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Journal of Mathematical Physics

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

Uwe K. Albertin, Harry L. Morrison

Published Jun 1, 1990• 1 citation
Publisher

Abstract

A number of representations of the group of volume preserving diffeomorphisms of M=S3 in the finite-volume case and M=R3 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 γ. Let Dif f γμ(M) be the group of volume preserving diffeomorphisms that map γ onto itself, Mγ the manifold M with γ removed, and Map(Mγ,T) the space of smooth maps of Mγ into the complex numbers of modulus unity. It is shown, using the formalism of Klauder [J. Math. Phys. 11, 233 (1970)], that an infinite number of inequivalent representations of Dif f γμ(M) exist in the infinite-volume limit using cyclic vectors, which are coherent states based on elements in Map(Mγ,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 f γμ(M) acts as gauge group. The representations of Dif f γμ(M) are then used to construct induced representations of the full group volume preserving diffeomorphisms of M, which are realized in the Hilbert space over the space of unparametrized loops in 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. 58, 2162 (1987)].

DOI
10.1063/1.528696

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A number of representations of the group of volume preserving diffeomorphisms of M=S3 in the finite-volume case and M=R3 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 γ. Let Dif f γμ(M) be the group of volume preserving diffeomorphisms that map γ onto itself, Mγ the manifold M with γ removed, and Map(Mγ,T) the space of smooth maps of Mγ into the complex numbers of modulus unity. It is shown, using the formalism of Klauder [J. Math. Phys. 11, 233 (1970)], that an infinite number of inequivalent representations of Dif f γμ(M) exist in the infinite-volume limit using cyclic vectors, which are coherent states based on elements in Map(Mγ,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 f γμ(M) acts as gauge group. The representations of Dif f γμ(M) are then used to construct induced representations of the full group volume preserving diffeomorphisms of M, which are realized in the Hilbert space over the space of unparametrized loops in 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. 58, 2162 (1987)].
The existence of inequivalent representations of the canonical commutation relations which describe a nonrelativistic infinite free Bose gas of uniform density is investigated, with a view to possible applications to statistical mechanics. The functional E(f, g)=(Ψ, eiφ(f) eiπ(g)Ψ) is used to describe the inequivalent representations. This functional is calculated for the free Bose gas in a box of volume V, and the limit is then taken as V → ∞. In this way we construct cyclic representations describing an infinite system of particles with a density distribution ρ(k) in momentum space. For a given ρ(k) the operator algebra generated by the φ(f), π(g) is reducible. For the ground-state representation (all particles in the zero-momentum state), the representation is a direct integral of irreducible representations (analogous to BCS theory). For finite temperatures the situation is complicated by the occurrence of representations which are not type I. The physical significance of the reducibility of the representations is discussed. It is argued that the thermal ensemble for the infinite system is a pure state at zero temperature, although there is some ambiguity as to which operators should belong to the algebra of observables. For finite temperatures, the thermal ensemble seems to be a mixture. The case of an interacting Bose gas is considered briefly.
In this paper we discuss various geometric aspects related to the Schrödinger and the Pauli equations. First we resume the Madelung–Bohm hydrodynamical approach to quantum mechanics and recall the Hamiltonian structure of the Schrödinger equation. The probability current provides an equivariant moment map for the group [Formula: see text] of volume-preserving diffeomorphisms of [Formula: see text] (rapidly approaching the identity at infinity) and leads to a current algebra of Rasetti–Regge type. The moment map picture is then extended, mutatis mutandis, to the Pauli equation and to generalized Schrödinger equations of the Pauli–Thomas type. A gauge theoretical reinterpretation of all equations is obtained via the introduction of suitable Maurer–Cartan gauge fields and it is then related to Weyl geometric and pilot wave ideas. A general framework accommodating Aharonov–Bohm and Aharonov–Casher effects is presented within the gauge approach. Furthermore, a kind of holomorphic geometric quantization can be performed and yields natural “coherent state” representations of [Formula: see text]. The relationship with the covariant phase space and density manifold approaches is then outlined. Comments on possible extensions to nonlinear Schrödinger equations, on Fisher-information theoretic aspects and on stochastic mechanics are finally made.
The dynamics of interacting quantized vortex filaments in a rotating Bose–Einstein condensate existing in the Thomas–Fermi regime at zero temperature and obeying the Gross–Pitaevskii equation has been considered in the hydrodynamic “nonelastic” approximation. A noncanonical Hamilton equation of motion for the macroscopically averaged vorticity has been derived for a smoothly inhomogeneous array of filaments (vortex lattice) taking into account spatial nonuniformity of the equilibrium density of the condensate, which is determined by the trap potential. The minimum of the corresponding Hamiltonian describes the static configuration of the deformed vortex lattice against the preset density background. The condition of minimum can be reduced to a nonlinear second-order partial differential vector equation for which some exact and approximate solutions are obtained. It has been shown that if the condensate density has an anisotropic Gaussian profile, the equation of motion for the averaged vorticity has solutions in the form of a vector exhibiting a nontrivial time dependence, but homogeneous in space. An integral representation has also been obtained for the matrix Green function that determines the nonlocal Hamiltonian of a system of several quantized vortices of an arbitrary shape in a Bose–Einstein condensate with the Gaussian density. In particular, if all filaments are straight and oriented along one of the principal axes of the ellipsoid, we have a finitedimensional reduction that can describe the dynamics of the system of pointlike vortices against an inhomogeneous background. A simple approximate expression is proposed for the 2D Green function with an arbitrary density profile and is compared numerically with the exact result in the Gaussian case. The corresponding approximate equations of motion, describing the long-wavelength dynamics of interacting vortex filaments in condensates with a density depending only on transverse coordinates, have been derived.
