Franz Hinterleitner

Canonical doubly special relativity theory

For a certain example of a “doubly special relativity theory” the modified space-time Lorentz transformations are obtained from the 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 volume in space and the energy-momentum contained in it, a solution of the problem of macroscopic bodies in doubly special relativity is suggested.

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.

Towards Loop Quantization of Plane Gravitational Waves

The polarized Gowdy model in terms of Ashtekar–Barbero variables is reduced with an additional constraint derived from the Killing equations for plane gravitational waves with parallel rays. The new constraint is formulated in a diffeomorphism invariant manner and, when it is included in the model, the resulting constraint algebra is first class, in contrast to the prior work done in special coordinates. Using an 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 these plane gravitational wave spacetimes accessible to loop quantization techniques.

Factor ordering in standard quantum cosmology

The Wheeler–DeWitt equation of Friedmann models with a massless scalar quantum field is formulated with arbitrary factor ordering of the Hamiltonian constraint operator. A scalar product of wavefunctions is constructed, giving rise to a probability interpretation and making comparison with the classical solution possible. In general the behaviour of the wavefunction 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 ordering the critical energy can be pushed down to zero. The essential features of the models, i.e. the dependence of the wavefunction on factor ordering and on the field energy, and the existence of a correct classical limit are the same for zero and nonzero cosmological constant.

A quantized closed Friedmann model

Dust-filled Friedmann universes have only one dynamical degree of freedom and one constraint (the Hamiltonian constraint), which plays the role of the generator of time evolution. In a corresponding quantum model the latter gives rise to a one-dimensional quantum mechanical wave equation. This equation is formulated in Ashtekar-type variables in two different factor orderings of the constraint operator, which turn out to require different Hilbert spaces. For each choice the spectra of the total mass and the maximal radius of the universe are discrete and for large quantum numbers both versions have acceptable classical limits.

Plane gravitational waves in real connection variables

Reduction of the cosmological Gowdy model to parallel-fronted plane gravitational waves by means of second class constraints and Dirac brackets

Remarks on doubly special relativity theories and gravity

Modifications of special relativity by the introduction of an invariant energy and/or momentum level (so-called doubly special relativity theories, DSR) or by an energy?momentum dependence of the Planck constant (generalized uncertainty principle, GUP) are compared with classical gravitational effects in an interaction process. For the low-energy limit of the usual formulations of DSR to be equivalent to Newtonian gravity, a restrictive condition is found. GUP yields an effective repulsion, in analogy to gravitational repulsion in loop quantum cosmology.Modifications of Special Relativity by the introduction of an invariant energy and/or momentum level (so-called Doubly Special Relativity theories, DSR) or by an energymomentum dependence of the Planck constant (Generalized Uncertainty Principle, GUP) are compared with classical gravitational effects in an interaction process. For the low energy limit of the usual formulations of DSR to be equivalent to Newtonian gravity, a restrictive condition is found. GUP yields an effective repulsion, in analogy to gravitational repulsion in loop quantum cosmology.

Quantum volume and length fluctuations in a midi-superspace model of Minkowski space

In a (1+1)-dimensional midi-superspace model for gravitational plane waves, a flat space–time condition is imposed with constraints derived from null Killing vectors. Solutions to a straightforward regularization of these constraints have diverging length and volume expectation values. Physically acceptable solutions in the kinematic Hilbert space are obtained from the original constraint by multiplying with a power of the volume operator and by a similar modification of the Hamiltonian constraint, which is used in a regularization of the constraints. The solutions of the modified Killing constraint have finite expectation values of geometric quantities. Further, the expectation value of the original Killing constraint vanishes, but its moment is non-vanishing. As the power of the volume grows, the moment of the original constraint grows, while the moments of volume and length both decrease. Thus, these states provide possible kinematic states for flat space, with fluctuations. As a consequence of the regularization of operators, the quantum uncertainty relations between geometric quantities such as length and its conjugate momentum do not reflect naive expectations from the classical Poisson bracket relations.

Plane Gravitational Waves and Flat Space in Loop Quantum Gravity

Classically a system of arbitrary plane gravitational waves propagating in the same or opposite directions can be restricted by first-class constraints to unidirectional waves, which travel without dispersion on a flat background. The unidirectionality constraints are formulated as well-defined Loop Quantum Gravity operators, together with criteria for an anomaly-free implantation, which is crucial for the occurrence or non-occurrence of dispersion, and more generally, of local Lorentz invariance violations due to (loop) quantum effects. By a set of further first-class constraints of the same kind we construct a quantum model of a no-wave state, i.e. of empty space.

Plane Gravitational Waves and Loop Quantization

Starting from the polarized Gowdy model in Ashtekar variables, the Killing equations characteristic for plane-fronted parallel gravitational waves are introduced in part as a set of first-class constraints, in addition to the standard ones of General Relativity. These constraints are expressed in terms of quantities that have an operator equivalent in Loop Quantum Gravity, making plane wave space-times accessible to loop quantization techniques.