This paper by E. P. Gross develops a theory for the elementary line vortex excitations in a system of weakly repelling bosons. The vortex state is characterized by a finite fraction of particles in a single state of integer angular momentum. The radial dependence of the highly occupied state is derived from a self-consistent field equation, with the radial function and particle density being essentially constant everywhere except inside a core, where they drop to zero. The core size is determined by the de Broglie wavelength associated with the mean interaction energy per particle. The velocity expectation value follows a classical vortex radial dependence, and in the Hartree approximation, vorticity is zero everywhere except on the vortex line. When zero-point oscillations of the phonon field are included, vorticity spreads out over the core. These results support the intuitive arguments of Onsager and Feynman, confirming that phonons moving perpendicular to the vortex line are coherent excitations with equal and opposite angular momentum relative to the moving particle substrate. The vortex motion resolves the degeneracy of Bogoljubov phonons with respect to the azimuthal quantum number.This paper by E. P. Gross develops a theory for the elementary line vortex excitations in a system of weakly repelling bosons. The vortex state is characterized by a finite fraction of particles in a single state of integer angular momentum. The radial dependence of the highly occupied state is derived from a self-consistent field equation, with the radial function and particle density being essentially constant everywhere except inside a core, where they drop to zero. The core size is determined by the de Broglie wavelength associated with the mean interaction energy per particle. The velocity expectation value follows a classical vortex radial dependence, and in the Hartree approximation, vorticity is zero everywhere except on the vortex line. When zero-point oscillations of the phonon field are included, vorticity spreads out over the core. These results support the intuitive arguments of Onsager and Feynman, confirming that phonons moving perpendicular to the vortex line are coherent excitations with equal and opposite angular momentum relative to the moving particle substrate. The vortex motion resolves the degeneracy of Bogoljubov phonons with respect to the azimuthal quantum number.