Why don't creation and destruction operators conmute?

[carllacan]

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 a_- a_+ \vert n \rangle = a_-\vert n+1 \rangle = \vert n \rangle

Thank you for your time.

 

[Lapidus]

I would say that since a+ and a- are not Hermitean, they thus represent no physical measurable observables. The number operator is a physical observable, though.

 

[PeterDonis]

Isn't destruction/creation a symmetric transformation?

No. For example, suppose you are starting in the ground state/vacuum state, with no particles. Creation, then destruction takes you to a 1-particle state, then back to the vacuum state. Destruction, then creation is physically impossible: the destruction operator applied to the vacuum state gives zero, which doesn't represent any physical state.

More generally, if you are in an n-particle state, there is no guarantee that creation + destruction will take you back to the *same* n-particle state as destruction + creation, because there are many possible n-particle states.

 

[Nugatory]

Mathematically a_- a_+ \vert n \rangle = a_-\vert n+1 \rangle = \vert n \rangle

Try this line of thinking on $a_+a_-\vert{0}\rangle$. What happens?

$a_+\vert n \rangle = \alpha \vert n+1 \rangle$ not $\vert n+1 \rangle$ and you can't ignore the value of $\alpha$.

 

[carllacan]

Ok, thank you guys, I get it now.

 

[homeomorphic]

This might be of interest if you can understand some of it.

http://math.ucr.edu/home/baez/qg-fall2003/

Quote from there:

"Andre Joyal invented his theory of "espèces des structures" - translated as "species" or "structure types" - in order to understand more deeply how people use generating functions to count structures on finite sets. It turns out that just as a natural number is a watered-down or "decategorified" version of a finite set, a generating function is a decategorified version of a structure type.

Recently, James Dolan and John Baez realized that structure types and more general "stuff types" can also be used to more deeply understand the role of annihilation and creation operators, Feynman diagrams and the like in quantum theory. It turns out that some of the mysteries of quantum mechanics are really just decategorified versions of simple facts about structures on finite sets. For example, the fact that position and momentum don't commute has a purely combinatorial interpretation! "

Here's the punchline, which you can meditate on as a combinatorics problem, whether or not you're interested in category theory or structure types:

"Ultimately, it boils down to the fact that there's one more way to put a ball in a box and then take one out than to take one out and then put one in."