# Matrix reppresentation of poincare group generators

1. Jun 26, 2009

### Bobhawke

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.

2. Jun 26, 2009

### George Jones

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

3. Jun 27, 2009

### 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?

4. Jun 27, 2009

### Fredrik

Staff Emeritus
A translation isn't a linear transformation on $\mathbb R^4$, 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 $a^\mu$ is $\exp(-ia_\mu P^\mu)$, where the $P^\mu$ 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.

5. Jun 28, 2009

### George Jones

Staff Emeritus
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.

6. Jun 28, 2009

### Bobhawke

Awesome, thanks for the replies guys.

7. Oct 6, 2010

### 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 .