p Although Boltzmann and Gibbs considered statistical law and chance in a subjectivist manner, their studies actually included the concept of chance in the category of a physical law; or rather they abandoned the concept of necessity as a category in physics isolated from chance and came to regard chance itself in connection with necessity, in spite of their mechanistic views on causality. The dialectical process of the approximating of the concepts of necessity and chance in physics that had begun with the creation of statistical mechanics was developed further in quantum theory. It was at the level of quantum physics, when nonclassical atomism became established, that the transitions between and the unity of statistical and dynamic laws were discovered, and the idea of quantum determined new approaches to statistical regularity. We shall leave discussing of the relevant points to the next sections, and consider here certain aspects of the problem of the nature of statistical laws.

p A mechanistic view according to which statistical laws were subjective in character was common among physicists of both the classical period in science and in modern times. This view was quite consistently applied by Heisenberg to both classical and quantum mechanics (his relevant statements were cited above). Max Born supported this view; he overstressed the idea of the subjective nature of statistical laws and rejected ‘determinism’ even in classical mechanics (let us recall that he employed the term ’ determinism’, like many other physicists, to denote the dynamic regularity that these physicists usually identify with objective regularity in general). It is of interest to consider his argument.

p From his analysis of the concept of determinism in physics Born concluded that the possibility of determinism is denned by our precise knowledge of state (taking it in the sense of classical mechanics). He tried to show that the situation in classical mechanics, contrary to the commonly held opinion, was also such that precise definition of state had no physical meaning and that the statistical method had therefore always to be used, even in the case of a single particle;⁸⁴ in other words, that classical mechanics was indeterminist (Born, of course, identified statistics with indeterminacy).

p His reasoning boils down to the following. Let us assume that a mass particle moves without friction along a straight line (the z-axis) under no forces, and is elastically reflected at the termini. In order to say where it will be at any moment of time, it is assumed that its velocity (and co- ordinate) are exactly known. If, however, there is even the least inaccuracy in the determination of velocity, the inaccuracy of the prediction of co-ordinate (A#) will increase with time and may attain a very large value. Thus, at time t_c = l/Av₀, where I is the distance between the termini, and Ai>₀ is the inaccuracy in the determination of velocity, Ax becomes equal to Z. From that he inferred that if there was the least inaccuracy in the determination of velocity, determinism turned into indeterminism.^^35^^

p One cannot agree with Born. Strictly speaking it is a matter here of the transition from a dynamic form of regularity to a statistical one, and that happens, according to him, because there is no absolute accuracy of measurement.

p In this case Born is saying in factjthat whether a regularity is dynamic (when the measurement is accurate) or statistical (when the measurement is inaccurate) depends on the accuracy of the measurement. This approach, however, does not agree with the content of physics. Both dynamic and statistical regularities are objective, and the fact that a falling of a stone is governed by a dynamic law and the Brownian motion of particles by a statistical law does not depend at all on the accuracy of the measurement of the quantities that characterise the object of measurement. It would be nonsensical, for example, to make the measurement of quantities describing the motion of a stone so precise that the precision exceeded the limits of a certain microscopic scale, because beyond that limit a qualitative change occurs in the quantity and it already acquires 165 another physical content. The concept of absolute accuracy of measurement is meaningless if it is employed without taking the concrete content of the measured quantity into account. When this content is allowed for, however, absolute accuracy of measurement becomes a concept with a definite meaning, and it is infinitely extensible refinement of the values of a quantity that gives an absolute, infinitely precise value to the measured quantity built up from an infinite number of values each of which is characterised by limited accuracy.

