What's the Normal Order of A + B?

[junjunxu]

Homework Statement

Hi, guys, I just wondering what's the normal order of A +B

Homework Equations

if N[A+B]=N[A]+N
then I have
$N[aa^{\dagger}]=N[1-a^{\dagger}a]=1-a^{\dagger}a$
however we know that
$N[aa^{\dagger}]=-a^{\dagger}a]$
what's wrong?

The Attempt at a Solution

another example,
$N[aa^{\dagger}b^{\dagger}b]=N[(1-a^{\dagger}a)b^{\dagger}b]=b^{\dagger}b+a^{\dagger}b^{\dagger}ab$

thank you!

 

[George Jones]

$N \left[ 1 \right] = N \left[ aa^\dagger + a^\dagger a \right]$ and linearity gives?