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Newton's expansion for non-commutative quantities

  1. Feb 3, 2014 #1

    ShayanJ

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    You probably know that for two commutative quantities x and y,we have:
    [itex] (x+y)^n=\sum_{r=0}^n \left( \begin{array}{c} n \\ r \end{array} \right) x^{n-r} y^r [/itex]
    Now I wanna know is there a similiar formula for the case when x and y don't commute and we have [itex] [x,y]=c [/itex] and [itex] [x,c]=[y,c]=0 [/itex]?
    Thanks
     
  2. jcsd
  3. Feb 4, 2014 #2

    maajdl

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    In the case of commutative types, the terms can be ordered as x^i y^j, as you showed, because x and y commute.
    In general, you will have all sorts of ordering, like xxy, xyx, yxx.
    Maybe you can work out the special cases n=2 and n=3, to guess an answer.
     
  4. Feb 4, 2014 #3

    maajdl

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