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That is also the reason why general relativity can easily be interpreted as delivering a model for the whole universe, whatever this would mean.

We know that quantum mechanics takes into account in an essential way the effect of the observer through the measuring apparatus on the state of the physical entity under study. In a theory generalizing quantum mechanics and relativity, such that both appear as special cases, this effect should certainly also appear in a fundamental way.

Of course Riemann is not equivalent to general relativity, a lot of detailed physics had to be known to apply Riemann resulting in general relativity. This is the same with operational quantum axiomatics, it has the potential to deliver the framework for the theory generalizing quantum mechanics and relativity theory. We want to remark that in principle a theory that describes the possible actions in the world, and a theory that delivers a model for the whole universe, should not be incompatible.

It should even be so that the theory that delivers a model of the whole universe should incorporate the theory of actions in the world, which would mean for the situation that exists now, general relativity should contain quantum mechanics, if it really delivers a model for the whole universe.

Non-locality means non-spatiality, which means that the reality of the micro-world, and hence the reality of the universe as a whole, is not time-space like. Time-space is not the global theatre of reality, but rather a crystallization and structuration of the macro-world.

This fact is the fundamental reason why general relativity, built on the mathematical geometrical Riemannian structure of time-space, cannot be the canvas for the new theory to be developed. A way to express this technically would be to say that the set of events cannot be identified with the set of time-space points as is done in relativity theory. Recourse will have to be taken to a theory that describes reality as a kind of pre-geometry, and where the geometrical structure arises as a consequence of interactions that collapse into the time-space context.

We believe that operational quantum axiomatics can deliver the framework as well as the methodology to construct and elaborate such a theory. In later work Piron made a stronger attempt to found operationally part of the axioms [8], and 4 this attempt was worked out further in [9], to arrive at a full operational foundation only recently [7].

Also mathematically the circle was closed only recently. There do exist a lot of finite dimensional generalized Hilbert spaces that are different from the three standard exam- ples, real, complex and quaternionic Hilbert space. But since a physical entity has to have at least a position observable, it follows that the generalized Hilbert space must be infinite dimensional. At the time when Piron gave his five axioms that lead to the representa- tion within a generalized Hilbert space, there only existed three examples of generalized Hilbert spaces that fitted all the axioms, namely real, complex and quaternionic Hilbert space.

Years later Hans Keller constructed the first counterexample, more specifically an example of an infinite dimensional generalized Hilbert space that is not isomorphic to one of the three standard Hilbert spaces [10].

She proved that an infinite dimensional generalized Hilbert space that contains an orthonormal base is isomorphic with one of the three standard Hilbert spaces [11].

An interesting and rather recent evolution is taking place, where quantum structures, as developed within this operational approach to quantum axiomatics, are used to model entities in regions of reality different of the micro-world [13, 14, 15, 16, 17, 18, 19, 20]. We believe that also this is a promising evolution in the way to understand deeper and more clearly the meaning of quantum mechanics in all of its aspects.

Primary References [1] Birkhoff, G. Towards a general operational and realistic framework for quantum mechanics and relativity theory.

Elitzur, S. Dolev and N.

Kolenda Eds. Possible Developments in Quantum Theory in the 21st Century pp.

Secondary References [2] von Neumann, J. Quantum games and quantum strategies. Physical Review Letters, 83, Quantum finance: A quantum approach to stock price fluctua- tions. Physica A, , Sakurai, Modern Quantum Mechanics, 3rd ed. Greiner, Quantum Mechanics, An Introduction, 4th ed.

Springer-Verlag, - intermediate C. Cohen-Tannoudji, B. Diu, F. I and Vol. Landau, L. Weinberg, Lectures on Quantum Mechanics, 1st ed. Schwinger, Quantum Mechanics, Springer, - advanced A.

Galindo, P. Sudbery, Quantum mechanics and the particles of nature, 3rd ed.

Black Body. Black-Body Radiation. Bohm Interpretation of Quantum Mechanics. Bohmian Mechanics. Bohr's Atomic Model. Bohr—Kramers—Slater Theory. Born Rule and its Interpretation. Bose-Einstein Condensation. Bose—Einstein Statistics. Brownian Motion. Bub—Clifton Theorem. Casimir Effect. Cathode Rays. Causal Inference and EPR. Cluster States. Coherent States.

Complementarity Principle. Complex-Conjugate Number.