Rotations in two Dimensions

As a first step into Group theory, we start with an easy, but impor-tant, example. What are transformations in two dimensions that leave the length of any vector unchanged? After thinking about it for a

while, we come up with⁸rotations and reflections. These transfor- ⁸Another kind of transformation that leaves the length of a vector unchanged are translations, which means we move every point a constant distance in a specified direction. These are described mathematically a bit different and we are going to talk about them later.

mations are of course the same ones that map the unit circle into the unit circle. This is an example of how one group may act on different kinds of objects: On the circle, which is a geometric shape and on a vector. Considering vectors, one can represent these transformations by rotation matrices⁹, which are of the form

9For an explicit derivation of these matrices have a look a look at ap-pendix A.2.

R_θ =

 cos(θ) sin(θ)

−sin(θ) cos(θ)



(3.3) and describe two-dimensional rotations about the origin by angleθ.

Reflections at the axes can be performed using the matrices:

P_x=

You can check that these matrices, together with the ordinary matrix multiplication as binary operation◦, satisfy the group axioms and therefore these transformations form a group.

We can formulate the task of finding "all transformations in two dimensions that leave the length of any vector unchanged" in a more abstract way. The length of a vector is given by the scalar product of the vector with itself. If the length of the vector is the same after the transformation a→a^, the equation

a^·a^{ !}=a·a (3.5)

must hold. We denote the transformation with O and write the trans-formed vector as a→a^ =Oa. Thus

a·a=a^Ta→a^Ta^ = (Oa)^TOa=a^TO^TOa=^! a^Ta=a·a, (3.6) where we can see the condition a transformation must fulfil to leave the length of a vector unchanged is

O^TO=I, (3.7)

where I denotes the unit matrix¹⁰. You can check that the

well-10I=^1₀ ⁰₁^

known rotation and reflection matrices we cited above fulfil exactly this condition¹¹. This condition for two dimensional matrices defines

11With the matrix from Eq. 3.3 we have R^T_θR =

12Every orthogonal 2×2 matrix can be written either in the form of Eq. 3.3, as in Eq. 3.4, or as a product of these matrices.

find a subgroup of this group that includes only rotations, by taking notice of the fact that it follows from the condition in Eq. 3.7 that

det(O^TO)=^! det(I) =1

→det(O^TO) =det(O^T)det(O)=^! det(I) =1

→ (det(^O))^{2 !}=1→det(^O)= ±^! 1 (3.8) The transformations of the group with det(O) =1 are rotations¹³and

13As can be easily seen by looking at the matrices in Eq. 3.3 and Eq. 3.4.

The matrices with det O = −1 are reflections.

the two conditions

O^TO=^I (3.9)

det O=1 (3.10)

define the SO(2)group, where the "S" denotes special and the "O"

orthogonal. The special thing about SO(2)is that we now restrict it to transformations that keep the system orientation, i.e., a

right-handed¹⁴coordinate system must stay right-handed. In the language ¹⁴If you don’t know the difference between a right-handed and a left-handed coordinate system have a look at the appendix A.5.

of linear algebra this means that the determinant of our matrices must be+1.

3.2.1 Rotations with Unit Complex Numbers

There is a quite different way to describe rotations in two dimen-sions that makes use of complex numbers: Rotations about the origin by angleθ can be described by multiplication with a unit complex

number ( z=a+ib which fulfils the condition¹⁵ |z|²=z^z=1). ^15Thesymbol denotes complex conjugation: z=a+ib→z^=a−ib

The unit complex numbers form a group, called¹⁶U(1)under

or-16The U stands for unitary, which means the condition U^U=1

dinary complex number multiplication, as you can check by looking at the group axioms. To make the connection with the group defini-tions for O(3)and SO(3)introduced above, we write the condition

as^{17 17}For more general information about

the definition of groups involving a complex product, have a look at the appendix in Sec. 3.10.

U^U=1. (3.11)

Another way to write a unit complex number is¹⁸

18This is known as Euler’s formula, which is derived in appendix B.4.2. For a complex number z=a+ib, a is called the real part of z: Re(z) =a and b the imaginary part: Im(z) =b. In Euler’s formula cos(θ)is the real part, and sin(θ)the imaginary part of R_θ.

Fig.3.5: The unit complex numbers lie on the unit circle in the complex plane.

R_θ =e^iθ =cos(θ) +i sin(θ), (3.12) because then

R^_θR_θ =e^−iθe^iθ=^cos(θ) −^{i sin}(θ)^cos(θ) +^{i sin}(θ)^=1 (3.13) Let’s take a look at an example: We rotate the complex number z=3+5i, by 90^◦, thus

z→z^ =e^i90◦z= (cos_{ } (90^◦)

=0

+i sin_{ } (90^◦)

=1

)(3+5i) =i(3+5i) =3i−5 (3.14) The two complex numbers are plotted in Fig. 3.6 and we see the multiplication with e^i90◦ does indeed rotate the complex number by 90^◦. In this description, the rotation operator e^i90◦ acts on complex numbers instead of on vectors. To describe a rotation in two dimen-sions, one parameter is necessary: the angle of rotationφ. A complex number has two degrees of freedom and with the constraint to unit complex numbers|z| =1, one degree of freedom is left as needed.

We can make the connection to the previous description by repre-senting complex numbers by real 2×2 matrices. We define

1=

1 0 0 1



, i=

0 −1

1 0



. (3.15)

You can check that these matrices fulfil

1²=1, i²= −1, 1i=i1=i. (3.16)

Fig.3.6: Rotation of a complex number, by multiplication with a unit complex number

So now, the complex representation of rotations of the plane reads

R_θ=cos(θ) +i sin(θ) =cos(θ)

we go back to the familiar representation of rotations of the plane.

Maybe you have noticed a subtle point: The familiar rotation matrix in 2-dimensions acts on vectors, but here we identified the complex unit i with a real matrix (Eq. 3.15), therefore, the rotation matrix will act on a 2×2 matrix, because the complex number we act on becomes a matrix, too.

A generic complex number in this description reads

z=a+ib=a

Let us take a look at how rotations act on such a matrix that repre-sents a complex number: which is the same result we get if we act with R_θon a column vector



We see that both representations do exactly the same thing and math-ematically speaking we have an isomorphism¹⁹between SO(2)and

19An isomorphism is a one-to-one map Π that preserves the product structure

Π(g1)Π(g2) =Π(g1g2) ∀g1, g₂∈G.

U(1). This is a very important discovery and we will elaborate on such lines of thoughts in the following chapters.

Next we want to describe rotations in three dimensions and find

similarly two descriptions for rotations in three dimensions²⁰. ²⁰Things are about to get really interest-ing! Analogous to the two-dimensional case we discussed in the preceding sec-tion, we will find a second description of rotations in three dimensions and this alternative description will reveal something fundamental about nature.