  In this study, the problem of the energy spectrum of a quantum vortex loop
moving in a thin long pipe is solved for the first time. The vortex filament is
described in the Local Induction Approximation. We quantize this dynamic system
using a new method, which leads to non-trivial results for circulation $\Gamma$
and energy values $E$. In the final form, we present the spectrum of the vortex
loop in the form of a ''Regge trajectory'' $E = E(\Gamma)$. The vortex
quantization problem is considered outside of two-fluid hydrodynamics and other
conventional approaches.
