Representations of lorentz group and transformations IN DETAIL

[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:
[itex]\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)[/itex]
we can define rotations [itex]L^{i}=\frac{1}{2}\epsilon^{ijk}J^{jk}[/itex] and boosts [itex]K^{i}=J^{0i}[/itex]. An infinitesimal Lorentz transformation can then be written [itex]\boldsymbol{\Phi}\rightarrow\left(1-i \boldsymbol{\theta}.\boldsymbol{L}-i \boldsymbol{\beta}.\boldsymbol{K}\right) \boldsymbol{\Phi}[/itex].
The combinations [itex]\boldsymbol{J_{+}}=\frac{1}{2}\left(\boldsymbol{L}+i\boldsymbol{K}\right)[/itex] and [itex]\boldsymbol{J_{-}}=\frac{1}{2}\left(\boldsymbol{L}-i\boldsymbol{K}\right)[/itex] 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, [itex]\left(j_{+},j_{-}\right)[/itex], corresponding to pairs of representations of the rotation group. Using the fact that [itex]\boldsymbol{J}=\boldsymbol{\sigma}/2[/itex] in the spin-1/2 representation of angular momentum, write explicitly the transformation according to the [itex]\left(\frac{1}{2},0\right)[/itex] and [itex]\left(0,\frac{1}{2}\right)[/itex] representations of the Lorentz group. You should find these correspond precisely to the transformations for the Weyl spinors:
[itex]\psi_{R,L}\rightarrow \left(i \boldsymbol{\theta}\!\frac{\boldsymbol{\sigma}}{2}\pm\boldsymbol{\beta}\!\frac{\boldsymbol{\sigma}}{2}\right)\psi_{R}[/itex].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 [itex]\boldsymbol{J_{+}}[/itex] and [itex]\boldsymbol{J_{-}}[/itex], 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 [itex]\left(j_{+},j_{-}\right)[/itex]. But why are they in pairs? Doesn't acting with one [itex]\boldsymbol{J_{+}}[/itex] or [itex]\boldsymbol{J_{-}}[/itex] produce one eigenvalue; [itex]j_{+}[/itex] or [itex]j_{-}[/itex]?
The [itex]\left(j_{+},j_{-}\right)[/itex] correspond to representations of the rotation group means that there are some pairs of representations of the rotations group that will give the same [itex]\left(j_{+},j_{-}\right)[/itex] as eigenvalues. By "pairs of representations" they mean one representations will give [itex]j_{+}[/itex] and the other will give [itex]j_{-}[/itex].
The [itex]\left(\frac{1}{2},0\right)[/itex] and [itex]\left(0,\frac{1}{2}\right)[/itex] 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 [itex]\boldsymbol{J_{+}}[/itex] and [itex]\boldsymbol{J_{-}}[/itex] to [itex]\boldsymbol{J}[/itex] and replace [itex]\boldsymbol{\Phi}[/itex] 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.

 

[_coyote_]

A:In a finite-dimensional representation of the Lorentz group, the generators $J^{\mu\nu}$ are represented by matrices. Each matrix will have a set of eigenvalues, which correspond to the allowed values for angular momentum. Since these generators satisfy the Lorentz commutation relations, they can be written as the linear combination of rotations and boosts given by$$J^{\mu\nu} = \frac{1}{2}\epsilon^{ijk} L^i J^{jk} + K^i J^{0i},$$where $L^i$ and $K^i$ are the generators of rotations and boosts, respectively.The eigenvalues of the $J^{\mu\nu}$ must obey the same commutation relations, so they can be written as a linear combination of eigenvalues of the $L^i$ and $K^i$ as$$j^{\mu\nu} = \frac{1}{2}\left(j_+^i L^i + j_-^i K^i\right).$$The eigenvalues of the $L^i$ and $K^i$ are integers or half-integers, corresponding to the allowed values of angular momentum. So the eigenvalues of $J^{\mu\nu}$ must also be integers or half-integers. Furthermore, since they must obey the same commutation relations, the eigenvalues must come in pairs, with one from $L^i$ and one from $K^i$. Thus, in a finite-dimensional representation of the Lorentz group, the allowed eigenvalues of $J^{\mu\nu}$ are given by pairs of integers or half-integers $(j_+,j_-)$.Now, for the spin-1/2 representation of the Lorentz group, the generator $J^{\mu\nu}$ is related to the Pauli matrices as $J^{\mu\nu}=\frac{1}{2}\sigma^{\mu\nu}$. An infinitesimal Lorentz transformation can then be written as$$\Phi \rightarrow \left(1 - i\theta^i L^i - i\beta^i K^i\right