Pauli Vectors and Matrices

Table of Contents

• Table of Contents
• Pauli Vectors
  • Reflecting the Pauli Vector
  • Rotation of the Pauli Vector
  • U(2) and SU(2) Groups
  • SU(2) as the Double Cover of SO(3)
• Equivalences to Quaternions
• Summary and Next Steps
• Appendix: Proof of Product of Orthogonal Vectors

Pauli Vectors

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 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.

Reflecting 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

which 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.

Rotation of the Pauli Vector

To rotate the Pauli vector, we leverage the fact that a rotation is just two successive reflections. More specifically, to rotate a vector by an angle , we reflect it twice, where the "mirrors" are at an angle of to the axis of rotation.

For example, consider a vector reflected along the -axis and then the -axis. Because they are at a right angle to each other, the resulting vector is rotated by . We can prove this by applying the transformation

to the vector , yielding

which is indeed the vector rotated by .

To generalize, we can define a rotation by an angle as follows. We need to reflect around two mirrors at an angle to the axis of rotation. Choose the first mirror to just be the -axis, and the second mirror to be at an angle to the -axis.

The normal vector to the first mirror is , and the normal vector to the second mirror is

Algebraically, we can write the rotation as

To see that this indeed rotates the Pauli vector, consider a Pauli vector . Applying the transformation gives us:

This matches the expected result of a rotation by an angle . There are mainly two approaches to derive this result. We can either use the algebraic approach as we just did, or we can use the matrix representations. Matrices typically make the derivation more straightforward, but they could be argued to be less pure. This will be important when we discuss Clifford algebras and Lie algebras in the future. Anyways, the matrix derivation is as follows.

First, the half-rotation on each side can be expressed as

Then, we can write the rotated Pauli vector as

In other words, the matrix elements of the rotated Pauli vector are equivalent to transforming and by a phase factor of . We also see from Equation that the and components transform as

This matches the regular rotation of a vector in the -plane by an angle

U(2) and SU(2) Groups

We have seen how to reflect and rotate Pauli vectors using the Pauli matrices. A summary of the transformations is given in the table below.

Transformation	Form
Reflection along
Reflection along
Reflection along
Reflection along an arbitrary axis , where
Rotation by an angle in the -plane
Rotation by an angle in the -plane
Rotation by an angle in the -plane

All these transformations are performed by conjugating the Pauli vector with the appropriate Pauli matrix or combination of Pauli matrices. These matrices are unitary, and are therefore within the unitary group . Additionally, the rotations also have a positive determinant, and are therefore within the special unitary group .

Now, let's prove that rotations must be performed by conjugating the Pauli vector with a matrix. The key point is that the norm of the Pauli vector must remain invariant under rotations. We shall now prove a few important lemmas in order to show this.

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Lemma. In the matrix representation of a Pauli vector, the determinant is the negative of the norm squared of the corresponding vector.

Proof. This is trivial.

Thus, the determinant of the Pauli vector is indeed the negative of the norm squared of the corresponding vector.

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We also know that all Pauli vectors are Hermitian, and therefore the rotated Pauli vector must also be Hermitian. This means that

As such, we can guess that and . Thus, the transformed Pauli vector takes the form . We just need to show that now.

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Lemma. Under the transformation , the norm of the Pauli vector is invariant if and only if .

Proof. Since the norm is just the determinant of the Pauli vector, we need to show that

The terms cancel out, and we are left with

This means that must lie on the unit circle, i.e., .

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We also show that is phase-ambiguous, meaning that we can multiply it by a phase factor without changing the transformation.

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Lemma. The matrix performs the same transformation as for any .

Proof. This is quite straightforward.

Thus, the transformation is indeed invariant under multiplication by a phase factor.

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To resolve the phase ambiguity, we just multiply the transformation by a certain phase factor such that . It can be shown that given , then . Finally,

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Theorem. The transformation preserves the norm of the Pauli vector if and only if .

