# Path-Integral of Charged Particle in Chern-Simons Gauge Fields

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From the paper "Fermi-Bose Transmutations Induced by Gauge Fields" by Polyakov,

http://inspirehep.net/record/22956/citations

the theory in 3D,

$$\mathcal{L}=\sum_{k=1}^{2}|\partial_{\mu}z_{k}+iA_{\mu}z_{k}|^{2}+\frac{\theta}{16\pi^{2}}\epsilon_{\mu\nu\rho}A^{\mu}\partial^{\nu}A^{\rho}$$

with a constraint $|z_{1}|^{2}+|z_{2}|^{2}=1$ for $z_{1}(x^{0},x^{1},x^{2})$ and $z_{2}(x^{0},x^{1},x^{2})$,

has the transition amplitude given by

$$G(x,y)=\int\mathcal{D}x \, e^{-m\int ds}\left\langle\exp\left\{i\int dx^{\mu}A_{\mu}\right\} \right\rangle$$

where the average is given by

$$\left\langle\exp\left\{i\int dx^{\mu}A_{\mu}\right\} \right\rangle=\int\mathcal{D}A \exp\left\{i\int dx^{\mu}A_{\mu}\right\}\exp\left\{ iS_{CS}[A]\right\}$$

How to derive this two-point function?

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