Rotations in three Dimensions

The standard methods to rotate vectors in three dimensions uses 3×3 rotation matrices:

R_x=

Analogous to the definition for SO(2), these matrices form a

ba-sis²¹for a group, called SO(3). This means an arbitrary element of ²¹This notion is explained in ap-pendix A.1.

SO(3)can be written as a linear combination of the above matrices.

If we want to rotate the vector

v=

around the z-axis²², we multiply it with the corresponding rotation ²²A general, rotated vector is derived explicitly in appendix A.2.

To get a second description for rotations in three dimensions, the first thing we have to do is find a generalisation of complex numbers in higher dimensions. A first guess may be to go from 2-dimensional complex numbers to 3-dimensional complex numbers, but it turns out that there are no 3-dimensional complex numbers. Instead, we can find 4-dimensional complex numbers, called quaternions. The quaternions will prove to be the correct second tool to describe rota-tions in 3-dimensions and the fact that this tool is 4-dimensional re-veals something deep about the universe. We could have anticipated this result, because to describe an arbitrary rotation in 3-dimensions, 3parameters are needed. Four dimensional complex numbers, with

the constraint to unit quaternions²³, have exactly 3 degrees of free- ²³Remember that we used the con-straint to unit complex numbers in the two dimensional case, too.

dom.

3.3.1 Quaternions

The 4-dimensional complex numbers can be constructed analogous to the 2-dimensional complex numbers. Instead of just one complex

"unit" we introduce three, named i, j, k. These fulfil

i²=j²=k²= −1. (3.25) Then a 4-dimensional complex number, called a quaternion, can be written as

q=a1+bi+cj+dk. (3.26)

We now need multiplication rules for ij =? etc., because products like this will occur when one multiplies two quaternions. The extra condition

ijk= −1 (3.27)

suffices to compute all needed relations, for example ij=k follows from multiplying Eq. 3.27 with k:

ij kk

=−1

= −k→ij=k. (3.28)

The set of unit quaternions q=a1+^bi+^cj+dk are those satisfy-ing the condition²⁴

24The symbol †, here is called "dagger"

and denotes transposition plus complex conjugation: a^† = (a^)^T. The ordinary scalar product always includes a trans-position a·b=a^Tb, because matrix multiplication requires that we multiply a row with a column. In addition, for complex entities we include complex conjugation that makes sure we get something real, which is important if we want to interpret things in terms of length.

q^†q=^! 1

→ (a1−bi−cj−dk)(a1+bi+cj+dk) =a²+b²+c²+d^{2 !}=1. (3.29) Exactly as the unit complex numbers formed a group under complex number multiplication, the unit quaternions form a group under quaternion multiplication.

Analogous to what we did for two-dimensional complex numbers, we represent each of the three complex units with a matrix. There are different possible ways of doing this, but one choice that does the job is as complex 2×2 matrices:

1=

1 0 0 1



, i=

 0 1

−1 0



j=

 0 −i

−i 0



, k=

 i 0 0 −i



. (3.30)

You can check that these matrices fulfil the defining conditions in Eq. 3.25 and Eq. 3.27. A generic quaternion can then be written, using

these matrices, as

q=a1+bi+cj+dk=



a+di −b−ci b−ci a−di



. (3.31)

Furthermore, we have

det(q) =a²+b²+c²+d². (3.32) Comparing this with Eq. 3.29 tells us that the set of unit quaternions is given by matrices of the above form with unit determinant. The unit quaternions, written as complex 2×2 matrices therefore fulfil the conditions

U^†U=1 and det(U) =1. (3.33)

In addition, the matrices in Eq. 3.30 are linearly independent²⁵and ²⁵This notion is explained in ap-pendix A.1.

therefore form a basis for the group called SU(2). Take note that the way we define SU(2)here, is analogous to how we defined SO(2). The S denotes special, which means det(U) = 1 and U stands for

unitary, which means the property²⁶U^†U=1. Every unit quaternion ²⁶For some more information about this, have a look at the appendix Sec. 3.10 at the end of this chapter.

can be identified with an element of SU(2).

Now, how is SU(2)related to rotations? Unfortunately, the map

between SU(2)and²⁷SO(3)is not as simple as the one between U(1) ²⁷Recall that SO(3)is the set of the usual rotation matrices acting on 3 dimensional vectors.

and SO(2).

In 2-dimensions the 2 parameters of a complex number z =a+ib could be easily identified with the two spatial axes, i.e. v = x+ iy. The restriction to unit complex numbers automatically makes

sure that the resulting matrix preserves the length of any vector^{28 28}Recall that here R is a unit complex number, because complex numbers can be rotated by multiplication with unit complex numbers. Therefore we have R^R=1, which is the defining condition for unit complex numbers.

(Rz)^Rz = z^R^Rz = z^z. The quaternions have 4 parameters, so an identification with the 3 coordinates of a usual three-dimensional vector is not obvious. If we define

v≡^xi+^yj+^zk (3.34)

we can compute, using the matrix representation of the quaternions det(v) =x²+y²+z². (3.35) Therefore, if we want to consider transformations that preserve the length of the vector(x, y, z), we must use matrix transformations that preserve determinants. The restriction to unit quaternions means that we must restrict to matrices with unit determinant. Everything seems straight forward, but now comes a subtle point. A first guess would be that a unit quaternion u induces a rotation on v simply by multiplication. This is not the case, because the product of u and v

may not belong toRi+Rj+Rk. Therefore, the transformed vec-tor can have a component we are not able to interpret. Instead the transformation that does the job is given by

v^=q⁻¹vq. (3.36)

It turns out that by making this identification unit quaternions can describe rotations in 3-dimensions.

