THE ORIGIN OF UNCERTAINTY IN QUANTUM MECHANICS AND THE PRINCIPLE OF CAUSALITY*

F. Galperin and M. Markov, Moscow

The Metaphysical Conception of Measurement

As is well known, one formulation of the problems of classical mechanics reads as follows:

“If, for some mechanical motion, the equations of motion and the initial conditions for some time \(t_0\)—the momentum \((p_0)\) and coordinate \((q_0)\)—are given, then one can specify, for any moment, the position and momentum of the particle.”

The initial \(t_0\), \(p_0\), \(q_0\) do not follow from the equations of motion, but must be determined by means of measurements.

By measuring the initial data “absolutely exactly,” we can specify with absolute precision the position and momentum of the particle for any \(t\).

In mechanics it has always been assumed that exact measurement of velocity and position encounters no difficulties of principle. By improving measuring instruments, we ultimately approach as closely as desired to “absolute accuracy.”

Let us undertake an analysis of the problem of absolutely exact measurement.

The refinement of measurements is a historical process. From measurements of length by means of Châtelet’s toise (toise de Chatelet, 1668) in France to the measurements of lengths by Michelson and Benoît (1894) by means of wavelengths of light, the distance is quite great.

Each epoch is characterized by a certain degree of measurement accuracy possible at the given stage of development of technology and science.

And therefore it is very important that, on the one hand, every problem has a character determined in some part by those

* A revised transcript of a report read by the authors at the Communist Academy on November 30, 1931.

physical concepts* that predominate at a given moment, but, on the other hand, the physical concepts that dominate in that epoch are most closely connected with, and in turn are also determined by, the degree of accuracy of the measurements of that time.

And the accuracy of measurement, as it improves, ultimately comes into conflict with the established physical concepts, insistently demanding that they be changed or even completely banished from science as having become “unscientific.” But thereby the task of measurement itself also changes.

Of course, every measurement concerns an object, a concrete thing. Of course, in the formulation of measurement problems in physics, technology and economics are the impelling and determining factors; but the concept of the given concrete thing that exists at that moment in science leaves its well-known imprint on the concrete task of measurement, on its anticipated results, and on the character of the formulation of the problem. The concept reflects only within certain limits the real thing that actually takes place in reality.

The study of this reality is accompanied by the development of our concepts of it; the concepts change, but then the task of measurement itself also changes.

At a certain stage of knowledge about the table, we introduce,

* As regards the principal determining factor, the economic one, we shall not dwell here on it in detail, since the aim of the present chapter is not the study of the methodology of measurement in general, but only to make several cursory remarks on this question, namely those that are necessary for elucidating the main tasks of the article. The reader will find a very vivid historical note on the role of economic factors in the question we are discussing, in part (as applied to electrical engineering), in one of Helmholtz’s speeches (Helmholtz, Vorträge und Reden, II B., S. 321).

“Electrical engineering has gradually developed so strongly that at the present moment enormous capital has been invested in it, and it constitutes an exceptionally lively industry.

Under these circumstances there can be no lack of disputed questions coming before the courts, and there is felt a special need for establishing a unit of measurement on the basis of which correct decisions can be rendered.

When a manufacturer undertakes to deliver wire for wiring, it is essential that the resistance of the wire not exceed certain limits, and that it be possible to arrive at a just decision as to whether the wire corresponds to the conditions of the contract.

The same applies to another manufacturer, who undertakes to build a dynamo-electric machine, which must rotate, upon applying a definite speed of rotation, produce a definite electromotive force; it is necessary, therefore, to arrive at a measure for the electromotive force of machines,” etc.

See specifically A. A. Maksimov’s article on the methodology of measurement, “Under the Banner of Marxism,” No. 7–8, 1929.

for example, the concept “width of the table”; this concept, at a certain stage of measurement, has a definite meaning and, generally speaking, emphasizes, encompasses, a certain aspect of the reality under study.

Subsequently it turns out that the concept of the width of the table, beginning from a certain point, loses its “exact” meaning (molecular motion), and consequently the original problem of measurement becomes devoid of meaning.

The experiments known in the history of physics concerning the ponderability of caloric, of course, corresponded to and were in part determined by those notions which prevailed in that epoch; within the framework of these concepts the question of whether caloric had weight or was weightless was appropriate, and the corresponding experimental problem was appropriate. In later epochs of the kinetic theory of heat these measurements lose all meaning; but with an increase in the precision of the corresponding measurements and on the basis of the principle of relativity \((E = mc^2)\), the experiment can again be posed in principle, though now on a different basis.

