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Pauli Spinors

Previously, we introduced the Pauli formalism, which involves using the Pauli sigma matrices as basis vectors. In this section, we will explore the Pauli spinors, which are two-component complex vectors that represent the state of a spin-1/2 particle. More specifically, we will focus on how Pauli vectors can be factored into two-component complex vectors, which are known as Pauli spinors.

Table of Contents​

Factoring Matrices​

Consider a matrix that can be factored into a column vector and a row vector , i.e.,

Suppose we are in a two-dimensional space, then we can write all of this as

This means that we can write the matrix as a product of two vectors, one column vector and one row vector. We have already seen that this operation, the outer product, is helpful in quantum mechanics.

Only certain matrices can be factored in this way, and they are known as rank-1 matrices. These matrices have zero determinant.

Now suppose we have successfully factored a matrix into a column vector and a row vector ,

There is ambiguity here, because we can insert into the middle and rewrite as any complex number :

Is there a unique, unambiguous way to factor a matrix into a column vector and a row vector? We can enforce this by requiring that the top of the column vector is . This gives us a unique factorization of the matrix,

Factoring Pauli Vectors​

Recall that a Pauli vector is a vector of the form

For us to be able to factor this into a column vector and a row vector, we need to ensure that the determinant of the Pauli vector is zero. Since the determinant is just the negative norm of the vector, we require that

Such a vector is known as a null vector or isotropic vector. It requires that either all three components are zero, or some of them are complex. Anyways, let's factor the Pauli vector into a column vector and a row vector,

We have, trivially, that

First, let's write in terms of and . This is a simple rearrangement of ,

From here we can choose and .

We can plug these values into :

This implies that . Finally, we can plug into :

This implies that . Therefore we have

We can summarize this as

We shall make a definition to make this more readable. Let

Then we can rewrite the Pauli vector as

The components , , and can be expressed in terms of and as follows:

And finally, we can remove the ambiguity by requiring that the top component of the column vector is . This gives us

The column vector is known as a Pauli spinor. Note that different Pauli spinors with the same ratio are considered different Pauli spinors. However, they correspond to the same Pauli vector.

Transformation of Pauli Spinors​

Recall that under a rotation, Pauli vectors transform by a negative conjugation with a matrix,

If we factor like we did above, we can see how the Pauli spinors transform;

We can group them like this:

This shows that the Pauli spinors transform with one matrix, while the Pauli vector transforms with two matrices. This finally creates a solid mathematical explanation for the half-rotation property of spinors, like we have seen with Jones vectors and quantum spin states.

Spinor Spaces​

We will now discuss the abstract algebraic structure of Pauli spinors. This discussion requires some prior knowledge of tensor products, which I included a brief overview of here.

Spinors are elements of a vector space known as a spinor space. The spinor space is a two-dimensional complex vector space, which can be denoted as . We write that

Dual spiniors are elements of the dual space of the spinor space, which is denoted as ,

Just like a vector, we can write a spinor as a linear combination of basis vectors. Similarly, we can write a dual spinor as a linear combination of dual basis vectors.

The spinor components and are covariant, while the dual spinor components and are contravariant.

Pauli Spinors as Tensor Products​

Now let's consider how we can write a Pauli vector using spinors. In abstract form, a Pauli vector's components are written as , where and are indices that can take values or .

We can thus write

In Einstein notation, we can write this as

or

We can perform this procedure for the basis Pauli matrices as well, which yield

Sigma Matrices as Linear Maps​

We can also view the Pauli matrices as linear maps from the 3D vector space to the spinor space. We define the map such that