Generalized Eulerian angles and the collective motion of many-particle system

Harry J. Lipkin

Harry J. Lipkin

Brian H Buck, Lawrence C. Biedenharn, R. Y. Cusson

Peter Holland

Quantum hydrodynamics represents quantum mechanics through two complementary models: the Eulerian picture, a direct transcription of wave mechanics, and the Lagrangian picture, in which the quantum state is represented by the collective motion of a continuum of fluid particles, the latter being obtained by continuously varying the particle label. The Lagrangian picture thus adds variables to the quantum formalism and exhibits a corresponding new quantum symmetry viz. a continuous particle-relabelling covariance group. Using Noether's theorem, the relabelling symmetry is derived as a component of a general investigation of symmetries and conservation laws in the quantum Lagrangian picture. Relations with symmetries and conservation laws in the Eulerian picture are explored in detail. Alongside the infinite relabelling group, the 12-parameter kinematical covariance group of the Schrödinger equation is derived. It is stressed that the role of label transformations extends beyond the class corresponding to Eulerian invariance. As an example, it is shown that the linear superposition of waves, a fundamental symmetry of the quantum Eulerian picture, can be generated by a deformation-dependent label transformation in the Lagrangian picture.

Jin Au Kong

Biaxial crystal optics is studied both classically and quantum mechanically. A light can, in general, be decomposed into two linearly polarized waves. By proper choice of one of the three eulerian angles, two characteristic waves in the crystal are determined. Quantization is facilitated by postulating commutation relations between D and B fields and Heisenberg equations of the motion or, equivalently, Maxwell’s equations. Two types of photons corresponding to the two characteristic waves are found and their energy states are calculated.

Peter Holland

We present an alternative Eulerian hydrodynamic model for the electromagnetic field in which the discrete vector indices in Maxwell's equations are replaced by continuous angular freedoms, and develop the corresponding Lagrangian picture in which the fluid particles have rotational and translational freedoms. This enables us to extend to the electromagnetic field the exact method of state construction proposed previously for spin 0 systems, in which the time-dependent wavefunction is computed from a single-valued continuum of deterministic trajectories where two spacetime points are linked by at most a single orbit. This is achieved by generalizing the spin 0 theory to a general Riemannian manifold from which the electromagnetic construction is extracted as a special case. The Lorentz covariance of the Eulerian field theory is obtained from the non-covariant Lagrangian-coordinate model as a kind of collective effect. The method makes manifest the electromagnetic analogue of the quantum potential that is tacit in Maxwell's equations. This implies a novel definition of the "classical limit" of Maxwell's equations that differs from geometrical optics.