Light-Ray Radon Transform for Abelianin and Nonabelian Connection in 3 and 4 Dimensional Space with Minkowsky Metric

We consider a real manifold of dimension 3 or 4 with Minkovsky metric, and with a connection for a trivial GL(n,C) bundle over that manifold. To each light ray on the manifold we assign the data of paralel transport along that light ray. It turns out that these data are not enough to reconstruct the connection, but we can add more data, which depend now not from lines but from 2-planes, and which in some sence are the data of parallel transport in the complex light-like directions, then we can reconstruct the connection up to a gauge transformation. There are some interesting applications of the construction: 1) in 4 dimensions, the self-dual Yang Mills equations can be written as the zero curvature condition for a pair of certain first order differential operators; one of the operators in the pair is the covariant derivative in complex light-like direction we studied. 2) there is a relation of this Radon transform with the supersymmetry. 3)using our Radon transform, we can get a measure on the space of 2 dimensional planes in 4 dimensional real space. Any such measure give rise to a Crofton 2-density. The integrals of this 2-density over surfaces in R^4 give rise to the Lagrangian for maps of real surfaces into R^4, and therefore to some string theory. 4) there are relations with the representation theory. In particular, a closely related transform in 3 dimensions can be used to get the Plancerel formula for representations of SL(2,R).Comment: We add an important discussion part, establishing the relation of our Radon transform with the self-dual Yang-Mills, string theory, and the represntation theory of the group SL(2,R

Fuzzy de Sitter Space from kappa-Minkowski Space in Matrix Basis

We consider the Lie group $\mathbb{R}^D_\kappa$ generated by the Lie algebra of $\kappa$-Minkowski space. Imposing the invariance of the metric under the pull-back of diffeomorphisms induced by right translations in the group, we show that a unique right invariant metric is associated with $\mathbb{R}^D_\kappa$. This metric coincides with the metric of de Sitter space-time. We analyze the structure of unitary representations of the group $\mathbb{R}^D_\kappa$ relevant for the realization of the non-commutative $\kappa$-Minkowski space by embedding into $(2D-1)$-dimensional Heisenberg algebra. Using a suitable set of generalized coherent states, we select the particular Hilbert space and realize the non-commutative $\kappa$-Minkowski space as an algebra of the Hilbert-Schmidt operators. We define dequantization map and fuzzy variant of the Laplace-Beltrami operator such that dequantization map relates fuzzy eigenvectors with the eigenfunctions of the Laplace-Beltrami operator on the half of de Sitter space-time.Comment: 21 pages, v3 differs from version published in Fortschritte der Physik by a note and references added and adjuste

Amplituhedron meets Jeffrey-Kirwan Residue

The tree amplituhedra A^(m)_n,k are mathematical objects generalising the notion of polytopes into the Grassmannian. Proposed for m=4 as a geometric construction encoding tree-level scattering amplitudes in planar N=4 super Yang-Mills theory, they are mathematically interesting for any m. In this paper we strengthen the relation between scattering amplitudes and geometry by linking the amplituhedron to the Jeffrey-Kirwan residue, a powerful concept in symplectic and algebraic geometry. We focus on a particular class of amplituhedra in any dimension, namely cyclic polytopes, and their even-dimensional conjugates. We show how the Jeffrey-Kirwan residue prescription allows to extract the correct amplituhedron volume functions in all these cases. Notably, this also naturally exposes the rich combinatorial and geometric structures of amplituhedra, such as their regular triangulations.Peer reviewedFinal Accepted Versio

Dynamical Structure Factors for Dimerized Spin Systems

We discuss the transition strength between the disordered ground state and the basic low-lying triplet excitation for interacting dimer materials by presenting theoretical calculations and series expansions as well as inelastic neutron scattering results for the material KCuCl_3. We describe in detail the features resulting from the presence of two differently oriented dimers per unit cell and show how energies and spectral weights of the resulting two modes are related to each other. We present results from the perturbation expansion in the interdimer interaction strength and thus demonstrate that the wave vector dependence of the simple dimer approximation is modified in higher orders. Explicit results are given in 10th order for dimers coupled in 1D, and in 2nd order for dimers coupled in 3D with application to KCuCl_3 and TlCuCl_3.Comment: 17 pages, 6 figures, part 2 is based on cond-mat/021133

