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About the Lie algebra of our Lorentz group

  1. Mar 25, 2013 #1

    I'm currently reading Ryder - Quantum Field Theory and am a bit confused about his discussion on the correpsondence between Lorentz transformations and SL(2,C) transformations on 2-spinor.
    He writes that the Lie algebra of Lorentz transformations can be satisfied by setting
    [tex]\vec{K} =\pm \frac{i \vec{\sigma}}{2}. [/tex]
    Here it seems as if the dimensions are mixed up. The Pauli matrices are 2 times 2 while the Loretnz generators are 4 times 4.

    Secondly he argues that the Lorentzgroup can be ''factorized'' into [tex]SU(2) \times SU(2)[/tex] but how come this goes along with the fact that the Loretnz group is non-compact.
    It seems as if we take the product group of two compact group the resulting group is compact?
    Am I wrong about this?
  2. jcsd
  3. Mar 25, 2013 #2


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    He means K and σ as abstract operators which satisfy a certain set of commutation relations, not a particular matrix representation of those operators.
    To go from one set of generators to the other you have to consider the complexified Lie algebra, A = (J + i K)/2 and B = (J − i K)/2. This does not preserve group compactness.
  4. Mar 25, 2013 #3
    Ah cool!
    Maybe I'm a bit under the level that this book is written on.. I come back with future confusions. :)
    Thanks Bill!
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