Eigenvalue Density, Li’s Positivity, and the Critical Strip

We rewrite the zero-counting formula within the critical strip of the Riemann zeta function as a cumulative density distribution; this subsequently allows us to formally derive an integral expression for the Li coefficients associated with the Riemann xi-function which, in particular, indicate that their positivity criterion is obeyed, whereby entailing the criticality of the non-trivial zeros. We conjecture the validity of this and related expressions without the need for the Riemann Hypothesis and also offer a physical interpretation of the result and discuss the Hilbert-Polya approach

From Veneziano to Riemann: A string theory statement of the Riemann hypothesis

We discuss a precise relation between the Veneziano amplitude of string theory, rewritten in terms of ratios of the Riemann zeta function, and two elementary criteria for the Riemann hypothesis formulated in terms of integrals of the logarithm and the argument of the zeta function. We also discuss how the integral criterion based on the argument of the Riemann zeta function relates to the Li criterion for the Riemann hypothesis. We provide a new generalization of this integral criterion. Finally, we comment on the physical interpretation of our recasting of the Riemann hypothesis in terms of the Veneziano amplitude

Background Independent Quantum Mechanics and Gravity

We argue that the demand of background independence in a quantum theory of gravity calls for an extension of standard geometric quantum mechanics. We discuss a possible kinematical and dynamical generalization of the latter by way of a quantum covariance of the state space. Specifically, we apply our scheme to the problem of a background independent formulation of Matrix Theory.Comment: 9 pages, LaTe

Deconstruction and Holography

It was recently pointed out that the physics of a single discrete gravitational extra dimension exhibits a peculiar UV/IR connection relating the UV scale to the radius of the effective extra dimension. Here we note that this non-locality is a manifestation of holography, encoding the correct scaling of the number of fundamental degrees of freedom of the UV theory. This in turn relates the Wilsonian RG flow in the UV theory to the effective gravitational dynamics in the extra dimension. The relevant holographic c-function is determined by the expression for the holographic bound. Holography in this context is a result of the requirements of unitarity and diffeomorphism invariance. We comment on the relevance of this observation for the cosmological constant problem.Comment: 11 pages, LaTe

Inflation as a probe of new physics

In this paper we consider inflation as a probe of new physics near the string or Planck scale. We discuss how new physics can be captured by the choice of vacuum, and how this leads to modifications of the primordial spectrum as well as the way in which the universe expands during inflation. Provided there is a large number of fields contributing to the vacuum energy -- as typically is expected in string theory -- we will argue that both types of effects can be present simultaneously and be of observational relevance. Our conclusion is that the ambiguity in choice of vacuum is an interesting new parameter in serious model building.Comment: 14 page

Turbulence and Holography

We examine the interplay between recent advances in quantum gravity and the problem of turbulence. In particular, we argue that in the gravitational context the phenomenon of turbulence is intimately related to the properties of spacetime foam. In this framework we discuss the relation of turbulence and holography and the interpretation of the Kolmogorov scaling in the quantum gravitational setting.Comment: 19 pages, LaTeX; version 2: reference adde

Abelian gerbes as a gauge theory of quantum mechanics on phase space

We construct a U(1) gerbe with a connection over a finite-dimensional, classical phase space P. The connection is given by a triple of forms A,B,H: a potential 1-form A, a Neveu-Schwarz potential 2-form B, and a field-strength 3-form H=dB. All three of them are defined exclusively in terms of elements already present in P, the only external input being Planck's constant h. U(1) gauge transformations acting on the triple A,B,H are also defined, parametrised either by a 0-form or by a 1-form. While H remains gauge invariant in all cases, quantumness vs. classicality appears as a choice of 0-form gauge for the 1-form A. The fact that [H]/2i\pi is an integral class in de Rham cohomology is related with the discretisation of symplectic area on P. This is an equivalent, coordinate-free reexpression of Heisenberg's uncertainty principle. A choice of 1-form gauge for the 2-form B relates our construction with generalised complex structures on classical phase space. Altogether this allows one to interpret the quantum mechanics corresponding to P as an Abelian gauge theory.Comment: 18 pages, 1 figure available from the authors upon reques

Alpha-states in de Sitter space

Field theory in de Sitter space admits a one-parameter family of vacua determined by a superselection parameter alpha. Of these vacua, the Euclidean vacuum uniquely extrapolates to the vacuum of flat Minkowski space. States which resemble the alpha-vacua can be constructed as excitations above the Euclidean vacuum. Such states have modes alpha(k) which decay faster that k^{(1-d)/2}. Fields in such states exhibit non-local correlations when examined from the perspective of fields in the Euclidean vacuum. The dynamics of such entangled states are fully consistent. If an alpha-state with properties that interpolate between an alpha-vacuum and the Euclidean vacuum were the initial condition for inflation, a signature for this may be found in a momentum dependent correction to the inflationary power spectrum. The functional formalism, which provides the tool for examining physics in an alpha-state, extends to fields of other spin. In particular, the extension to spin-2 may proffer a new class of infrared modifications to gravitational interactions. The implications of superselection sectors for the landscape of string vacua are briefly discussed.Comment: 28 pages, LaTe

Aspects of Quantum Gravity in de Sitter Spaces

In these lectures we give a review of recent attempts to understand quantum gravity on de Sitter spaces. In particular, we discuss the holographic correspondence between de Sitter gravity and conformal field theories proposed by Hull and by Strominger, and how this may be reconciled with the finite-dimensional Hilbert space proposal by Banks and Fischler. Furthermore we review the no-go theorems that forbid an embedding of de Sitter spaces in string theory, and discuss how they can be circumvented. Finally, some curious issues concerning the thermal nature of de Sitter space are elucidated.Comment: 36+1 pages, 5 Postscript figures, introduction and section 6 extended, further references, final version to appear in JCA