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This paper investigates the evolution and interaction of the quantum vortexloops with the small radius of the core $a$. The quantization scheme of theclassical vortex system is based on the approach proposed by the authorearlier. We consider the small variations of the ring-shaped loops only, whichinclude both helical-type shape variations and small excitations of the flow inthe vortex core. The quantization of both an initial radius of the vortex ring$R$ and the circulation $\Gamma$ is deduced from the quantum theory firstprinciples but is not postulated separately. The constructed model leads to theappearance of a spatio-temporal scale hierarchy in the quantum description ofthe vortices. The method of random Hamiltonians is used to describe theinteraction of vortex loops. The specific type of these loops interaction ischosen in way that the constructed theory can serve as the model of turbulentflow at the critical mode.

The evolution of a quantum vortices system: the spatio-temporal scale hierarchy and the turbulence in the critical mode