Canonical DSR

For a certain example of a "doubly special relativity theory" the modified space-time Lorentz transformations are obtained from momentum space transformations by using canonical methods. In the sequel an energy-momentum dependent space-time metric is constructed, which is essentially invariant under the modified Lorentz transformations. By associating such a metric to every Planck cell in space and the energy-momentum contained in it, a solution of the problem of macroscopic bodies in doubly special relativity is suggested.Comment: 11 page

Isotropic Loop Quantum Cosmology with Matter II: The Lorentzian Constraint

The Lorentzian Hamiltonian constraint is solved for isotropic loop quantum cosmology coupled to a massless scalar field. As in the Euclidean case, the discreteness of quantum geometry removes the classical singularity from the quantum Friedmann models. In spite of the absence of the classical singularity, a modified DeWitt initial condition is incompatible with a late-time smooth behavior. Further, the smooth behavior is recovered only for positive or negatives times but not both. An important feature, which is shared with the Euclidean case, is a minimal initial energy of the order of the Planck energy required for the system to evolve dynamically. By forming wave packets of the matter field an explicit evolution in terms of an internal time is obtained.Comment: 19 pages, 4 figure

Factor ordering in standard quantum cosmology

The Wheeler-DeWitt equation of Friedmann models with a massless quantum field is formulated with arbitrary factor ordering of the Hamiltonian constraint operator. A scalar product of wave functions is constructed, giving rise to a probability interpretation and making comparison with the classical solution possible. In general the bahaviour of the wave function of the model depends on a critical energy of the matter field, which, in turn, depends on the chosen factor ordering. By certain choices of the ordering the critical energy can be pushed down to zero.Comment: 15 pages, 3 figure

On plane gravitational waves in real connection variables

We investigate using plane fronted gravitational wave space-times as model systems to study loop quantization techniques and dispersion relations. In this classical analysis, we start with planar symmetric space-times in the real connection formulation. We reduce via Dirac constraint analysis to a final form with one canonical pair and one constraint, equivalent to the metric and Einstein equations of plane-fronted with parallel rays waves. Due to the symmetries and use of special coordinates general covariance is broken. However, this allows us to simply express the constraints of the consistent system. A recursive construction of Dirac brackets results in non-local brackets, analogous to those of self-dual fields, for the triad variables chosen in this approach.Comment: v2: Matches published version, up to minor stylistic change

Towards Loop Quantization of Plane Gravitational Waves

The polarized Gowdy model in terms of Ashtekar-Barbero variables is further reduced by including the Killing equations for plane-fronted parallel gravitational waves with parallel rays. The resulting constraint algebra, including one constraint derived from the Killing equations in addition to the standard ones of General Relativity, are shown to form a set of first-class constraints. Using earlier work by Banerjee and Date the constraints are expressed in terms of classical quantities that have an operator equivalent in Loop Quantum Gravity, making space-times with pp-waves accessible to loop quantization techniques.Comment: 14 page

Isotropic Loop Quantum Cosmology with Matter

A free massless scalar field is coupled to homogeneous and isotropic loop quantum cosmology. The coupled model is investigated in the vicinity of the classical singularity, where discreteness is essential and where the quantum model is non-singular, as well as in the regime of large volumes, where it displays the expected semiclassical features. The particular matter content (massless, free scalar) is chosen to illustrate how the discrete structure regulates pathological behavior caused by kinetic terms of matter Hamiltonians (which in standard quantum cosmology lead to wave functions with an infinite number of oscillations near the classical singularity). Due to this modification of the small volume behavior the dynamical initial conditions of loop quantum cosmology are seen to provide a meaningful generalization of DeWitt's initial condition.Comment: 18 pages, 4 figure

Local and global aspects of separating coordinates for the Klein-Gordon equation

summary:The author considers the Klein-Gordon equation for $(1+1)$-dimensional flat spacetime. He is interested in those coordinate systems for which the equation is separable. These coordinate systems are explicitly known and generally do not cover the whole plane. The author constructs tensor fields which he can use to express the locus of points where the coordinates break down

On a new set of orthogonal polynomials

summary:An orthogonal system of polynomials, arising from a second-order ordinary differential equation, is presented