Spectral dimension from nonlocal dynamics on causal sets

Abstract

We investigate the spectral dimension obtained from non-local continuum d'Alembertians derived from causal sets. We find a universal dimensional reduction to 2 dimensions, in all dimensions. We conclude by discussing the validity and relevance of our results within the broader context of quantum field theories based on these nonlocal dynamics.

Quantum gravity signatures in the Unruh effect

“…Since the theory is Lorentz invariant, G M can only depend on (x − x ) 2 . Using the relation 20) with ∆τ = τ − τ , the Rindler Green's function has a τ dependence of the form G R (cosh a∆τ ). Focusing for simplicity on τ = 0, a Wick rotation t = it E will induce, through t = a −1 sinh aτ , a corresponding Wick rotation in Rindler time, τ = iτ E .…”

“…A prototypical example is provided by Causal Set Theory. Here G(p 2 ) interpolates between the standard propagator for a massive scalar field for momenta p 2 below the discretization scale and a nonlocal expression without giving rise to additional poles in the complex p 0 -plane [19,20]. In these cases it is still possible to obtain an explicit formula for the profile function F(E) based on the Källen-Lehmann representation of the two-point function.…”

“…The prototypical example for this mechanism is provided by Causal Dynamical Triangulations [3] where a random walk sees a two-dimensional spacetime at short distances while long walks exhibit a four-dimensional behavior [4]. Similar features are encountered in Asymptotic Safety [5][6][7][8][9], Loop Quantum Gravity [10][11][12][13][14][15], Hořava-Lifshitz gravity [16,17], Causal Set Theory [18][19][20], κ-Minkowski space [21][22][23], non-commutative geometry [24,25], non-local gravity theories [26,27], minimal length models [28], and based on the Hagedorn temperature seen by a gas of strings [29].…”

Causal set theory, non-locality and phenomenology

“…It is possible that the two quantities carry different physical meaning given that they are obtained with different methods and in different regimes of the theory (albeit having the same name), or that the spectral dimension computed in Ref. 15 ceases to be valid at scales when the continuum approximation is no longer reliable. Note moreover that additional hints towards dimensional reduction in causal set theory were recently presented.…”

Dimensional reduction in manifoldlike causal sets

“…Based in part on the properties of very small causal sets, it was suggested in [10] that causal set theory might exhibit similar phenomenon of dimensional reduction. Added evidence has come from the short-distance behavior of the causal set d'Alembertian [11] and perhaps from the appearance of "asymptotic silence" at short distances [12]. But while there are some hints from earlier investigations of small subsets of manifold-like causal sets [13], the question has not yet been systematically investigated.…”