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Gerald A. Goldin - One of the best experts on this subject based on the ideXlab platform.
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Diffeomorphism Group Representations in Relativistic Quantum Field Theory
Trends in Mathematics, 2019Co-Authors: Gerald A. Goldin, David H. SharpAbstract:We explore the role played by the Diffeomorphism Group and its unitary representations in relativistic quantum field theory. From the quantum kinematics of particles described by representations of the Diffeomorphism Group of a space-like surface in an inertial reference frame, we reconstruct the local relativistic neutral scalar field in the Fock representation. An explicit expression for the free Hamiltonian is obtained in terms of the Lie algebra generators (mass and momentum densities). We suggest that this approach can be generalized to fields whose quanta are spatially extended objects.
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quantum configuration spaces of extended objects Diffeomorphism Group representations and exotic statistics
30th Workshop on Geometric Methods in Physics 2011, 2013Co-Authors: Gerald A. GoldinAbstract:A fundamental approach to quantum mechanics is based on the unitary representations of the Group of Diffeomorphisms of physical space (and correspondingly, self-adjoint representations of a local current algebra). From these, various classes of quantum configuration spaces arise naturally, as well as the usual exchange statistics for point particles in spatial dimensions \( d\,\,\geq\, 3\), induced by representations of the symmetric Group. For \( d\,\,=\, 3\), this approach led to an early prediction of intermediate or “anyon” statistics induced by unitary representations of the braid Group. I review these ideas, and discuss briefly some analogous possibilities for infinite-dimensional configuration spaces, includinga nyonic statistics for extended objects in three-dimensional space.
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Diffeomorphism Group REPRESENTATIONS AND NONLINEAR QUANTUM THEORIES
Modern Group Theoretical Methods in Physics, 1995Co-Authors: Gerald A. GoldinAbstract:Quantum theories with nonlinear time-evolution equations are suggested by certain continuous unitary representations of Diffeomorphism Groups. Their physical interpretation is discussed in the light of nonlinear gauge transformations.
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properties of nonlinear schrodinger equations associated with Diffeomorphism Group representations
Journal of Physics A, 1994Co-Authors: H D Doebner, Gerald A. GoldinAbstract:The authors recently derived a family of nonlinear Schrodinger equations on R3 from fundamental considerations of generalized symmetry: ih(cross) delta t psi =-(h(cross)2/2m) Del 2 psi +F( psi , psi ) psi +ih(cross)D( Del 2 psi +( mod Del psi mod 2/ mod psi mod 2) psi ), where F is an arbitrary real functional and D a real, continuous quantum number. These equations, descriptive of a quantum mechanical current that includes a diffusive term, correspond to unitary representations of the Group Diff(M) parametrized by D, where M=R3 is the physical space. In the present paper we explore the most natural ansatz for F, which is labelled by five real coefficients. We discuss the invariance properties, describe the stationary states and some non-stationary solutions, and determine the extra, dissipative terms that occur in the Ehrenfest theorem. We identify an interesting, Galilean-invariant subfamily whose properties we investigate, including the case where the dissipative terms vanish.
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THE Diffeomorphism Group APPROACH TO NONLINEAR QUANTUM SYSTEMS
International Journal of Modern Physics B, 1992Co-Authors: Gerald A. GoldinAbstract:Unitary representations of Diffeomorphism Groups predict some unusual possibilities in quantum theory, including non-standard statistics and certain nonlinear effects. Many of the fundamental physical properties of “anyons” were first derived from their study by R. Menikoff, D.H. Sharp, and the author. This paper surveys new applications in two other domains: first (with Menikoff and Sharp) some surprising conclusions about the nature of quantum vortex configurations in ideal, incompressible fluids; second (with H.-D. Doebner) a natural description of dissipative quantum mechanics by means of a nonlinear Schrödinger equation different from the sort usually studied. Our equation follows from including a diffusion current in the equation of continuity.
Boris Kolev - One of the best experts on this subject based on the ideXlab platform.
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Right-invariant Sobolev metrics of fractional order on the Diffeomorphism Group of the circle
Journal of Geometric Mechanics, 2014Co-Authors: Joachim Escher, Boris KolevAbstract:In this paper, we study the geodesic flow of a right-invariant metric induced by a general Fourier multiplier on the Diffeomorphism Group of the circle and on some of its homogeneous spaces. This study covers in particular right-invariant metrics induced by Sobolev norms of fractional order. We show that, under a certain condition on the symbol of the inertia operator (which is satisfied for the fractional Sobolev norm $H^{s}$ for $s \ge 1/2$), the corresponding initial value problem is well-posed in the smooth category and that the Riemannian exponential map is a smooth local Diffeomorphism. Paradigmatic examples of our general setting cover, besides all traditional Euler equations induced by a local inertia operator, the Constantin-Lax-Majda equation, and the Euler-Weil-Petersson equation.
