Physics Beyond the Limits of Uncertainty Relations

4. Additional Remarks

We saw that quantum description is understandable as an approximation of the subquantum one. Subquantum states of the system do not distinguish subquantum variables that coincide modulo the elements from the commutant

of .

of .

All phenomenas related to causality are expressible in terms of . Nevertheless, the complete description is given by a non-commutative algebra not less non-commutative than the quantum one.

One of the principal questions a coherent subquantum theory must answer is how dynamics looks like. There exists an interesting class of subquantum theories [D4] where the subquantum space is equiped with a symplectic manifold structure, such that the quantum evolution can be obtained from one-parametric flow generated by a smooth function on , playing the role of the subquantum hamiltonian. In this sense, Shrodinger equation can be viewed as a statistical version of classical Hamiltonian equations.

The algebra of quantum observables does not admit dispersion-free states (characters). This is a purely algebraic formulation of the mentioned no-go theorems, and can be viewed as a consequence of uncertainty relations.

It is important to mention that C*-algebraic extensions considered here are also very interesting from the point of view of non-commutative geometry [C].

At first, there exist powerful techniques [BDF] to apply C*-algebraic extensions (of the types similar to our subquantum extensions) to study topological properties of classical topological spaces . Secondly, noncommutative C*-algebras give us examples of quantum spaces--the main objects of study of non-commutative geometry. Let us observe that, from the non-commutative geometric viewpoint, subquantum states (as characters) play the role of points of the quantum space associated to the algebra .