Simple field theoretical models on noncommutative manifolds

We review recent progress in formulating two-dimensional models over noncommutative manifolds where the space-time coordinates enter in the formalism as non-commuting matrices. We describe the Fuzzy sphere and a way to approximate topological nontrivial configurations using matrix models. We obtain an ultraviolet cut off procedure, which respects the symmetries of the model. The treatment of spinors results from a supersymmetric formulation; our cut off procedure preserves even the supersymmetry.Comment: 15 pages, TeX (based on lectures given by H. Grosse at Workshops in Clausthal (Germany) and Razlog (Bulgaria) in August 1995

Field Theory on a Supersymmetric Lattice

A lattice-type regularization of the supersymmetric field theories on a supersphere is constructed by approximating the ring of scalar superfields by an integer-valued sequence of finite dimensional rings of supermatrices and by using the differencial calculus of non-commutative geometry. The regulated theory involves only finite number of degrees of freedom and is manifestly supersymmetric.Comment: 31 pages, LaTe

Towards Finite Quantum Field Theory in Non-Commutative Geometry

We describe the self-interacting scalar field on the truncated sphere and we perform the quantization using the functional (path) integral approach. The theory posseses a full symmetry with respect to the isometries of the sphere. We explicitely show that the model is finite and the UV-regularization automatically takes place.Comment: 19 pages, LaTe

Topologically nontrivial field configurations in noncommutative geometry

In the framework of noncommutative geometry we describe spinor fields with nonvanishing winding number on a truncated (fuzzy) sphere. The corresponding field theory actions conserve all basic symmetries of the standard commutative version (space isometries and global chiral symmetry), but due to the noncommutativity of the space the fields are regularized and they contain only finite number of modes.Comment: 27 pages, LaTeX (normalization coefficients in Eqs. (93) corrected

Space-Time Noncommutativity, Discreteness of Time and Unitarity

Violation of unitarity for noncommutative field theory on compact space-times is considered. Although such theories are free of ultraviolet divergences, they still violate unitarity while in a usual field theory such a violation occurs when the theory is nonrenormalizable. The compactness of space-like coordinates implies discreteness of the time variable which leads to appearance of unphysical modes and violation of unitarity even in the absence of a star-product in the interaction terms. Thus, this conclusion holds also for other quantum field theories with discrete time. Violation of causality, among others, occurs also as the nonvanishing of the commutation relations between observables at space-like distances with a typical scale of noncommutativity. While this feature allows for a possible violation of the spin-statistics theorem, such a violation does not rescue the situation but makes the scale of causality violation as the inverse of the mass appearing in the considered model, i.e., even more severe. We also stress the role of smearing over the noncommutative coordinates entering the field operator symbols.Comment: 10 pages, plain LaTe

Proučavanje abelizacijskog prijelaza u SU(2) gluodinamici na konačnoj temperaturi

We discuss the problem of an effective description of the phase transition phenomena in the pure gluodynamics in SU(2) symmetric QCD. The calculation method is choosen following the conjecture that the infrared sector of the theory possesses the same confinement characteristic as the full theory. It is shown that the analytic description of this phenomena is beyond the Gaussian method of evaluations of functional integrals. We point to a non-perturbative method of the evaluation of functional integral for two dimensional Wiener integral for x4 theory which could solve this problem.Raspravljamo problem učinkovitog opisa pojava faznih prijelaza u čistoj gluodinamici u simetričnom SU(2) QCD. Odabrali smo računalni pristup slijedeći slutnju da infracrveni sektor teorije ima jednake značajke zarobljivanja kao potpuna teorija. Pokazuje se da analitički opis problema nije moguć primjenom Gaussove metode računanja funkcionalnih integrala. Ukazujemo na neperturbativan način izračunavanja funkcionalnih integrala za dvodimenzijski Wienerov integral za x 4 teoriju koji bi mogao riješiti problem