OK, maybe I'm like a literal paranoia, but what I don't understand is absolutely related to what description we use in the group representation.
Let me make it clearer:
when we say "the representation of group SO(3)", the image in my mind is a field full of infinite numbers of 3x3 matrices. Every matrix can operate on a vector, these matrices of course represent different rotation transformations; and here, when we multiply two matrices through the normal matrix multiplication rules, the new matrix we get is still can operate on a vector, and represent a new space rotation. So I inclined to say these matrices can represent the group SO(3) instead of the claiming "group SO(3) is represented by vector". Similar things happen about the spinor and SU(2), I think it's reasonable to call these 2x2 complex matrices as the representation of SU(2), not the spinor on which rotation matrices operated.
Do I clarify myself? Maybe this is only a trial question about nomenclature, but I'm really confused on it. |
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