Small-scale Structure Of Spacetime Research Articles

Relics of the quantum spacetime: from Synge’s world function as the fundamental probe of spacetime architecture to the emergent description of gravity

All our observations that characterise space and time are expressed in terms of non-local, bi-tensorial objects such as geodesic intervals between events and two-point (Green) functions. In this contribution, I highlight the importance of characterising spacetime geome-try in terms of such non-local objects, focusing particularly on two important bi-tensors that play a particular fundamental role – Synge’s World function and the van Vleck determinant. I will first discuss how these bi-tensors help capture information about spacetime geometry, and then describe their role in characterising quantum spacetime endowed with a lower bound, say ℓ 0, on spacetime intervals. Incorporating such a length scale in a Lorentz covariant manner necessitates a description of spacetime geometry in terms of above bi-tensors, and naturally replaces the conventional description based on the metric tensor gab (x) with a description in terms of a non-local bi-tensor qab (x, y). The non-analytic structure of qab (x, y) which renders a perturbative expansion in ℓ 0 meaningless, also generically leaves a non-trivial “relic” in the limit ℓ 0 → 0. I present some results where such a relic term is manifest; specifically, I will discuss how this: (i) suggests a description of gravitational dynamics different from the one based on Einstein-Hilbert lagrangian, (ii) implies dimensional reduction to 2 at small scales, (iii) connects with the notion of cosmological constant itself being a non-local vestige of the small scale structure of spacetime, (iv) helps address the issues of spacetime singularities. I will conclude by discussing the ramifications of these ideas for quantum gravity.

Kinematics of arbitrary spin matter fields in loop quantum gravity

Loop quantum gravity envisions a small scale structure of spacetime that is markedly different from that of the classical spacetime continuum. This has ramifications for the excitation of matter fields and for their coupling to gravity. There is a general understanding of how to formulate scalar fields, spin $\frac{1}{2}$ fields, and gauge fields in the framework of loop quantum gravity. The goal of the present work is to investigate kinematical aspects of this coupling. We will study implications of the Gau\ss{} and diffeomorphism constraints for the quantum theory: We define and study a less ambiguous variant of the Baez-Krasnov path observables, and we investigate symmetry properties of spin network states imposed by diffeomorphism group averaging. We will do this in a setting which allows for matter excitations of spin $\frac{1}{2}$ and higher. In the case of spin $\frac{1}{2}$, we will also discuss extensions of it by introducing an electromagnetic field and antiparticles. We finally discuss how far the picture with matter excitations of higher spin can be obtained from classical actions for higher spin fields.

Small-scale structure of spacetime and its implications

If there exists a lower bound [Formula: see text] to spacetime intervals which is Lorentz-invariant, then the effective description of spacetime that incorporates such a lower bound must necessarily be nonlocal. Such a nonlocal description can be derived using standard tools of differential geometry, but using as basic variables certain bi-tensors instead of the conventional metric tensor [Formula: see text]. This allows one to construct a qmetric [Formula: see text], using the Synge’s world function [Formula: see text] and the van Vleck determinant [Formula: see text], that incorporates the lower bound on spacetime intervals. The same nonanalytic structure of the reconstructed spacetime which renders a perturbative expansion in [Formula: see text] meaningless, will then also generically leave a non-trivial “relic” in the limit [Formula: see text]. We present specific results derived from [Formula: see text] where such a relic term manifests, and discuss several implications of the same. Specifically, we will discuss how these results: (i) suggest a description of gravitational dynamics different from the conventional one based on the Einstein–Hilbert Lagrangian, (ii) imply a dimensional reduction to [Formula: see text] at small scales and (iii) can be significant for the idea that the cosmological constant itself might be related to some nonlocal vestige of the small-scale structure of spacetime. We will conclude by discussing the ramifications of these ideas in the context of quantum gravity.

Small scale structure of spacetime: The van Vleck determinant and equigeodesic surfaces

It has recently been argued that if spacetime $\mathcal M$ possesses non-trivial structure at small scales, an appropriate semi-classical description of it should be based on non-local bi-tensors instead of local tensors such as the metric $g_{ab}$. Two most relevant bi-tensors in this context are Synge's World function $\Omega(p,p_0)$ and the van Vleck determinant (VVD) $\Delta(p,p_0)$, as they encode the metric properties of spacetime and (de)focussing behaviour of geodesics. They also characterize the leading short distance behavior of two point functions of the d'Alembartian $_{p_0} \square_p$. We begin by discussing the intrinsic and extrinsic geometry of equi-geodesic surfaces $\Sigma_{G,p_0}$ defined by $\Omega(p,p_0)=constant$ in a geodesically convex neighbourhood of an event $p_0$, and highlight some elementary identities relating the VVD with geometry of these surfaces. As an aside, we also comment on the contribution of $\Sigma_{G,p_0}$ to the surface term in the Einstein-Hilbert (EH) action and show that it can be written as a volume integral of $\square \ln \Delta$. We then study the small scale structure of spacetime in presence of a Lorentz invariant short distance cut-off $\ell_0$ using $\Omega$ and $\Delta$, based on some recently developed ideas. We derive a 2nd rank bi-tensor $q_{ab}$ which naturally yields geodesic intervals bounded from below, and present a general and mathematically rigorous analysis of short distance structure of spacetime based on (a) geometry of $\Sigma_{G,p_0}$, (b) structure of the non local d'Alembartian associated with $q_{ab}$, and (c) properties of the VVD. In particular, we show that the Ricci bi-scalar of $q_{ab}$ is completely determined by $\Sigma_{G,p_0}$, the tidal tensor, and first two derivatives of the van Vleck determinant, and has a non-trivial "classical" limit given by (constant) $R_{ab} q^a q^b$ (see text).

