13. M. Heller, W. Sasin
Differential Groupoids and Their Application to the Theory of Spacetime Singularities
International Journal of Theoretical Physics, vol. 41, pp. 919-937 (2002).

Abstract:

14. M. Heller, W. Sasin, Z. Odrzygóźdź
State Vector Reduction as a Shadow of Noncommutative Dynamics
Journal of Mathematical Physics, vol. 41, pp. 5168-5179 (2000).

Abstract:
Abstract

15. M. Heller, W. Sasin
Noncommutative Unification of General Relativity and Quantum Mechanics
International Journal of Theoretical Physics, vol. 38, pp. 1619-1642 (1999).

Abstract:

16. M. Heller, W. Sasin
Origin of Classical Singularities
General Relativity and Gravitation, vol. 31, pp. 555-570 (1999).

Abstract:

17. Michael Heller, Wiesław Sasin, and Dominique Lambert
Groupoid approach to noncommutative quantization of gravity
J. Math. Phys., vol. 38, p. 5840 (1997).
[abstract] [preprint] [journal]

Abstract:
We propose a new scheme for quantizing gravity based on a noncommutative geometry. Our geometry corresponds to a noncommutative algebra A = Gc∞(G,C) of smooth compactly supported complex functions (with convolution as multiplication) on the groupoid G = E◁Γ being the semidirect product of a structured space E of constant dimension (or a smooth manifold) and a group Γ. In the classical case E is the total space of the frame bundle and Γ is the Lorentz group. The differential geometry is developed in terms of a Z(A)-submodule V of derivations of A and a noncommutative counterpart of Einstein’s equation is defined. A pair (A,math), where math is a subset of derivations of A satisfying the noncommutative Einstein’s equation, is called an Einstein pair. We introduce the representation of A in a suitable Hilbert space, by completing A with respect to the corresponding norm change it into a C∗-algebra, and perform quantization with the help of the standard C∗-algebraic method. Hermitian elements of this algebra are interpreted as quantum gravity observables. We introduce dynamical equation of quantum gravity which, together with the noncommutative counterpart of Einstein’s equation, forms a noncommutative dynamical system. For a weak gravitational field this dynamical system splits into ordinary Einstein’s equation of general relativity and Schrödinger’s equation (in Heisenberg’s picture) of quantum mechanics. Some interpretative questions are considered.

18. Marek Szydłowski, Michael Heller, and Wiesław Sasin
Geometry of spaces with the Jacobi metric
J. Math. Phys., vol. 37, p. 346 (1996).
[abstract] [journal]

Abstract:
The generalized Maupertuis principle is formulated for systems with the natural Lagrangian and an indefinite form of the kinetic energy. The generalization is applied to the theory of gravity and cosmology. For such systems, the metric determined by the kinetic energy form has a Lorentz signature. The theorem is proved concerning the behavior of trajectories in a neighborhood of the boundary of the region admissible for motion. This region is not a smooth manifold but turns out to be a differential space of constant differential dimension. This fact allows us to use geometric methods analogous to those elaborated for smooth manifolds. It is shown that singularities of the Jacobi metric are not dangerous for the motion; its trajectories are smooth in the sense of the theory of differential spaces.