# Homework Help: Representations of lorentz group and transformations IN DETAIL!

1. Dec 14, 2011

### Onamor

From Peskin and Schroeder:
The finite-dimentional representations of the rotation group correspond precisely to the allowed values for the angular momentum: integers or half integers.
From the Lorentz commutation relations:
$\left[J^{\mu \nu},J^{\rho \sigma}\right]=i \left(g^{\nu \rho}J^{\mu \sigma}-g^{\mu \rho}J^{\nu \sigma}-g^{\nu \sigma}J^{\mu \rho}+g^{\mu \sigma}J^{\mu \rho}\right)$
we can define rotations $L^{i}=\frac{1}{2}\epsilon^{ijk}J^{jk}$ and boosts $K^{i}=J^{0i}$. An infinitesimal Lorentz transformation can then be written $\boldsymbol{\Phi}\rightarrow\left(1-i \boldsymbol{\theta}.\boldsymbol{L}-i \boldsymbol{\beta}.\boldsymbol{K}\right) \boldsymbol{\Phi}$.
The combinations $\boldsymbol{J_{+}}=\frac{1}{2}\left(\boldsymbol{L}+i\boldsymbol{K}\right)$ and $\boldsymbol{J_{-}}=\frac{1}{2}\left(\boldsymbol{L}-i\boldsymbol{K}\right)$ commute and satisfy the commutation relations of angular momentum.
This implies that all finite dimensional representations of the Lorentz group correspond to pairs of integers or half integers, $\left(j_{+},j_{-}\right)$, corresponding to pairs of representations of the rotation group. Using the fact that $\boldsymbol{J}=\boldsymbol{\sigma}/2$ in the spin-1/2 representation of angular momentum, write explicitly the transformation according to the $\left(\frac{1}{2},0\right)$ and $\left(0,\frac{1}{2}\right)$ representations of the Lorentz group. You should find these correspond precisely to the transformations for the Weyl spinors:
$\psi_{R,L}\rightarrow \left(i \boldsymbol{\theta}\!\frac{\boldsymbol{\sigma}}{2}\pm\boldsymbol{\beta}\!\frac{\boldsymbol{\sigma}}{2}\right)\psi_{R}$.

I don't know how to answer the question as I don't understand some of the preamble, so I will try and motive the (colour-coded) sentances in detail, and I'd be very grateful if someone could tell me where my reasoning is incorrect:

The "representations of the Lorentz group" means the $\boldsymbol{J_{+}}$ and $\boldsymbol{J_{-}}$, as they satisfy the Lorentz commutation relations. Where it says "correspond to" pairs of integers or half integers, it means these representations (ie matrices) "produce the eigenvalues" denoted $\left(j_{+},j_{-}\right)$. But why are they in pairs? Doesn't acting with one $\boldsymbol{J_{+}}$ or $\boldsymbol{J_{-}}$ produce one eigenvalue; $j_{+}$ or $j_{-}$?
The $\left(j_{+},j_{-}\right)$ correspond to representations of the rotation group means that there are some pairs of representations of the rotations group that will give the same $\left(j_{+},j_{-}\right)$ as eigenvalues. By "pairs of representations" they mean one representations will give $j_{+}$ and the other will give $j_{-}$.
The $\left(\frac{1}{2},0\right)$ and $\left(0,\frac{1}{2}\right)$ are spinors, ie a "reduced" representation of the Lorentz group. The designation of half integers seems somewhat arbitrary to me, I don't know what the significance of these numbers is...

To do the question I'm guessing I have to somehow relate $\boldsymbol{J_{+}}$ and $\boldsymbol{J_{-}}$ to $\boldsymbol{J}$ and replace $\boldsymbol{\Phi}$ in the infinitesimal Lorentz transformation with the spinors.
But as shown there is some understanding missing, and I wouldn't really know what I'm doing.

Sorry for the excruciating detail, and many thanks to anyone that can help. If I haven't been clear or you think I should split this up into more than one post please let me know.