It is known to the physicist better than to anyone else that measurement is a problem not only quantitative but also qualitative. What is always measured is not some abstract quantity, but in every case the quantity “of something.”*

Every problem of measurement is posed not only in a definite historical epoch, but also within the concrete setting of a definite form of the motion of matter (the problems of classical mechanics, thermodynamics, electrodynamics, etc.), and at the point where we, while “refining” measurements, pass into the domain of another form of the motion of matter, the problem becomes devoid of meaning, for it is blood-bound to its own “medium,” to that form of the motion of matter which has remained behind somewhere in the stages of measurement already traversed. (The same examples: the measurement of the temperature of an electron, or the measurement of the “speed of sound” in an iron atom.)

To study phenomena, as the history of physics also shows, means, in the final analysis, to study the formation of phenomena; but any problem of measuring the phenomenon under study, brought to the moment, to the place, where the given phenomenon becomes such—there and beyond these limits—becomes indefinite and simply devoid of meaning, for here there is only being formed, or has not yet even been formed, that which is subject to measurement.

For every phenomenon that arises, at the sources of its aris—

* Maksimov, Methodology of Measurement, PZM, No. 7–8, 1929; Marx, Capital, vol. I; Hegel, Logic.

...of emergence at first possesses very vague, unclearly expressed properties, like the laws of heat studied on two or three moving molecules, and the problems of measurement that arise where these properties have already developed into full-blooded physical categories here usually become indeterminate; and further, having crossed the threshold of formation into a new form of matter which becomes, if one may so put it, the arena of the emergence of the phenomenon under study, they lose all meaning.

It is senseless to ask whether an atom of a substance is “liquid,” “solid,” or “gaseous.”

By all this we wish to say that the problem of mechanics posed at the beginning of this paragraph, i.e., the problem of determining exactly, for example, the motion of an electron from exactly measured initial data and the law of motion, with the further progress of physics may, beginning with a certain stage of measurement (the measurement of the initial \(p\) and \(q\)), become indeterminate—which would not be at all surprising in view of the rich material which contemporary physics provides us concerning “absolutely exact” measurement.

\[ * \quad * \]
\[ * \]

In the methodology of measurement one can also trace how the hyperbolic exaggeration of some one aspect in the general problem of measurement leads to a corresponding philosophical error. The fact that a measurement problem, once posed, becomes continuously, within broad limits, more and more accurately executable, is made so habitual that the possibility of attaining some “absolute ideal” seems almost obvious; there appears a habit of regarding measurement merely as a qualitative problem, or else the “qualitative content” of which (the problem) begins to be considered independent, so to speak, of quantitative formulation, i.e., it is tacitly assumed that what is measured can be taken in arbitrarily small quantities.

But no less dangerous in methodological respect is the excessive emphasizing of the “failure” of absolutely exact measurement,* which can lead to other “idealistic vacillations.”

From the fact that it is impossible to measure exactly the width of a table, one may come to the conclusion that in general this concept has no meaning, forgetting that at a known stage of measurement it grasps and characterizes certain features of objective real-

* I.e., to fall into philosophical relativism.

...ness; this concept may be interpreted as “unscientific,” for it is “not exact,” it is relative, at a known stage only “apparent.”

But since such is the fate of every physical measurement, the entire objective world also begins to become apparent, merely relative, opening the way to all sorts of philosophical speculations.

• *  *

Thus, the metaphysical “absolutely exact measurement” is impossible, not because it is in principle inaccessible to cognition, but because, beginning from a certain moment, the objectively concrete physical problem—the problem of refining a measurement—loses, in the nature of things, its former definiteness.

But, on the other hand, every problem of measurement, once posed, is refined more and more, tending toward a certain limit, where all “exactness” is exhausted to the end, i.e. where measurement “with greater exactness” ceases to characterize the object under study, objectively loses meaning; by this very fact the question of any “inexactness” in things themselves is removed, and the task is posed of striving toward certain limiting, genuinely “absolutely exact measurements,” but now in a new, real, not metaphysical, but “physical” sense, i.e. toward such “limiting” measurements beyond which it is impossible to measure more exactly in the given concrete problem—we emphasize, not “impossible,” but “nothing”*.