Small scale structure of spacetime in presence of a minimal length

I present some recent results which yields some specific consequences of having a minimal spacetime length L0 while maintaining Lorentz covariance. In particular: (i) the resultant description of geometry must be in terms of a non-local bi-tensor qab(P,p, L20) instead of the conventional metric tensor gab(p), (ii) qab and gab are related disformally through the Synge world function, (iii) the resultant non-local d'Alembartian yields Green's functions with a natural covariant short distance cut-off, (iv) the Ricci bi-scalar R(P,p, L20) corresponding to qab yields a local scalar S(P, L20) = R(P, P, L20) in the coincidence such thatwhere ta are arbitrary normalized vectors representing gauge degrees of freedom at each spacetime event; this limit turns out to be independent of the precise details of the short distance modification, (v) the Gibbons-Hawking-York surface term for qab yields a residual entropy for each spacetimeevent

Quantum fluctuations and the small scale structure of spacetime

We show that vacuum fluctuations of the stress-energy tensor in two-dimensional dilaton gravity lead to a sharp focusing of light cones near the Planck scale, effectively breaking space up into a large number of causally disconnected regions. This phenomenon, called “asymptotic silence” when it occurs in cosmology, might help explain several puzzling features of quantum gravity, including evidence of spontaneous dimensional reduction at short distances. While our analysis focuses on a simplified two-dimensional model, we argue that the qualitative features should still be present in four dimensions.

Minimal length and small scale structure of spacetime

Many generic arguments support the existence of a minimum spacetime interval $L_0$. Such a "zero-point" length can be naturally introduced in a locally Lorentz invariant manner via Synge's world function bi-scalar $\Omega(p,P)$ which measures squared geodesic interval between spacetime events $p$ and $P$. I show that there exists a \emph{non-local} deformation of spacetime geometry given by a \emph{disformal} coupling of metric to the bi-scalar $\Omega(p,P)$, which yields a geodesic interval of $L_0$ in the limit $p \rightarrow P$. Locality is recovered when $\Omega(p,P) >> L_0^2/2$. I discuss several conceptual implications of the resultant small-scale structure of spacetime for QFT propagators as well as spacetime singularities.

Vacuum Fluctuations and the Small Scale Structure of Spacetime

We show that vacuum fluctuations of the stress-energy tensor in two-dimensional dilaton gravity lead to a sharp focusing of light cones near the Planck scale, effectively breaking space up into a large number of causally disconnected regions. This phenomenon, called "asymptotic silence" when it occurs in cosmology, might help explain several puzzling features of quantum gravity, including evidence of spontaneous dimensional reduction at short distances. While our analysis focuses on a simplified two-dimensional model, we argue that the qualitative features should still be present in four dimensions.

Spacetime foam and high-energy photons

It is shown that high-energy astrophysics can provide information on the small-scale structure of spacetime.

Discrete quantum gravity and causal sets

This paper provides a thorough introduction to the physical and conceptual need for a theory of quantum gravity; some knowledge of general relativity and nonrelativistic quantum mechanics is assumed. A theory of quantum gravity would have wide-ranging implications for high-energy physics, astrophysics, and cosmology. The paper goes on to describe an important approach to quantum gravity that is not well known outside of the quantum gravity research community — causal sets. The causal-set approach falls within the framework of discrete quantum gravity, which considers the possibility that the small-scale structure of spacetime might be discrete rather than continuous. Herein, I elucidate the arguments for why a discrete causal structure might be appropriate for a theory of quantum gravity. The logical and formal development of a causal-set theory as well as a few illuminating examples are also provided. PACS Nos.: 04.60-m, 04.60Nc

Gravitational instantons and quantisation of the cosmological constant

Complex gravitational instantons, introduced recently into quantum cosmology, may be important in the small scale structure of spacetime. If the cosmological constant is negative, then there is a complex instanton which breaks the electromagnetic gauge symmetry unless the cosmological constant is quantised, Λ= −3e 2 N 2 , where e is the unit of charge and N is a natural number.

Small-scale structure of spacetime as the origin of the gravitational constant

We suggest a means of incorporating the Planck length as a fundamental constant determined by the structure of spacetime. In this scheme the spacetime symmetry group is taken as the de Sitter group with radius of curvature proportional to the Planck length, but we argue that the effects of this curvature are not apparent at elementary-particle length scales. To make the connection with gravity we present a formulation of the gravitational interaction as the gauge theory of the de Sitter group. We obtain an action containing the Einstein-Cartan action, but in which the dimensional gravitational constant $G$ appears naturally as the consequence of the commutation relation of the de Sitter group. We also find a cosmological term, higherderivative couplings of the gravitational field, and a propagating torsion field, which we discuss.