Weyl Spinors

We have previously seen in quantum mechanics that Pauli matrices are a set of matrices that are used to represent the spin operators for spin-1/2 particles. Pauli vectors (which are actually matrices) are used to represent the spin states of particles, and they can be factored into the tensor product of two 2-component vectors. We call these vectors Pauli spinors.

Normally, a vector is rotated using a transformation. When we transform a Pauli vector, we instead place two transformationson the left and right of the vector, where one is the Hermitian conjugate of the other. When the Pauli vector is factored, each spinor takes one of the transformations. This is why spinors transform half as much as the vector.

Pauli vectors are associated with three-dimensional space. However, if we want to make quantum mechanics relativistic, we need a four-dimensional object. One such object is the Weyl spinor.

Table of Contents

• Table of Contents
• Review of Lorentz Transformations
• Weyl Vectors
  • Transformations of Weyl Vectors
  • Matrix Representations of SL(2, C) Transformations
  • Abstract Form of SL(2, C) Transformations
• Weyl Spinors
• Inner Product of Weyl Spinors
• Chirality of Weyl Spinors
• Weyl Vector Space as a Tensor Product Space
  • Infeld-Van der Waerden Notation
• Summary and Next Steps

Review of Lorentz Transformations

Recall that Lorentz transformations are transformations that preserve the Minkowski metric . In Minkowski geometry, we define the square-magnitude of a vector as

If we apply a Lorentz transformation to a vector , and assert that the square-magnitude is invariant, we have

We know that the transpose of a product of matrices is the product of the transposes in reverse order. Thus, we can write the left-hand side as . This implies that

The group of transformations satisfying this equation is , the group of orthogonal transformations in one time and three space dimensions. If we forbid spatial reflections, we have the group , the group of ** orthogonal transformations in one time and three space dimensions. Finally, if we also forbid time reflections, we have the group , the group of **proper orthochronous transformations in one time and three space dimensions. This has the effect of making positive.

We summarize the groups as follows:

1. :
2. :
3. :

There are six basic Lorentz transformations, defined by pairs of coordinate axes. These are shown in the table below.

Rotation in -plane	Rotation in -plane	Rotation in -plane
Boost in -plane	Boost in -plane	Boost in -plane

Weyl Vectors

Suppose we have a 4-vector . Let the time basis vector be (to be interpreted as the identity matrix), and the rest be Pauli matrices. Then we can write the vector as

This is the Weyl vector. There are a few interesting properties about Weyl vectors. First, the Weyl vector is Hermitian, meaning that . Second, the determinant of a Weyl vector is the spacetime interval of its corresponding 4-vector. This can be proven easily:

Transformations of Weyl Vectors

Normally, we transform a 4-vector by multiplying it by a Lorentz transformation (). What are its corresponding transformation pairs?

In other words, given a 4-vector that transforms to , we want to find a transformation pair such that corresponds to . To find out, let's first leverage the fact that the Weyl vector is Hermitian; :

We postulate that this implies and .

Next, we leverage the determinant property of the Weyl vector, :

This means that

This means that is a complex number of unit modulus, which can be read as . We can thus arbitrarily introduce a phase factor into the transformation. Plugging this back into the transformed Weyl vector, we have:

In other words, adding a phase factor to the transformation does not change the Weyl vector. This means that there are multiple matrices that represent a single transformation. To pick one, we can enforce the condition that .

The transformation that we just derived is a member of the group , the group of complex matrices with determinant 1. stands for special, and stands for linear.

Matrix Representations of SL(2, C) Transformations

We now seek a specific form of the transformation matrix. A matrix can be written as:

Without any conditions, this matrix has four complex degrees of freedom, or eight real degrees of freedom. However, we can impose the condition that :

Given that the determinant can be written as , we reduce the degrees of freedom to six ().

The six degrees of freedom correspond to the six transformations. They are given by the following matrices:

Boost in -plane	Boost in -plane	Boost in -plane
Rotation in -plane	Rotation in -plane	Rotation in -plane

Abstract Form of SL(2, C) Transformations

Alternatively, if we do not want to use matrices, we can use the abstract form of the transformation. The rotations are the same as Pauli matrices, while the boosts use hyperbolic trigonometric functions instead of the circular ones.

Weyl Spinors

With the fundamental properties of Weyl vectors and transformations in mind, we can now define Weyl spinors. Recall that Pauli spinors were obtained by factoring the Pauli vector as a tensor product of two 2-component vectors. We can do the same with Weyl vectors.

First, we can write the Weyl vector as

In this case, the tensor product is just an outer product. The resulting matrix is

leading to

Given that we know all Weyl vector components are real, these products must also be real. As such, no matter what is, for example, must be a multiple of . We can thus write for some real . The same applies to and ; for some real .

For the last two equations, since and are complex conjugates,

The left-hand side is , while the right-hand side is , so

This means that , or . But since both and are real, we have . Summarizing, we have

Letting for simplicity, we can write the Weyl vector as

The reverse conversions are given by

Then, the individual components of the Weyl vector can be expressed in terms of the Weyl spinors as

Finally, in order to actually find and , we first write them in Euler form:

Skipping the algebra, we can find that

We will now use the symbols and to denote the Weyl spinors.

The term is a Weyl spinor, and the term is the Hermitian conjugate of the Weyl spinor. It exists in the dual space of the Weyl spinor. Just like with Pauli spinors, each Weyl spinor takes half of the transformation;

Inner Product of Weyl Spinors

We have seen that for Pauli spinors, the inner product is defined as . It is defined this way because it is invariant under transformations;

However, for Weyl spinors, we have a different transformation group, . If we try to define the inner product as , we have

However, we cannot cancel because is not necessarily unitary. The solution is to introduce a new symbol between the two spinors,

such that . This is similar to how we add the Minkowski metric to the inner product of 4-vectors. However, it turns out that there is no that satisfies this condition. Instead, we need to replace the Hermitian conjugate with a simple transpose;

It can be shown that the simplest choice of that satisfies this condition is

The inner product defined this way is antisymmetric, meaning that . The technical term for is a symplectic form.

Chirality of Weyl Spinors

Weyl spinors also exhibit a unique property known as chirality. Chirality, or handedness, is a geometric property in which an object cannot be superimposed onto its mirror image. A common example of chirality is the human hand; the left hand cannot be superimposed onto the right hand. Similarly, many chiral molecules exist in nature, such as amino acids (except glycine) and sugars.

In the context of Weyl spinors, chirality refers to the property of a Weyl spinor being either left-handed or right-handed. The transformations can be divided into two groups: the left representations and the right representations. They are the complex conjugates of each other.

To realize why this is the case, we need to consider parity transformations. Suppose we have two rotation matrices and . rotates a vector in the -plane, while rotates a vector in the -plane. While they are different rotations, one can instead think of them as the same rotation, but in different coordinate systems. In other words, the rotation in the -plane is equivalent to a rotation in the -plane, but with a different coordinate system.

For a vector that transforms under , if we want to transform it under , we first apply a transformation that changes the coordinate system from to . Then, we apply the rotation , and finally, we apply the inverse transformation to change the coordinate system back. This can be written as

Suppose and are complex conjugates of each other, and are both members of . We will now label these as and , respectively. Then, we can write the transformation as

It turns out that if we transform the vector normally, with , there is no way to write it as . In other words, there is no such that . This is the source of chirality. As such, up to a complex conjugation, there are two different representations of Weyl spinors, which we call left-handed and right-handed Weyl spinors.

This means that spinors can be classified into two types, which we denote as and . Spinor indices are also classified as left-handed and right-handed. There are quite a lot of notations for left-handed and right-handed Weyl spinors, many of which are summarized in the table below (copied from this video).

Notation	Left-handed	Left-handed dual	Right-handed	Right-handed dual
O'Donnell, Peter. Introduction to 2-Spinors for General Relativity
Wald. General Relativity
Wikipedia + Zee. Quantum Field Theory in a Nutshell + Müller-Kirsten, Wiedemann. Introduction to Supersymmetry
Schwichtenberg, Jakob. No Nonsense Quantum Field Theory
Penrose, Roger. The Road to Reality		?	?
Eigenchris. Spinors for Beginners

If is a left-handed Weyl spinor, the dual spinor is equal to

Dual spinors transform with the inverse transformation, so if transforms as , then the dual spinor transforms as . This is such that the inner product is invariant under the transformation;

If we take the complex conjugate of , we get a right-handed dual spinor . It transforms as . Taking the expression for the inner product of the left-handed dual spinor, we can take a complex conjugate to get the inner product of the right-handed dual spinor;

It is this expression that is why we define the inner product of right-handed Weyl spinors as . In other words, instead of just taking the transpose, we take the Hermitian conjugate of the spinor.

A summary of all spinor types is given in the table below, once again copied from this video.

Spinor Type	Transformation
Left-handed Weyl spinor
Left-handed dual Weyl spinor
Right-handed dual Weyl spinor
Right-handed Weyl spinor

Weyl Vector Space as a Tensor Product Space

Finally, we conclude that the Weyl vector space is a tensor product space of two Weyl spinor spaces.

This is because we can decompose any Weyl vector into individual matrix components. A null Weyl vector can be expressed as the tensor product of two Weyl spinors. One of them is left chiral, and the other is right chiral.

For a non-null Weyl vector, we can no longer express it as a tensor product of two Weyl spinors. Instead, we can express it as a sum of tensor products of Weyl spinors.

where we have used Einstein summation notation to sum over the indices and . From this, we can clearly see that

Infeld-Van der Waerden Notation

Can we convert an ordinary four-vector into a Weyl vector conveniently?

This matrix in the middle is a linear map whose components are known as the Infeld-Van der Waerden symbols. Generally we have

Notice that every one tensor component () corresponds to a pair of Weyl spinor components ().

Summary and Next Steps

In this section, we went through a deep dive into Weyl vectors and Weyl spinors.

Here are the key points to remember:

• Lorentz transformations are represented by matrices in three groups: , , and .

• Weyl vectors are constructed from 4-vectors and have properties such as Hermiticity and determinant equal to the spacetime interval.

• Weyl vectors transform as , where is a member of , the special linear group of complex matrices.

• Weyl vectors can be expressed as tensor products of Weyl spinors. They transform as

• The inner product of Weyl spinors is defined as , where is a symplectic form.

• Weyl spinors exhibit chirality because there is no such that .

• They can be classified into left-handed and right-handed Weyl spinors, whose notation are summarized in this table.

• Weyl spinors transform based on their chirality. This is summarized in this table.

• The Weyl vector space can be expressed as a tensor product space of two Weyl spinor spaces, .

• The Infeld-Van der Waerden symbols allow us to convert an ordinary 4-vector into a Weyl vector;

In the next section, we will explore Dirac spinors, which are constructed from a pair of Weyl spinors. This leads us to discuss more advanced topics such as spin groups, Clifford algebras, and Lie groups.