# Inverting Operators in 2+1 Dimensions

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 $$\mathcal{O}^{\mu\nu}(\mathcal{O}^{-1})_{\nu\rho} = \delta^\mu_\rho$$

$\left\{\left\{A\to \frac{\alpha }{\alpha ^2+\gamma ^2 k^2},B\to \frac{\gamma ^2-\alpha \beta }{\left(\alpha +\beta k^2\right) \left(\alpha ^2+\gamma ^2 k^2\right)},C\to -\frac{\gamma }{\alpha ^2+\gamma ^2 k^2}\right\}\right\}$
• Did you use my result for the constraints or did you invert the operator yourself? I do find $A = \frac{-\gamma}{\alpha^2 - \gamma^2 k^2}$, $B = \frac{1}{\alpha^2 - \gamma^2 k^2} \frac{\gamma(\beta - \alpha)}{\alpha + \beta k^2}$, $C = \frac{\alpha}{\alpha^2 - \gamma^2 k^2}$ from my constraint equations. – JgL Jun 9 '17 at 11:18
• Ah, you are completely right, I did some very stupid things while solving them by hand and decided it was time to get some coffee. Using the right values for $A$, $B$ and $C$ everything works out fine. Thanks for pointing out my mistakes. – JgL Jun 9 '17 at 12:45