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Geodesic completeness for Sobolev H s -metrics on the Diffeomorphism Group of the circle
Journal of Evolution Equations, 2014Co-Authors: Joachim Escher, Boris KolevAbstract:We prove that the weak Riemannian metric induced by the fractional Sobolev norm H s on the Diffeomorphism Group of the circle is geodesically complete, provided that s > 3/2.
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Geometrical Methods for Equations of Hydrodynamical Type
Journal of Nonlinear Mathematical Physics, 2012Co-Authors: Boris Kolev, Joachim EscherAbstract:We describe some recent results for a class of nonlinear hydrodynamical approximation models where the geometric approach gives insight into a variety of aspects. The main contribution concerns analytical results for Euler equations on the Diffeomorphism Group of the circle for which the inertia operator is a non local operator.
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Remarks on the geometry of the Diffeomorphism Group of the circle
2012Co-Authors: Boris Kolev, Adrian ConstantinAbstract:We discuss some of the possibilities of endowing the Diffeomorphism Group of the circle with Riemannian structures arising from right-invariant metrics.
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On the geometry of the Diffeomorphism Group of the circle
Number Theory Analysis and Geometry, 2011Co-Authors: Adrian Constantin, Boris KolevAbstract:We discuss some of the possibilities of endowing the Diffeomorphism Group of the circle with Riemannian structures arising from right-invariant metrics.
Janusz Grabowski - One of the best experts on this subject based on the ideXlab platform.
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a poisson lie structure on the Diffeomorphism Group of a circle
Letters in Mathematical Physics, 1994Co-Authors: Janusz GrabowskiAbstract:Starting from the Gelfand-Fuks-Virasoro cocycle on the Lie algebraX(S1) of the vector fields on the circleS1 and applying the standard procedure described by Drinfel'd in a finite dimension, we obtain a classicalr-matrix (i.e. an elementr ∈X(S1) ∧X(S1) satisfying the classical Yang-Baxter equation), a Lie bialgebra structure onX(S1), and a sort of Poisson-Lie structure on the Group\(\widetilde{Diff}(S^1 )\) of Diffeomorphisms. Quantizations of such Lie bialgebra structures may lead to ‘quantum Diffeomorphism Groups’.
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A poisson—lie structure on the Diffeomorphism Group of a circle
Letters in Mathematical Physics, 1994Co-Authors: Janusz GrabowskiAbstract:Starting from the Gelfand-Fuks-Virasoro cocycle on the Lie algebraX(S1) of the vector fields on the circleS1 and applying the standard procedure described by Drinfel'd in a finite dimension, we obtain a classicalr-matrix (i.e. an elementr ∈X(S1) ∧X(S1) satisfying the classical Yang-Baxter equation), a Lie bialgebra structure onX(S1), and a sort of Poisson-Lie structure on the Group\(\widetilde{Diff}(S^1 )\) of Diffeomorphisms. Quantizations of such Lie bialgebra structures may lead to ‘quantum Diffeomorphism Groups’.
Qhan Park - One of the best experts on this subject based on the ideXlab platform.
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gravitation as gauge theory of the Diffeomorphism Group
Physics Letters B, 1992Co-Authors: Y M Cho, Kwangsup Soh, J H Yoon, Qhan ParkAbstract:Abstract The (m+n)-dimensional Einstein theory of gravitation is identified with an m-dimensional generally invariant gauge theory of DiffN, where N is an n-dimensional manifold. This means that the four-dimensional Einstein gravity can be identified as a lower dimensional gauge theory of an infinite dimensional Group of Diffeomorphism. We discuss the physical implications of the results.
Vladimir L. Popov - One of the best experts on this subject based on the ideXlab platform.
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Finite subGroups of Diffeomorphism Groups
Proceedings of the Steklov Institute of Mathematics, 2015Co-Authors: Vladimir L. PopovAbstract:We prove the following: (1) the existence, for every integer n ≥ 4, of a noncompact smooth n-dimensional topological manifold whose Diffeomorphism Group contains an isomorphic copy of every finitely presented Group; (2) a finiteness theorem for finite simple subGroups of Diffeomorphism Groups of compact smooth topological manifolds.
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Finite subGroups of Diffeomorphism Groups
arXiv: Group Theory, 2013Co-Authors: Vladimir L. PopovAbstract:We prove:(1) the existence, for every integer n > 3, of a noncompact smooth n-dimensional topological manifold whose Diffeomorphism Group contains an isomorphic copy of every finitely presented Group; (2) a finiteness theorem on finite simple subGroups of Diffeomorphism Groups of compact smooth topological manifolds.