Pauli Spinors

Pauli spinors are a mathematical representation of spin-1/2 particles in quantum mechanics. We have previously introduced spinors in the context of physical systems (e.g., the Stern-Gerlach experiment, light polarization). Now we will take a purely mathematical approach to spinors, focusing on their algebraic properties and how they relate to the rotation group.

Table of Contents

• Table of Contents
• Introduction
  • Transforming the Pauli Vector
• Appendix: Proof of Product of Orthogonal Vectors

Introduction

Recall that the spin operators are defined as:

Also recall that the matrix element of an operator is given by . In this case, the general form of the matrix element is:

The matrices are known as the Pauli matrices. Their explicit forms are:

These matrices are traceless () and Hermitian (). Squaring any of the Pauli matrices gives the identity operator, and they anticommute with each other:

The last two properties can be combined into the following identity:

Lastly, they satisfy the following commutation relations:

with an implicit sum over .

Review: Levi-Civita Symbol

The term is known as the Levi-Civita symbol. It is defined as:

Recall that in classical physics, we define a vector as a linear combination of basis vectors:

In the Pauli formalism, we replace the basis vectors with the Pauli matrices, , , and . This gives us the following expression for a vector:

This form of the vector is known as the Pauli vector. The fundamental benefit of using matrices for basis vectors is that we can now directly multiply vectors with each other.

Transforming the Pauli Vector

First, consider reflecting the Pauli vector along . In other words, we want to find the transformation :

In the Pauli formalism, we do this by negative conjugating the Pauli vector with the matrix. By conjugating we mean to multiply the Pauli vector by on the left and on the right. We will see how this transformation applies to each basis Pauli matrix:

For :

And for :

Hence, we can see that it takes to and leaves the other two matrices unchanged, precisely what we wanted. To see this more concretely, consider a Pauli vector and apply the transformation:

This is the transformation we wanted to achieve. To reflect the Pauli vector along an arbitrary axis, we choose an arbitrary unit vector (where ) and apply the transformation:

To see that this indeed transforms the Pauli vector, we can write as a sum of its perpendicular and parallel components to :

Because is parallel to , it can be written as a scalar multiple of :

To multiply the perpendicular component by , we can use the identity:

This is proven in the appendix.

Putting this all together, we have:

As such, it is clear that the transformation we have defined indeed transforms the Pauli vector as desired. It flips the parallel component of the vector while leaving the perpendicular component unchanged.

Appendix: Proof of Product of Orthogonal Vectors

We want to prove the following theorem:

where and are orthogonal vectors. Recall that they are orthogonal if:

We can write both as linear combinations of the Pauli matrices:

Multiplying these two vectors gives us:

where we have used implicit summation over and .

Note that only applies to . However, because and are orthogonal, we have . As such, the only terms that survive are those where , which gives us the desired result.