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The Rotation Group

Previously, we discussed spinors as members of the complex projective space . This is a mathematical structure that describes the behavior of particles with spin. We now turn our attention to the rotation group, which is a mathematical group that describes rotations in three-dimensional space.

Table of Contents​

Rotations​

Rotations in three-dimensional space are described with matrices. We will go through a few properties of these matrices and how they relate to the rotation group.

Orthogonality​

To rotate a vector , we can use the rotation matrix . This means that the vector is rotated by the matrix to give a new vector . Because is just a rotation, it does not change the length of the vector (). We can hence write:

From linear algebra we know that the transpose of a product of matrices is the product of the transposes in reverse order. This hence yields:

Hence , where is the identity matrix. This property is known as orthogonality. We have shown in a previous page that this implies that the columns form an orthonormal basis.

Commutation​

There are two important facts to note about these matrices:

  1. Rotations about the same axis commute with each other.
  2. Rotations about different axes do not commute with each other.

To further understand this, consider as a rotation, applied to a vector :

Before writing an explicit form of , note that different conventions exist for the rotation matrix:

  • Active rotation: where physical systems are rotated but the coordinate system is fixed.
  • Passive rotation: where the coordinate system is rotated but the physical system is fixed.

Switching between these two conventions flips the sign of the rotation angle. In the active rotation, we can write that for a rotation about the -axis, we have:

Next, we can write the rotation about the -axis and -axis as:

Finally, consider the Taylor expansion of the sine and cosine functions— and . This means that for an infinitesimal rotation of an angle , we can write:

We can also combine these rotations by matrix multiplication. For example, we can write:

If we switch the order of the rotations, we get:

Notice that if we ignore the terms, the two matrices are equal. In other words, infinitesimal rotations of order in different axes commute with each other. If we do not ignore the terms, we can see that the two matrices are not equal. We can write this as a commutation relation:

Rotations in Quantum Mechanics​

Recall from a previous page that the Jones vector is a two-component vector that describes the polarization of light. As a complex vector, it rotates with unitary matrices. Similarly, a quantum state rotates with a unitary operator.

Let be a classical rotation matrix. We can associate it with a quantum rotation operator . We have seen previously that infinitesimal versions of these operators are given by:

where is a Hermitian operator.

For the translation operator, as we have . For the time-evolution operator, as we have . The pattern is that the operator is the generator of the transformation.

Because angular momentum is the generator of rotations, we can define the angular momentum operator such that:

Substituting these into the infinitesimal rotation operator, we have:

We can write as , where is a unit vector in the direction of the axis of rotation. Then, we can write:

Finally, for a finite rotation, we can apply the infinitesimal rotation operator times where :

To put these in context, recall that for both the Jones vector and the spin-1/2 particle, we have two separate spaces. One is the physical space, and the other is an abstract space (the Poincaré sphere for the Jones vector and the Bloch sphere for the spin-1/2 particle). The classical rotation operator is a operator that rotates the physical space, whereas the abstract rotation operator is a operator that rotates the abstract space.