Quantum Theory of Superfluid Vortices. I. Liquid Helium II

A Hamiltonian formulation is constructed for the classical dynamical equations of slightly deformed rectilinear vortices. The system is then quantized by interpreting the conjugate variables as quantum-mechanical operators that obey canonical commutation relations. A linear canonical transformation diagonalizes the Hamiltonian in terms of operators that create and destroy single quanta of vortex vibrations. The theory is applied to two distinct configurations in He II: a single vortex and a rotating vortex lattice. The specific heat associated with the vortex waves varies approximately as $T^{\frac{1}{2}}$ at low temperatures. Quantum-mechanical and thermal fluctuations produce a finite mean-square displacement of the vortex core, which is studied both at $T=0\mathrm{}$ and at $T>\mathrm{}0\mathrm{}$.