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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. This allows us to generalize the infinitesimal rotation operator to any axis of rotation :

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

The commutation relations of the angular momentum operators are given by

where is the Levi-Civita symbol. This will be important when we introduce the concept of Lie groups and Lie algebras in the future.

Finally, let's try to apply the z-rotation (without loss of generality) operator to a given state by examining the value of . The rotated ket is simply . Then the expectation value is

Recalling the Baker-Campbell-Hausdorff formula, we can write this as

This shows that the expectation value of the angular momentum operator is rotated by an angle about the -axis.

Rotation Operators for Spin-1/2 Systems​

For the commutation relations of angular momentum to hold, the dimensionality of the representation must be at least 2. In this context, "spin" is just another name for the angular momentum in the special case of a spin-1/2 system. Recall that the spin-1/2 operators are defined as

These operators satisfy the commutation relations of

similar to the angular momentum operators. Also, just like the angular momentum operators, the expectation values for spins are given by

As such, the behavior of under rotations is the same as that of a classical angular momentum vector.

Where Spinors Come From​

Now we can see how spinors arise from the rotation group. Suppose we have any spin-1/2 system represented by a state . In the spin-z basis, we can insert a completeness relation,

If we apply a z-rotation operator to this state, we have

Notice that the angle is halved on the right-hand side. In other words, if we apply to the state, we get

This means that rotating by radians results in a sign change of the state. We need to instead rotate by radians to get back to the original state.