9,855 research outputs found
Matrix models on the fuzzy sphere
Field theory on a fuzzy noncommutative sphere can be considered as a
particular matrix approximation of field theory on the standard commutative
sphere. We investigate from this point of view the scalar theory. We
demonstrate that the UV/IR mixing problems of this theory are localized to the
tadpole diagrams and can be removed by an appropiate (fuzzy) normal ordering of
the vertex. The perturbative expansion of this theory reduces in the
commutative limit to that on the commutative sphere.Comment: 6 pages, LaTeX2e, Talk given at the NATO Advanced Research Workshop
on Confiment, Topology, and other Non-Perturbative Aspects of QCD, Stara
Lesna, Slovakia, Jan. 21-27, 200
The \beta-function in duality-covariant noncommutative \phi^4-theory
We compute the one-loop \beta-functions describing the renormalisation of the
coupling constant \lambda and the frequency parameter \Omega for the real
four-dimensional duality-covariant noncommutative \phi^4-model, which is
renormalisable to all orders. The contribution from the one-loop four-point
function is reduced by the one-loop wavefunction renormalisation, but the
\beta_\lambda-function remains non-negative. Both \beta_\lambda and
\beta_\Omega vanish at the one-loop level for the duality-invariant model
characterised by \Omega=1. Moreover, \beta_\Omega also vanishes in the limit
\Omega \to 0, which defines the standard noncommutative \phi^4-quantum field
theory. Thus, the limit \Omega \to 0 exists at least at the one-loop level.Comment: 11 pages, LaTe
Noncommutative QFT and Renormalization
Field theories on deformed spaces suffer from the IR/UV mixing and
renormalization is generically spoiled. In work with R. Wulkenhaar, one of us
realized a way to cure this disease by adding one more marginal operator. We
review these ideas, show the application to models and use the heat
kernel expansion methods for a scalar field theory coupled to an external gauge
field on a -deformed space and derive noncommutative gauge field
actions.Comment: To appear in the proceedings of the Workshop "Noncommutative Geometry
in Field and String Theory", Corfu, 2005 (Greece
On Finite 4D Quantum Field Theory in Non-Commutative Geometry
The truncated 4-dimensional sphere and the action of the
self-interacting scalar field on it are constructed. The path integral
quantization is performed while simultaneously keeping the SO(5) symmetry and
the finite number of degrees of freedom. The usual field theory UV-divergences
are manifestly absent.Comment: 18 pages, LaTeX, few misprints are corrected; one section is remove
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
Geometry of the Grosse-Wulkenhaar Model
We define a two-dimensional noncommutative space as a limit of finite-matrix
spaces which have space-time dimension three. We show that on such space the
Grosse-Wulkenhaar (renormalizable) action has natural interpretation as the
action for the scalar field coupled to the curvature. We also discuss a natural
generalization to four dimensions.Comment: 16 pages, version accepted in JHE
The Luttinger-Schwinger Model
We study the Luttinger-Schwinger model, i.e. the (1+1) dimensional model of
massless Dirac fermions with a non-local 4-point interaction coupled to a
U(1)-gauge field. The complete solution of the model is found using the
boson-fermion correspondence, and the formalism for calculating all gauge
invariant Green functions is provided. We discuss the role of anomalies and
show how the existence of large gauge transformations implies a fermion
condensate in all physical states. The meaning of regularization and
renormalization in our well-defined Hilbert space setting is discussed. We
illustrate the latter by performing the limit to the Thirring-Schwinger model
where the interaction becomes local.Comment: 19 pages, Latex, to appear in Annals of Physics, download problems
fixe
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
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