Inverse of an operator does not exist, can't see why

1. Jan 25, 2012

Elwin.Martin

I feel kind of lame, but here's my situation:
We start with the operator $g_{\mu \nu} \Box - \partial_{\mu}\partial_{\nu}$ and convert to momentum space to get $-g_{\mu \nu} k^{2} - k_{\mu}k_{\nu}$.

Apparently it's easy to see that this has no inverse?

I'm told that if it *did* it would be of the form
$Ag^{\nu \lambda} +B k^{\nu}k^{\lambda}$
but I don't see why it would be in this form, to start with. Are these A's and B's just constant coefficients...?

I understand that given the form above, we can simply multiply our inverse and original operator and we get that $-Ak^{2}\delta_{\mu}^{\lambda} +A k_{\mu}k^{\lambda}=\delta_{\mu}^{\lambda}$
. . . but I don't see why this is an issue, or rather, I can't see why this equation has no solution.

Thanks for any and all help,
E_Martin

2. Jan 25, 2012

Elwin.Martin

:P then you're in agreement with Ryder...I don't see it though. I have the feeling it doesn't, but I cannot show it.

3. Jan 25, 2012

Mentz114

Sorry, I deleted my post because I'm looking for a solution and it might be on.

Have you found a solution ?
 Any solution leads to a contradiction, so there isn't one. Try writing out all 16 equations.

Last edited: Jan 25, 2012
4. Jan 25, 2012

Elwin.Martin

OH, WOW...thanks!

Just writing out the first component works, haha...fail.

Thanks again!

5. Jan 25, 2012

samalkhaiat

Non-trivial projection operator has no inverse; $P^{2}=P$. Consider
$$P_{ab} = g_{ab} - \frac{k_{a}k_{b}}{k^{2}},$$
$$P^{ac}= g^{ac} - \frac{k^{a}k^{c}}{k^{2}}.$$
then
$$P_{ab}P^{ac} = \delta^{c}_{b} - \frac{k_{b}k^{c}}{k^{2}} \equiv P^{c}_{b}$$