57 research outputs found
The Kirillov picture for the Wigner particle
We discuss the Kirillov method for massless Wigner particles, usually
(mis)named "continuous spin" or "infinite spin" particles. These appear in
Wigner's classification of the unitary representations of the Poincar\'e group,
labelled by elements of the enveloping algebra of the Poincar\'e Lie algebra.
Now, the coadjoint orbit procedure introduced by Kirillov is a prelude to
quantization. Here we exhibit for those particles the classical Casimir
functions on phase space, in parallel to quantum representation theory. A good
set of position coordinates are identified on the coadjoint orbits of the
Wigner particles; the stabilizer subgroups and the symplectic structures of
these orbits are also described.Comment: 19 pages; v2: updated to coincide with published versio
Moyal Planes are Spectral Triples
Axioms for nonunital spectral triples, extending those introduced in the
unital case by Connes, are proposed. As a guide, and for the sake of their
importance in noncommutative quantum field theory, the spaces endowed
with Moyal products are intensively investigated. Some physical applications,
such as the construction of noncommutative Wick monomials and the computation
of the Connes--Lott functional action, are given for these noncommutative
hyperplanes.Comment: Latex, 54 pages. Version 3 with Moyal-Wick section update
The Fuzzy Sphere Star-Product and Spin Networks
We analyze the expansion of the fuzzy sphere non-commutative product in
powers of the non-commutativity parameter. To analyze this expansion we develop
a graphical technique that uses spin networks. This technique is potentially
interesting in its own right as introducing spin networks of Penrose into
non-commutative geometry. Our analysis leads to a clarification of the link
between the fuzzy sphere non-commutative product and the usual deformation
quantization of the sphere in terms of the star-product.Comment: 21 pages, many figure
'Schwinger Model' on the Fuzzy Sphere
In this paper, we construct a model of spinor fields interacting with
specific gauge fields on fuzzy sphere and analyze the chiral symmetry of this
'Schwinger model'. In constructing the theory of gauge fields interacting with
spinors on fuzzy sphere, we take the approach that the Dirac operator on
q-deformed fuzzy sphere is the gauged Dirac operator on fuzzy
sphere. This introduces interaction between spinors and specific one parameter
family of gauge fields. We also show how to express the field strength for this
gauge field in terms of the Dirac operators and alone. Using the path
integral method, we have calculated the point functions of this model and
show that, in general, they do not vanish, reflecting the chiral non-invariance
of the partition function.Comment: Minor changes, typos corrected, 18 pages, to appear in Mod. Phys.
Lett.
Quantum Black Hole in the Generalized Uncertainty Principle Framework
In this paper we study the effects of the Generalized Uncertainty Principle
(GUP) on canonical quantum gravity of black holes. Through the use of modified
partition function that involves the effects of the GUP, we obtain the
thermodynamical properties of the Schwarzschild black hole. We also calculate
the Hawking temperature and entropy for the modification of the Schwarzschild
black hole in the presence of the GUP.Comment: 11 pages, no figures, to appear in Physical Review
Noncommutative geometry and physics: a review of selected recent results
This review is based on two lectures given at the 2000 TMR school in Torino.
We discuss two main themes: i) Moyal-type deformations of gauge theories, as
emerging from M-theory and open string theories, and ii) the noncommutative
geometry of finite groups, with the explicit example of Z_2, and its
application to Kaluza-Klein gauge theories on discrete internal spaces.Comment: Based on lectures given at the TMR School on contemporary string
theory and brane physics, Jan 26- Feb 2, 2000, Torino, Italy. To be published
in Class. Quant. Grav. 17 (2000). 3 ref.s added, typos corrected, formula on
exterior product of n left-invariant one-forms corrected, small changes in
the Sect. on integratio
Local covariant quantum field theory over spectral geometries
A framework which combines ideas from Connes' noncommutative geometry, or
spectral geometry, with recent ideas on generally covariant quantum field
theory, is proposed in the present work. A certain type of spectral geometries
modelling (possibly noncommutative) globally hyperbolic spacetimes is
introduced in terms of so-called globally hyperbolic spectral triples. The
concept is further generalized to a category of globally hyperbolic spectral
geometries whose morphisms describe the generalization of isometric embeddings.
Then a local generally covariant quantum field theory is introduced as a
covariant functor between such a category of globally hyperbolic spectral
geometries and the category of involutive algebras (or *-algebras). Thus, a
local covariant quantum field theory over spectral geometries assigns quantum
fields not just to a single noncommutative geometry (or noncommutative
spacetime), but simultaneously to ``all'' spectral geometries, while respecting
the covariance principle demanding that quantum field theories over isomorphic
spectral geometries should also be isomorphic. It is suggested that in a
quantum theory of gravity a particular class of globally hyperbolic spectral
geometries is selected through a dynamical coupling of geometry and matter
compatible with the covariance principle.Comment: 21 pages, 2 figure
On Spectral Triples in Quantum Gravity II
A semifinite spectral triple for an algebra canonically associated to
canonical quantum gravity is constructed. The algebra is generated by based
loops in a triangulation and its barycentric subdivisions. The underlying space
can be seen as a gauge fixing of the unconstrained state space of Loop Quantum
Gravity. This paper is the second of two papers on the subject.Comment: 43 pages, 1 figur
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