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Vector represenation

  1. Mar 10, 2012 #1
    The vector representation of the Lorentz algebra in 4 dimensions can be very explicitly given by six 4x4 matrices. Peskin/Schroeder has it on page 39, formula 3.18, for example

    But then a four-vector is also a tensor product of a left-handed and a right-handed Weyl spinor!

    Knowing the Weyl spinor represenation of the Lorentz algebra, how do I arrive at these explicit matrices for the vector representation?

    Strangely, no book explains that. Though, many make a lot of effort showing how vectors, Dirac spinors and tensors are direct sums and/ or products of Weyl spinors, they just give the explicit formula for the vector representation.
     
  2. jcsd
  3. Mar 10, 2012 #2

    fzero

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    You can map a 4-vector to a bispinor via

    [tex] V^\mu \rightarrow V^\mu (\sigma_\mu)_{\alpha \dot{\alpha}}.[/tex]

    More generally, you can use this to map any representation of [itex]SO(3,1)[/itex] to a representation of [itex]SU(2)\times SU(2)[/itex]. In principle you can work out the representations from one side to the other via

    [tex] {\Lambda^\mu}_\nu V^\nu (\sigma_\mu)_{\alpha \dot{\alpha}} = {M_\alpha}^\beta V^\mu (\sigma_\mu)_{\beta \dot{\beta}} {M^{\dot{\beta}}}_{\dot{\alpha}}. [/tex]
     
  4. Mar 11, 2012 #3
    Thank fzero!

    I saw that in Srednicki's book, too, but could not quite decipher what he means. But that's probably because I have not bothered yet to learn this funny dot notation....

    Luckily I discovered that in this new book Symmetries and the Standard Model by Robinson, that there is a lovely section on Lorentz symmetry and all its representation. By the way the whole book is great!
     
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