Infinitesimal Rotations and Generators of SO(2)

Sooo, from the last post 'The Generator of SO(2) and Lie Algebra ' (Sep. 16), we differentiated the representation matrix of SO(2) and set the parameter as 0 and called it the generator of SO(2), and we even did something with the exponential function. It seems nonsense and so confusing because we don't know any reason why we differentiated and so on. But we know that the generator was some kind of matrix that's very close with the identity element of SO(2), $I=R(0)$. Thus we can think that $\mathcal{J}$, the generator of SO(2) is related with infinitesimal rotations. From this notion, let's define an infinitesimal rotation matrix by using approximation.

\[ \mathcal{R}(\theta) \simeq I+A \quad \textrm{and} \quad A\sim\mathcal{O}(\theta) \]

Using the above approximation, we neglect $\mathcal{O}(\theta^2)$ term of $A$. As the SO(2) group is the 'rotation group' in $R^2$ and the name of it says that it is an 'orthogonal' group, $\mathcal{R}(\theta)$ which is an element of SO(2), needs to satisfy orthogonality as a matrix. (Every matrix mentioned on this post has a dimension of 2)

\[ \mathcal{R}(\theta)^{T}\mathcal{R}(\theta) = \mathcal{R}(\theta)\mathcal{R}(\theta)^{T} = I \]

And using the approximation of $\mathcal{R}(\theta)$,

\[ \mathcal{R}(\theta)^{T}\mathcal{R}(\theta) \simeq (I+A)^{T}(I+A)=(I+A^T)(I+A) \]

\[ (I+A^T)(I+A) = I+A+A^T+A^{T}A \]

And as $A^{T}A \sim \mathcal{O}(\theta^2)$, and as we only want the terms of $\mathcal{O}(\theta)$, we can neglect $A^{T}A$.

\[ I+A+A^T+A^{T}A \simeq I+A+A^T \]

Thus, the below equation should be satisfied for the orthogonality of $\mathcal{R}(\theta)$.

\[ \mathcal{R}(\theta)^{T}\mathcal{R}(\theta) \simeq I+A+A^T = I \]

So, we can draw a new condition for $A$ which is $A^T=-A$, and this condition is known as the anti-symmetric condition of a matrix. As the matrix $A$ is a $2 \times 2$ anti-symmetric matrix, we can easily find $A$ by comparing the components.

\[ \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}^{T}=-\begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}\]

\[ \begin{pmatrix} A_{11} & A_{21} \\ A_{12} & A_{22} \end{pmatrix}=\begin{pmatrix} -A_{11} & -A_{12} \\ -A_{21} & -A_{22} \end{pmatrix}\]

By comparing the components we can find that $tr(A)=0$, and $A_{12}=-A_{21}$. Let's regard $A_{12}$ as 1 for convenience and define the matrix as $\mathcal{J}$.

\[ \mathcal{J} \equiv \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \]

So, we can express $A$ as $\theta\mathcal{J}$ when $\theta$ is an arbitrary real number.

\[ A=\theta\mathcal{J}=\begin{pmatrix}0 & \theta \\ -\theta & 0 \end{pmatrix} \]

And as we seen before, the $\mathcal{J}$ is known as the generator of SO(2). Using what all we have derived, we can express $\mathcal{R}(\theta)$ as below.

\[ \mathcal{R}(\theta) \simeq I + \theta\mathcal{J} = \begin{pmatrix}1 & \theta \\ -\theta & 1 \end{pmatrix} \sim \mathcal{O}(\theta)\]

Now, let's consider a rotation of non-infinitesimal angle, or let's just call it a finite angle $\phi$. We can infinitesimally divide $\phi$, thus making $\phi$ as a sum of infinite infinitesimal angles. With this idea, we can express a finite rotation $R(\phi)$ with $\mathcal{R}(\theta)$.

\[ R(\phi) = \lim_{N \rightarrow \infty} \mathcal{R}(\theta)^N = \lim_{N \rightarrow \infty} \mathcal{R}\Big(\frac{\phi}{N}\Big)^N \]

And using the final expression of $\mathcal{R}(\theta)$,

\[ \lim_{N \rightarrow \infty}\mathcal{R}\Big(\frac{\phi}{N}\Big)^N =\lim_{N \rightarrow \infty} \mathcal{R}\Big(\ I+\frac{\phi\mathcal{J}}{N}\Big)^N  \]

The final expression looks familiar... It looks like the definition of the exponential function! And now we start to realize how exponential functions and generators are related.

\[ \lim_{N \rightarrow \infty} \mathcal{R}\Big(\ I+\frac{\phi\mathcal{J}}{N}\Big)^N = e^{\phi\mathcal{J}} \]

Thus, 

\[ R(\phi)=e^{\phi\mathcal{J}} \]

As we know derivatives of a certain point of a analytic function, we can 'construct' the function by what is well known as the 'Taylor series'. When we set the certain point as 0, it seems to be very similar with the logic of generating a group. Knowing an element near the identity element can 'generate' the group, just as the 'Taylor series'.