THE UNCERTAINTY RELATION

Quantum mechanics proceeds from the fact that in the region of the microcosm measurement always introduces essential changes into the state of the observed object. As an example, observation of a bound electron revolving around the nucleus of an atom is sometimes cited. Consider, for example, a hydrogen atom in the unexcited state. In this case the size of the atom is equal to \(10^{-8}\) cm. Suppose we determine the position of the electron in its orbit, for example, with an accuracy up to \(10^{-9}\) cm. But a single quantum of light of this same order of wavelength is sufficient for the electron, by virtue of the Compton effect, to be ejected beyond the limits of the atom.

Thus, the next observation of the same state is impossible. We emphasize that here the point is not only the destruction of the system under observation. What is essential is that the changes

* To measure “nothing” not “in general,” but for the given problem; in general, however, other problems appear here with their own tasks of measurement.

F. GALPERIN AND M. MARKOV

into states that occur during observation, within known limits, are not controllable.

Quantum mechanics treats the electron as a material particle.*

But, on the other hand, in broad areas, for the “description” of phenomena associated with the electron, the concept of a wave is used.

If we have a free electron with momentum \(mv\), then, according to de Broglie,

\[ \lambda=\frac{h}{mv}, \]

where \(\lambda\) is precisely the corresponding wavelength. And from this side, a purely wave description of the phenomenon gives absolutely no right to speak exactly, for example, about the position of the electron, for plane monochromatic waves begin nowhere and end nowhere, but fill “all space,” not singling out in any way a point or region in which the electron is located. The only possibility of singling out such a region is to choose a set of waves in such a way that, as a result of superposition, they “cancel” one another everywhere except in a narrow region where, on the contrary, their amplitudes add. Such a formation in quantum mechanics is called a wave packet. A wave packet has the form shown in Fig. 1. But in order to obtain a packet, it is necessary that the wavelengths composing the packet differ somewhat from one another, and the packet will be, as it is proved, the narrower the greater the difference in the wavelengths of the waves from which it is built.

Fig. 1.

The particle itself, the electron, may be, for example, at point \(A\) or at point \(B\), etc., lying inside the packet, but where exactly is not known.** Consequently, the uncertainty in determining the location of the particle is deter—

* The grounds for such an interpretation will be discussed below.

** Quantum mechanics, in contrast to classical mechanics, does not assign exact values of the positions and momenta of the electron, but assigns their distribution. For example, from \(n\) observations made on the momenta and positions of the electron, \(m_1\) observations refer to a certain result \(v_1\); \(m_2\) to a certain result \(v_2\), and so on. If this is represented graphically, i.e. plotted on a drawing, then a curve of the distribution of positions and momenta is obtained. Thus here the fundamental difference between the approach of quantum and classical mechanics is visible. The former, from the very beginning, takes the standpoint of a probabilistic specification of position and momentum, and not of an exact one, as in classical mechanics.

is the width of the packet, the interval \(\Delta q\). But does this mean that it is impossible to specify the position, the coordinate of a particle with any accuracy in the sense of classical mechanics? No, it does not. For this it is only necessary that the width of the packet be as small as desired. And this means that among the wavelengths composing the packet there must also be some that differ very greatly in length from one another.

Let us introduce* a relation indicating the order of magnitude of the product of the two inaccuracies. This relation is called the “relation of inaccuracies,” or the “uncertainty relation.”

We have already seen that a wave packet is a collection of waves whose wavelengths differ by a small amount. Let us form a wave packet. Let its width be \(\Delta q\). Suppose that we have waves whose wavelength is \(\lambda_0\) and which fit into the interval \(\Delta q\) in the number \(n\). But in order that outside this interval the wave field be equal to zero, it is obvious that other waves besides those indicated earlier must also fit into \(\Delta q\), with which the latter interfere in such a way that outside this interval the wave field is absent. From the construction one can establish that the necessary condition for carrying out this kind of interference consists in these other waves, having, say, wavelength equal to \(\lambda_1\), fitting into this interval, at least, in the number \(n+1\). Consequently, the necessary condition for constructing the packet is the following:

\[ \Delta q=\lambda_0 n=\lambda_1(n+1);\quad \lambda_1=\lambda_0\frac{n}{n+1}. \tag{1} \]

From this it is easy to derive the “relation of inaccuracies.” According to de Broglie’s theory, to each electron there corresponds a wavelength

\[ \lambda=\frac{h}{p}. \tag{2} \]

Consequently, one may write

\[ \lambda_1=\frac{h}{p_1};\quad n\lambda_0(p_1-p_0)=h;\quad \text{or}\quad \Delta q\,\Delta p=h. \tag{3} \]

Here \(\Delta q\) is the width of the packet and represents the inaccuracy in determining the coordinate of the electron; \(\Delta p\) is the inaccuracy in determining its momentum.