p To illustrate our point, let us consider Born’s example above. In it the inaccuracy in prediction of co-ordinate at a certain time grows with time and is Az = t&v₀, where Ai;₀ is the inaccuracy in the determination of the initial velocity (which, according to Born, cannot be eliminated). After the critical moment t_c—l/hv_a is reached, where I is the distance between the termini, the inaccuracy or deviation Ax becomes greater than I, and the particle is located somewhere in the interval 0 < x < I (where 0 is the initial point of the interval, and x is the particle’s co-ordinate). When Aw₀ decreases, the critical moment t_c is only moved back, but remains finite for each finite Av₀. Born notes that t_c— oo only for Av₀ = 0, but he excludes an absolutely accurate" value of velocity. In reality, however, Ai>₀ = 01 means passage of Ay₀ to the zero limit, so that his interpretation of the absolutely accurate value of a quantity and his conclusion about the indeterminacy of the final position lose their point.

p Let us now consider the uncertainty principle from a different angle: does it actually disprove determinism, as certain authors state (examples have been given above). To make our discussion more specific, let us consider the uncertainty relation for position and momentum.

p The uncertainty principle is a relation between quantum momentum and quantum position. This relation is a fundamental quantum mechanical law and, like every fundamental physical proposition, has great heuristic value. It was possible, by means of it, for instance, to establish that there are no electrons in the atomic nucleus, and that particles are not in a state of rest at a temperature of absolute zero. Modern experimental technique is sufficiently refined to confirm the truth of this relation, as Blokhintsev has pointed out.^^36^^

p The fact that the uncertainty principle is a relation between quantum momentum and quantum position seems to us to be a crucial point in quantum mechanics for correct interpretation of its physical meaning. The various real and imaginary paradoxes associated with it are then automatically removed. As L. I. Mandelstam stressed: ’the uncertainty relation troubles us just because we call x and p position and momentum and think that it is a matter of the corresponding classical quantities. Let us call x and p quasi-position and quasi-momentum. Then the relation between them will trouble us as little as that between v and ? (frequency and time in wave optics).^^37^^

p The Copenhagen interpretation of course makes use only of classical concepts in describing atomic phenomena. According to Heisenberg it is impossible in general ’to construct the whole physical description from a new "quantum- theory" system of concepts’.^^38^^

p The question arises: why is it impossible? Heisenberg, citing Weizsacker (as was mentioned in Chapter II in connection with another matter), points out that the role of classical concepts in the interpretation of quantum theory is similar to that of the a priori forms of contemplation in Kant’s philosophy. Just as Kant declared the concepts of space, time, and causality (which are a prerequisite of every experience in his philosophy) to be a priori concepts, so (Heisenberg says) the concepts of classical physics are the a priori basis of a quantum-theoretical experiment. At the same time, in his opinion, the a priori concepts of classical physics can be employed to describe quantum-mechanical experiments only with a certain inaccuracy. He arrives, as a result, at the conclusion that it is a question of not nature itself in quantum mechanics but of nature comprehended and described by man through classical concepts that are either innate or obtained through contemplation.

p In this case Heisenberg ignores in his reasoning the dialectics of concepts, which reflects the dialectics of things. The dialectical idea of the mobility and development of concepts has already been employed in science for a long time (unconsciously in the past). Whereas the concept of number used to mean what is now called ’a positive integer’, it was extended and its conlent’enriched by the development of mathematics, which introduced negative numbers, rational numbers, irrational numbers, real numbers, 167 and imaginary numbers. The same thing is happening in physics; its development, for instance, altered an initial meaning of the words ‘light’ and ‘sound’: there is invisible light, and sounds that cannot be heard, the concepts of mass and energy have also been altered, retaining something in common with the initial concepts while acquiring a deeper content.

p Non-classical physics is no exception. As physics penetrates into the realm of phenomena occurring at enormous speeds approaching the velocity of light, and into the atomic world, the classical concepts are necessarily being altered and becoming subject to new, broader, more meaningful concepts expressing the fact that another step has been taken along the path of understanding nature. A well-known example is the behaviour of an electron in an atom, which to some extent resembles the motion of macro-particles; the electron, however, at the same time has wave properties that make it on the whole dissimilar either to a moving macro-particle or a propagating wave.

