Jacques Maritain Center: Thomistic Institute

Quantum mechanics and realistic view of nature

Milos Lokajicek
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

Questions:

Is our world real or is it a mere fiction of our mind? What is meant by reality at all? How may be the world evolution influenced by our mind? Can our mind or our decision change the reality as we have been taught by quantum mechanics, which introduced a fundamental duality into the microscopic world?

Abstract

Quantum mechanics influenced fundamentally the thinking of this century and contributed significantly to spreading postmodern ideas in the human community. However, one must put opposite questions: How much did the postmodern philosophy contribute to the formulation of the quantum mechanics and the corresponding interpretation of the world? Does not represent the quantum mechanics with its paradoxes and duality a kind of religion?

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.

Introduction

Quantum mechanics proposed by N. Bohr changed significantly our view of world in the beginning of this century. It was accepted practically by the whole physical community even if it was denoted as an incomplete theory by A. Einstein, who never ceased from such a conviction. And it is necessary to introduce that the old famous controversy between Einstein and Bohr has not been satisfactorily solved until now.

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).

Quantum-mechanical model and reality requirement

However, one should say, too, that the role of mathematical models was overvalued and practically reversed (or misused) in this century. They have not been more picturing the physical reality but they started to play a primary role determining very often what may or may not exist in the nature; or: what are the properties of reality. That is quite proper to quantum-mechanical models describing the behavior of microscopic objects that cannot be followed directly and continuously by our senses, in contradistinction to the classical physics. The properties of microparticles may be derived only from some data obtained with the help of macroscopic devices, which leaves always a piece of arbitrariness in our conclusions.

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.

Controversy between Einstein and Bohr

These paradoxes are usually being closely related to the orthodox or Copenhagen interpretation of the quantum-mechanical model. One expects often that such problems might be removed if the so called ensemble interpretation were made use of. Then of course, some additional parameters would have to be introduced to characterize physical states before a measurement is performed. And Einstein's criticism [1] that quantum mechanics should be considered incomplete would be fully justified.

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]).

Mathematical picture of physical processes

Before passing to basic assumptions the quantum mechanics is based on we must say several words more about mathematical models representing behavior of physical systems. According to the classical physics their behavior is determined by motions of individual matter objects under corresponding forces. Knowing a physical state at a given moment then all states at any time in the past as well as in the future may be derived with the help of differential equations describing the influence of the forces acting in a given physical system. However, the state of a matter (structureless) object is fully determined not only by its position but also by its momentum (velocity); i.e., by six parameters that may be represented by a vector in a 6-dimensional abstract space. The state of a physical system consisting of N objects may be then represented by a vector in the so called (6N-dimensional) phase space. And the time evolution of such a system is represented by a trajectory in such a phase space, which may be given, e.g., by solutions of corresponding Hamilton equations. Each trajectory is characterized by the values of quantities that are conserved during the whole evolution. It is evident that the individual trajectories corresponding, e.g., to different values of energy are represented by vector sets being fully disjoint.

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.

Schroedinger equation and Hilbert space

Having passed from the classical physics to quantum mechanics the Hamilton equations have been substituted by time-dependent Schroedinger equation. The time-dependent psi-functions obtained by its solutions may represent evolution trajectories in individual Hilbert subspaces. However, actual physical characteristics of a given system are not determined by such psi-functions only; they are fundamentally influenced by the choice of the structure of the corresponding Hilbert subspace and by the type of correspondence between individual vectors and physical states.

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.

Extended quantum-theoretical model

The extension of the corresponding Hilbert subspace may be exemplified with the help of a two-particle system. In such a case the given subspace being represented by general solutions of a corresponding time-dependent Schroedinger equation is to be separated further into two mutually orthogonal subspaces, distinguishing particles with different momentum directions being in the same positions. If there is not any particle change the trajectory goes from one subspace to another (once for an open system and performing many mutual transitions for a bound system). If particle changes are possible the evolution may go to other Hilbert subspaces conserving corresponding conservation laws.

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.

Differences between the standard and extended models

The matter world described by the standard quantum mechanics must be denoted as a fictive world derived from the mathematical model proposed by N. Bohr; a greater number of different physical states being represented by one mathematical symbol (one vector in a Hilbert space). That was a direct consequence of representing physical states by Hamiltonian eigenfunctions and by introducing the so called superposition principle. According to it all linear superpositions of these eigenfunctions (represented again by one vector) correspond to other physical states forming a system of all possible pure states; i.e., of simple elementary states. It is then necessary to distinguish between these states and the so called mixed states that represent normal standard statistical combinations of different pure states (statistical combination of two different vectors).

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.

Problem of the phase

The impossibility of the standard quantum-mechanical model to represent a real word should be related to the fact that it was not able to solve the problem of the phase in its framework. By identifying basic physical states of a given physical system with eigenfunctions of an Hamiltonian Bohr limited the model to determining, e.g., energy levels of an atom without taking any care of instantaneous positions of electrons running around an atom nucleus. He even declared the electron as having not an instantaneous position.

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.

EPR (Einstein-Podolsky-Rosen) experiment

Until now, we have discussed the problem of one-to-one correspondence between the model and physical reality. It follows from our analysis that for some regions of reality the extended and standard models may give the same predictions while for other regions these predictions should be different.

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].

Falsification of the standard quantum-mechanical model

There are, of course, regions of reality for which the two models give very different predictions. While both the model can be brought to agreement with experimental data for the light transmission through two polarizers (generalized Malus law) the predictions concerning the light transmission through three polarizers were found to exhibit fundamental differences for suitable chosen angles between polarizer axes. And corresponding experiments [19,20] showed then decisive discrepancy with the predictions of the standard quantum-mechanical model (Jones' matrices). These experimental discrepancies are of such a kind that they may be interpreted as a falsification of the quantum- mechanical model, even if a definite answer to some further questions should be denoted still as open.

Concluding remarks

The extended model enables to interpret the evolution of microscopic systems in a semi-classical deterministic way provided any particle change does not occur. The decay and collision processes involving creation or annihilation of some new particles have to combine a deterministic behavior with a probabilistic one, leaving a detailed evaluation of processes in corresponding short time transition periods for a more detailed exploration in the future. However, an absolute chance has not more any place in the interpretation of microscopic phenomena.

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.

Notes

[1] A. Einstein, B. Podolsky, N. Rosen: Can quantum-mechanical description of physical reality be considered complete?;Phys.Rev.47 (1935), 777--780.

[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.