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I Finite and infinitesimal Rotations

  1. Aug 8, 2017 #1
    Hi,

    I'm not sure about where I should post this question, so sorry in advance if I posted it in the wrong place.

    My question is basically this screenshot. So I really have some difficulty in understanding the two equations. I mean how can it not be equal? I understand that rotations are non-commutative, but I really don't see why mathematically these two lines are not equal. Doesn't that violate the properties of matrix exponentials?

    Thanks!!!

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  2. jcsd
  3. Aug 8, 2017 #2

    TeethWhitener

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    Gold Member

    In general, if ##A## and ##B## are non-commuting operators, ##e^{A+B} \neq e^A e^B##. You can see this most easily by expanding each side in a Taylor series. The general solution is known as the Baker-Campbell-Hausdorff formula.

    EDIT: actually, probably the easiest way to see this is to note that ##e^{A+B}=e^{B+A}## but ##e^A e^B \neq e^B e^A## for noncommuting ##A## and ##B##.
     
  4. Aug 8, 2017 #3
    Ah...I see that now...didn't think it carefully enough

    Thanks TeethWhitener!
     
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