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

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Discussion Overview

The discussion revolves around the existence of the inverse of a specific operator in momentum space, represented as g_{\mu \nu} \Box - \partial_{\mu}\partial_{\nu}. Participants explore the implications of this operator's form and whether an inverse can be constructed, delving into theoretical aspects of operator algebra in physics.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about why the proposed inverse of the operator would take the form Ag^{\nu \lambda} + B k^{\nu}k^{\lambda}, questioning whether A and B are constant coefficients.
  • Another participant agrees that the equation -Ak^{2}\delta_{\mu}^{\lambda} + A k_{\mu}k^{\lambda} = \delta_{\mu}^{\lambda} does not seem to have a solution but admits an inability to prove it at the moment.
  • A later reply suggests that any solution leads to a contradiction and recommends writing out all 16 equations to clarify the situation.
  • One participant reflects on their earlier confusion and acknowledges that writing out the first component of the equations reveals the issue, indicating a realization about the problem's complexity.
  • Another participant introduces the concept of a non-trivial projection operator, stating that it has no inverse and providing an example involving P_{ab} and P^{ac} to illustrate their point.

Areas of Agreement / Disagreement

Participants generally express uncertainty about the existence of a solution to the equations, with some agreeing that contradictions arise when attempting to find one. However, there is no consensus on the matter, as different perspectives and interpretations are presented.

Contextual Notes

Participants note that the problem may involve complex algebraic manipulations, and the discussion highlights the potential for contradictions in the equations derived from the operator's form.

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

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

I'm told that if it *did* it would be of the form
[itex]Ag^{\nu \lambda} +B k^{\nu}k^{\lambda}[/itex]
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 [itex]-Ak^{2}\delta_{\mu}^{\lambda} +A k_{\mu}k^{\lambda}=\delta_{\mu}^{\lambda}[/itex]
. . . 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
 
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Mentz114 said:
I don't think these equations have a solution

[tex] -Ak^{2}\delta_{\mu}^{\lambda} +A k_{\mu}k^{\lambda}=\delta_{\mu}^{\lambda}[/tex]

but I can't prove it right now.

: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.
 
Elwin.Martin said:
: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.

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

Have you found a solution ?
[Edit] Any solution leads to a contradiction, so there isn't one. Try writing out all 16 equations.
 
Last edited:
Mentz114 said:
Sorry, I deleted my post because I'm looking for a solution and it might be on.

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

OH, WOW...thanks!

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

Thanks again!
 
Non-trivial projection operator has no inverse; [itex]P^{2}=P[/itex]. Consider
[tex]P_{ab} = g_{ab} - \frac{k_{a}k_{b}}{k^{2}},[/tex]
[tex]P^{ac}= g^{ac} - \frac{k^{a}k^{c}}{k^{2}}.[/tex]
then
[tex]P_{ab}P^{ac} = \delta^{c}_{b} - \frac{k_{b}k^{c}}{k^{2}} \equiv P^{c}_{b}[/tex]
 

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