Matrix reppresentation of poincare group generators

[Bobhawke]

Does anyone know what a matrix representation of the poincare group generators looks like (specifically the translation parts)? I've been trying to figure this out but I get matrices that are dependent on the group parameters.

 

[George Jones]

Does anyone know what a matrix representation of the poincare group generators looks like (specifically the translation parts)? I've been trying to figure this out but I get matrices that are dependent on the group parameters.

Is translation by a fixed 4-vector a linear transformation?

 

[Bobhawke]

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?

 

[Fredrik]

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.

 

[George Jones]

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.

 

[Bobhawke]

Awesome, thanks for the replies guys.

 

[msnopesrunner]

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 .