# Galiliei transformations explicit proof

1. Oct 23, 2015

### ma18

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. Oct 24, 2015

### Orodruin

Staff Emeritus
I suggest expanding the exponentials up to order $\epsilon^2$ and then simply checking that the expression reduces to the given one.

3. Oct 25, 2015

### ma18

4. Oct 25, 2015

### Orodruin

Staff Emeritus
Dont let wolfram do it for you, just multiply the terms together and keep only terms up to order two in epsilon.

5. Oct 26, 2015

### ma18

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

which doesn't lead to the correct answer, perhaps I am making an algebraic mistake

6. Oct 26, 2015

### Orodruin

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