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Prove Commutator Exponentials Algebra

  1. Sep 9, 2012 #1
    1. The problem statement, all variables and given/known data

    Prove the following for operators A and B.

    e^A B e^-A = B + [A,B] + (1/2!) * [A,[A,B]] + (1/3!) * [A,[A,[A,B]]] + ...

    2. Relevant equations

    e^A = 1 + A + (1/2!)A^2 + (1/3!)A^3 + ...

    3. The attempt at a solution

    I have no clue how to start.

    For the highly special case of [A,B] = constant and [A,B] commutes with A and B, we can prove that e^A B e^-A = B + [A,B]

    through the following:

    Take the given identity [A,F(B)] = [A,B] dF/dB and set F = e^B

    [A,e^B] = Ae^B - e^B*A = e^B * [A,B]

    multiply both sides by e^-B to left.

    e^-B A e^B - A = [A,B]

    e^-B A e^B = A + [A,B]

    This is the first 2 terms of the series and if we take [A,B] = c, then the other terms are zero. How do I prove it for general [A,B] then?
     
  2. jcsd
  3. Sep 10, 2012 #2
    any help with this? i am seriously stuck and everything i've looked up involves Lie algebras or some abstract math stuff i've never learned in physics before =(
     
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