TITLE: Inverting Operators in 2+1 Dimensions
QUESTION [0 upvotes]: If I have a 2+1 dimensional operator of the form
$$
     \mathcal{O}^{\mu\nu} = \alpha g^{\mu\nu} + \beta k^\mu k^\nu + \gamma \epsilon^{\mu\nu\kappa} k_\kappa
$$
and want to find it's inverse
$$
(\mathcal{O}^{-1})_{\nu\rho} = A g_{\nu\rho} + B k_\nu k_\rho + C \epsilon_{\nu\rho\lambda} k^\lambda
$$
I could multiply them with each other and use the constraint that (here I used a Minkowski metric with $(+,-,-)$ signature, resulting in the property that $\epsilon^{\mu\nu\rho} = \epsilon^{\;\;\nu\rho}_\mu = \epsilon_{\mu\nu\rho}$ )
$$\begin{align}
     \mathcal{O}^{\mu\nu}(\mathcal{O}^{-1})_{\nu\rho} &= \left( \alpha A - \gamma C k^2 \right)\delta^\mu_\rho \\
     & + \left( \gamma C + A \beta + \alpha B + \beta B k^2 \right)k^\mu k_\rho\\
     & + \left( \gamma A + \alpha C \right)\epsilon_{\nu\rho\lambda}k^\lambda g^{\mu\nu} = \delta^\mu_\rho
\end{align}$$
However, this gives me the constraints
$$\begin{align*}
    &\alpha A - \gamma C k^2 = 1   &   A &= \frac 1\alpha (1-\gamma C k^2)\\
    &\gamma A + \alpha C = 0   &   A &= - \frac{\alpha C}{\gamma}\\
    &\gamma C + A \beta + \alpha B + \beta B k^2 = 0   &   B &= \frac{-(\gamma C + A\beta)}{\alpha + \beta k^2} \\
\end{align*}$$
that do not seem to be self-consistent. For example, calculating $C$ from the first two constraints, and then calculating the corresponding values for $A$ gives different values for $A$. In addition, when I multiply the resulting inverse operator
$$\begin{equation}
     (\mathcal{O}^{-1})_{\nu\rho} = \frac{1}{\alpha^2 - \gamma^2 k^2} \left[ -\gamma g_{\nu\rho} + \frac{\gamma(\beta - \alpha)}{\alpha + \beta k^2} k_\nu k_\rho + \alpha \epsilon_{\nu\rho\lambda} k^\lambda \right]
\end{equation}$$
with the original, I do not find
$$
\mathcal{O}^{\mu\nu}(\mathcal{O}^{-1})_{\nu\rho} = \delta^\mu_\rho
$$

REPLY [1 votes]: Maybe I have done some error but I have this expressions for the coefficients:
$
\left\{\left\{A\to \frac{\alpha }{\alpha ^2+\gamma ^2 k^2},B\to \frac{\gamma
   ^2-\alpha  \beta }{\left(\alpha +\beta  k^2\right) \left(\alpha ^2+\gamma ^2
   k^2\right)},C\to -\frac{\gamma }{\alpha ^2+\gamma ^2 k^2}\right\}\right\}
$
Are they the same of yours?
{"set_name": "stack_exchange", "score": 0, "question_id": 338329}