Realistic Approach of the Relations of Uncertainty of Heisenberg

Due to the requirements of the principle of causality in the theory of relativity, one cannot make a device for the simultaneous measuring of the canonical conjugate variables in the conjugate Fourier spaces. Instead of admitting that a particle’s position and its conjugate momentum cannot be accurately measured at the same time, we consider the only probabilities which can be determined when working at subatomic level to be valid. On the other hand, based on Schwinger's action principle and using the quadridimensional form of the unitary transformation generator function of the quantum operators in the paper, the general form of the evolution equation for these operators is established. In the nonrelativistic case one obtains the Heisenberg's type evolution equations which can be particularized to derive Heisenberg's uncertainty relations. The analysis of the uncertainty relations as implicit evolution equations allows us to put into evidence the intrinsic nature of the correlation expressed by these equations in straight relations with the measuring process. The independence of the quantisation postulate from the causal evolution postulate of quantum mechanics is also put into discussion.

A new characterization ofB-bounded semigroups with application to implicit evolution equations

We consider the one-parameter family of linear operators that A. Belleni Morante recently introduced and calledB-bounded semigroups. We first determine all the properties possessed by a couple(A,B)of operators if they generate aB-bounded semigroup(Y(t))t≥0. Then we determine the simplest further property of the couple(A,B)which can assure the existence of aC0-semigroup(T(t))t≥0such that for allt≥0,f∈D(B)we can writeY(t)f=T(t)Bf. Furthermore, we compare our result with the previous ones and finally we show how our method allows to improve the theory developed by Banasiak for solving implicit evolution equations.

B-bounded semigroups and implicit evolution equations

The aim of this paper is to show an application of the recently introducedB-bounded semigroups in the theory of implicit and degenerate evolution equations. The most interesting feature of this approach is its applicability to problems with noncloseable operators.