Instantons and the spectral function of electrons in the half-filled Landau level

We calculate the instanton-anti-instanton action $S_{M {\bar M}} (\tau)$ in the gauge theory of the half-filled Landau level. It is found that $S_{M {\bar M}} (\tau) = (3 - \eta) \left [ \Omega_0 (\eta) \ \tau \right ]^{1 / (3 - \eta)}$ for a class of interactions $v ({\bf q}) = V_0 / q^{\eta} \ ( 0 \leq \eta < 2 )$ between electrons. This means that the instanton-anti-instanton pairs are confining so that a well defined `charged' composite fermion can exist. It is also shown that $S_{M {\bar M}} (\tau)$ can be used to calculate the spectral function of electrons from the microscopic theory within a semiclassical approximation. The resulting spectral function varies as $e^{ - \left [ \Omega_0 (\eta) / \omega \right ]^{1 / ( 2 - \eta ) } }$ at low energies.