Quantum Hall Effect on the Hyperbolic Plane

In this paper, we study both the continuous model and the discrete model of the Quantum Hall Effect (QHE) on the hyperbolic plane. The Hall conductivity is identified as a geometric invariant associated to an imprimitivity algebra of observables. We define a twisted analogue of the Kasparov map, which enables us to use the pairing between $K$-theory and cyclic cohomology theory, to identify this geometric invariant with a topological index, thereby proving the integrality of the Hall conductivity in this case.Comment: AMS-LaTeX, 28 page

On the problem of interactions in quantum theory

The structure of representations describing systems of free particles in the theory with the invariance group SO(1,4) is investigated. The property of the particles to be free means as usual that the representation describing a many-particle system is the tensor product of the corresponding single-particle representations (i.e. no interaction is introduced). It is shown that the mass operator contains only continuous spectrum in the interval $(-\infty,\infty)$ and such representations are unitarily equivalent to ones describing interactions (gravitational, electromagnetic etc.). This means that there are no bound states in the theory and the Hilbert space of the many-particle system contains a subspace of states with the following property: the action of free representation operators on these states is manifested in the form of different interactions. Possible consequences of the results are discussed.Comment: 35 pages, Late

Could Only Fermions Be Elementary?

In standard Poincare and anti de Sitter SO(2,3) invariant theories, antiparticles are related to negative energy solutions of covariant equations while independent positive energy unitary irreducible representations (UIRs) of the symmetry group are used for describing both a particle and its antiparticle. Such an approach cannot be applied in de Sitter SO(1,4) invariant theory. We argue that it would be more natural to require that (*) one UIR should describe a particle and its antiparticle simultaneously. This would automatically explain the existence of antiparticles and show that a particle and its antiparticle are different states of the same object. If (*) is adopted then among the above groups only the SO(1,4) one can be a candidate for constructing elementary particle theory. It is shown that UIRs of the SO(1,4) group can be interpreted in the framework of (*) and cannot be interpreted in the standard way. By quantizing such UIRs and requiring that the energy should be positive in the Poincare approximation, we conclude that i) elementary particles can be only fermions. It is also shown that ii) C invariance is not exact even in the free massive theory and iii) elementary particles cannot be neutral. This gives a natural explanation of the fact that all observed neutral states are bosons.Comment: The paper is considerably revised and the following results are added: in the SO(1,4) invariant theory i) the C invariance is not exact even for free massive particles; ii) neutral particles cannot be elementar