# What does "transform like" mean?

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".

• In my experience, this is a term the physicists use more frequently than the mathematicians, so you might ask over on the physics Stackexchange. Apr 11, 2020 at 14:39
• Please reference that pdf Apr 11, 2020 at 14:39
• Or maybe describe the document for what it contains rather than its file format. Apr 11, 2020 at 14:39
• indeed I found this statement in a pdf from a physics book: Quantum Field Theory, Ryder (page 33) Apr 11, 2020 at 14:42
• @SimoBartz, include the link. May 10, 2020 at 8:11

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