Quantum Field Theory, Momentum Space Commutation Relations

In summary, the conversation discusses deriving the commutation relation in momentum space using the canonical commutation relation in position space. The approach involves Fourier transforming the commutation relation and solving the resulting integral. The final answer should be i(2π)^3δ^3(p+q).
  • #1
Arcturus7
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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 $$[φ^*(\vec p), π^*(\vec q)]= i(2π)^3δ^3(\vec p+\vec q) $$
 
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  • #2

Related to Quantum Field Theory, Momentum Space Commutation Relations

1. What is quantum field theory?

Quantum field theory is a theoretical framework that combines the principles of quantum mechanics and special relativity to describe the behavior of particles at a subatomic level.

2. What is momentum space in quantum field theory?

Momentum space is a mathematical representation of the momentum of particles in quantum field theory. It is a three-dimensional space where each point represents a particular momentum value.

3. What are commutation relations in quantum field theory?

Commutation relations are mathematical rules that describe how two operators in quantum field theory interact with each other. They determine how the order of operations affects the final outcome of a calculation.

4. How do momentum space commutation relations differ from position space commutation relations?

Momentum space commutation relations involve the momentum operators of particles, while position space commutation relations involve the position operators. These two types of commutation relations have different mathematical forms and play different roles in quantum field theory calculations.

5. What is the importance of momentum space commutation relations in quantum field theory?

Momentum space commutation relations are essential for understanding the behavior of particles at a subatomic level. They allow us to calculate various properties of particles, such as their energy and momentum, and make predictions about their behavior in different situations.

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