Hi. I was wondering why creation and destruction operators a+ and a- do not conmute. Of course, I can show that they don't conmute by computing the conmutator [a+, a-] = -1. But I want to know the "physical" meaning of this. Isn't destruction/creation a symmetric transformation? We "go up the ladder" with a+ and we "go down the ladder" with a-. Shouldn't they therefore cancel each other, i.e. a+a-=a-a+ = I? Mathematically [itex] a_- a_+ \vert n \rangle = a_-\vert n+1 \rangle = \vert n \rangle [/itex] Thank you for your time.