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Representations of lorentz group and transformations IN DETAIL!

  1. Dec 14, 2011 #1
    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.
     
  2. jcsd
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