p When non-classical physics was being constructed and its concepts formed, the following feature acquired essential significance. In a non-classical theory, say quantum theory, the classical concepts are not excluded but are retained; they figure, however, not like those in the conceptual system of classical physics but as an element of a quantum concept being formed. The concept of quantum momentum, for instance, retrospectively, so to say, retains classical momentum in the form of the eigenvalue of the momentum operator.

p This kind of law of the moulding of concepts when a theory that is broader and more meaningful grows out of another one is valid not only for the transition from classical to non-classical physics but also within classical physics itself (development of the concepts of radiation and mass), in mathematics (development of the concepts of number and set), and in many other sciences.

p It is not the business of this chapter to go fully into this question.  [167•*  What we have said is enough to conclude that quantum concepts, as new ones reflecting laws of the atomic sphere, assume the existence of corresponding classical 168 concepts as an element of them; the link between the concepts of quantum theory and instrument readings that inform about real atomic objects is effected through classical concepts. Heisenberg’s argumentation above about classical concepts as the a priori basis of quantum theory in fact records the same thing, but in a mistaken, idealistically Kantian form.

p Let us consider in greater detail whether the uncertainty relation confirms or disproves determinism in physics.

p By itself this relation elucidates the connection between certain quantum quantities and consequently confirms determinism. When, however, it is said that it disagrees with determinism it is not this aspect of the relation that is implied, but something else, namely, that the uncertainty relation renders Laplacian determinism meaningless when the behaviour of atoms is being considered, and that because of this determinism is allegedly bankrupt in quantum mechanics.

p Determinism, however, does not only have a Laplacian form (which corresponds to the motion of macro-particles according to the laws of Newton’s mechanics). Furthermore, neither rejection nor confirmation of Laplacian determinism ,-which is associated with the concept of dynamic regularity in classical mechanics, follows from the uncertainty relation. The latter clarifies the content of the concept of quantum state, i. e. it states that a quantum state is such that the eigenvalues of operators of momentum and position do not exist simultaneously in it. And the quantum state itself, of course, is described mathematically by a wave function that satisfies the Schrodinger equation, and this signifies in the final count that quantum state at a certain moment of time is necessarily connected with a preceding quantum state (i.e. that determinism is valid in quantum mechanics).

p The uncertainty relation would only violate Laplacian determinism in the case when it were instrumental in proving the following: viz., that the initial state (characterised by position and velocity) of a system subject to forces does not unambiguously, in accordance with the laws of Newton’s mechanics, determine the state of the system at any other moment of time. Such proof, however, has nothing to do with the uncertainty relation in the same way as the latter has nothing to do with demonstration of the validity of 169 Laplacian determinism (within the limits of classical mechanics).

p A typical feature of quantum state, namely, its radical difference from classical state, is thus expressed in the uncertainty relation (principle). Although quantum mechanics is a determinist theory, its determinism is not identical with Laplacian determinism or in general with the determinism of classical physics. The uniqueness of the laws and relations of quantum mechanics is determined not by the behaviour of micro-objects excluding determinism’and causality but by their dual particle-wave nature.

p The interpretation of the uncertainty principle (relation) considered here, which assumes acceptance of determinism in quantum mechanics, rests on the fact that quantummechanical concepts are new ones differing qualitatively from those in classical physics. If, on the other hand, it is interpreted from the classical point of view (which is often met in the literature), then grounds are created for idealistic ghosts in quantum theory, including indeterminism. Let us dwell, in conclusion, on this point.

