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Register a new userExotic galilean symmetry and non-commutative mechanics
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Some aspects of the exotic particle, associated with the two-parameter central extension of the planar Galilei group are reviewed. A fundamental property is that it has non-commuting position coordinates. Other and generalized non-commutative models are also discussed. Minimal as well as anomalous coupling to an external electromagnetic field is presented. Supersymmetric extension is also considered. Exotic Galilean symmetry is also found in Moyal field theory. Similar equations arise for a semiclassical Bloch electron, used to explain the anomalous/spin/optical Hall effects.
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Logarithmic representations of the conformal Galilean algebra (CGA) and the Exotic Conformal Galilean algebra ({sc ecga}) are constructed. This can be achieved by non-decomposable representations of the scaling dimensions or the rapidity indices, specific to conformal galilean algebras. Logarithmic representations of the non-exotic CGA lead to the expected constraints on scaling dimensions and rapidities and also on the logarithmic contributions in the co-variant two-point functions. On the other hand, the {sc ecga} admits several distinct situations which are distinguished by different sets of constraints and distinct scaling forms of the two-point functions. Two distinct realisations for the spatial rotations are identified as well. The first example of a reducible, but non-decomposable representation, without logarithmic terms in the two-point function is given.
The six-dimensional exotic Galilean algebra in (2+1) dimensions with two central charges $m$ and $theta$, is extended when $m=0$, to a ten-dimensional Galilean conformal algebra with dilatation, expansion, two acceleration generators and the central charge $theta$. A realisation of such a symmetry is provided by a model with higher derivatives recently discussed in cite{peterwojtek}. We consider also a realisation of the Galilean conformal symmetry for the motion with a Coulomb potential and a magnetic vortex interaction. Finally, we study the restriction, as well as the modification, of the Galilean conformal algebra obtained after the introduction of the minimally coupled constant electric and magnetic fields.
We investigate the planar anisotropic harmonic oscillator with explicit rotational symmetry as a particle model with non-commutative coordinates. It includes the exotic Newton-Hooke particle and the non-commutative Landau problem as special, isotropic and maximally anisotropic, cases. The system is described by the same (2+1)-dimensional exotic Newton-Hooke symmetry as in the isotropic case, and develops three different phases depending on the values of the two central charges. The special cases of the exotic Newton-Hooke particle and non-commutative Landau problem are shown to be characterized by additional, so(3) or so(2,1) Lie symmetry, which reflects their peculiar spectral properties.
In this paper we have successfully established (from first principles) that anyons do live in a 2-dimensional {it{noncommutative}} space. We have directly computed the non-trivial uncertainty relation between anyon coordinates, ${sqrt{Delta x^2Delta y^2}}=Thetasigma$, using the recently constructed anyon wave function [J. Majhi, S. Ghosh and S.K. Maiti, Phys. Rev. Lett. textbf{123}, 164801 (2019)] cite{jan}, in the framework of I. Bialynicki-Birula and Z. Bialynicka-Birula, New J. Phys. textbf{21}, 07306 (2019) cite{bel}. Furthermore we also compute the Heisenberg uncertainty relation for anyon and as a consistency check, show that the results of cite{bel} prove that, as expected, electrons live in 3-dimensional commutative space.
General non-commutative supersymmetric quantum mechanics models in two and three dimensions are constructed and some two and three dimensional examples are explicitly studied. The structure of the theory studied suggest other possible applications in physical systems with potentials involving spin and non-local interactions.
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