Proof. We know that is always traceless, so

The only way for to be zero is if is a scaled identity operator; for some scalar . This is because the trace of a matrix is the sum of its eigenvalues, and the only way for this sum to be zero is if all eigenvalues are zero. Next, we take the determinant of both sides,

This means that . However, since we know that is positive along the diagonal (because it is a rotation), we have . Therefore we have . This is the definition of a unitary matrix, and therefore .

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SU(2) as the Double Cover of SO(3)

Going back to the transformations we defined, we can see that they are all performed by conjugating the Pauli vector with a matrix in ,

Notice that even if we restrict the determinant of to be (i.e., ), there is one last ambiguity. This is because we can multiply by without changing the transformation, because

Recall that is the Pauli-vector representation of a regular transformation. This means that for every transformation, there are two transformations that perform the same transformation. As such, we say that SU(2) is the double cover of SO(3).

The topological interpretation of this is as follows. The group can be represented as a 3-sphere, , since it has three degrees of freedom (the three axes of rotation). On the other hand, the group can instead be represented as the real projective space , which is the 3-sphere with antipodal points identified. Don't worry if this doesn't make sense yet; we will explore this in much more detail when we finish discussing Weyl and Dirac spinors, from which we discuss , , and so on.

Equivalences to Quaternions

If you have ever worked with quaternions, you may have noticed that the Pauli matrices are very similar to the quaternion units , , and .

Quaternions are a number system that extends complex numbers. While complex numbers have two dimensions (real and imaginary), quaternions have four dimensions, represented as , where are real numbers and are the quaternion units. Th e unit quaternions , , and are defined such that

The operations of quaternions are defined very similarly to complex numbers. The conjugate of a quaternion is defined as .

Multiplying by its conjugate gives us the norm squared, as

The key is that the Pauli matrices themselves behave slightly differently than the quaternion units, but the set works the exact same way as the quaternion units.

Pauli Vector	Quaternion Unit

The set of unit quaternions is denoted as , which is the group of unit quaternions under multiplication. Given that there is a one-to-one correspondence between the Pauli vectors and the unit quaternions, we can say that the set is isomorphic to . This means that we can perform all the operations on the Pauli vectors as we would on the unit quaternions, and vice versa.

The transformation rules are the same as those for the Pauli vectors, and we can perform reflections and rotations in the same way. For quaternions, we express a 3D vector by replacing the , , and components with the quaternion units:

Then, transformations can be performed by conjugating the quaternion with the appropriate unit quaternion. For a rotation by an angle around an axis, we can express the transformation as

This is similar to the transformation we derived for the Pauli vectors,

Now, given that the Pauli matrices transform with , and the quaternions transform with unit quaternions, is there a group that unifies these two? This group is known as the spin group , which is the double cover of .

The question remains how we construct higher-dimensional spin groups like . This must be done using a mathematical structure known as a Clifford algebra. We will explore this in the future.

Summary and Next Steps

In this chapter, we have explored the transformations of Pauli vectors, including reflections and rotations.

Here are the key things to remember:

• Pauli sigma matrices are used to represent spin-1/2 particles. They are defined as

  satisfy the commutation relations

  and act as the basis for a vector space.

• A "regular" vector can be expressed as a linear combination of the Pauli matrices:

  In matrix form, this is

• Reflections and rotations of Pauli vectors can be performed by negative conjugating the Pauli vector with appropriate matrices in or . The transformations are given in this table.

• Quaternions are a number system that extends complex numbers, and the Pauli matrices can be related to the quaternion units. The set is isomorphic to the unit quaternions .

• is a double cover of , meaning that for every transformation in , there are two corresponding transformations in . This is topologically represented as being the double cover of .

• The spin group is the double cover of , and it is represented by for Pauli vectors and unit quaternions for quaternions.

In the next section, we explore how Pauli vectors can be split into a pair of spinors, and how spinors transform.

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.