p According to Heisenberg, the uncertainty relation establishes the impossibility of simultaneous determination of the position and momentum of an atomic particle with an arbitrary accuracy. Either its position can be measured with great accuracy, but then knowledge of its momentum is lost to some extent because of the interference of the instrument; or, on the contrary, knowledge of position is lost because of the measurement of momentum. There is therefore a lower boundary of the product of the two inaccuracies, which is determined by Planck’s constant.^^39^^

p By interpreting the uncertainty relation in this way Heisenberg reached the conclusion that ’incomplete knowledge of a system must be an intrinsic ingredient of every formulation of quantum theory’ and because of that ’ quantum-theory laws must be of a statisticalnature’.^^40^^ Furthermore, he stated, the statistical element of atomic physics is incompatible with determinism.^^41^^ This interpretation of the uncertainty relation, which starts from the idea that new quantum concepts do not, in principle, exist in quantum mechanics, cannot be regarded as consistent, as was brought out in the foregoing exposition. Let us now summarise the position.

p (1) The uncertainty relation does not so much establish the limit of errors in simultaneous measurements of the position and momentum of a micro-particle as the boundary of the applicability of simultaneously the concepts of position and momentum (or of the concept of a classical particle) to (say) the electron. This boundary exists not by virtue of the uncontrollable interaction of the micro-object and the macroscopic instrument, but because the uncertainty relation is a relation between quantum momentum and quantum position. In fact, it follows from it in operator form that there is a limit to the applicability of the classical particle concept (from the commutation relation

p P_XX—XP_X = Uli one can mathematically deduce a relation AzAjD,. ^H).

p (2) To cite the fact that the data discovered by atomic experiments do not fit into the theoretical schemes of classical physics in order to assert that knowledge of a microparticle is incomplete (in principle), that the behaviour of a micro-particle is without cause, and so to revise the concept of objective reality, means to commit a philosophical error. The content of the concepts of objective reality, determinism and causality, and cognition cannot be ^r reduced to the content these concepts are associated with in classical theory, or in general in any physical theory whatsoever. These philosophical concepts are immeasurably broader, and no single scientific theory, or science as a whole, can do without them. Niels Bohr, who in one of his last articles did not use the concept of ’uncontrollability in principle’ that he had employed in his earlier work, was inclined in his own way to support this idea. It is relevant to cite again here his idea that we quoted in Chapter II in another connection: ’It may be stressed that, far from involving any arbitrary renunciation of the ideal of causality, the wider frame of complementarity directly expresses our position as regards the account of fundamental properties of matter presupposed in classical physical description, but outside its scope.’^^42^^

p (3) The novel statistical nature of quantum-mechanical laws is by no means the result of our incomplete knowledge of the system being an inseparable part of the formulation of the laws of quantum mechanics (differing in this respect from statistical mechanics, which, according to the erroneous subjective view discussed above,recognised incomplete 171 knowledge of a system only in practice but not in principle). Quantum statistical laws, like those of statistical mechanics, are objective laws. The connections between quantum quantities relating to change of state in time do not have simply a statistical character or only dynamic regularity; and that constitutes the novel feature of the statistical (probability) laws of quantum mechanics. It is incorrect to reduce the laws of quantum mechanics just to statistical laws. The point here is not one of accepting some hidden aspect of statistical laws that are governed by a dynamic regularity. That (i.e. recognition of the primacy of dynamic laws) is as wrong as recognition of the primacy of statistical laws. In quantum mechanics there is a synthesis of statistical and dynamic conceptions of atomic laws, and not their reduction to each other. This feature of quantum laws finds adequate expression in the mathematical apparatus of quantum mechanics. In quantum mechanics it makes sense, for instance, if the eigenvalue of the operator of an electron’s position is determined with absolute accuracy. This also means that the electron (in such and such conditions) occupies a certain position. It would be wrong to consider such examples as only special cases of a more general statistical relation. Without them the very concept of a quantum quantity represented by a mathematical oporator would become quite meaningless.

The uncertainty principle, by bringing out the content of the concept of quantum state, thus poses the question of the statistical nature of physical laws and the nature of probability in physics on a much deeper level than they were posed and solved after its fashion in classical physics. In quantum mechanics the concepts of possibility, probability, and chance are included in the category of the fundamental law, and became integrally interwoven with the concepts of reality, necessity, and regularity.