Geometry of the central limit theorem in the nonextensive case

We uncover geometric aspects that underlie the sum of two independent stochastic variables when both are governed by q-Gaussian probability distributions. The pertinent discussion is given in terms of random vectors uniformly distributed on a p-sphere.     

Geometry of the quantum universe

A quantum universe with the global shape of a (Euclidean) de Sitter spacetime appears as dynamically generated background geometry in the causal dynamical triangulation (CDT) regularisation of quantum gravity. We investigate the micro- and macro-geometry of this universe, using geodesic shell decompositions of spacetime. More specifically, we focus on evidence of fractality and global anisotropy, and on how they depend on the bare coupling constants of the theory.     

Geometry of the string equations

The string equations of hermitian and unitary matrix models of 2D gravity are flatness conditions. These flatness conditions may be interpreted as the consistency conditions for isomonodromic deformation of an equation with an irregular singularity. In particular, the partition function of the matrix model is shown to be the tau function for isomonodromic deformation. The physical parameters defining the string equation are interpreted as moduli of meromorphic gauge fields, and the compatibility conditions can be interpreted as defining a quantum analog of a Riemann surface. In the latter interpretation, the equations may be viewed as compatibility conditions for transport on quantum moduli space of correlation functions in a theory of free fermions. We discuss how the free fermion field theory may be deduced directly from the matrix model integral. As an application of our formalism we discuss some properties of the BMP solutions of the string equations. We also mention briefly a possible connection to twistor theory.     

Geometry of the de-Sitter universe

By making use of the fact that the de-Sitter metric corresponds to a hyperquadric in a five-dimensional flat space, it is shown that the three Robertson-Walker metrics for empty spacetime and positive cosmological constant, corresponding to 3-space of positive, negative and zero curvative, are geometrically equivalent. The 3-spaces correspond to intersections of the hyperquadric by hyperplanes, and the time-like geodesics perpendicular to them correspond to intersections by planes, in all three cases.     

Brillouin's theorem and the Hellmann-Feynman theorem for Hartree-Fock wave functions

It is shown that there is an intimate relation between the Hellmann-Feynman theorem and Brillouin's theorem. A more general form of Brillouin's theorem is provided, which applies to excited states of arbitrary symmetry and multiplicity. This new form leads to a simple proof of the Hellmann-Feynman theorem. This theorem is valid when all the orbitals that occur in the wave function are determined by a complete, and not a partial, variational procedure. Arguing in the opposite direction it is shown that the complete satisfaction of the generalized Brillouin's theorem provides an alternative scheme for obtaining the Hartree-Fock orbitals.     

Converse of the Eilers-Horst theorem

We generalize the theorem of Eilers and Horst, showing that any finite as well as any-finite measure on a quantum logic of all closed subspaces of a Hilbert spaceH of dimension 2 is a Gleason one iff the dimension ofH is a nonmeasurable cardinal.     

Scattering the Geometry of Weighted Graphs

Given two weighted graphs (X, b_k, m_k), k =?1,2 with b₁ ～ b₂ and m₁ ～ m₂, we prove a weighted L¹-criterion for the existence and completeness of the wave operators W_±(H₂, H₁, I_1,2), where H_k denotes the natural Laplacian in ?²(X, m_k) w.r.t. (X, b_k, m_k) and I_1,2 the trivial identification of ?²(X, m₁) with ?²(X, m₂). In particular, this entails a general criterion for the absolutely continuous spectra of H₁ and H₂ to be equal.     

