# Transformations under some representation

I have a profound question. We call spinor an element of a space which transforms with the irreps with semi-integer $$j$$ of $$SU(2)$$. But i can't find the meaning of this. How could it transform in other ways? I mean, i am working with $$SU(2)$$ matrices so it MUST transfom with them. If i use a random matrix it would transform in other ways, so what's the meaning? A tensor is defined by how it transforms under some action of a group, but how could it transform in other ways if we apply only that group?

In 3D space we impose the distance to be not-modified, so we found the group of transformation is $$SO(3)$$. Why we say "vectors are quantities that transforms with $$SO(3)$$ matrices"? I can use a general matrix and it would be still a coordinate in the space.

For example Lorentz group, we impose the $$s^2$$ to rest the same, so we get transformations left-right spinors like $$(1/2,0)$$ or $$(1,1)$$ but what does this all mean?

• This is a very good question. It deserves to be up-voted rather than down-voted. It is true that we use $SU(2)$ representation matrices for the integer as well as the half integer spins. But for the integer spins (and only the integer ones), the $SU(2)$ representation matrices are unitarily equivalent to $SO(3)$ matrices, i.e.,$M_{SU(2)} = V M_{SO(3)} V^{-1}$, where $V$ is unitary. $M_{SO(3)}$ is a representation of $SO(3)$, in particular all of its matrix elements are real. Jan 30 '18 at 16:37

## 1 Answer

1. In the context of $$SU(2)$$, it is only the $$j=\frac{1}{2}$$ representation that is called a spinor representation. It is also called the fundamental or defining representation. Let's call it $$V$$. It is 2-dimensional.

The higher irreducible representations $$j\in\frac{1}{2}\mathbb{N}$$ with dimension $$2j+1$$ can in turn be realized as the $$2j$$'th symmetric tensor product $$V^{\odot 2j}$$, cf. e.g. Ref. 1.

2. That $$SU(2)$$ is the double cover of $$SO(3)$$ is discussed in e.g. this, this, this & this Phys.SE posts.

3. For irreducible representations of the Lorentz group, see e.g. this Phys.SE post.

References:

1. G. 't Hooft, Introduction to Lie Groups in Physics, lecture notes, p. 35-37. The pdf file is available here.