Hmm, good question. Would we say the "SO(3) group can be represented by a vector"?
quoting Griffiths, "Introduction to Elementary Particles"
"Thus spin-1/2 particles transform under rotations according to the fundamental, two-dimensional representation of SU(2). Similarly, particles of spin 1, described by vectors, belong to the three-dimensional representation of SU(2).... Particles of different spin, then belong to different representations of the rotation group."
So particles of spin 1/2, desribed by spinors, belong to the 2D representation of SU(2). This seems to make sense, if we take this analogy to classical mechanics, points in space are described by vectors and belong to the 3D representation of S0(3).
So I think what this means is that a spinor is a representation of the SU(2) group but to me this means the same thing as you said a "SU(2) group can be represented by a spinor" So yes you are correct it is NOT "a spinor is represented by the SU(2) group but the other way around so yes you are right. Cheers.
Ben
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