Related papers: Functional Integration Over Geometries

The space-time geometry is considered to be a physical geometry, i.e. a geometry described completely by the world function. All geometrical concepts and geometric objects are taken from the proper Euclidean geometry. They are expressed via…

Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by \[ I(\mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y), \] and set $M(X) =…

The partition function corresponding to the "polytopic" action, a new action for the gravitational interaction which we have proposed recently, is computed in the simplest two-dimensional geometries of genus zero and one. The functional…

In this paper, we introduce and study the Fourier transform of functions which are integrable with respect to a vector measure on a compact group (not necessarily abelian). We also study the Fourier transform of vector measures. We also…

We consider the relational approach to construct gauge-invariant observables in cosmological perturbation theory using synchronous coordinates. We construct dynamical synchronous coordinates as non-local scalar functionals of the metric…

We study invariant measures and thermodynamic formalism for a class of endomorphisms $F_T$ which are only piecewise differentiable on countably many pieces and non-conformal. The endomorphism $F_T$ has parametrized countably generated limit…

Diffeomorphisms and an internal symmetry (e.g., local Lorentz invariance) are typically regarded as the symmetries of any geometrical gravity theory, including general relativity. In the first-order formalism, diffeomorphisms can be thought…

I examine the relationship between $(d+1)$-dimensional Poincar\'e metrics and $d$-dimensional conformal manifolds, from both mathematical and physical perspectives. The results have a bearing on several conceptual issues relating to…

One-dimensional topological gravity is defined as a Gaussian integral as its partition function. The Gaussian integral supplies a toy model as a simpler version of one-matrix model that is well known to provide a description of…

We show that there exists a divergent correlation length in 2d quantum gravity for the matter fields close to the critical point provided one uses the invariant geodesic distance as the measure of distance. The corresponding…

The method of dimensional recurrences proposed by one of the authors [1,2] is applied to the evaluation of the pentagon-type scalar integral with on-shell external legs and massless internal lines. For the first time, an analytic result…

We extend Fourier analysis to curved spaces by defining a Generalized Fourier Transform (GFT) on any Riemannian manifold $\Sigma$ via spectral decomposition. Under minimal requirements that the transform is an isometric isomorphism and has…

A geometric picture of conformally invariant mechanics is presented. Although the standard form of the model is recovered, the careful analysis of global geometry of phase space leads to the conclusion that, in the attractive case, the…

In this paper, we define new functionals generalizing scientometric indices proposed by Mesiar and G\k{a}golewski in 2016 to overcome some limitations of h-index. These functionals are integrals with respect to a monotone measure as well as…

The result of performing integrations over connection type variables in the path integral for the discrete field theory may be poorly defined in the case of non-compact gauge group with the Haar measure exponentially growing in some…

The primary objective of the present paper is to develop the theory of quantization dimension of an invariant measure associated with an iterated function system consisting of finite number of contractive infinitesimal similitudes in a…

This is the second in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars.…

In this paper we consider conformal symmetry in the context of manifolds with general affine connection. We extend the conformal transformation law of the metric to a general metric compatible affine connection, and find that it is a…

We construct a generalization of the theory of symmetric functions involving functions of commuting and anticommuting (Grassmannian) variables. These new functions, called symmetric functions in superspace, are invariant under the diagonal…

In the functional integral approach to quantum gravity, the quantum configurations are usually treated to order hbar through a stationary phase approximation around the saddle point of the action where spacetime is flat. We show that from…