Representations of the diffeomorphism group describing an infinite Bose gas in the presence of ideal vortex filaments

Journal Article · Fri Jun 01 04:00:00 EDT 1990 · Journal of Mathematical Physics (New York); (USA)

DOI:https://doi.org/10.1063/1.528696· OSTI ID:6941258

Albertin, U K; Morrison, H L ^[1]

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1. Department of Physics, University of California, Berkeley, California 94720 (USA)

A number of representations of the group of volume preserving diffeomorphisms of {ital M}={ital S}{sup 3} in the finite-volume case and {ital M}={ital R}{sup 3} in the infinite-volume case that describe a Bose gas in the presence of a vortex filament are examined. The core of the vortex is taken to lie along a curve {gamma}. Let Dif {ital f} {sup {gamma}}{sub {mu}}({ital M}) be the group of volume preserving diffeomorphisms that map {gamma} onto itself, {ital M}{sub {gamma}} the manifold {ital M} with {gamma} removed, and Map({ital M}{sub {gamma}},{ital T}) the space of smooth maps of {ital M}{sub {gamma}} into the complex numbers of modulus unity. It is shown, using the formalism of Klauder (J. Math. Phys. {bold 11}, 233 (1970)), that an infinite number of inequivalent representations of Dif {ital f} {sup {gamma}}{sub {mu}}({ital M}) exist in the infinite-volume limit using cyclic vectors, which are coherent states based on elements in Map({ital M}{sub {gamma}},{ital T}). It is found, however, that these states have questionable physical significance since they appear to violate the continuity equation for quantum probability flow. This leads to the postulation that Dif {ital f} {sup {gamma}}{sub {mu}}({ital M}) acts as gauge group. The representations of Dif {ital f} {sup {gamma}}{sub {mu}}({ital M}) are then used to construct induced representations of the full group volume preserving diffeomorphisms of {ital M}, which are realized in the Hilbert space over the space of unparametrized loops in {ital M} in a natural way. These latter representations are found to be essentially the ones under current study in quantum hydrodynamics (G. A. Goldin, R. Menikoff, and D. H. Sharp, Phys. Rev. Lett. {bold 58}, 2162 (1987)).

• OSTI ID: 6941258
• Journal Information: Journal of Mathematical Physics (New York); (USA), Journal Name: Journal of Mathematical Physics (New York); (USA) Vol. 31:6; ISSN JMAPA; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

640450* -- Fluid Physics-- Superfluidity
75 CONDENSED MATTER PHYSICS
SUPERCONDUCTIVITY AND SUPERFLUIDITY
BANACH SPACE
BOSONS
CANONICAL DIMENSION
COMMUTATION RELATIONS
FLUID MECHANICS
HILBERT SPACE
HYDRODYNAMICS
IRREDUCIBLE REPRESENTATIONS
MATHEMATICAL MANIFOLDS
MATHEMATICAL SPACE
MECHANICS
METRICS
QUANTUM MECHANICS
SCALE DIMENSION
SPACE
SYMMETRY GROUPS
TENSORS
VECTORS
VOLUME
VORTICES