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Matrix reppresentation of poincare group generators

  1. Jun 26, 2009 #1
    Does anyone know what a matrix representation of the poincare group generators looks like (specifically the translation parts)? Ive been trying to figure this out but I get matrices that are dependent on the group parameters.
     
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  3. Jun 26, 2009 #2

    George Jones

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    Is translation by a fixed 4-vector a linear transformation?
     
  4. Jun 27, 2009 #3
    No it isnt.

    But matrix multiplication is linear. So is the correct conclusion that the translation group operators could be represented as matrices only if we define a funky new way of multiplying matrices together that allows non-linearity?
     
  5. Jun 27, 2009 #4

    Fredrik

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    A translation isn't a linear transformation on [itex]\mathbb R^4[/itex], but if we're talking about a representation on a Hilbert space, there is of course a linear operator that corresponds to the translation. A translation by a four-vector with components [itex]a^\mu[/itex] is [itex]\exp(-ia_\mu P^\mu)[/itex], where the [itex]P^\mu[/itex] are the momentum operators. These momentum operators can't be written as matrices when we're working with this representation.

    Hmm...I just remembered that the Poincaré group can be interpreted as a group of 5x5 matrices. (I read that here). So I guess the generators can be expressed as 5x5 matrices too, but I don't think that fact is very useful in QM.
     
  6. Jun 28, 2009 #5

    George Jones

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    For a detailed treatment of the irreducible representation of the Poincare group, see Group Theory in Physics by Wu-Ki Tung. For treatments more amenable to mathematicians, see some combination of: Theory of Group Representations and Applications by A. O. Barut and Ryszard Raczka; The Dirac Equation by Bernd Thaller; Lie Groups and Quantum Mechanics by D. J. Simms.
     
  7. Jun 28, 2009 #6
    Awesome, thanks for the replies guys.
     
  8. Oct 6, 2010 #7
    Translations are generated by momentum matrices, just as rotations are generated by angular momentum matrices (spin matrices). Momentum matrices are arrays of Clebsch Gordon coefficients. For formulas see `A Derivation of Vector and Momentum Matrices', arXiv:math-ph/0401002 .
     
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