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Proving an identity and some interesting maths stuff

  1. Oct 27, 2014 #1
    So, I would like to prove that


    where the matrix gamma is a totally antisymmetric matrix defined as [itex]\gamma^{\mu_{1}...\mu_{r}}=\gamma^{[\mu_{1}}\gamma^{\mu_{2}}...\gamma^{\mu_{r}]}[/itex]

    What I have done is to prove that


    by simply commuting all the matrices past each other until their order is reversed (picking up just the minus sign as they are antisymmetrised, so we can take [itex]\mu_{i}\neq\mu_{j}[/itex] for [itex]i\neq j[/itex]).

    What's a nice way to see that [itex](r-1)+(r-2)+...+1=r(r-1)/2[/itex]? It works for some values of r, which one can see by substituting in.

    ALSO - PART 2

    I am aware of [itex]\sum_{n=1}^{\infty}n=\frac{x(x+1)}{2}=-\frac{1}{12}[/itex],

    but I found out that

    Any comments or clarifications on this relationship between [itex]\frac{x(x-1)}{2}[/itex] and [itex]\frac{x(x+1)}{2}[/itex].
  2. jcsd
  3. Oct 27, 2014 #2


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