Symmetry properties charge conjugation matrices in even dimension.

+ 4 like - 0 dislike
219 views

While reading a paper on supersymmetry (by Peter West) i faced the following problem. Its about the symmetry property of charge conjugation matrix in different space time dimension. The charge conjugation matrix is defined as

C^{T} = -\epsilon C

Problem is to find out the $\epsilon$ as a function of space-time dimension.
If the spacetime dimension $\mathit{D}$ is even, the finite group generated by Clifford algebra $\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=2\eta_{\mu\nu}$ (Where , and $\eta_{\mu\nu}=\text{Diag}(-,_,+,\cdots,+)$) consists of elements

I, \gamma_{m}, \gamma_{m_{1}m_{2}},\gamma_{{m_{1}m_{2}}m_{3}},\dots\gamma_{{m_{1}m_{2}}m_{3}\dots m_{D}}

There are altogether $2^D$ such matrices.

Following little algebra it can be shown that

(C\gamma_{{m_{1}m_{2}}m_{3}\dots m_{D}})^{T} = \epsilon (-1)^{\frac{(p-1)(p-2)}{2}} C\gamma_{{m_{1}m_{2}}m_{3}\dots m_{D}}

Which means $C\gamma_{{m_{1}m_{2}}m_{3}\dots m_{D}}$ either symmetric or anti symmetric.
It says number of anti symmetric matrices in this set is equal to

\sum_{p=0}^{D}\frac{1}{2}(1-\epsilon(-1)^{\frac{(p-1)(p-2)}{2}})\binom{D}{p}

I know how to prove this result by calculating the (anti)symmetry property of $C\gamma_{m_{1}m_{2}\cdots m_{D}}$ for even dimensions by explicit calculations. But unable to prove the result mentioned.

Can anyone help to derive the last expression? It will be a great help. Thanks in advance.

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$y$\varnothing$icsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.