\bmit{\SO(2)}

Section 1.1 \(\bmit{\SO(2)}\)

Subsection 1.1.1 Representations

The rotation group in two Euclidean dimensions is known as \(\SO(2)\text{.}\) How many representations of this group can you think of?

The first representation of \(\SO(2)\) we consider is in terms of \(2\times2\) matrices, of the form

\begin{equation} M(\phi) = \begin{pmatrix} \cos\phi & -\sin\phi \\ \sin\phi & \cos\phi \end{pmatrix} .\tag{1.1.1} \end{equation}

The “O” in \(\SO(2)\) stands for orthogonal. Orthogonal matrices satisfy

\begin{equation} M^T M = \one\tag{1.1.2} \end{equation}

where \(T\) denotes matrix transpose, and where we write \(\one\) for the identity matrix (rather than \(I\text{,}\) for which we will have another use). Such matrices preserve the (squared) magnitude

\begin{equation} |v|^2 = v^T v\tag{1.1.3} \end{equation}

of a vector \(v\in\RR^2\text{,}\) since

\begin{equation} (Mv)^T (Mv) = (v^TM^T) (Mv) = v^T v .\tag{1.1.4} \end{equation}

The “S” in \(\SO(2)\) stands for special, and refers to the additional condition that

\begin{equation} |M| = \det(M) = 1 .\tag{1.1.5} \end{equation}

Orthogonal matrices \(M\) with \(|M|=1\) are rotations; if \(|M|=-1\text{,}\) the only other possibility, they are reflections.

Our second representation of \(\SO(2)\) is in terms of the complex numbers, of the form

\begin{equation} w(\phi) = e^{i\phi} .\tag{1.1.6} \end{equation}

Such complex numbers have norm 1, that is

\begin{equation} |w(\phi)|^2 = w\,\bar{w} = 1\tag{1.1.7} \end{equation}

and preserve the magnitude \(|z|\) of any complex number \(z\in\CC\text{,}\) since

\begin{equation} |wz| = |w|\,|z| = |z| .\tag{1.1.8} \end{equation}

Sound familiar?

Our third representation of \(\SO(2)\) is purely geometric. Rotations are rigid transformations of \(\RR^2\text{,}\) obtained by, well, rotating the plane through a given angle \(\phi\text{.}\) In other words, the rotations in \(\SO(2)\) are in one-to-one correspondence with the angles in the (unit) circle, that is, with the circle itself. Thus, \(\SO(2)\) can be thought of as the circle \(\SS^1\text{.}\)

Take a moment to compare and contrast these various representations of \(\SO(2)\text{.}\) What are their properties?

Subsection 1.1.2 Properties

The geometric representation makes clear that \(\SO(2)\) is a group; the composition of two rotations is another rotation. In matrix language, we have

\begin{equation} M(\alpha+\beta) = M(\alpha) M(\beta)\tag{1.1.9} \end{equation}

and similarly

\begin{equation} e^{i(\alpha+\beta)} = e^{i\alpha}e^{i\beta}\tag{1.1.10} \end{equation}

for complex numbers. Setting \(\alpha=0\) corresponds to the identity element, and setting \(\beta=-\alpha\) leads immediately to inverse elements.

Our two algebraic representations are clearly closely related. The identification of \(M(\phi)\) with \(e^{i\phi}\) suggests the further identification

\begin{equation} x+iy \longleftrightarrow \begin{pmatrix}x & -y \\ y & x\end{pmatrix} .\tag{1.1.11} \end{equation}

This identification seems even more reasonable after writing

\begin{equation} \begin{pmatrix}x & -y \\ y & x\end{pmatrix} = x \one + y \Omega\tag{1.1.12} \end{equation}

where again \(\one\) denotes the identity matrix, and

\begin{equation} \Omega = \begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix} .\tag{1.1.13} \end{equation}

Notice that \(\Omega^2=-1\text{!}\)

We note for future reference both that

\begin{equation} \Omega = M'(0)\tag{1.1.14} \end{equation}

and that

\begin{equation} M(\phi) = e^{\Omega\phi}\tag{1.1.15} \end{equation}

where matrix exponentiation is formally defined in terms of a power series (which always converges).

Returning to our geometric representation, since \(\SO(2)\) can be thought of as \(\SS^1\text{,}\) it is a smooth manifold, that is, it is a smooth 1-surface (i.e. a curve), on which one can introduce coordinates (e.g. \(\phi\)). Thus, \(\SO(2)\) is both a group and a manifold; it is our first example of a Lie group.

In the language of vector calculus, we can introduce a vector field that is tangent to \(\SS^1\text{.}\) One possible choice would be the unit vector tangent to the circle, often written \(\Hat\phi\text{.}\) In the language of differential geometry, however, vector fields are interpreted as directional derivative operators, so that

\begin{equation} \vv(f) = \vv\cdot\grad f .\tag{1.1.16} \end{equation}

Since

\begin{equation} \Hat\phi\cdot\grad f = \frac{1}{r}\frac{\partial f}{\partial \phi}\tag{1.1.17} \end{equation}

we choose instead the tangent vector

\begin{equation} r\,\Hat\phi = x\,\Hat{y} - y\,\Hat{x}\tag{1.1.18} \end{equation}

Equivalently, as differential operators we choose

\begin{equation} \partial_\phi = x\,\partial_y - y\,\partial_x\tag{1.1.19} \end{equation}

where we have introduced the notation \(\partial_q\) for \(\frac{\partial}{\partial q}\text{.}\)

What do tangent vectors look like in the complex representation of \(\SO(2)\text{?}\) Take the derivative! We have

\begin{equation} \frac{dw}{d\phi} = iw = ie^{i\phi}\tag{1.1.20} \end{equation}

What does this result mean geometrically?

Evaluate this derivative first at the identity element, where \(\phi=0\text{.}\) At the point \(z=1\text{,}\) this derivative is \(i\text{.}\) But the \(i\)-direction is vertical; this direction is tangent to the circle at \(z=1\text{.}\) A similar argument works at any point on the circle; \(iw\) always represents the direction rotated \(\frac{\pi}{2}\) counterclockwise from \(w\)—precisely the direction tangent to the circle.

Finally, consider the matrix representation of \(\SO(2)\text{.}\) Again, take the derivative, yielding

\begin{equation} A = M'(0) = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} .\tag{1.1.21} \end{equation}

At other points on the circle, we have

\begin{equation} M'(\alpha) = \begin{pmatrix} -\sin\phi & -\cos\phi \\ \cos\phi & -\sin\phi \end{pmatrix} = M(\alpha) A .\tag{1.1.22} \end{equation}

This relationship between the derivative of a path in the group at any point and its derivative at the identity element is a hallmark of the study of Lie groups, and allows us to study such groups by studying their derivatives at the identity element, a much simpler process.