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 quantum mechanical principle according to Werner Heisenberg as it is shown in his scientific paper [

This principle known as the Heisenberg uncertainty principle can be expressed as follows:

After Heisenberg, two other scientists, H. P. Robertson and E. Schrödinger, developed his uncertainty principle. They expanded it in 1930, by adding other pairs of variables that cannot be simultaneously known, such as: energy and position, angular position and angular momentum. The uncertainty principle continues to be refined and brought up to date.

Generally, Heisenberg's uncertainty principle, states that incompatible dynamic variables in relation to the measuring process, satisfy the relations:

The above relations can be written, as it is known in the form of commutation relationships, between corresponding incompatible observables according to the theorem: being given two hermitic operators

The conclusions drawn from the study of the universality of particle-wave dualism in the context of relativistic causality, and other observations resulting from mathematical formalisms of quantum theory led the authors to a new approach of Heisenberg's uncertainty relationships as they put into question the possibility of simultaneous determination in the measurement process of the uncertainties involved by these relationships, showing the incompatibility of “simultaneity” with relativistic causality requirements and proposing a reformulation of the principle of uncertainty.

From the classical theory of wave propagation, it is known that the width of a wave packet

Similarly, in the case of finite duration disturbances

In quantum theory, relations (

The incompatible quantum observables are correlated with Heisenberg's uncertainty principle, but as we have shown [

From another point of view, the analytical expressions of quantum observables can be obtained by replacing the classical canonically conjugate variables within a symmetrised expression of the classical variable

Switching, within the measuring process, from the direct variable space to the canonical conjugate variable space which we will be denoted by

Let

If the functions do not satisfy the conditions of Fourier transform, then we get the generalized Fourier transform.

It is known that an optical system suitable for a two-dimensional Fourier transform may be a lens. If the focal plane

Heisenberg's uncertainty relations are usually established on the basis of the experimental observations. It is well known in this regard that Heisenberg imagined an experience of “simultaneous” measuring of the electron position and momentum by using “ray microscope.” Other examples of experimental facts from which analysis suggest that the existence of uncertainty relations is as follows:

experience concerning diffraction of electrons through a slit;

experience concerning the deviation of a particle in a magnetic field;

compton collision processes of particles with a photon, and so forth.

From the experiments of this type we can conclude that there can be no question of simultaneous measurement of incompatible observables due to the finite speed of propagation of interactions.

Let us assume the following hypotheses derived from the study presented in the previous paragraphs:

canonical conjugate variables, that is, incompatible variables belong to reciprocal Fourier spaces;

Fourier transforms assume the existence of physical systems as information processing;

the deduction of uncertainty relations experimentally by using physical systems that perform Fourier transforms to reveal two canonical conjugate spaces and which can thus perform measurements on quantities which are incompatible.

Therefore, the sizes of the input and output of the system that makes Fourier transforms and conjugates canonical variables or its incompatible observables are considered to perform simultaneous measurements, in the ordinary sense of the formulation of Heisenberg's uncertainty relations.

Reflecting the uncertainty relations, the wave-particle dualism still remains true, but its content is not changing the principle of uncertainty with respect to a simultaneous measurement.

The cause and the effect cannot be measured simultaneously as a consequence of the used measurement device (Fourier transformer), through which the signals pass at a finite speed. Far from contradicting the principle of causality, uncertainty relations come to confirm it, because if we admit Einstein relativity principles when analyzing the measurement process, one must take into account the finite speed of propagation of interactions.

Let us consider, for example, a moving quantum particle, such as electron or photon through a slot machine that is performing the Fourier transform, due to diffraction phenomena. Signal function at slot plane is the wave function of the particle. Its Fourier transform is pulse wave function in space which may be considered a simultaneous determination of the electron coordinate passing through the slot.

According to the interpretation presented in this paper, it results in the impossibility of localizing inside the volume element of the phase space, of a quantum particle at a given time moment, because the measurements of position and momentum of a quantum particle cannot be simultaneously made.

The canonically conjugate variables being correlated by uncertainty relations one must pay more attention to the problem of the nature of these correlations.

Let

According to Schwinger's action principle, the change of the action operator is given by the following expressions:

The operators

Accordingly, for different

Let us to consider the particular cases

The time interval of the Fourier transformation processing by the measuring apparatus to put into evidence the correlated

(a) The temporal evolution equation (

From (

(b) The interpretation of uncertainty Heisenberg's relations (

(c) The results can be generalized for the analysis of any group of two canonically conjugate dynamical variables, the equations implying these variables showing an intrinsic correlation being implicitly present during the measuring process.

(d) One can assert the importance of the noncommutative operators which are principled, closed, and related to the evolution of the quantum system, contrary to the commutative ensembles of operators which define maximal the quantum states of the corresponding system.

The “standard experiments” are not meant for simultaneous measurements of position and momentum of a quantum system, as it is usually admitted, being a consequence of the used measurement device (Fourier transformer), through which the signals pass having a finite speed.

Any quantum system is subject to uncertainty relations, proving its dual nature.

Due to the requirements of the principle of causality in the theory of relativity, one cannot make a device for simultaneous measuring of the canonical conjugate variables in the conjugate Fourier spaces, so that these interpretations of the measurements are nonsensical.

According to the interpretation we have presented in this paper, the result is the impossibility of localizing inside the volume element of the phase space of a quantum system at a given time, because the measurements of position and momentum of a quantum particle cannot be simultaneously performed.

Uncertainty relations must be regarded as statistical correlations between the results of measurements of the dynamic variables in Fourier conjugate spaces.

Therefore, Heisenberg's uncertainty principle could be formulated as follows: “the results of measurements in reciprocal Fourier spaces of the canonical conjugate dynamic variables of the quantum systems are statistically correlated, the uncertainties product being of the order of magnitude of the constant of Planck.”

The concordance between uncertainty relations interpretations based on that understanding of the measuring process and the implicit evolution equations derived in this paper is relevant for the coherence of the theory developed above.

The treatise of the uncertainly relations as implicit evolution equations allowed us to put into evidence the intrinsic nature of the correlation expressed by these equations in straight relations with the measuring process.

In the context of the above results, the quantisation postulate of quantum mechanics is not independent from the causal evolution postulate, that is, these two postulates are connected in the axiomatic development of the quantum theory. This shows the principled role of the incompatible observables in the evolution of the quantum systems.

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