Deformed Lorentz Symmetry and Corresponding Geometry in Ultra-High Energy Astrophysics

This paper purports to develop a consistent microscopic theory of deformed Lorentz symmetry and the corresponding deformed geometry. Among the key geometric predictions of this approach, one lies in both the deformed line element (DLE) and the deformed maximum attainable velocity (DMAV) of a particle leading to potentially observable signatures in ultra-high energy astrophysics. In particular, the DMAV has in the past often been tested in a phenomenological approach to cosmic-ray and astrophysical-photon physics in order to extract constraints on those velocities. To this aim, we develop the theory of, so-called, master space (MS\({}_{p}\)) induced supersymmetry, subject to certain rules. We derive the Standard Lorentz Code (SLC) in a new perspective of global double MS\({}_{p}\)-SUSY transformations in terms of Lorentz spinors (\(\underline{\theta},\underline{\bar{\theta}}\)) referred to MS\({}_{p}\). The MS\({}_{p}\), embedded in the background 4D-space, is an unmanifested indispensable individual companion to the particle of interest as the intrinsic property devoid of any external influence. While all particles are living on \(M_{4}\), their superpartners can be viewed as living on MS\({}_{p}\). In the sequel, we turn to the deformation of these spinors: \(\underline{\theta}\to\underline{\tilde{\theta}}=\lambda^{1/2}\,\underline{\theta}\), etc., where \(\lambda\) appears as a deformation scalar function of the Lorentz invariance (LIDF). This yields both the DLE and DMAV, respectively, in the form \(\tilde{ds}=\lambda ds\) and \(\tilde{c}=\lambda c\), provided the invariance of DLE, and the same value of DMAV in free space holds for all inertial systems. Thus the LID (Lorentz invariance deformation) generalization of global MS\({}_{p}\)-SUSY theory formulates the generalized relativity postulates in a way that preserve the relativity of inertial frames, in spite of the appearance of modified terms in the LID dispersion relations. We complement this conceptual investigation with testing of various LIDFs in the UHECR- and TeV-\(\gamma\) threshold anomalies by implications for several scenarios: the Coleman and Glashow-type perturbative extension of SLC, the LID extension of standard model, the LID in quantum gravity motivated space-time models, the LID in loop quantum gravity models, and the LID for the models preserving the relativity of inertial frames.