What does "transform like" mean?

Question

I read on a pdf that considering $SU(2)$ the spinor $(\xi_1, \xi_2)^T$ transform the same way as $(-\xi_2^*, \xi_1^*)^T$. What does it mean that they transform the same way? I don't know what's the meaning of "two things transform in the same way".

Answer 1

In the context of group actions (in this case, the group $SU(2)$ acting on spinors), “$A$ transforms like $B$” means that $A$ and $B$ afford the same linear representation of the group, so the same matrices can be used to transform them if an element of the group is applied; in this sense, $A$ and $B$ are subject to the same transformation law. In the present case, the spinor $\xi=\pmatrix{\xi_1\\\xi_2}$ transforms as $\xi\to U\xi$ and its Hermitian conjugate $\xi^\dagger$ transforms as $\xi^\dagger\to\xi^\dagger U^\dagger$. This is a different transformation law, so the book says that “$\xi$ and $\xi^\dagger$ transform in different ways”, but it then goes on to show that $\xi$ and $\pmatrix{-\xi_2^*\\\hphantom-\xi_1^*}$ have the same linear transformation law under the action of $SU(2)$, with

$$ \xi\to\pmatrix{a&b\\-b^*&a^*}\xi $$

and

$$ \pmatrix{-\xi_2^*\\\hphantom-\xi_1^*}\to\pmatrix{a&b\\-b^*&a^*}\pmatrix{-\xi_2^*\\\hphantom-\xi_1^*}\;, $$

so they afford the same representation of $SU(2)$ (in this case, the defining representation).

Another usage example that perhaps makes clearer that different types of objects can be compared in this manner is “The $p$ orbitals transform like a $3$-vector”. That is, in an atom for every principal quantum number $n$ there are three $p$ orbitals, $p_x$, $p_y$, and $p_z$. Together they afford a three-dimensional representation of the rotation group $SO(3)$ that obeys the same transformation law as a spatial vector. This is a non-trivial observation, as other sets of orbitals transform according to other representations, e.g. the one $s$ orbital transforms according to the trivial representation and the five $d$ orbitals transform according to the $5$-dimensional representation; there just happens to be one set of three orbitals that transform like a spatial vector (and can be chosen to have their symmetry axes aligned with the coordinate axes; hence their names).