Convergent expansions for properties of the Heisenberg model for CaV4_4O9_9

We have carried out a wide range of calculations for the $S=1/2$ Heisenberg model with nearest- and second-neighbor interactions on a two-dimensional lattice which describes the geometry of the vanadium ions in the spin-gap system CaV$_4$O$_9$. The methods used were convergent high-order perturbation expansions (``Ising'' and ``Plaquette'' expansions at $T=0$, as well as high-temperature expansions) for quantities such as the uniform susceptibility, sublattice magnetization, and triplet elementary excitation spectrum. Comparison with the data for CaV$_4$O$_9$ indicates that its magnetic properties are well described by nearest-neighbor exchange of about 200K in conjunction with second-neighbor exchange of about 100K.Comment: Uses REVTEX macros. Four pages in two-column format, five postscript figures. Files packaged using uufile

Ground State and Elementary Excitations of the S=1 Kagome Heisenberg Antiferromagnet

Low energy spectrum of the S=1 kagom\'e Heisenberg antiferromagnet (KHAF) is studied by means of exact diagonalization and the cluster expansion. The magnitude of the energy gap of the magnetic excitation is consistent with the recent experimental observation for \mpynn. In contrast to the $S=1/2$ KHAF, the non-magnetic excitations have finite energy gap comparable to the magnetic excitation. As a physical picture of the ground state, the hexagon singlet solid state is proposed and verified by variational analysis.Comment: 5 pages, 7 eps figures, 2 tables, Fig. 4 correcte

On the relation between p-adic and ordinary strings

The amplitudes for the tree-level scattering of the open string tachyons, generalised to the field of p-adic numbers, define the p-adic string theory. There is empirical evidence of its relation to the ordinary string theory in the p_to_1 limit. We revisit this limit from a worldsheet perspective and argue that it is naturally thought of as a continuum limit in the sense of the renormalisation group.Comment: 13 pages harvmac (b), 2 eps figures; v2: revtex, shortened, published versio

Tree-level scattering amplitudes from the amplituhedron

7 pages, 2 figures, to be published in the Journal of Physics: Conference Series. Proceedings for the "7th Young Researcher Meeting", Torino, 2016A central problem in quantum field theory is the computation of scattering amplitudes. However, traditional methods are impractical to calculate high order phenomenologically relevant observables. Building on a few decades of astonishing progress in developing non-standard computational techniques, it has been recently conjectured that amplitudes in planar N=4 super Yang-Mills are given by the volume of the (dual) amplituhedron. After providing an introduction to the subject at tree-level, we discuss a special class of differential equations obeyed by the corresponding volume forms. In particular, we show how they fix completely the amplituhedron volume for next-to-maximally helicity violating scattering amplitudes.Peer reviewe

Quantum field theory with a fundamental length: A general mathematical framework

We review and develop a mathematical framework for nonlocal quantum field theory (QFT) with a fundamental length. As an instructive example, we reexamine the normal ordered Gaussian function of a free field and find the primitive analyticity domain of its n-point vacuum expectation values. This domain is smaller than the usual future tube of local QFT, but we prove that in difference variables, it has the same structure of a tube whose base is the (n-1)-fold product of a Lorentz invariant region. It follows that this model satisfies Wightman-type axioms with an exponential high-energy bound which does not depend on n, contrary to the claims in the literature. In our setting, the Wightman generalized functions are defined on test functions analytic in the complex l-neighborhood of the real space, where l is an n-independent constant playing the role of a fundamental length, and the causality condition is formulated with the use of an analogous function space associated with the light cone. In contrast to the scheme proposed by Bruning and Nagamachi [J. Math. Phys. 45 (2004) 2199] in terms of ultra-hyperfunctions, the presented theory obviously becomes local as l tends to zero.Comment: 25 pages, v2: updated to match J. Math. Phys. versio