Covariant Information-Density Cutoff in Curved Space-Time

Abstract

In information theory, the link between continuous information and discrete information is established through well-known sampling theorems. Sampling theory explains, for example, how frequency-filtered music signals are reconstructible perfectly from discrete samples. In this Letter, sampling theory is generalized to pseudo-Riemannian manifolds. This provides a new set of mathematical tools for the study of space-time at the Planck scale: theories formulated on a differentiable space-time manifold can be completely equivalent to lattice theories. There is a close connection to generalized uncertainty relations which have appeared in string theory and other studies of quantum gravity.PACS numbers: 04.60.-m, 03.67.-a, 02.90.+p It is generally assumed that the notion of distance loses operational meaning at the Planck scale, l P ≈ 10 −35 m (assuming 3+1 dimensions), due to the combined effects of general relativity and quantum theory. Namely, if one tried to resolve a spatial region with an uncertainty of less than a Planck length, then the corresponding momentum uncertainty should randomly curve and thereby significantly disturb the very region in space that was meant to be resolved. It is expected, therefore, that the existence of a smallest possible length, area or volume, at the Planck scale or above, plays a central role in the yet-to-be-found theory of quantum gravity.In the literature, no consensus has been reached as to whether this implies that space-time is discrete. On the one hand, quantization literally means discretization, and space-time discreteness is indeed naturally accommodated within the functional analytic framework of quantum theory, see, e.g., [1]. Also, most interacting quantum field theories are mathematically well-defined only on lattices. On the other hand, within the mathematical framework of general relativity, space-time is naturally described as a differentiable manifold and deep principles such as local Lorentz invariance would appear to be violated if space-time were discrete.There is the possibility that the cardinality of spacetime is between discrete and continuous, but it is strongly restricted by results of Gödel and Cohen. In [2], they proved that both are consistent with conventional (ZF) set theory: to adopt an axiom claiming the existence of sets with intermediate cardinality or to adopt an axiom claiming their non-existence. Therefore, it is not possible to explicitly construct any set of cardinality between discrete and continuous infinity from the axioms of conventional set theory. If space-time is describable as a set and if this set is of intermediate cardinality, then its description cannot be constructive and requires mathematics beyond conventional set theory.In this Letter, we consider a simpler possibility. In a concrete sense, space-time could be simultaneously discrete and continuous. Namely, in the simplest case, physical fields could be differentiable functions which possess merely a finite density of degrees of freedom. If such a field's amplitude is sam...

Information Theory, Spectral Geometry, and Quantum Gravity

“…Sampling theory is much less developed for the case of bandlimited functions in R n , though Landau [5] established the average spacing which sample points {t n } must have if the sample values {f (t n )} are to allow the stable reconstruction of bandlimited functions f (t) for all t ∈ R n . Of interest for our purposes is of course the generalization of sampling theory for functions on generic non-compact curved spaces, a field that is so far only at its beginning, [6,7]. In this Letter, we find a powerful new tool for developing sampling theory on generic non-compact curved spaces, namely a deep relationship between sampling theory and spectral geometry.…”

“…This UV cutoff could arise in various ways depending on the underlying theory of quantum gravity. For example, the full effective action could contain a power series in the Laplacian with a finite radius of convergence, see [7]. The subspace of covariantly Ω−bandlimited functions on a curved manifold M…”

“…This UV cutoff could arise in various ways depending on the underlying theory of quantum gravity. For example, the full effective action could contain a power series in the Laplacian with a finite radius of convergence, see [7]. The subspace of covariantly Ω−bandlimited functions on a curved manifold M is then defined as B(M, Ω) : While, therefore, it is straightforward to define bandlimitation covariantly, the fundamental problem is to show that these bandlimited functions can be reconstructed from their discrete samples if those samples are taken at a suitable average spacing that is finite.…”

Minimum length cutoff in inflation and uniqueness of the action

“…[12,13,14,15]. In fact, any theory with this type of ultraviolet cutoff can be written, equivalently, as a continuum theory and as a lattice theory, see [16]. While in the continuum formulation the theory displays unbroken external symmetries, the theory's ultraviolet regularity is displayed in its lattice formulation.…”

A fully covariant information-theoretic ultraviolet cutoff for scalar fields in expanding Friedmann Robertson Walker spacetimes

“…This ultraviolet cutoff on fields in flat space is naturally generalizable to scalar fields on curved space. 9 Given an arbitrary curved space, or Riemannian manifold, we can assume its Laplacian to be self-adjoint. If it possesses boundaries, we assume suitable boundary conditions have been chosen.…”