- #1
Haorong Wu
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- TL;DR Summary
- What is the relation between tensors and group SO(3)?
I am reading Group Theory in a Nutshell for Physicists by A. Zee.
I have big problems when learning chapter IV.1 Tensors and Representations of the Rotation Groups SO(N).
It reads
I can understand why ##D\left ( R \right )## is a representation of SO(3), but I hardly can see why the tensor T can be a representation. I thought a representation should be scalar or square matrix, but why a column consisting of tensors can be a representation, as well.
Also, I do not understand why the tensor came in. If I am correct, elements of groups should be some transformations that leave some objects invariant, but I can hardly imagin how tensors become transformations.
I became more frustrated when the following sections are full of tensors, and I got totally lost.
I have big problems when learning chapter IV.1 Tensors and Representations of the Rotation Groups SO(N).
It reads
Mentally arrange the nine objects ##T^{ij}## in a column ##
\begin{pmatrix}
T^{11} \\
T^{12} \\
\vdots \\
T^{33}
\end{pmatrix}
##. The linear transformation on the nine objects can then be represetned by a 9-by-9 matrix ##D\left ( R \right )## acting on this column.
For every rotation, specified by a 3-by-3 matrix R, we can thus associate a 9-by-9 matrix ##D\left ( R \right )## transforming the nine objects ##T^{ij}## linearly among themselves.
...
The tensor T furnishes a 9-dimensional representation of the rotation group SO(3).
I can understand why ##D\left ( R \right )## is a representation of SO(3), but I hardly can see why the tensor T can be a representation. I thought a representation should be scalar or square matrix, but why a column consisting of tensors can be a representation, as well.
Also, I do not understand why the tensor came in. If I am correct, elements of groups should be some transformations that leave some objects invariant, but I can hardly imagin how tensors become transformations.
I became more frustrated when the following sections are full of tensors, and I got totally lost.