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Galiliei transformations explicit proof

  1. Oct 23, 2015 #1
    1. The problem statement, all variables and given/known data

    Show explicitly that

    ei*ε*κu * ei*ε*κv * e-i*ε*κu* e-i*ε*κv = Identity + ε2 [Kv,Ku + O (ε3)

    3. The attempt at a solution

    Kv,Ku = Kv*Ku - Ku*Kv

    I'm not sure exactly how to approach this problem. I know that

    U (tau) = ∏ ei*su*Ku

    and that for operators O --> O' = U O U

    I have this information but I don't know how to put it together, any help would be greatly appreciated
     
  2. jcsd
  3. Oct 24, 2015 #2

    Orodruin

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    I suggest expanding the exponentials up to order ##\epsilon^2## and then simply checking that the expression reduces to the given one.
     
  4. Oct 25, 2015 #3
  5. Oct 25, 2015 #4

    Orodruin

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    Dont let wolfram do it for you, just multiply the terms together and keep only terms up to order two in epsilon.
     
  6. Oct 26, 2015 #5
    Alright, If I do that I get

    (1+i*e*v-e^2*v^2/2 +i*ex-e^2*x*v-e^2*x^2/2)(1-i*e*v-e^2*v^2/2-i*e*x-e^2*v*x-e^2*x^2/2)

    then expanding that leads to many terms

    upload_2015-10-26_15-47-1.png
    which doesn't lead to the correct answer, perhaps I am making an algebraic mistake

     
  7. Oct 26, 2015 #6

    Orodruin

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    Go order by order. First check that all the linear terms cancel.
     
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