Physical Mathematics

We investigate the Weyl-Wigner-Gr•oenewold-Moyal, the Stratonovich, and the Berezin group quantization schemes for the space-space noncommutative Heisenberg-Weyl group. We show that the ?-product for the deformed algebra of Weyl functions for the rst scheme is di erent than that for the other two, even though their respective quantum mechanics' are equivalent as far as expectation values are concerned, provided that some additional criteria are imposed on the implementation of this process. We also show that it is the ?-product associated with the Stratonovich and the Berezin formalisms that correctly gives the Weyl symbol of a product of operators in terms of the deformed product of their corresponding Weyl symbols. To conclude, we derive the stronger ?-valued equations for the 3 quantization schemes considered and discuss the criteria that are also needed for them to exist.

This paper shows how a gravitational eld generated by a given energy-momentum distribution (for all realistic cases) can be represented by distinct geometrical structures (Lorentzian, teleparallel, and nonnull nonmetricity spacetimes) or that we even can dis-pense all those geometrical structures and simply represent the gravitational eld as a eld in Faraday's sense living in Minkowski spacetime. The explicit Lagrangian density for this theory is given, and the eld equations (which are Maxwell's like equations) are shown to be equivalent to Einstein's equations. Some examples are worked in detail in order to convince the reader that the geometrical structure of a manifold (modulus some topological constraints) is conventional as already emphasized by Poincare long ago, and thus the realization that there are distinct geometrical representations (and a physical model related to a deformation of the continuum supporting Minkowski spacetime) for any realistic gravitational eld strongly suggests that we must investigate the origin of its physical nature. We hope that this paper will convince readers that this is indeed the case.

A symmetry group classi cation for fourth-order reaction-di usion equations, allow-ing for both second-order and fourth-order di usion terms, is carried out. The fourth-order equations are treated, rstly, as systems of second-order equations that bear some resemblance to systems of coupled reaction-di usion equations with cross di usion, sec-ondly, as systems of a second-order equation and two rst-order equations. The paper generalizes the results of Lie symmetry analysis derived earlier for particular cases of these equations. Various exact solutions are constructed using Lie symmetry reductions of the reaction-di usion systems to ordinary di erential equations. The solutions include some unusual structures as well as the familiar types that regularly occur in symmetry reductions, namely, self-similar solutions, decelerating and decaying traveling waves, and steady states.

We discuss the Hilbert program for the axiomatization of physics in the context of what Hilbert and von Neumann came to call the analytical apparatus and its conditions of reality. We suggest that the idea of a physical logic is the basis for a physical mathematics and we use quantum mechanics as a paradigm case for axiomatics in the sense of Hilbert. Finite probability theory requires nite derivations in the measurement theory of QM and we give a polynomial formulation of local complementation for the metric induced on the topology of the Hilbert space. The conclusion hints at a constructivist physics.

Braiding operators corresponding to the third Reidemeister move in the theory of knots and links are realized in terms of parametrized unitary matrices for all dimensions. Two distinct classes are considered. Their (nonlocal) unitary actions on separable pure product states of three identical subsystems (i.e., the spin projections of three particles) are explicitly evaluated for all dimensions. This, for our classes, is shown to generate entangled superposition of four terms in the base space. The 3-body and 2-body en-tanglements (in three 2-body subsystems), the 3 tangles, and 2 tangles are explicitly evaluated for each class. For our matrices, these are parametrized. Varying parameters they can be made to sweep over the domain (0; 1). Thus, braiding operators correspond-ing to over- and undercrossings of three braids and, on closing ends, to topologically entangled Borromean rings are shown, in another context, to generate quantum entan-glements. For higher dimensions, starting with di erent initial triplets one can entangle by turns, each state with all the rest. A speci c coupling of three angular momenta is brie y discussed to throw more light on three body entanglements.