- #1
Tio Barnabe
I would kindly appreciate any corrections to my conclusions, because I need to get this subject straight for learning QFT in a satisfactory way.
From what I have been reading about Lie groups so far, I have concluded the following:
1 - A group is independent of a representation, but we usually define a group using some representation. We do that in order to get the Lie Algebra of the group.
2 - Once we get the Lie Algebra of the group, we can derive what it looks like in any representation.
3 - A given group has the same number of generators in any representation, because of 1 above.
[I realized that these properties of groups are analogous to those of topological spaces, in the sense that we usually define a topological space, for instance, the 2-sphere in ##\mathbb{R}^3##, because it seems the only reasonable way to define it. After defining, the 2-sphere becomes totally independent of ##\mathbb{R}^3##, i.e., we don't need to see it as embedded in ##\mathbb{R}^3##. Other examples are the cylinder, the torus... The analog of various representations of a group would be various possible metrics for a given topological space.]
Now, comes a part that I still don't understand.
When we are using the ##2 \otimes 2## representation of ##SU(2)##, i.e., the direct sum of two 2-dimensional representations of ##SU(2)##, it seems that the correct way to operate with the transformation matrices, is to operate with them on a matrix ##M## from the left & from the right of ##M##. This is in contradiction with the assumption that the transformation matrices would operate on vectors, only from the left.
From what I have been reading about Lie groups so far, I have concluded the following:
1 - A group is independent of a representation, but we usually define a group using some representation. We do that in order to get the Lie Algebra of the group.
2 - Once we get the Lie Algebra of the group, we can derive what it looks like in any representation.
3 - A given group has the same number of generators in any representation, because of 1 above.
[I realized that these properties of groups are analogous to those of topological spaces, in the sense that we usually define a topological space, for instance, the 2-sphere in ##\mathbb{R}^3##, because it seems the only reasonable way to define it. After defining, the 2-sphere becomes totally independent of ##\mathbb{R}^3##, i.e., we don't need to see it as embedded in ##\mathbb{R}^3##. Other examples are the cylinder, the torus... The analog of various representations of a group would be various possible metrics for a given topological space.]
Now, comes a part that I still don't understand.
When we are using the ##2 \otimes 2## representation of ##SU(2)##, i.e., the direct sum of two 2-dimensional representations of ##SU(2)##, it seems that the correct way to operate with the transformation matrices, is to operate with them on a matrix ##M## from the left & from the right of ##M##. This is in contradiction with the assumption that the transformation matrices would operate on vectors, only from the left.
