Abstract:
The inconsistence of Dirac-Weyl field equations with the universal U(1) gauge invariance of neutrinos in quantum mechanics led to generalize the special relativity to the generic relativity, which was composed of the special relativity and its three analogues with different signatures of metrics. The combination of the universal U(1) gauge invariance and the generic relativity naturally deduced the strong gauge symmetries $% SU(3)$ and the electroweak ones $SU(2)\times U(1)$ in the Standard Model. The universal U(1) gauge invariance of elementary fermions is attributed to its nature of complex line in three dimensional projective geometry.

Abstract:
It is shown that when the Einstein-Hilbert Lagrangian is considered without any non-covariant modifications or change of variables, its Hamiltonian formulation leads to results consistent with principles of General Relativity. The first-class constraints of such a Hamiltonian formulation, with the metric tensor taken as a canonical variable, allow one to derive the generator of gauge transformations, which directly leads to diffeomorphism invariance. The given Hamiltonian formulation preserves general covariance of the transformations derivable from it. This characteristic should be used as the crucial consistency requirement that must be met by any Hamiltonian formulation of General Relativity.

Abstract:
Different aspects of relativity, mainly in a canonical formulation, relevant for the question "Is spacetime nothing more than a mathematical space (which describes the evolution in time of the ordinary three-dimensional world) or is it a mathematical model of a real four-dimensional world with time entirely given as the fourth dimension?" are presented. The availability as well as clarity of the arguments depend on which framework is being used, for which currently special relativity, general relativity and some schemes of quantum gravity are available. Canonical gravity provides means to analyze the field equations as well as observable quantities, the latter even in coordinate independent form. This allows a unique perspective on the question of dimensionality since the space-time manifold does not play a prominent role. After re-introducing a Minkowski background into the formalism, one can see how distinguished coordinates of special relativity arise, where also the nature of time is different from that in the general perspective. Just as it is of advantage to extend special to general relativity, general relativity itself has to be extended to some theory of quantum gravity. This suggests that a final answer has to await a thorough formulation and understanding of a fundamental theory of space-time. Nevertheless, we argue that current insights into quantum gravity do not change the picture of the role of time obtained from general relativity.

Abstract:
A new canonical transformation is found that enables the direct canonical treatment of the conformal factor in general relativity. The resulting formulation significantly simplifies the previously presented conformal geometrodynamics. It provides a further theoretical basis for the conformal approach to loop quantum gravity and offers a generic framework for the conformal analysis of spacetime dynamics.

Abstract:
We propose a new interpretation of doubly special relativity (DSR) based on the distinction between the momentum and the translation generators in its phase space realization. We also argue that the implementation of DSR theories does not necessarily require a deformation of the Lorentz symmetry, but only of the translation invariance.

Abstract:
We propose a new interpretation of doubly special relativity based on the distinction between the momenta and the translation generators in its phase space realization. We also argue that the implementation of the theory does not necessarily require a deformation of the Lorentz symmetry, but only of the translation invariance.

Abstract:
It has long been recognized that lattice gauge theory formulations, when applied to general relativity, conflict with the invariance of the theory under diffeomorphisms. Additionally, the traditional lattice field theory approach consists in fixing the gauge in a Euclidean action, which does not appear appropriate for general relativity. We analyze discrete lattice general relativity and develop a canonical formalism that allows to treat constrained theories in Lorentzian signature space-times. The presence of the lattice introduces a ``dynamical gauge'' fixing that makes the quantization of the theories conceptually clear, albeit computationally involved. Among other issues the problem of a consistent algebra of constraints is automatically solved in our approach. The approach works successfully in other field theories as well, including topological theories like BF theory. We discuss a simple cosmological application that exhibits the quantum elimination of the singularity at the big bang.

Abstract:
Doubly Special Relativity is usually formulated in momentum space, providing the explicit nonlinear action of the Lorentz transformations that incorporates the deformation of boosts. Various proposals have appeared in the literature for the associated realization in position space. While some are based on noncommutative geometries, others respect the compatibility of the spacetime coordinates. Among the latter, there exist several proposals that invoke in different ways the completion of the Lorentz transformations into canonical ones in phase space. In this paper, the relationship between all these canonical proposals is clarified, showing that in fact they are equivalent. The generalized uncertainty principles emerging from these canonical realizations are also discussed in detail, studying the possibility of reaching regimes where the behavior of suitable position and momentum variables is classical, and explaining how one can reconstruct a canonical realization of doubly special relativity starting just from a basic set of commutators. In addition, the extension to general relativity is considered, investigating the kind of gravity's rainbow that arises from this canonical realization and comparing it with the gravity's rainbow formalism put forward by Magueijo and Smolin, which was obtained from a commutative but noncanonical realization in position space.

Abstract:
According to the theory of unimodular relativity developed by Anderson and Finkelstein, the equations of general relativity with a cosmological constant are composed of two independent equations, one which determines the null-cone structure of space-time, another which determines the measure structure of space-time. The field equations that follow from the restricted variational principle of this version of general relativity only determine the null-cone structure and are globally scale-invariant and scale-free. We show that the electromagnetic field may be viewed as a compensating gauge field that guarantees local scale invariance of these field equations. In this way, Weyl's geometry is revived. However, the two principle objections to Weyl's theory do not apply to the present formulation: the Lagrangian remains first order in the curvature scalar and the non-integrability of length only applies to the null-cone structure.