Geometric Quantization of Riemann Rotors

Abstract

Geometric quantization enables the unification of the classical and quantum the ories of rotating fluids. The Riemann hydrodynamic equations of motion form a Hamiltonian dynamical system on co-adjoint orbits of a Lie subgroup GCM(3) of the noncompact Lie group Sp(3,R). The general collective motion group GCM(3) is a 15-dimensional semidirect product of a six-dimensional abelian normal subgroup R 6 with the motion group G1(3,R). The Lie algebra of R 6 is spanned by the in ertia tensor. The quantum theory of rotating fluids is constructed from irreducible unitary representations of GCM(3). The Kelvin circulation of the fluid is quantized to nonnegative integer multiples of h. Riemann's ModelRotating systems of particles are a ubiquitous form of collectivity that is found in nature on a continuum scale spanning 35 orders of magnitude from galaxies (period T~ 200 million years ~ 10 15 s), stars (T~ 10 6 s), and fluid droplets (T~ Is) to atomic nuclei (T~ lO" 2^) . 1 ' 2 A rotating system may be either classical (L>> h) or quantum mechanical (L~ h) depending upon the value of its angular momentum L compared to h. The moment of inertia may attain the limiting cases of rigid rotation or irrotational flow or it may fall at some intermediate value.Amidst this dynamical complexity, there is one simplifying common denominator. The shape of a rotating system is usually ellipsoidal. The size, deformation, and orientation of an ellipsoid is characterized completely by the inertia tensor, Q% ="£m a X ai X aj ,a where m^ is the mass of particle a located at the position vector X a with respect to an inertial center of mass frame. For a continuous fluid, the sum is replaced by an integral over the mass density distribution. With respect to the body-fixed principal axis frame, the inertia tensor is, by definition, diagonal,n 0 Quantization and Coherent States Methods Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 08/20/15. For personal use only.