SOME ASPECTS OF QUANTUM MECHANICS AND FIELD THEORY IN A LORENTZ INVARIANT NONCOMMUTATIVE SPACE

We obtained the Feynman propagators for a noncommutative (NC) quantum mechanics defined in the recently developed Doplicher–Fredenhagen–Roberts–Amorim (DFRA) NC background that can be considered as an alternative framework for the NC space–time of the early universe. The operators' formalism was revisited and we applied its properties to obtain an NC transition amplitude representation. Two examples of DFRA's systems were discussed, namely, the NC free particle and NC harmonic oscillator. The spectral representation of the propagator gave us the NC wave function and energy spectrum. We calculated the partition function of the NC harmonic oscillator and the distribution function. Besides, the extension to NC DFRA quantum field theory is straightforward and we used it in a massive scalar field. We had written the scalar action with self-interaction ϕ4 using the Weyl–Moyal product to obtain the propagator and vertex of this model needed to perturbation theory. It is important to emphasize from the outset, that the formalism demonstrated here will not be constructed by introducing an NC parameter in the system, as usual. It will be generated naturally from an already existing NC space. In this extra dimensional NC space, we presented also the idea of dimensional reduction to recover commutativity.

NOTES ON NONCOMMUTATIVE SUPERSYMMETRIC GAUGE THEORY ON THE FUZZY SUPERSPHERE

In this paper we review Klimčík's construction of noncommutative gauge theory on the fuzzy supersphere. This theory has an exact SUSY gauge symmetry with a finite number of degrees of freedom and thus in principle it is amenable to the methods of matrix models and Monte Carlo numerical simulations. We also write down in this paper a novel fuzzy supersymmetric scalar action on the fuzzy supersphere.

FERMION MASS IN THE WILSON-YUKAWA APPROACH FOR CHIRAL YUKAWA THEORY

We consider a modification of the Wilson-Yukawa model to overcome the difficulty that the fermion mass is not proportional to the Higgs vacuum expectation value. In the modification scalar and fermionic regulator fields are introduced so that all the physical fermion fields possess shift symmetry when the Yukawa coupling vanishes. With the fermionic hopping parameter expansion it is shown that the fermion mass is proportional to the Higgs vacuum expectation value. We find, however, that the coupling of fermion to the external gauge field is always vectorlike in the continuum limit and that further modifications to the scalar action cannot change this undesirable conclusion.

Scalar Actions

The subject of this paper arises from the familiar process whereby an automorphism of a field generates new representations from old. One may think of that process spatially, as a change of vector space structure in the representation space by means of the automorphism. The operators of the representation acting in the “new“ space then constitute the new representation. This point of view makes visible an algebraic structure we call a scalar action. A scalar action f of a ring R (with unity) in an abelian group Kis a ring homomorphism f:R → End(V) taking the unity element of R to the identity operator in End(V). If f is a scalar action of a field F and ϕ is an automorphism of F then f ∘ ϕ is another scalar action of F, and it is this construction which is used to define the “new” representation space mentioned above. But the variety of scalar actions goes rather beyond that construction.

Commutative Endomorphism Rings

The problem of classifying the torsion-free abelian groups with commutative endomorphism rings appears as Fuchs’ problems in [4, Problems 46 and 47]. They are far from solved, and the obstacles to a solution appear formidable (see [4; 5]). It is, however, easy to see that the only dualizable abelian group with a commutative endomorphism ring is the infinite cyclic group. (An R-module Miscalled dualizable if HomR(M, R) ≠ 0.) Motivated by this, we study the class of prime rings R which possess a dualizable module M with a commutative endomorphism ring. A characterization of such rings is obtained in § 6, which as would be expected, places stringent restrictions on the ring and the module.Throughout we will write homomorphisms of modules on the side opposite to the scalar action. Rings will not be assumed to contain identity elements unless otherwise indicated.