174 research outputs found
Construction of 'Support Vector' Machine Feature Spaces via Deformed Weyl-Heisenberg Algebra
This paper uses deformed coherent states, based on a deformed Weyl-Heisenberg
algebra that unifies the well-known SU(2), Weyl-Heisenberg, and SU(1,1) groups,
through a common parameter. We show that deformed coherent states provide the
theoretical foundation of a meta-kernel function, that is a kernel which in
turn defines kernel functions. Kernel functions drive developments in the field
of machine learning and the meta-kernel function presented in this paper opens
new theoretical avenues for the definition and exploration of kernel functions.
The meta-kernel function applies associated revolution surfaces as feature
spaces identified with non-linear coherent states. An empirical investigation
compares the deformed SU(2) and SU(1,1) kernels derived from the meta-kernel
which shows performance similar to the Radial Basis kernel, and offers new
insights (based on the deformed Weyl-Heisenberg algebra)
Mass-Gaps and Spin Chains for (Super) Membranes
We present a method for computing the non-perturbative mass-gap in the theory
of Bosonic membranes in flat background spacetimes with or without background
fluxes. The computation of mass-gaps is carried out using a matrix
regularization of the membrane Hamiltonians. The mass gap is shown to be
naturally organized as an expansion in a 'hidden' parameter, which turns out to
be : d being the related to the dimensionality of the background
space. We then proceed to develop a large perturbation theory for the
membrane/matrix-model Hamiltonians around the quantum/mass corrected effective
potential. The same parameter that controls the perturbation theory for the
mass gap is also shown to control the Hamiltonian perturbation theory around
the effective potential. The large perturbation theory is then translated
into the language of quantum spin chains and the one loop spectra of various
Bosonic matrix models are computed by applying the Bethe ansatz to the one-loop
effective Hamiltonians for membranes in flat space times. Apart from membranes
in flat spacetimes, the recently proposed matrix models (hep-th/0607005) for
non-critical membranes in plane wave type spacetimes are also analyzed within
the paradigm of quantum spin chains and the Bosonic sectors of all the models
proposed in (hep-th/0607005) are diagonalized at the one-loop level.Comment: 36 Page
Curvature decomposition of G_2 manifolds
Explicit formulas for the -components of the Riemannian curvature tensor
on a manifold with a structure are given in terms of Ricci contractions.
We define a conformally invariant Ricci-type tensor that determines the
27-dimensional part of the Weyl tensor and show that its vanishing on compact
manifold with closed fundamental form forces the three-form to be
parallel. A topological obstruction for the existence of a structure with
closed fundamental form is obtained in terms of the integral norms of the
curvature components. We produce integral inequalities for closed
manifold and investigate limiting cases. We make a study of warped products and
cohomogeneity-one manifolds. As a consequence every Fern\'andez-Gray type
of structure whose scalar curvature vanishes may be realized such that
the metric has holonomy contained in .Comment: LaTeX 2e, 26 pages, 2 tables. Changes in version 2: shortened,
reorganized, misprints corrected, several remarks and new introduction. A
formula in the proof of Theorem 1.2a has been corrected. Submitte
A review of quantum cellular automata
Discretizing spacetime is often a natural step towards modelling physical systems. For quantum systems, if we also demand a strict bound on the speed of information propagation, we get quantum cellular automata (QCAs). These originally arose as an alternative paradigm for quantum computation, though more recently they have found application in understanding topological phases of matter and have been proposed as models of periodically driven (Floquet) quantum systems, where QCA methods were used to classify their phases. QCAs have also been used as a natural discretization of quantum field theory, and some interesting examples of QCAs have been introduced that become interacting quantum field theories in the continuum limit. This review discusses all of these applications, as well as some other interesting results on the structure of quantum cellular automata, including the tensor-network unitary approach, the index theory and higher dimensional classifications of QCAs. © 2020 Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften. All rights reserved
Time and a Temporally Statistical Quantum Geometrodynamics
This paper is an exposition of the author's recent work (arXiv:1001.3382
[hep-th], arXiv:1005.5430 [cond-mat.dis-nn], arXiv:1212.4956 [quant-ph]) on
modeling M-theory vacua and quantum mechanical observers in the framework of a
temporally statistical description of quantum geometrodynamics, including
measurement processes based on the canonical theory of quantum gravity. In this
paper we deal with several fundamental issues of time: the time-less problem in
canonical quantum gravity; the physical origin of state reductions; and
time-reversal symmetry breaking. We first model the observers and consider the
time-less problem by invoking the time reparametrization symmetry breaking in
the quantum mechanical world as seen by the observers. We next construct the
hidden time variable theory, using a model of the gauged and affinized
S-duality symmetry in type IIB string theory, as the statistical theory of time
and explain the physical origin of state reductions using it. Finally, by the
extension of the time reparametrization symmetry to all of the temporal hidden
variables, we treat the issue of time reversal symmetry breaking as the
spontaneous breaking of this extended time reparametrization symmetry. The
classification of unitary time-dependent processes and the geometrizations of
unitary and non-unitary time evolutions using the language of the derived
category are also investigated.Comment: Review, 89 pages, version to be published, v
Grassmannian Integrals in Minkowski Signature, Amplitudes, and Integrability
We attempt to systematically derive tree-level scattering amplitudes in
four-dimensional, planar, maximally supersymmetric Yang-Mills theory from
integrability. We first review the connections between integrable spin chains,
Yangian invariance, and the construction of such invariants in terms of
Grassmannian contour integrals. Building upon these results, we equip a class
of Grassmannian integrals for general symmetry algebras with unitary
integration contours. These contours emerge naturally by paying special
attention to the proper reality conditions of the algebras. Specializing to
psu(2,2|4) and thus to maximal superconformal symmetry in Minkowski space, we
find in a number of examples expressions similar to, but subtly different from
the perturbative physical scattering amplitudes. Our results suggest a subtle
breaking of Yangian invariance for the latter, with curious implications for
their construction from integrability.Comment: 44 pages, 2 figures; v2: published version, minor change
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