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Proving an identity and some interesting maths stuff

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

    [tex]\gamma^{\mu_{1}...\mu_{r}}=(-)^{r(r-1)/2}\gamma^{\mu_{r}...\mu_{1}}[/tex]

    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

    [tex]\gamma^{\mu_{1}...\mu_{r}}=(-)^{(r-1)+(r-2)+...+1}\gamma^{\mu_{r}...\mu_{1}}[/tex]

    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
    [tex]\int^{1}_{0}\frac{x(x-1)}{2}dx=-\frac{1}{12}[/tex]

    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

    Avodyne

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