Simple model of pointlike spacetime defects and implications for photon propagation

M. Schreck, F. Sorba, and S. Thambyahpillai

Phys. Rev. D 88, 125011 – Published 6 December 2013

Abstract

A model in which pointlike defects are randomly embedded in Minkowski spacetime is considered. The distribution of spacetime defects is constructed to be Lorentz invariant. Since it is based on a sprinkling process, it does not introduce a preferred reference frame. A field-theoretic action for the photon and a fermion is set up, in which the photon is assumed not to couple to the defects directly, but via a scalar field. We are interested in signs for Lorentz violation caused by the spacetime defects, which are expected to reveal themselves in the photon sector. A modification of the photon dispersion relation may result as a quantum effect, and we compute it at leading order perturbation theory. The outcome of the calculation is that the photon dispersion law remains conventional, if the defect distribution is dense, homogeneous, and isotropic. This result sheds some new light on Lorentz violation in the framework of a small-scale structure of spacetime. It shows that Lorentz invariance can be preserved even in the presence of a spacetime structure that is supposed to emerge at the Planck scale. This conclusion has already been drawn on general grounds in other publications, where the current paper delivers a demonstration by a direct computation in a simple model.

Received 1 April 2013

DOI:https://doi.org/10.1103/PhysRevD.88.125011

Authors & Affiliations

M. Schreck^*, F. Sorba^†, and S. Thambyahpillai^‡

Institute for Theoretical Physics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany

^*marco.schreck@kit.edu
^†fabrizio.sorba@kit.edu
^‡shiyamala.thambyahpillai@kit.edu

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Vol. 88, Iss. 12 — 15 December 2013

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Images

Figure 1
Example of sprinkling in a finite region of a two-dimensional spacetime ( $time \times 1 D$ space) as it looks in two different inertial frames. Since the defects are instantaneous, they are illustrated as points. The second frame (b) is boosted along the positive spatial axis with a Lorentz boost factor $β = 0.7$ with respect to the first frame (a). This boost changes the shape of the spacetime region. However, the mean density $⟨ ϱ_{obs} ⟩$ of defects is the same in both frames. An illustration compares the two enlarged regions of both distributions. Note that in the boosted distribution not all defects are shown.Reuse & Permissions
Figure 2
Possible corrections to the bare scalar field $ϕ (k)$ . (a) One-loop photon self-energy contribution to the scalar field. (b) Correction to the scalar field originating from the scattering of $ϕ$ at the defects.Reuse & Permissions
Figure 3
Each panel in the current figure shows a distribution of spacetime defects where a photon with a typical wavelength is symbolically illustrated by a wiggly line below each panel. The scales of the panels are chosen to make clear that the distances between the defects in the left square correspond to multiples of the Planck scale (i.e., they are much smaller than the photon wavelength), whereas in the right square they may be in the order of (and even larger than) the wavelength of the photons to be considered. Hence the right panel (b) illustrates a foam in which individual defects are separated by distances that lie many orders of magnitude above the typical distances in the left panel (a).Reuse & Permissions
Figure 4
The first panel (a) illustrates an anisotropic distribution defining a single preferred direction $ζ$ , whereas the second panel (b) contains a regular lattice of defects defining two preferred directions $ζ_{1}$ and $ζ_{2}$ . The third panel (c) depicts a section of an inhomogeneous defect distribution, where in the center of the region shown the density is higher than near the margin.Reuse & Permissions
Figure 5
Several data points obtained for the coefficient $α (γ)$ in Eq. (9.2) are compared with the expected behavior for the order parameter: $α (γ) \propto (1 - γ_{c} / γ)^{β}$ . The exponent used is $β = 0.64$ , which is also obtained in the context of a four-dimensional lattice percolation. (See Table 2 on p. 52 in 35 and Table 1.2 on p. 81 in 36 for a slightly different value. Besides, in 37 continuous percolation is studied, and it is observed that the critical exponents are the same for continuous and lattice percolation, at least in two and three dimensions.) From the plot it becomes evident that $γ_{c} \approx 2$ . The latter point depends on the renormalization scale we choose.Reuse & Permissions
Figure 6
Additional contribution to the photon field with more than two scalar field lines attached to a defect vertex [see (a)]. From this and Eq. (b1h) we obtain the Feynman rule where three scalar fields are attached to such a vertex, corresponding to the mathematical expression $ϱ δ^{(4)} (k + q + p)$ [see (b)]. Additional Feynman rules with even more external scalars can be derived analogously.Reuse & Permissions

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