- #1

- 760

- 69

## Homework Statement

Given that U upon acting on the creation operator gives a creation operator for the transformed momentum $$U(\Lambda) a_p^\dagger U(\Lambda)^\dagger = a_{\boldsymbol{\Lambda} \mathbf{p}}^\dagger $$

and ##\Lambda ## is a pure boost, that is ## U(\Lambda) = e^{i \boldsymbol{\phi} \cdot \mathbf{K}}##, find

$$ \left[\mathbf{K}, \, a_p^\dagger\right] $$

## Homework Equations

## The Attempt at a Solution

The first idea we had was to expand the left side giving

$$ a_p^\dagger + i\left[\boldsymbol{\phi} \cdot \mathbf{K}, \, a_p^\dagger\right] + \dots = a_{\boldsymbol{\Lambda} \mathbf{p}}^\dagger$$

However we are not able to meaningfully expand the right hand side, so we have hit a dead end. We don't know how to assign meanings to derivatives of ## a_{\mathbf{p}}^\dagger##. Is this the wrong approach?

Last edited: