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Lorentz Algebra in Boosts for the spin-1/2 Dirac Field

  1. Jul 14, 2009 #1
    Hi,

    What is the origin of the following commutation relation in Lorentz Algebra:

    [tex][J^{\mu\nu}, J^{\alpha\beta}] = i(g^{\nu\alpha}J^{\mu\beta}-g^{\mu\alpha}J^{\nu\beta}-g^{\nu\beta}J^{\mu\alpha}+g^{\mu\beta}J^{\nu\alpha})[/tex]

    This looks a whole lot similar to the commutation algebra of angular momentum in O(4). But how does the Metric Tensor enter here? I encountered this while studying about boosts in the context of the Dirac Equation.

    Thanks.
     
  2. jcsd
  3. Jul 14, 2009 #2

    Hans de Vries

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    Science Advisor


    This is the "Master commutation" rule of the Lorentz section of the Poincaré group

    http://en.wikipedia.org/wiki/Poincaré_group

    [itex]J^{\mu\nu}[/itex] can be both a rotation generator as well as a boost generator, so this is the most
    general way of defining the commutation rules between these generators.


    It's far simpler to look at the individual rules like [itex][J^i,J^i][/itex] and [itex][K^i,K^i][/itex] or [itex][J^i,K^i][/itex] where
    J and K are the rotation and boost generators respectively.


    You can find these in many books like Ryder, Weinberg (vol 1) or P&S


    Regards, Hans
     
  4. Jul 14, 2009 #3
    Thanks.
     
  5. Jul 18, 2009 #4
    Lorentz algebra is SO(4) algebra, synonymous almost. To be exact, Lorentz algebra is SO(3,1) algebra. So you can derive the commutation relations for SO(4), and try to turn SO(4) into SO(3,1) by converting Krockner deltas into metric tensors by raising or lowering.
     
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