Clifford Algebras: Matrix Representations & Higher Order

[sjhanjee]

Can matrix representations of any higher order Clifford Alebras be found ?

 

[RedX]

Well for an even-dimensional representation, an obvious choice is to generalize the Weyl basis. For example in 8 dimensions have:

$$\gamma^0 = \begin{pmatrix} 0 & 0 & 0 & I \\ 0 & 0 & I & 0 \\ 0 & I & 0 & 0 \\ I & 0 & 0 & 0 \end{pmatrix}$$

$$\gamma^k = \begin{pmatrix} 0 & 0 & 0& \sigma^k \\ 0 & 0 & -\sigma^k & 0 \\ 0 & \sigma^k & 0& 0\\ -\sigma^k & 0 & 0& 0 \end{pmatrix}$$

which obey the Clifford algebra. Similarity transformation or constructing reducible matrices can get you more variety, of the trivial type.

As for odd-dimensional representations, you're on your own, as my math isn't any good: I have no clue as to how you would construct those.

 

[sjhanjee]

Thanks for the reply. Can you elaborate the even order case. I think you are using pauli matrices but I know only 3 of them. Can you clarify? Or can you tell me where to look for them.

 

[RedX]

Thanks for the reply. Can you elaborate the even order case. I think you are using pauli matrices but I know only 3 of them. Can you clarify? Or can you tell me where to look for them.

The $$\sigma$$ matrices are the 3 Pauli matrices. The I is the 2x2 identity matrix. So the matrices in this case would be 8x8.

Here is the Wikipedia link for the 4x4 matrices:

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

 

[sjhanjee]

Another question? My clifford algebras are Cl(0,n) (of negative signature) , not the space time algebra, so all the gamma matrices should square to -1.

 

[RedX]

Another question? My clifford algebras are Cl(0,n) (of negative signature) , not the space time algebra, so all the gamma matrices should square to -1.

All the matrices I gave above square to -1, except the 1st matrix. You can multiply every entry of the 1st matrix by the square root of -1. Then all the matrices square to -1, and they all anticommute with each other. So they form an 8-dimensional matrix representation of a Clifford algebra with 4 elements.

 

[sjhanjee]

Yes ,I am getting there. Another (silly) question. Can you give matrix representations of Cl(0,6) or Cl(0,8) similarily ( or for that matter any Cl(0,2n) )?

 

[RedX]

Yes ,I am getting there. Another (silly) question. Can you give matrix representations of Cl(0,6) or Cl(0,8) similarily ( or for that matter any Cl(0,2n) )?

I don't know. My math is not really any good, so I'm only interested in Cl(0,4) and Cl(1,3), which are useful for spacetime. Perhaps if you try another board, they would be better able to help.

Here is a general list of boards:

https://www.physicsforums.com/

I don't know which one would be good though, but I don't believe this stuff is really seen in classical physics (I could be wrong though), so maybe try quantum mechanics or beyond the standard model.

 

[sjhanjee]

Thanks for your reply. I am coming from pure mathematics backgrond. And I don't know where to look for these representations , I will try other boards.Thks