Peter C. Stichel

Galilean-Invariant (2+1)-Dimensional Models with a Chern–Simons-Like Term andD=2 Noncommutative Geometry☆

Abstract We consider a new D =2 nonrelativistic classical mechanics model providing via the Noether theorem the (2+1)-Galilean symmetry algebra with two central charges: mass m and the coupling constant k of a Chern–Simons-like term. In this way we provide the dynamical interpretation of the second central charge of the (2+1)-dimensional Galilean algebra. We discuss also the interpretation of k as describing the noncommutativity of D =2 space coordinates. The model is quantized in two ways: using the Ostrogradski–Dirac formalism for higher order Lagrangians with constraints and the Faddeev–Jackiw method which describes constrained systems and produces nonstandard symplectic structures. We show that our model describes the superposition of a free motion in noncommutative D =2 spaces as well as the “internal” oscillator modes. We add a suitably chosen class of velocity-dependent two-particle interactions, which is described by local potentials in D =2 noncommutative space. We treat, in detail, the particular case of a harmonic oscillator and describe its quantization. It appears that the indefinite metric due to the third order time derivative term in the field equations, even in the presence of interactions, can be eliminated by the imposition of a subsidiary condition.

Exotic Galilean conformal symmetry and its dynamical realisations

Abstract The six-dimensional exotic Galilean algebra in ( 2 + 1 ) dimensions with two central charges m and θ , is extended when m = 0 , to a ten-dimensional Galilean conformal algebra with dilatation, expansion, two acceleration generators and the central charge θ . A realisation of such a symmetry is provided by a model with higher derivatives recently discussed in [P.C. Stichel, W.J. Zakrzewski, Ann. Phys. 310 (2004) 158]. 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.

Acceleration-extended Galilean symmetries with central charges and their dynamical realizations

Abstract We add to Galilean symmetries the transformations describing constant accelerations. The corresponding extended Galilean algebra allows, in any dimension D = d + 1 , the introduction of one central charge c while in D = 2 + 1 we can have three such charges: c , θ and θ ′ . We present nonrelativistic classical mechanics models, with higher order time derivatives and show that they give dynamical realizations of our algebras. The presence of central charge c requires the acceleration square Lagrangian term. We show that the general Lagrangian with three central charges can be reinterpreted as describing an exotic planar particle coupled to a dynamical electric and a constant magnetic field.

Noncommutative planar particle dynamics with gauge interactions.

We consider two ways of introducing minimal Abelian gauge interactions into the model presented in [Ann. Phys. 260 (1997) 224]. These two approaches are different only if the second central charge of the planar Galilei group is nonzero. One way leads to the standard gauge transformations and the other one to a generalised gauge theory with gauge transformations accompanied by time-dependent area-preserving coordinate transformations. Both approaches, however, are related to each other by a classical Seiberg–Witten map supplemented by a noncanonical transformation of the phase space variables for planar particles. We also formulate the two-body problem in the model with our generalised gauge symmetry and consider the case with both CS and background electromagnetic fields, as it is used in the description of fractional quantum Hall effect.

Galilean exotic planar supersymmetries and nonrelativistic supersymmetric wave equations

Abstract We describe the general class of N-extended D = ( 2 + 1 ) Galilean supersymmetries obtained, respectively, from the N-extended D = 3 Poincare superalgebras with maximal sets of central charges. We confirm the consistency of supersymmetry with the presence of the ‘exotic’ second central charge θ. We show further how to introduce a N = 2 Galilean superfield equation describing nonrelativistic spin 0 and spin 1 2 free particles.

A new type of conformal dynamics

Abstract We consider the Lagrangian particle model introduced in [Ann. Phys. 260 (1997) 224] for zero mass but nonvanishing second central charge of the planar Galilei group. Extended by a magnetic vortex or a Coulomb potential the model exhibits conformal symmetry. In the former case we observe an additional SO(2,1) hidden symmetry. By either a canonical transformation with constraints or by freezing scale and special conformal transformations at t=0 we reduce the six-dimensional phase-space to the physically required four dimensions. Then we discuss bound states (bounded solutions) in quantum dynamics (classical mechanics). We show that the Schrodinger equation for the pure vortex case may be transformed into the Morse potential problem thus providing us with an explanation of the hidden SO(2,1) symmetry.

Can cosmic acceleration be caused by exotic massless particles

To describe dark energy we introduce a fluid model with no free parameter on the microscopic level. The constituents of this fluid are massless particles which are a dynamical realization of the unextended

Acceleration-enlarged symmetries in nonrelativistic space-time with a cosmological constant

D=(3+1)

From Gauging Nonrelativistic Translations to N-Body Dynamics

Galilei algebra. These particles are exotic as they live in an enlarged phase space. Their only interaction is with gravity. A minimal coupling to the gravitational field, satisfying Einsteins equivalence principle, leads to a dynamically active gravitational mass density of either sign. A two-component model containing matter (baryonic and dark) and dark energy leads, through the cosmological principle, to Friedmann-like equations. Their solutions show a deceleration phase for the early universe and an acceleration phase for the late universe. We predict the Hubble parameter

Self-gravitating darkon fluid with anisotropic scaling

H(z)/{H}_{0}