1

I'm currently studying Jakob Schwichtenberg's "Physics from Symmetry" where on subsection 3.7.8 he proves that the $(1/2,1/2)$ representation of the Lorentz Group corresponds to 4-vectors.

I have a few questions as to why we can do a few of the things he does. For context, this method heavily uses Van der Waerden notation (dot vs undotted indices), which is introduced before getting the 4-vector representation:

It is clear that the representation is 4 dimensional. He chooses to study 2x2 complex matrices. Specifically, it is clear why general matrices would not be an irreducible representation, so we pick Hermitian matrices (which have 4 free parameters, exactly what we want). Let $v$ be the object that's being acted on by the representation.

My questions have to do with the following portion of the text: (page 81)

enter image description here

Can we add a lower undotted index and an upper doted index into $v$ just because we have the $(1/2,1/2)$ representation? As in, we know beforehand that $(1/2,0)$ corresponds to lower undotted and $(0,1/2)$ corresponds to upper undotted. But I don't see how this is an argument to simply put the lower undotted index and an the upper doted index into $v$.

I understand that we have to copies of $\mathfrak{s}\mathfrak{u}2$, and that they are independent of each other, and that \begin{equation} \left( \frac{1}{2},\frac{1}{2} \right) = \left( \frac{1}{2},0 \right) \otimes \left( 0,\frac{1}{2} \tag{1}\right) \end{equation} But I'm unsure what Eq. 1 really implies/means.

Furthermore, why can't we use the Pauli matrices:

enter image description here

If we have the lower undotted and the upper dotted index?

Qmechanic
  • 201,751
Fernando Garcia Cortez
  • 137

1 Answers1

1

The main points are as follows:

  1. Ref. 1 defines the left-handed $(\frac{1}{2},0)$ Weyl spinor $\chi_{L,a}$ with a lower undotted index in eq. (3.196) and the right-handed $(0,\frac{1}{2})$ Weyl spinor $\chi_R^{\dot{a}}$ with upper dotted index in eq. (3.197).

  2. If we lower the dotted index $\chi_{R,\dot{a}}=\epsilon_{\dot{a}\dot{b}}\chi_R^{\dot{b}}$, then we get an equivalent representation that is the complex conjugate representation of the left-handed Weyl spinor $\chi_{L,a}$, cf. eqs. (3.199) & (3.205).

  3. The left-handed Weyl spinor is the fundamental/defining representation $\chi\mapsto g\chi$ of $g\in G:=SL(2,\mathbb{C})$, cf. eq. (3.209).

  4. $(\frac{1}{2},\frac{1}{2})\cong (\frac{1}{2},0)\otimes (0,\frac{1}{2})$, cf e.g. this related Phys/SE post.

  5. The vector space for the $(\frac{1}{2},\frac{1}{2})$ representation is the set $u(2)$ of Hermitian $2\times 2$ matrices, which is the real linear span of the 3 Pauli matrices (3.80) and the identity matrix, cf. e.g. this related Phys.SE post.

  6. There is a group action $\rho: SL(2,\mathbb{C})\times u(2) \to u(2)$ of the $(\frac{1}{2},\frac{1}{2})$ representation given by $$g\quad \mapsto\quad\rho(g)v~:= ~gv g^{\dagger}, \qquad g\in SL(2,\mathbb{C}),\qquad v~\in~ u(2),$$ which precisely corresponds to the left-handed Weyl representation times its complex conjugate representation if $v_{a\dot{b}}$ has lower indices. (The complex conjugate $\bar{g}$ turns into a Hermitian conjugate $g^{\dagger}$ since acting from right transposes the matrix.)

References:

  1. J. Schwichtenberg, Physics from Symmetry, 2nd edition, 2018.
Qmechanic
  • 201,751