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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.)