Institute of Physics, Acad. of Sciences of the Czech Republic,

CZ-18040 Prague, Czech Republic

and

Center of Interdisciplinary Studies (Christian Academic Forum)

CZ-12000 Prague, Jecna 2

It will be shown that all the strange properties of the microworld have been based on the mathematical model which does not fulfill necessary requirements. An extended model will be mentioned being free of all mysterious properties and responding in principle to Einstein's criticism from 1935. Some predictions of the extended model differ significantly from those of the standard model. Experiments concerning light transmission through three polarizers (initiated on such a basis) have led to the results which may be interpreted as a falsification of the standard quantum mechanical model. All these results bring us to a new view of the world reality and to a new look at chance and determinism in physical processes.

One may say that the substance of such a controversy has its beginning already with Galileo when he showed the importance of mathematical models for physical theories, which helped to solve many problems in the past two centuries enabling to represent the behavior of the matter world and its causal relations by a mathematical picture (Newton, Maxwell, Hamilton, Lagrange, etc).

Any mathematical model must be able to provide us with a representation of all possible physical states in a one-to-one correspondence in agreement with the actual evolution of a given physical system, which corresponds to Einstein's requirement of a complete theory. However, the standard quantum-mechanical model does not fulfill such a requirement, which is the source of all quantum- mechanical paradoxes.

It is then fully understandable that the controversy between Einstein and Bohr [2] lasting for more than 60 years (see [3]) has been still continuing even if the physical community have preferred Bohr's view all the time. In the first half of this century the given controversy could be hardly solved as it was philosophical, and not physical, in principle. All discussions concerned always the justification (or not) of the standard quantum-mechanical model; any alternative physical model has not been ever presented. The problem may have been put into a quite new light only when the extended model proposed recently is taken into account (see [4,5,6]).

In the quantum mechanics, i.e., for the description of physical systems where the number or the nature of matter objects may change, a more general representation space has to be made use of. Any state is represented by a vector in a Hilbert space, which consists in principle of many mutually orthogonal subspaces each of them corresponding to a physical system of a fixed number of objects. In the case of a particle change the evolution goes from one Hilbert orthogonal subspace to another.

In the standard quantum mechanics this structure has been given by choosing the eigenfunctions of the corresponding Hamiltonian (and eventually of other mutually commuting operators) for an orthogonal vector basis. In such a Hilbert space the trajectories belonging, e.g., to different values of energy have some common points, which means that more physical states are to belong to the same vector in the corresponding Hilbert space. It is evident that the basic requirement of one-to-one correspondence between the model and the reality cannot be fulfilled. And all quantum-mechanical paradoxes follow from this fact. Being aware of these paradoxes some physicists have resigned to world reality and attempted to interpret the world as a picture of our mind, only.

To obtain the mentioned one-to-one correspondence it is necessary to extend each Hilbert subspace in an appropriate way. Such an extension was proposed for the first time by Lax and Phillips in 1967 [7] (see also [8,9]), being applied to a semi-classical description of acoustic and optical waves. The given structure was then derived independently by Alda et al. [10] in solving the problem of exponential decay of unstable particles; the given extended model being related for the first time to a quantum problem. And later on, it was considered also by Newton [11] and Bauer[12] in trying to solve the time-operator problem for harmonic oscillator.

Only in such a case the one-to-one correspondence between the model and the real world may exist. That does not mean, however, that we can describe the evolution of a physical system exactly in any time point. Such a possibility exists only when individual particles move in the ordinary space (no particle change). The process when a particle type is changed (i.e., in case of a decay or in the moment of a collision) may be described only in a phenomenological way as a probability of a given transition. However, as some additional parameters (denoted as hidden-variables in the discussions about incompleteness of the standard quantum-mechanical model) may be defined in the time before and after the particle change we cannot speak more about an absolute chance. It is the chance as defined by Aristotle and Thomas Aquinus. And the problem of particle changes is open to a further detailed exploration.

Thus, the extended model enables to describe the real world on the level of our contemporary knowledge: the deterministic evolution (characterized be solutions of Schroedinger equations) of matter objects when their nature has not been changed and the random transitions (with the help of probabilistic parameters) when any object change has occurred. However, as already mentioned these probabilities depend on some other parameters (being not considered in the standard quantum mechanics), which gives a possibility of more detailed studies of corresponding transition processes.

In the extended model the eigenfunctions of Hamiltonian do not belong to the given Hilbert space and do not represent any physical states. A simple physical state is represented always by a psi-function derived from a corresponding Schroedinger equation and belonging to a set of values being conserved during the whole evolution (e.g. energy, etc.). Any superposition of such different vectors represents always a statistical mixture of basic states.

Basic states are represented by all psi-functions (i.e., functions of space coordinates obtained at different values of the time parameter) corresponding to all possible quantities being conserved during the evolution). Consequently, there is not any substantial difference between the current selection rules and the so called superselection rules. Any linear combination of two vectors (even if it is represented formally again by a vector in the total Hilbert space) represents always a statistical mixture.

