- #1
Arcturus7
- 16
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Homework Statement
Derive, using the canonical commutation relation of the position space representation of the fields φ(x) and π(y), the corresponding commutation relation in momentum space.
Homework Equations
[φ(x), π(y)] = iδ3(x-y)
My Fourier transforms are defined by: $$ φ^*(\vec p)=\int φ(\vec x)e^{-i\vec p⋅\vec x}d^3\vec x $$
The Attempt at a Solution
I have essentially taken the approach of Fourier transforming my canonical position space commutation relation, but after evaluating: $$[φ^*(\vec p), π^*(\vec q)] =i\int δ^3(\vec x-\vec y)e^{-i(\vec p⋅\vec x+\vec q⋅\vec y)}d^3\vec x d^3\vec y $$ I am left with an integral that I don't know how to solve, namely: $$ i\int e^{-i(\vec p+\vec q)⋅x}d^3\vec x $$
1) Am I right up till here?
and:
2) Any hints on how to solve this integral? I tried using polar coordinates but didn't get very far. I figured there was probably a symmetry I was missing, but I can't for the life of me see what it would be. I know that the answer should be simply $$[φ^*(\vec p), π^*(\vec q)]= i(2π)^3δ^3(\vec p+\vec q) $$