The SO(3) group in Group Theory

In summary: Yeah, I've had to take the latter approach as well. Maybe I'll have time in the future to really learn group theory and then revisit the concepts in more depth.
  • #1
sophiatev
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TL;DR Summary
Confusion about the elements of the SO(3) group
In Griffith's Introduction to Elementary Particles, he provides a very cursory introduction to group theory at the start of chapter four, which discusses symmetries. He introduces SO(n) as "the group of real, orthogonal, n x n matrices of determinant 1 is SO(n); SO(n) may be thought of as the group of all rotations in a space of n dimensions", pg. 118. Based on this description, it seems that the elements of the SO(3) group would be the set of 3 x 3 orthogonal rotation matrices. Then later, on page 119 he asserts that "Actually, you have already encountered several examples of group representations, probably without realizing it: an ordinary scalar belongs to the one-dimensional representation of the rotation group, SO(3), and a vector belongs to the three-dimensional representation". What exactly does he mean by this? Clearly neither is a matrix, so I don't quite see how this is true. And how can there be a one-dimensional representation of a group that seems to inherently be in 3D space? (Sorry if this is a question with a very simple answer, this is my first time reading about group theory.)
 
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  • #2
A representation here means a vector space on which the group acts (linearly). The word might make you think that it means a way of representing (as in expressing it in some way) the group itself, but it only means way the group acts on the space. For example SO(3) rotates vectors in three space, so the 3D space is a representation of the group. The group acts by multiplication on the space of all 3x3 matrices. And many others.
 
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  • #3
sophiatev said:
What exactly does he mean by this? Clearly neither is a matrix, so I don't quite see how this is true. And how can there be a one-dimensional representation of a group that seems to inherently be in 3D space?
A (linear) representation of a group ##G## on a vector space ##V## is a group homomorphism ##\varphi\, : \,G\longrightarrow GL(V)##.

There is no a priori restriction for the dimension of ##V##. In the one dimensional case we could e.g. consider the left multiplication with the determinant: ##\varphi(g)\, : \,(\lambda \longmapsto \det(g)\cdot \lambda)##. Not very interesting for ##g\in G=SO(3)##, admitted, but it fulfills the required conditions.

In case ##\dim V=3## we have the matrix application on vectors as natural representation (= group homomorphism = operation) on ##V=\mathbb{R}^3##, namely ##\varphi(g)(\vec{v})=g \cdot \vec{v}##.

There is always the trivial representation ##\varphi(g)=\operatorname{id}_V## which maps any group element (here orthogonal matrix) on the identity in ##GL(V)##. This shows that for any dimension of ##V## there is at least one representation. The one dimensional example above with the determinant is indeed the trivial representation, since all determinants are equal ##1##, so ##\lambda## maps onto ##\lambda##, regardless which ##g\in SO(3)## we choose.
 
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  • #4
Thank you, this makes more sense to me now!
 
  • #5
sophiatev said:
Thank you, this makes more sense to me now!
I'm working my way through that book. In my opinion there is too little in the book to give you a working knowledge of the group theory underpinning the subject. Either you take time out to learn that separately or, what i did, was focus on the specific symmetries inherent in the material. And not worry about the abstract maths behind the scenes.
 
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  • #6
PeroK said:
I'm working my way through that book. In my opinion there is too little in the book to give you a working knowledge of the group theory underpinning the subject. Either you take time out to learn that separately or, what i did, was focus on the specific symmetries inherent in the material. And not worry about the abstract maths behind the scenes.
Yeah, I've had to take the latter approach as well. Maybe I'll have time in the future to really learn group theory and then revisit the concepts in more depth.
 
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What is the SO(3) group in Group Theory?

The SO(3) group, also known as the special orthogonal group in three dimensions, is a mathematical group that consists of all rotations in three-dimensional Euclidean space that preserve orientation and distance. It is an important concept in group theory and has numerous applications in physics, chemistry, and computer graphics.

How is the SO(3) group represented?

The SO(3) group is commonly represented by a 3x3 orthogonal matrix, which is a square matrix with real entries that satisfies the condition ATA = I, where AT is the transpose of A and I is the identity matrix. Each element of the matrix represents a rotation in three-dimensional space.

What are the properties of the SO(3) group?

The SO(3) group has several important properties, including closure, associativity, identity, and inverse. This means that the composition of two rotations in the group is also a rotation in the group, rotations can be performed in any order, there exists an identity rotation that leaves all points unchanged, and every rotation has an inverse rotation within the group.

What are some examples of the SO(3) group in real life?

The SO(3) group has numerous applications in the real world, including in physics, chemistry, and computer graphics. Some examples include the rotation of a rigid body in space, the orientation of a molecule, and the rotation of 3D objects in computer-generated images and animations.

What is the significance of the SO(3) group in mathematics?

The SO(3) group is an important concept in mathematics as it is an example of a Lie group, which is a group that is also a smooth manifold. It has applications in various fields of mathematics, including algebra, geometry, and topology, and has connections to other important mathematical concepts such as quaternions and Euler angles.

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