511 research outputs found
Ultralocal Fields and their Relevance for Reparametrization Invariant Quantum Field Theory
Reparametrization invariant theories have a vanishing Hamiltonian and enforce
their dynamics through a constraint. We specifically choose the Dirac procedure
of quantization before the introduction of constraints. Consequently, for field
theories, and prior to the introduction of any constraints, it is argued that
the original field operator representation should be ultralocal in order to
remain totally unbiased toward those field correlations that will be imposed by
the constraints. It is shown that relativistic free and interacting theories
can be completely recovered starting from ultralocal representations followed
by a careful enforcement of the appropriate constraints. In so doing all
unnecessary features of the original ultralocal representation disappear.
The present discussion is germane to a recent theory of affine quantum
gravity in which ultralocal field representations have been invoked before the
imposition of constraints.Comment: 17 pages, LaTeX, no figure
Weak UCP and perturbed monopole equations
We give a simple proof of weak Unique Continuation Property for perturbed
Dirac operators, using the Carleman inequality. We apply the result to a class
of perturbations of the Seiberg-Witten monopole equations that arise in Floer
theory.Comment: 22 pages LaTeX, one .eps figur
Orthosymplectically invariant functions in superspace
The notion of spherically symmetric superfunctions as functions invariant
under the orthosymplectic group is introduced. This leads to dimensional
reduction theorems for differentiation and integration in superspace. These
spherically symmetric functions can be used to solve orthosymplectically
invariant Schroedinger equations in superspace, such as the (an)harmonic
oscillator or the Kepler problem. Finally the obtained machinery is used to
prove the Funk-Hecke theorem and Bochner's relations in superspace.Comment: J. Math. Phy
Learning Incoherent Subspaces: Classification via Incoherent Dictionary Learning
In this article we present the supervised iterative projections and rotations (s-ipr) algorithm, a method for learning discriminative incoherent subspaces from data. We derive s-ipr as a supervised extension of our previously proposed iterative projections and rotations (ipr) algorithm for incoherent dictionary learning, and we employ it to learn incoherent sub-spaces that model signals belonging to different classes. We test our method as a feature transform for supervised classification, first by visualising transformed features from a synthetic dataset and from the âirisâ dataset, then by using the resulting features in a classification experiment
Classification of Static Charged Black Holes in Higher Dimensions
The uniqueness theorem for static charged higher dimensional black hole
containing an asymptotically flat spacelike hypersurface with compact interior
and with both degenerate and non-degenerate components of event horizon is
proposed. By studies of the near-horizon geometry of degenerate horizons one
was able to eliminate the previous restriction concerning the inequality
fulfilled by the charges of the adequate components of the aforementioned
horizons.Comment: 9 pages, RevTex, to be published in Phys.Rev. D1
The Affine Quantum Gravity Program
The central principle of affine quantum gravity is securing and maintaining
the strict positivity of the matrix \{\hg_{ab}(x)\} composed of the spatial
components of the local metric operator. On spectral grounds, canonical
commutation relations are incompatible with this principle, and they must be
replaced by noncanonical, affine commutation relations. Due to the partial
second-class nature of the quantum gravitational constraints, it is
advantageous to use the recently developed projection operator method, which
treats all quantum constraints on an equal footing. Using this method,
enforcement of regularized versions of the gravitational operator constraints
is formulated quite naturally by means of a novel and relatively well-defined
functional integral involving only the same set of variables that appears in
the usual classical formulation. It is anticipated that skills and insight to
study this formulation can be developed by studying special, reduced-variable
models that still retain some basic characteristics of gravity, specifically a
partial second-class constraint operator structure. Although perturbatively
nonrenormalizable, gravity may possibly be understood nonperturbatively from a
hard-core perspective that has proved valuable for specialized models. Finally,
developing a procedure to pass to the genuine physical Hilbert space involves
several interconnected steps that require careful coordination.Comment: 16 pages, LaTeX, no figure
Surfaces Meeting Porous Sets in Positive Measure
Let n>2 and X be a Banach space of dimension strictly greater than n. We show
there exists a directionally porous set P in X for which the set of C^1
surfaces of dimension n meeting P in positive measure is not meager. If X is
separable this leads to a decomposition of X into a countable union of
directionally porous sets and a set which is null on residually many C^1
surfaces of dimension n. This is of interest in the study of certain classes of
null sets used to investigate differentiability of Lipschitz functions on
Banach spaces
On bulk singularities in the random normal matrix model
We extend the method of rescaled Ward identities of Ameur-Kang-Makarov to
study the distribution of eigenvalues close to a bulk singularity, i.e. a point
in the interior of the droplet where the density of the classical equilibrium
measure vanishes. We prove results to the effect that a certain "dominant part"
of the Taylor expansion determines the microscopic properties near a bulk
singularity. A description of the distribution is given in terms of a special
entire function, which depends on the nature of the singularity (a
Mittag-Leffler function in the case of a rotationally symmetric singularity).Comment: This version clarifies on the proof of Theorem
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