Open FRW model in Loop Quantum Cosmology is under consideration. The left and right invariant vector fields and holonomies along them are studied. It is shown that in the hyperbolic geometry of k = −1 itis possible to construct a suitable loop which provides us with quantum scalar constraint originally introduced by Vandersloot [19]. The quantum scalar constraint operator with negative cosmolog...

Open FRW model in Loop Quantum Cosmology is under consideration. The left and right invariant vector fields and holonomies along them are studied. It is shown that in the hyperbolic geometry of k = −1 it is possible to construct a suitable loop which provides us with quantum scalar constraint originally introduced by Vandersloot [19]. Such an operator has correct geometrical interpretation. The...

We exhibit a purely quantum mechanical carrier of the imprints of gravitation by identifying for a relativistic system a property which (i) is independent of its mass and (ii) expresses the Poincare invariance of spacetime in the absence of gravitation. This carrier consists of the phase and amplitude correlations of waves in oppositely accelerating frames. These correlations are expressed as a...

The “D” matrices for all states of the two fundamental representations and octet are shown in the generalized Euler angle parameterization. The raising and lowering operators are given in terms of linear combinations of the left invariant vector fields of the group manifold in this parameterization. Using these differential operators the highest weight state of an arbitrary irreducible represen...

The Field-wise Objective Function is an important concept in eldwise wind retrieval. A discussion on the purpose of the eld-wise objective function is presented. Several variations of the eld-wise objective function are compared. The gradient vectors and Hessian matrices are derived for each variation. 1 An Overview of Field-wise Objective Functions An objective function is an error metric, pro...