chiral—there are left-handed Weyl spinors and right-handed Weyl spinors. In this section, we will explore Dirac spinors, which are four-component complex vectors that can represent both left-handed and right-handed Weyl spinors. Then, we will discuss various group theoretical aspects of spinors, including double covers and spin groups.

Table of Contents

• Table of Contents
• Introduction
• Double Covers
  • Correspondence Between S³, RP³, SU(2), and SO(3)
  • Simply Connectedness
• SL(2, C) Double Cover of SO⁺(1, 3)

Introduction

When we deal with spin-1/2 particles, we need to consider both left-handed and right-handed Weyl spinors. Is there a way to keep the two Weyl spinors together in a single object?

Recall that we have left-handed Weyl spinors and right-handed Weyl spinors . They transform by

To combine them, we first need to make both Weyl spinors column vectors. We can do this by taking the transpose of the right-handed Weyl spinors to get

Finally, we can combine the two Weyl spinors into a single four-component complex vector, known as a Dirac spinor:

As shown above, the Dirac spinor is a four-component complex vector that combines both left-handed and right-handed Weyl spinors. The top two components represent the left-handed Weyl spinor and transform with , while the bottom two components represent the right-handed Weyl spinor and transform with .

The matrix

is a Dirac spinor transformation matrix in the Weyl basis. Evidently, it is a block diagonal matrix, where the top-left block corresponds to the left-handed Weyl spinor and the bottom-right block corresponds to the right-handed Weyl spinor. We say that it belongs to the direct sum of the two Weyl spinor representations;

The Dirac spinor is often denoted as a single letter . It appears in the Dirac equation,

which describes the behavior of spin-1/2 particles in quantum field theory.

Double Covers

We will now formalize our understanding of double covers. In the past, we have seen that for every rotation, there are two rotations that correspond to it. This is because Pauli vectors transform by , and so adding a negative sign to the matrix does not change the Pauli vector. This means that the group is a double cover of the group.

Similarly, we have seen that for every Lorentz transformation, there are two transformations that correspond to it. This is because Weyl vectors transform by , and so adding a negative sign to the matrix does not change the Weyl vector. This means that the group is a double cover of the Lorentz group .

The idea of double covers has a topological perspective. is roughly the same as the 3-sphere , while is roughly the same as the real projective space . We will begin in one dimension to build up to this idea.

Recall that for , the real projective space in one dimension, we can think of it as a semicircle with the endpoints identified. This is because for every point on the line, there exists two points on a circle that correspond to it, in which they are antipodal to each other.

As such, we say that the 1-sphere is a double cover of the real projective space . This means that we can remove half of the circle and instead consider only the top semicircle. As the point walks off one endpoint of the semicircle, it reappears on the other endpoint.

The act of projection maps every point on the semicircle to the point on the line.

Now let's generalize to two dimensions. For , the real projective space in two dimensions, we can think of it as a hemisphere with the boundary circle identified. Just like in the one-dimensional case, for every point on the plane, there exists two points on a sphere that correspond to it, in which they are antipodal to each other. What this means is that we can remove half of the sphere and instead consider only the top hemisphere.

In two dimensions, projection maps every point on the hemisphere to the point on the plane.

For three dimensions, we have , the real projective space in three dimensions. There is no way to visualize this in three dimensions, but we can still analyze it algebraically.

The 3-sphere is defined as

The result of this is that we have three independent parameters, which we can think of as the angles of a rotation in three dimensions. Then, the last parameter is fixed by the constraint that the sum of the squares equals one.

Following the same pattern, we expect projection to act on a vector as

Correspondence Between S³, RP³, SU(2), and SO(3)

With this in mind, we now consider the correspondence between these spaces and the groups we have seen so far. is homeomorphic to the group , which is the group of rotations in three dimensions. Likewise, is homeomorphic to the group , which is the group of unitary transformations in two dimensions. This means that is a double cover of , as we have seen before.

We can now show that is homeomorphic to . In other words, is equivalent to the set of all such that .

One can easily verify that unitary matrices in have the form

where . Since all unitary matrices have a determinant of one, it follows that

We can replace and with their real and imaginary parts, such that and , where . Then, we have

As such, we can see that the set of all unitary matrices in is equivalent to the set of all such that . This is precisely the definition of , which means that is homeomorphic to .

Simply Connectedness

Another property of the groups is that is simply connected, while is not.

A simply connected space has the geometric interpretation that it has no holes. More formally, if, for any loop in the space, we can continuously shrink it to a point without leaving the space, then the space is simply connected. For example, the surface of a sphere is simply connected, while the surface of a torus is not.

For , all spheres are simply connected. For , the circle is not simply connected, as it has a hole in the middle. On the other hand, the real projective space is not simply connected. This is because when we deform a loop in , when we transform one part of the loop, the other end of the loop is also transformed.

SL(2, C) Double Cover of SO⁺(1, 3)

In four-dimensional spacetime, we have the Lorentz group , which is the group of Lorentz transformations that preserve the spacetime interval.

As we have seen, for any , there are two transformations that correspond to it. This is because for any Weyl vector , we have , and so adding a negative sign to the matrix does not change the Weyl vector. This means that the group is a double cover of the Lorentz group .