5,832 research outputs found
Unimodular graphs and Eisenstein sums
Motivated in part by combinatorial applications to certain sum-product
phenomena, we introduce unimodular graphs over finite fields and, more
generally, over finite valuation rings. We compute the spectrum of the
unimodular graphs, by using Eisenstein sums associated to unramified extensions
of such rings. We derive an estimate for the number of solutions to the
restricted dot product equation over a finite valuation ring.
Furthermore, our spectral analysis leads to the exact value of the
isoperimetric constant for half of the unimodular graphs. We also compute the
spectrum of Platonic graphs over finite valuation rings, and products of such
rings - e.g., . In particular, we deduce an improved lower
bound for the isoperimetric constant of the Platonic graph over
.Comment: V2: minor revisions. To appear in the Journal of Algebraic
Combinatoric
The Quantum Configuration Space of Loop Quantum Cosmology
The article gives an account of several aspects of the space known as the
Bohr compactification of the line, featuring as the quantum configuration space
in loop quantum cosmology, as well as of the corresponding configuration space
realization of the so-called polymer representation. Analogies with loop
quantum gravity are explored, providing an introduction to (part of) the
mathematical structure of loop quantum gravity, in a technically simpler
context.Comment: 14 pages. Minor changes, typos corrected, 1 reference added. To
appear in Class. Quantum Gra
Projective Techniques and Functional Integration
A general framework for integration over certain infinite dimensional spaces
is first developed using projective limits of a projective family of compact
Hausdorff spaces. The procedure is then applied to gauge theories to carry out
integration over the non-linear, infinite dimensional spaces of connections
modulo gauge transformations. This method of evaluating functional integrals
can be used either in the Euclidean path integral approach or the Lorentzian
canonical approach. A number of measures discussed are diffeomorphism invariant
and therefore of interest to (the connection dynamics version of) quantum
general relativity. The account is pedagogical; in particular prior knowledge
of projective techniques is not assumed. (For the special JMP issue on
Functional Integration, edited by C. DeWitt-Morette.)Comment: 36 pages, latex, no figures, Preprint CGPG/94/10-
Non-commutative flux representation for loop quantum gravity
The Hilbert space of loop quantum gravity is usually described in terms of
cylindrical functionals of the gauge connection, the electric fluxes acting as
non-commuting derivation operators. It has long been believed that this
non-commutativity prevents a dual flux (or triad) representation of loop
quantum gravity to exist. We show here, instead, that such a representation can
be explicitly defined, by means of a non-commutative Fourier transform defined
on the loop gravity state space. In this dual representation, flux operators
act by *-multiplication and holonomy operators act by translation. We describe
the gauge invariant dual states and discuss their geometrical meaning. Finally,
we apply the construction to the simpler case of a U(1) gauge group and compare
the resulting flux representation with the triad representation used in loop
quantum cosmology.Comment: 12 pages, matches published versio
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