58 research outputs found
Quantum Gravity coupled to Matter via Noncommutative Geometry
We show that the principal part of the Dirac Hamiltonian in 3+1 dimensions
emerges in a semi-classical approximation from a construction which encodes the
kinematics of quantum gravity. The construction is a spectral triple over a
configuration space of connections. It involves an algebra of holonomy loops
represented as bounded operators on a separable Hilbert space and a Dirac type
operator. Semi-classical states, which involve an averaging over points at
which the product between loops is defined, are constructed and it is shown
that the Dirac Hamiltonian emerges as the expectation value of the Dirac type
operator on these states in a semi-classical approximation.Comment: 15 pages, 1 figur
Vector supersymmetry in topological field theories
We present a simple derivation of vector supersymmetry transformations for
topological field theories of Schwarz- and Witten-type. Our method is similar
to the derivation of BRST-transformations from the so-called horizontality
conditions or Russian formulae. We show that this procedure reproduces in a
concise way the known vector supersymmetry transformations of various
topological models and we use it to obtain some new transformations of this
type for 4d topological YM-theories in different gauges.Comment: 19 page
Noncommutative spin-1/2 representations
In this letter we apply the methods of our previous paper hep-th/0108045 to
noncommutative fermions. We show that the fermions form a spin-1/2
representation of the Lorentz algebra. The covariant splitting of the conformal
transformations into a field-dependent part and a \theta-part implies the
Seiberg-Witten differential equations for the fermions.Comment: 7 pages, LaTe
IR-Singularities in Noncommutative Perturbative Dynamics?
We analyse the IR-singularities that appear in a noncommutative scalar
quantum field theory on . We demonstrate with the help of the
quadratic one-loop effective action and an appropriate field redefinition that
no IR-singularities exist. No new degrees of freedom are needed to describe the
UV/IR-mixing.Comment: 6 pages, amsLaTe
Noncommutative Lorentz Symmetry and the Origin of the Seiberg-Witten Map
We show that the noncommutative Yang-Mills field forms an irreducible
representation of the (undeformed) Lie algebra of rigid translations, rotations
and dilatations. The noncommutative Yang-Mills action is invariant under
combined conformal transformations of the Yang-Mills field and of the
noncommutativity parameter \theta. The Seiberg-Witten differential equation
results from a covariant splitting of the combined conformal transformations
and can be computed as the missing piece to complete a covariant conformal
transformation to an invariance of the action.Comment: 20 pages, LaTeX. v2: Streamlined proofs and extended discussion of
Lorentz transformation
Spin Foams and Noncommutative Geometry
We extend the formalism of embedded spin networks and spin foams to include
topological data that encode the underlying three-manifold or four-manifold as
a branched cover. These data are expressed as monodromies, in a way similar to
the encoding of the gravitational field via holonomies. We then describe
convolution algebras of spin networks and spin foams, based on the different
ways in which the same topology can be realized as a branched covering via
covering moves, and on possible composition operations on spin foams. We
illustrate the case of the groupoid algebra of the equivalence relation
determined by covering moves and a 2-semigroupoid algebra arising from a
2-category of spin foams with composition operations corresponding to a fibered
product of the branched coverings and the gluing of cobordisms. The spin foam
amplitudes then give rise to dynamical flows on these algebras, and the
existence of low temperature equilibrium states of Gibbs form is related to
questions on the existence of topological invariants of embedded graphs and
embedded two-complexes with given properties. We end by sketching a possible
approach to combining the spin network and spin foam formalism with matter
within the framework of spectral triples in noncommutative geometry.Comment: 48 pages LaTeX, 30 PDF figure
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