It has recently been argued that if spacetime 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 (p). Two most relevant bi-tensors in this context are Synge's World function Ω(p, p0) and the van Vleck determinant (VVD) ∆(p, p0), 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 p.We begin by discussing the intrinsic and extrinsic geometry of equi-geodesic surfaces ΣG,p 0 ≡ {p ∈ M|Ω(p, p0) = constant} in a geodesically convex neighbourhood of an event p0, and highlight some elementary identities relating the VVD with geometry of ΣG,p 0 . As an aside, we also comment on the contribution of ΣG,p 0 to the surface term in the Einstein-Hilbert (EH) action and show that it can be written as a volume integral of ln ∆.We then proceed to study the small scale structure of spacetime in presence of a Lorentz invariant short distance cut-off ℓ0 using Ω(p, p0) and ∆(p, p0), based on some recently developed ideas. We derive a 2nd rank bi-tensor q ab (p, p0; ℓ0) = q ab [g ab , Ω, ∆] which naturally yields geodesic intervals bounded from below and reduces to g ab for Ω ≫ ℓ 2 0 /2. We present a general and mathematically rigorous analysis of short distance structure of spacetime based on (a) geometry of equi-geodesic surfaces ΣG,p 0 of g ab , (b) structure of the non local d'Alembartian p 0 p associated with q ab , and (c) properties of VVD. In particular, we prove the following: (i) The Ricci bi-scalar Ric(p, p0) of q ab is completely determined by ΣG,p 0 , the tidal tensor and first two derivatives of ∆(p, p0), and has a non-trivial classical limit (see text for details):(ii) The GHY term in EH action evaluated on equi-geodesic surfaces straddling the causal boundaries of an event p0 acquires a non-trivial structure.These results strongly suggest that the mere existence of a Lorentz invariant minimal length ℓ0 can leave unsuppressed residues independent of ℓ0 and (surprisingly) independent of many precise details of quantum gravity. For e.g., the coincidence limit of Ric(p, p0) is finite as long as the modification of distances S ℓ 0 : 2Ω → 2 Ω satisfies (i) S ℓ 0 (0) = ℓ 2 0 (the condition of minimal length), (ii) S0(x) = x, and (iii) |S ℓ 0 |/S ′2 ℓ 0 (0) < ∞. In particular, the function S ℓ 0 (x), which should eventually come from a complete framework of quantum gravity, need not admit a perturbative expansion in ℓ0.Finally, we elaborate on certain technical and conceptual aspects of our results in the context of entropy of spacetime and classical description of gravitational dynamics based on Noether charge of Diff invariance instead of the EH lagrangian.

We investigate the random motion of a mirror in (1 + 1)-dimensions that is immersed in a thermal bath of massless scalar particles which are interacting with the mirror through a boundary condition. Imposing the Dirichlet or the Neumann boundary conditions on the moving mirror, we evaluate the mean radiation reaction force on the mirror and the correlation function describing the fluctuations in the force about the mean value. From the correlation function thus obtained, we explicitly establish the fluctuation-dissipation theorem governing the moving mirror. Using the fluctuation-dissipation theorem, we compute the mean-squared displacement of the mirror at finite and zero temperature. We clarify a few points concerning the various limiting behavior of the mean-squared displacement of the mirror. While we recover the standard result at finite temperature, we find that the mirror diffuses logarithmically at zero temperature, confirming similar conclusions that have been arrived at earlier in this context. We also comment on a subtlety concerning the comparison between zero temperature limit of the finite temperature result and the exact zero temperature result.

Assuming that high energy effects may alter the standard dispersion relations governing quantized fields, the influence of such modifications on various phenomena has been studied extensively in the literature. In different contexts, it has generally been found that, while super-luminal dispersion relations hardly affect the standard results, sub-luminal relations can lead to (even substantial) modifications to the conventional results. A polymer quantized scalar field is characterized by a series of modified dispersion relations along with suitable changes to the standard measure of the density of modes. Amongst the modified dispersion relations, one finds that the lowest in the series can behave sub-luminally over a small domain in wavenumbers. In this work, we study the response of a uniformly rotating Unruh-DeWitt detector that is coupled to a polymer quantized scalar field. While certain sub-luminal dispersion relations can alter the response of the rotating detector considerably, in the case of polymer quantization, due to the specific nature of the dispersion relations, the modification to the transition probability rate of the detector does not prove to be substantial. We discuss the wider implications of the result.

We investigate the observable consequences of Planck scale effects in the advanced gravitational wave detector by polymer quantizing the optical field in the arms of the interferometer. For large values of polymer energy scale, compared to the frequency of photon field in the interferometer arms, we consider the optical field to be a collection of infinite decoupled harmonic oscillators, and construct a new set of approximated polymer-modified creation and annihilation operators to quantize the optical field. Employing these approximated polymer-modified operators, we obtain the fluctuations in the radiation pressure on the end mirrors and the number of output photons. We compare our results with the standard quantization scheme and corrections from the Generalized Uncertainty Principle.

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