103. M. Heller, W. Sasin
Noncommutative Unification of General Relativity and Quantum Mechanics
International Journal of Theoretical Physics, vol. 38, pp. 1619-1642 (1999).
| Abstract:
| 104. M. Heller, W. Sasin
Origin of Classical Singularities
General Relativity and Gravitation, vol. 31, pp. 555-570 (1999).
| Abstract:
| 105. Guido Magnano, Leszek M. Sokołowski
Can the local stress-energy conservation laws be derived solely from field equations?
Gen. Rel. Grav., vol. 30, pp. 1281-1288 (1998).
[abstract] [preprint] | Abstract:
According to a recent suggestion [1], the energy--momentum tensor for gravitating fields can be computed through a suitable rearrangement of the matter field equations, without relying on the variational definition. We show that the property observed by Accioly et al. in [1] is the consequence of a general identity, which follows from the covariance of the matter Lagrangian in much the same way as (generalized) Bianchi identities follow from the covariance of the purely gravitational Lagrangian. However, we also show that only in particular cases can this identity be used to obtain the actual form of the stress-energy tensor, while in general the method leads to ambiguities and possibly to wrong results. Moreover, in nontrivial cases the computations turn out to be more difficult than the standard variational technique. | 106. Zdzislaw Pogoda, Leszek M. Sokołowski
Does Mathematics Distinguish Certain Dimensions of Spaces? Part II.
The American Mathematical Monthly, vol. 105, pp. 456-463 (1998).
[journal] | Abstract:
Abstract | 107. 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. | 108. Zdzislaw Pogoda, Leszek M. Sokołowski
Does Mathematics Distinguish Certain Dimensions of Spaces? Part I.
The American Mathematical Monthly, vol. 104, pp. 860-869 (1997).
[journal] | Abstract:
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