MG15 - Talk detail

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Existence of a lower bound, say L, to spacetime intervals arises very generically when one attempts to combine the basic principles of quantum theory and general relativity. I argue that incorporating such a length scale in a Lorentz covariant manner requires a description of spacetime geometry in terms of non-local bi-tensors rather than the conventional metric tensor. I show how such a description can be achieved by reconstructing the spacetime using the Synge world function and the van Vleck determinant. Using these, I demonstrate that the same non-analytic structure of the reconstructed spacetime which renders a perturbative expansion in L meaningless, will also generically leave a non-trivial “relic” in the limit L → 0. I present specific results where such a relic term is manifest, and discuss several implications of the same. Specifically, I will discuss how these results: (i) suggest a different description of gravitational dynamics from the conventional one based on the Einstein-Hilbert lagrangian, (ii) imply a dimensional reduction to D = 2 at small scales, (iii) support the idea that the cosmological constant Λ might be a non-local vestige of the small scale structure of spacetime. I will conclude by discussing the ramifications of these ideas in the context of quantum gravity, and what it implies for the (semi-)classical limit of any quantum gravity framework. The talk will be essentially based on work that has been systematically developed in: 1307.5618, 1405.4967, 1408.3963, 1503.03793, 1507.05669.

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I present a new framework for euclidean quantum gravity that replaces the conventional procedure of Wick rotation by a covariant, observer dependent class of metrics. Given a Lorentzian spacetime (M,g) and a non-vanishing −1 −1 timelike vector field u(λ) with level surfaces Σ, one can construct on M a class of metrics q = g − Θ(λ) u ⊗ u with an arbitrary function Θ that interpolates between the Euclidean (Θ = −2) and Lorentzian (Θ = 0) regimes, separated by the codimension one hypersurface Θ = −1. I will show that the curvature tensor and Einstein-Hilbert action associated with q and show that these have a rich mathematical structure: (i) The curvature acquires additional terms arise in the Euclidean regime Θ → −2 of q, thereby yielding a euclideanisation that is distinct from Wick rotation t → it. (ii) For the simplest choice of a step-profile for Θ, the Ricci scalar Ric[q] of q reduces to the complete Einstein-Hilbert lagrangian with the correct Gibbons-Hawking-York boundary term; the latter arises as a delta-function of strength 2K supported on Θ = −1. I discuss the several implications of the results for Euclidean quantum gravity, quantum cosmology, and small scale structure of spacetime. The talk will be essentially based on results presented in: arXiv: 1705.02504.

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The conventional formulation of Euclidean Quantum Gravity is based on Wick rotation, a procedure that is ill-defined and ambiguous. I give a local description of the Euclidean regime (M,g,u) of Lorentzian spacetimes (M,g) based on timelike geodesics u passing through an arbitrary event p0 ∈ M. I show that, to leading order, the Euclidean Einstein-Hilbert action S is proportional to the Einstein tensor G[g](u, u). The positivity of S follows if G[g](u,u) > 0 holds. I suggest an interpretation of this result in terms of the amplitude A[Σ0] = exp[−S] for a single space-like hypersurface Σ0 ∈ I+(p0) to emerge at a constant geodesic distance λ0 from p0. Implications for classical and quantum gravity are discussed. I discuss several implications of the results for classical and quantum gravity, as well as for the small scale structure of spacetime. In particular, the relevance of the result for the emergence of Einstein equations in the classical limit, through the euclidean action, is highlighted. The talk will be essentially based on the results presented in: arXiv: 1802.07055.

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