However, as a free parameter was involved in the complex psi- function, i.e., its phase, some authors (London [13], Dirac [14]) attempted to define a corresponding operator in the Hilbert space and to make a measurable quantity from the phase. However, the used definitions have not solved the phase problem (see [15]). It was not possible to solve the problem even if, e.g., in the case of harmonic oscillator the phase was defined with the help of tan (q/p), as two different physical states are still identified. These states may be distinguished only if suitable operators corresponding to "sinus" and "cosinus" are introduced. They can be regularly defined in an extended space only. Consequently, it is possible do describe the motion of microscopic particles in a semi-classical way when the evolution is represented with the help of the mentioned extended model.

The same predictions have been obtained for values of all quantities being conserved during time evolution. They have been obtained, however, also, e.g., for the values measured in the EPR experiments with two coinciding photons (see, e.g., [16]). Thus, the agreement of experimental data with the predictions of the standard model should be regarded as insignificant. The problem might be seen only in that these data violate Bell's inequalities [16], and especially in the fact that Bell's inequalities themselves are in a contradiction already to the Schroedinger equation, as stressed by d' Espagnat [17].

It may be shown, however, that Bell's inequalities are not a consequence of a mere locality conditions. There is an additional latent condition being involved in their derivation, which can hardly be brought to agreement with reality. It has been assumed that an actual space structure of a measuring device does not play any role, which is in disagreement with a realistic measurement theory. The problem has been analyzed to a greater detail and the mathematical formulation of the corresponding condition has been given in [18]. A more detailed description of the EPR problem in the light of the extended model may be then found in [6].

All claims of different philosophers (including Bohr's proclamation in 1938 - see [21]) that quantum mechanics must be interpreted in agreement with far-eastern ideas are based on the incompleteness of the standard quantum-mechanical model as stressed already by Einstein [1]. In contradiction to numerous requirements concerning the introduction of many-valued logic the extended model represents a full return to traditional logic having been developed by Aristotle and Thomas Aquinas.

[2] N. Bohr: Can quantum-mechanical description of physical reality be considered complete?; Phys.Rev.48 (1935), 696--702.

[3] The Dilemma of Einstein, Podolsky and Rosen - 60 Years Later (eds. A. Mann, M.Revzen), Inst. of Phys. Publish. Techno House, Bristol 1996.

[4] M.Lokajicek: Limitations of the standard quantum mechanics caused by its current mathematical model and a way out; New Theories in Physics (eds. Z.Ajduk et al.), World Scient. Publ. Co., Singapore 1989, 534-543.

[5] M.Lokajicek, V. Kundrat: Time operator in quantum mechanics; Hadronic J. 12 (1989), 31-4.

[6] J.Krasa, V.Kundrat, M.Lokajicek: A new solution of hidden- variable and measurement problems; l.c. [3], 87-90.

[7] P.D. Lax, R.S. Phillips: Scattering theory; Academic Press, New York - London 1967.

[8] P.D. Lax, R.S. Phillips: Scattering theory for automorphic functions; Princeton 1976.

[9] P.D. Lax, R.S. Phillips: Scattering theory (revised edition); Academic Press, Inc. 1989.

[10] V.Alda, V.Kundrat, M.Lokajicek: Exponential decay and irreversibility of decay and collision processes; Aplikace matematiky 19 (1974), 307-15.

[11] R.G.Newton: Quantum action-angle variables for harmonic oscillators; Ann. Phys. 124 (1980), 327-64.

[12] M.Bauer: A Time Operator in Quantum Mechanics; Ann. Phys. 150 (1983), 1-21.

[13] F. London: Winkelvariable und kanonische Transformationen in der Undulationsmechanik; Zschr. f. Phys. 40 (1927), 193-210.

[14] P.A.M. Dirac: The quantum theory of the emission and absorption of radiation; Proc. Roy. Soc. (London) A 114 (1927), 243-65.

[15] R. Lynch: The quantum phase problem, a critical review; Phys. Rep. 256 (1995), 367-437.

[16] A.Aspect, P.Grangier, G.Roger: Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A new violation of Bell's inequalities; Phys. Rev. Lett. 49 (1982), 91--94.

[17] B.d'Espagnat: Are the quantum rules exact? The case of imperfect measurements; Found. Phys. 16, 351-7 (1986).

[18] M.Lokajicek: Bell's inequalities and locality condition; submitted to Phys. Rev. Lett.

[19] J.Krasa, J.Jiricka, M.Lokajicek: Depolarization of the light by imperfect polarizers, Phys.Rev. E48 (1993), 3184--3186.

[20] J.Krasa, M.Lokajicek, J.Jiricka: Transmittance of laser beam through a pair of crossed polarizers, Phys.Lett. A186 (1994), 279--282.

[21] N.Bohr: Atomphysik und menschliche Erkenntnis; F. Vieweg, Braunschweig 1958, p. 19.