Abstract Berezin and Weyl quantization are renowned procedures for mapping commutative Poisson algebras of observables to their non-commutative, quantum counterparts. The latter is famous for its use on Weyl algebras, while the former is more appropriate for continuous functions decaying at infinity. In this work, we define a variant of the Berezin quantization map, which acts on the classical Weyl algebra W(E,0)\mathcal {W}(E,0)W(E,0) W ( E , 0 ) and constitutes a positive strict deformation quantization . This construction provides a natural framework to compare classical and quantum thermal equilibrium states of a Bose gas through the computation of their semi-classical limit. To this end, we first introduce a purely algebraic notion of KMS states for the classical Weyl algebra and establish that, in finite volume, there exists a unique such state, which can be interpreted as the Fourier transform of a Gibbs measure on a Hilbert space. We then construct a new class of classical KMS states that realize representations of the canonical commutation relations with infinite local density. These states arise as the semi-classical high-density limit of the quantum equilibrium states originally studied by Araki and Woods [5]. A key feature of our approach is that it preserves the macroscopic ground-state occupation of the Bose gas in the classical regime. Finally, we demonstrate that the infinite-volume classical states can be obtained as thermodynamic limits of finite-volume Gibbs states.
Geometric quantum hydrodynamics merges geometric hydrodynamics with quantum hydrodynamics to study the geometric properties of vortex structures in superfluid states of matter. Here the vortex line acts as the fundamental building block and is a topological defect of the fluid medium about which the otherwise irrotational fluid circulates. In this thesis, we show that except for the simplest fluid flows, a vortex line seeks to decompose localized regions of curvature into helical configurations. The simplest flow, known as the local induction approximation, is also an integrable one. Integrability makes the transfer of energy into helical modes impossible. In the following, we demonstrate that any arclength conserving correction to this approximation defines a non-Hamiltonian evolution of the vortex geometry, which is capable of supporting dissipative solitons and helical wavefronts. Quantum turbulence in ultracold vortex tangles relies on energy transfer between helical or Kelvin modes to decay. Thus, models of vortex lines beyond the lowest order integrable cases are vitally important to our mathematical description of free decay of turbulent tangles. To motivate the results of this thesis we connect our theory of vortex line dynamics to continuum fluid mechanics. The Navier-Stokes equations are a statement of momentum balance for a fluid whose response to shear stress is proportional to the fluid's velocity gradients. Building off of Onsager's rather obscure work in fluid turbulence, others have shown that solutions to Navier-Stokes limit to Euler evolutions of distributional velocity profiles for large Reynolds number. Using this as our context, we show that the Euler equation can be transformed, in an inverse Madelung sense, to the Gross-Pitaevskii equation associated with the mean-field quantum dynamics of a dilute Bose gas. This theory predicts that an irrotational fluid is capable of circulating around regions of density depletion known as vortex lines. Furthermore, if a vortex line is used as the ansatz for the Gross-Pitaevskii equation then it is possible to show that the Biot-Savart integral over the vortex source results. This connection between the vortex line geometry and the induced velocity field provides the basis for the application of geometric quantum hydrodynamics to Bose-Einstein condensates. If the Biot-Savart integral is the basis of geometric quantum hydrodynamics, then Hasimoto's transformation is the structure built on top. The fundamental theorem of space curves states that up to rotations and translations, a curve is defined by its curvature and torsion. Through the Hasimoto transformation, it is possible to map flows defined by the Biot-Savart integral to scalar partial differential equations evolving the curvature and torsion of a vortex line. In this thesis, we conduct an asymptotic expansion of the Biot-Savart integral and apply the Hasimoto transformation to show that vortex lines prefer to relax curved abnormalities through the excitation of helical waves along the vortex. Thus, our…
In this work, the topological and geometrical structure of the evolution equations of physical systems is treated and analyzed from the point of view of the set of solutions, in particular the singular ones. To this end, a generalization of the Wigner function is introduced for the case of multidimensional quadratic Hamiltonians of the type of our previous work [A. I. Aptekarev, Yu. G. Rykov and D. J. Cirilo-Lombardo, Diffeomorphic structure of evolution equations, J. Geom. Phys. 202 (2024) 105234]. The phase space solutions from the generalized Wigner equation are compared with those from the Schrodinger-type equation where the Madelung transformation was introduced. We show that singular delta-type solutions coming from the Wigner–Liouville system certainly exist, intuiting accordingly that they could be mapped to a hydrodynamic representation, in sharp contrast with the Schrodinger (wave mechanical) case, where delta-like solutions become inconsistent. This is achieved by considering a topological limit to the coherent subspace of solutions of the Wigner–Liouville equation, where the kernel (acting as quasidistribution) of these new coherent functions (coherent states) takes the main role. Our results are compared in the context of the wave mechanical approach to cosmic structure formation considering the Zeldovich approach of [P. Coles and K. Spencer, A wave-mechanical approach to cosmic structure formation, Mon. Not. R. Astron. Soc. 342 (2003) 176]. The possibility of obtaining a one-to-one functional relationship between the singular solution space of the hydrodynamic system and the Wigner–Liouville one is briefly discussed.