Let’s take a look at an explicit example: To make the connection to our example in two dimensions, we will define u as a unit vector in Ri+Rj+Rk and denote a unit quaternion with

t=cos(θ) +sin(θ)u. (3.37) Using Eq. 3.34 a generic vector can be written

^v= (^vx, vy, vz)^T=^vxi+^vyj+^vzk_ =

Eq. 3.31

 iv_z −^vx−^ivy

v_x−iv_y −iv_z

 . (3.38) With the identifications made above, we want to rotate, as an ex-ample, a vectorv = (1, 0, 0)^Taround the z-axis. We will make a particular choice for the vector and for the quaternion representing the rotation and show that it works. We write, using quaternions in their matrix representation (Eq. 3.30)

v= (1, 0, 0)^T →v=1i+0j+0k=

 0 −1

1 0



(3.39)

R_z(θ) =cos(θ)1+sin(θ)k=



cos(θ) +i sin(θ) 0 0 cos(θ) −i sin(θ)

 , (3.40) which we can rewrite using Euler’s formula²⁹

29For a derivation have a look at

ap-pendix B.4.2. e^ix =cos(x) +i sin(x)

⇒^Rz(θ) =

e^iθ 0 0 e^−iθ



. (3.41)

Inverting the quaternion rotation matrix yields

R_z(θ)⁻¹ =

cos(θ) −^{i sin}(θ) 0 0 cos(θ) +i sin(θ)



=

e^−iθ 0 0 e^iθ

 . (3.42) The rotated vector is then using Eq. 3.36

v^=R_z(θ)⁻¹vR_z(θ) =

e^−iθ 0 0 e^iθ

 0 −1

1 0

 e^iθ 0 0 e^−iθ



=



0 −e^−i2θ e^2iθ 0



=



0 −cos(2θ) +i sin(2θ) cos(2θ) +i sin(2θ) 0



(3.43) On the other hand, an arbitrary vector can be written in this quaternion notation (Eq. 3.38)

v^=

 iv^_z −^v_x−^iv_y v^_x−iv^_y −iv^_z



(3.44)

which we now compare with Eq. 3.43. This yields

v^_x=cos(2θ) , v^_y= −sin(2θ) , v^_z =0. (3.45) Therefore, written again in the conventional vector notation

→ v^ = (cos(2θ),−sin(2θ), 0)^T. (3.46)

The identifications do indeed induce rotations³⁰, but something ³⁰See the example in Eq. 3.24 where we rotated the vector, using the conven-tional rotation matrix

needs our attention. We haven’t rotatedv byθ, but by 2θ. There-fore, we defineφ≡2θ because then φ really represents the angle we rotate and rewrite Eq. 3.37, which yields

t=cos(^φ

2) +sin(^φ

2)u. (3.47)

We can now see that the identifications we made are not one-to-one, but rather we have two unit-quaternions describing the same

rotation. For example^{31 31}Because a rotation byπ is the same as

a rotation by 3π=2π+π for ordinary vectors, because 2π =360^◦is a full rotation. In other words: We can see that two quaternions u and−u can be used to rotate a vector byπ.

t_φ=π =^u

((

t_φ=3π = −^u

uu

Vector Rotation by ^π₂

This is the reason SU(2)is called the double-cover of SO(3). It is always possible to go unambiguously from SU(2)to SO(3)but not vice versa. One may think this is just a mathematical side-note, but we will understand later that groups which cover other groups are

indeed more fundamental³². ³²To spoil the surprise: We will use

the double cover of the Lorentz group, instead of the Lorentz group itself, because otherwise we miss something important: Spin. Spin is some kind of internal momentum and one of the most important particle labels. This is discussed in detail in Sec. 4.5.4 and Sec. 8.5.5.

In order to reveal the group that covers a given group, we need to introduce the most important tool of Lie theory: Lie algebras, which is the topic of the next section.

Take note that the fact we had one quaternion parameter too much, may be interpreted as a hint towards relativity. One may argue that a more natural identification would have been, as in the

two-dimensional case, v =t1+xi+yj+zk. We see that pure math-ematics pushes us towards the idea of using a 4th component and what could it be, if not time? If we now want to describe rotations in 4-dimensions, because we know that the universe we live in is 4-dimensional, we have two choices:

• We could search for even higher dimensional complex numbers or

• we could again try working with quaternions.

From the last paragraph it may seem that quaternions have some-thing to say about rotations in 4 dimensions, too. An arbitrary ro-tation in 4 dimensions is described by³³6parameters. There is no

33Using ordinary matrices, we need in four dimensions 4×4 matrices. The two conditions O^TO=1 and det(O) = 1 reduce the 16 components of an arbitrary 4×matrix to six independent components.

7-dimensional generalisation of complex numbers, which together with the constraint to unit objects would have 6 free parameters, but we see that two unit quaternions have exactly 6 free parameters.

Therefore, maybe it’s possible to describe a rotation in 4-dimensions by two quaternions? We will learn later that there is indeed a close connection between two copies of SU(2)and rotations in four dimen-sions.

In document Jakob Schwichtenberg - Physics from Symmetry.pdf (Page 46-51)