- #1

CAF123

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## Homework Statement

Given the mode expansion of the quantum field ##\phi## and the conjugate field one can derive $$\mathbf P = \int \frac{d^3 \mathbf p}{(2\pi)^3 2 \omega(\mathbf p)} \mathbf p a(\mathbf p)^{\dagger} a(\mathbf p)$$ By writing $$e^X = \text{lim}_{n \rightarrow \infty} \left(1+\frac{X}{n}\right)^n$$ calculate ##[\mathbf P, a(\mathbf q)^{\dagger}]## and hence evaluate $$\exp(-i \mathbf P \cdot \mathbf z) a(\mathbf q)^{\dagger} \exp(i \mathbf P \cdot \mathbf z),$$ with ##\mathbf z## a constant vector.

## Homework Equations

##[AB,C] = A[B,C] + [A,C]B##

Commutation relations for the creation and annihilation operators.

## The Attempt at a Solution

I think I can show that ##[\mathbf P, a(\mathbf q)^{\dagger}] = \mathbf q a(\mathbf q)^{\dagger}## but why in particular is this definition of the exponential useful? I can use it to rewrite the exponential factors appearing in that expression but that gives me $$\text{lim}_{n \rightarrow \infty} \text{lim}_{r \rightarrow \infty} \left(1- i \frac{\mathbf P \cdot \mathbf z}{n}\right)^n a(\mathbf q)^{\dagger} \left(1+ i \frac{\mathbf P \cdot \mathbf z}{r}\right)^r$$ but I can't see an easy way to proceed.

It seems to me that I could make more progress using the Baker Campbell Hausdorff formula for non commuting exponentials