10,204 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
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
Functional Renormalization of Noncommutative Scalar Field Theory
In this paper we apply the Functional Renormalization Group Equation (FRGE)
to the non-commutative scalar field theory proposed by Grosse and Wulkenhaar.
We derive the flow equation in the matrix representation and discuss the theory
space for the self-dual model. The features introduced by the external
dimensionful scale provided by the non-commutativity parameter, originally
pointed out in \cite{Gurau:2009ni}, are discussed in the FRGE context. Using a
technical assumption, but without resorting to any truncation, it is then shown
that the theory is asymptotically safe for suitably small values of the
coupling, recovering the result of \cite{disertori:2007}. Finally, we
show how the FRGE can be easily used to compute the one loop beta-functions of
the duality covariant model.Comment: 38 pages, no figures, LaTe
Noncommutative Chiral Anomaly and the Dirac-Ginsparg-Wilson Operator
It is shown that the local axial anomaly in dimensions emerges naturally
if one postulates an underlying noncommutative fuzzy structure of spacetime .
In particular the Dirac-Ginsparg-Wilson relation on is shown to
contain an edge effect which corresponds precisely to the ``fuzzy''
axial anomaly on the fuzzy sphere . We also derive a novel gauge-covariant
expansion of the quark propagator in the form where
is the lattice spacing on , is
the covariant noncommutative chirality and is an effective
Dirac operator which has essentially the same IR spectrum as
but differes from it on the UV modes. Most remarkably is the fact that both
operators share the same limit and thus the above covariant expansion is not
available in the continuum theory . The first bit in this expansion
although it vanishes as it stands in the continuum
limit, its contribution to the anomaly is exactly the canonical theta term. The
contribution of the propagator is on the other hand
equal to the toplogical Chern-Simons action which in two dimensions vanishes
identically .Comment: 26 pages, latex fil
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
Generalized local interactions in 1D: solutions of quantum many-body systems describing distinguishable particles
As is well-known, there exists a four parameter family of local interactions
in 1D. We interpret these parameters as coupling constants of delta-type
interactions which include different kinds of momentum dependent terms, and we
determine all cases leading to many-body systems of distinguishable particles
which are exactly solvable by the coordinate Bethe Ansatz. We find two such
families of systems, one with two independent coupling constants deforming the
well-known delta interaction model to non-identical particles, and the other
with a particular one-parameter combination of the delta- and (so-called)
delta-prime interaction. We also find that the model of non-identical particles
gives rise to a somewhat unusual solution of the Yang-Baxter relations. For the
other model we write down explicit formulas for all eigenfunctions.Comment: 23 pages v2: references adde
Spectral noncommutative geometry and quantization: a simple example
We explore the relation between noncommutative geometry, in the spectral
triple formulation, and quantum mechanics. To this aim, we consider a dynamical
theory of a noncommutative geometry defined by a spectral triple, and study its
quantization. In particular, we consider a simple model based on a finite
dimensional spectral triple (A, H, D), which mimics certain aspects of the
spectral formulation of general relativity. We find the physical phase space,
which is the space of the onshell Dirac operators compatible with A and H. We
define a natural symplectic structure over this phase space and construct the
corresponding quantum theory using a covariant canonical quantization approach.
We show that the Connes distance between certain two states over the algebra A
(two ``spacetime points''), which is an arbitrary positive number in the
classical noncommutative geometry, turns out to be discrete in the quantum
theory, and we compute its spectrum. The quantum states of the noncommutative
geometry form a Hilbert space K. D is promoted to an operator *D on the direct
product *H of H and K. The triple (A, *H, *D) can be viewed as the quantization
of the family of the triples (A, H, D).Comment: 7 pages, no figure
Renormalisation of \phi^4-theory on noncommutative R^2 in the matrix base
As a first application of our renormalisation group approach to non-local
matrix models [hep-th/0305066], we prove (super-)renormalisability of Euclidean
two-dimensional noncommutative \phi^4-theory. It is widely believed that this
model is renormalisable in momentum space arguing that there would be
logarithmic UV/IR-divergences only. Although momentum space Feynman graphs can
indeed be computed to any loop order, the logarithmic UV/IR-divergence appears
in the renormalised two-point function -- a hint that the renormalisation is
not completed. In particular, it is impossible to define the squared mass as
the value of the two-point function at vanishing momentum. In contrast, in our
matrix approach the renormalised N-point functions are bounded everywhere and
nevertheless rely on adjusting the mass only. We achieve this by introducing
into the cut-off model a translation-invariance breaking regulator which is
scaled to zero with the removal of the cut-off. The naive treatment without
regulator would not lead to a renormalised theory.Comment: 26 pages, 44 figures, LaTe
Absence of a fuzzy phase in the dimensionally reduced 5d Yang-Mills-Chern-Simons model
We perform nonperturbative studies of the dimensionally reduced 5d
Yang-Mills-Chern-Simons model, in which a four-dimensional fuzzy manifold,
``fuzzy S'', is known to exist as a classical solution. Although the
action is unbounded from below, Monte Carlo simulations provide an evidence for
a well-defined vacuum, which stabilizes at large , when the coefficient of
the Chern-Simons term is sufficiently small. The fuzzy S prepared as an
initial configuration decays rapidly into this vacuum in the process of
thermalization. Thus we find that the model does not possess a ``fuzzy S
phase'' in contrast to our previous results on the fuzzy S.Comment: 11 pages, 2 figures, (v2) typos correcte
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