Geometry of the electron local observables

We show that the complete set of the second power identities for the electron local observables consists of 36 equations. The identities connect the products of the electron bilinear forms and, being considered as geometrically meaningful equations in 3D Euclidean space, are separated over the groups of equations for the scalar, vector and tensor quantities. Considering the complete set of identities as a set of the second power equations, we solve the equations and find the irreducible representation for the electron local observables. The representation defines the 16 electron local observables as functions of 7 basic parameters and can be formulated in 6 various forms. The basic parameters include scalar and pseudoscalar, the time components of a 4-vector and a 4-pseudovector, and three Euler angles which define the angular position of a local 3D frame with respect to the 3D laboratory frame. These 7 parameters completely define the space components of the 4-vector and the 4-pseudovector, as well as the polar and axial vectors. The developed representation shows that the analysis of the any electron wave packet can be considerably simplified by the reduction of the number of analyzed real functions from 16 to 7. As an example, we present the structure of the local observables defined by the irreducible representation in a case of a traveling electron wave.     

Extension of the fluctuation theorem

Heat fluctuations are studied in a dissipative system with both deterministic and stochastic components for a simple model: a Brownian particle dragged through water by a moving potential. An extension of the stationary state fluctuation theorem is derived. For infinite time, this reduces to the conventional fluctuation theorem only for small fluctuations; for large fluctuations, it gives a much larger ratio of the probabilities of the particle to absorb rather than supply heat. This persists for finite times and should be observable in experiments similar to a recent one carried out by Wang et al.     

The Geometry of the Dirac Equation

We discuss the problem of the derivation and the interpretation of metric tensors and generalized equations of motion for test particles from quasilinear spinor equations.     

Generalization of the Hellmann-Feynman theorem

The well-known Hellmann-Feynman theorem of quantum mechanics connected with the derivative of the eigenvalues with respect to a parameter upon which the Hamiltonian depends, is generalized to include cases in which the domain of definition of the Hamiltonian of the system also depends on that parameter.     

Geometry of Skyrmions

A Skyrmion may be regarded as a topologically non-trivial map from one Riemannian manifold to another, minimizing a particular energy functional. We discuss the geometrical interpretation of this energy functional and give examples of Skyrmions on various manifolds. We show how the existence of conformal transformations can cause a Skyrmion on a 3-sphere to become unstable, and how this may be related to chiral symmetry breaking.     

Causal Geometry

Information geometry has offered a way to formally study the efficacy of scientific models by quantifying the impact of model parameters on the predicted effects. However, there has been little formal investigation of causation in this framework, despite causal models being a fundamental part of science and explanation. Here, we introduce causal geometry, which formalizes not only how outcomes are impacted by parameters, but also how the parameters of a model can be intervened upon. Therefore, we introduce a geometric version of “effective information”—a known measure of the informativeness of a causal relationship. We show that it is given by the matching between the space of effects and the space of interventions, in the form of their geometric congruence. Therefore, given a fixed intervention capability, an effective causal model is one that is well matched to those interventions. This is a consequence of “causal emergence,” wherein macroscopic causal relationships may carry more information than “fundamental” microscopic ones. We thus argue that a coarse-grained model may, paradoxically, be more informative than the microscopic one, especially when it better matches the scale of accessible interventions—as we illustrate on toy examples.     

Evading the Goldstone theorem

Peter W. Higgs describes in detail his interest from 1960 in the work of Nambu and Goldstone and the background events leading to the papers of 1964 that showed how one can evade the Goldstone theorem using a gauge theory. He also discusses how these ideas have been developed by others to culminate in the electroweak theory of the Standard Model. 1     

Geometry of torsion

We show that the metric of internal space has an interpretation as a potential for the torsion in a vector bundle (E, M, A); the relations among the spin, the local metric of internal space, and the torsion are briefly discussed.     

Geometry anomalies

When a global symmetry G breaks down to some subgroup H, the low-energy phenomenological lagrangian is given by a σ-model defined on the homogeneous space G/H. We show that when chiral fermions are coupled to this σ-model, there are in general anomalies in the Ward identities which express the non-conservation of the G-currents, thus breaking the standard current algebra predictions based on G/H. We discuss the anomaly cancellation conditions in detail. We then consider the extension of these results to the possible presence of anomalies under arbitrary reparametrizations of the Goldstone boson manifold. We relate these to anomalies in the holonomy group of the manifold and again exhibit the conditions under which these anomalies are cancelled.