The meaning of this relation, as quantum mechanics asserts, is that it is impossible simultane—

* The derivation is simplified. A more rigorous one is given in Die Physikalischen Prinzipien der Quantentheorie by Heisenberg.

...an exactly simultaneous determination of both momentum and coordinate. The determination of these quantities is associated with inaccuracies whose product is of order \(h\).

If relation (3) is rewritten in terms of other canonically conjugate quantities, for example \(E\) (energy) and \(t\) (time), then it is rewritten as:

\[ \Delta E\, \Delta t \geq h, \tag{4} \]

It is customary, in order to illustrate the correctness of this relation, to cite a whole series of so-called “thought” experiments. We shall consider only two of them.

Microscope

An electron \(e\) is located under the objective of a microscope, as shown in Fig. 2. It is illuminated by quanta of light of frequency \(\nu\).

Fig. 2.          Fig. 3.

Having struck the electron, a quantum of light changes its initial momentum. The direction of the new momentum of the electron is not known exactly, for the direction of the recoil quantum scattered by the electron is not known exactly. In order to see the electron, it is necessary that at least one scattered quantum pass through the tube of the microscope and reach the observer’s eye. The direction of the momentum of this quantum lies within the angle \(\varepsilon\).

The uncertainty in the determination of the momentum of the electron is equal to:

\[ \Delta p = \frac{h\nu}{c}\sin \varepsilon. \]

According to the laws of optics, the coordinate \(x\) can be measured, in the very best case, with an accuracy

\[ \Delta q = \Delta x = \frac{\lambda}{\sin \varepsilon}. \]

This relation gives the resolving power of the microscope. Hence it is clear that the product of the two uncertainties is a quantity of order \(h\).

Diffraction Slit

A moving electron with known momentum \(p\) passes through a slit \(d\) (Fig. 3). A narrow slit, the more precisely...

we can fix the position of the electron*. Thus the inaccuracy in determining the coordinate of the electron is given by the width of the slit:

\[ \Delta q = d. \]

But a narrow slit will cause diffraction of the electron waves. The inaccuracy in determining the momentum of the electron in the direction \(d\) is determined by the magnitude of the component of the momentum in the direction \(d\) and is equal to:

\[ \Delta p = p \sin \alpha. \]

It is known from optics that \(\sin \alpha = \frac{\lambda}{d}\). Consequently, \(\Delta p\) is equal to \(\frac{h}{d}\), and the product of the two inaccuracies is equal to:

\[ \Delta p \Delta q = h, \]

i.e. we have again obtained the “uncertainty relation.”

Here is an example of the reasoning of certain physicists:

“Uncertainties entering into this relation are fundamentally different from those with which classical physics dealt. The latter considered uncertainties of a technical order, dependent on the imperfection of measuring technique. It was assumed that the accuracy of measurement would become ‘absolute’ if the technique of experimentation became perfect. However, quantum mechanics asserts that there exists a fundamental limit to the accuracy of measurement, which follows from the very essence of physical processes”**. “If in classical mechanics the exact determination of the initial conditions was unattainable practically, there was no doubt that, in principle, the conditions could be determined with any accuracy whatever. In the new quantum mechanics, the determination of the initial conditions becomes not only practically difficult, but also fundamentally impossible”***.

In macroscopic measurements uncertainties of both kinds occur. But why are fundamental uncertainties not found in experiment? The point is that they are covered over by measurement uncertainties****. De Broglie***** calculated the fundamental uncertainty connected with the measurement of a macroscopic body—a moving ball weighing 1 mg. To determine its state at some moment, it is necessary to know the coordinate of its center of gravity and its velocity,

* See more on this below.
** See De Broglie, Einführung in die Wellenmechanik.
*** Schrödinger. Speech on the occasion of his election as a member of the Prussian Academy of Sciences.
**** See the previously cited work by de Broglie.
***** Ibid.

Suppose that the coordinate of the center of gravity is determined with an accuracy of up to \(0.001\) mm. This is enormous accuracy. Substituting these quantities into the “uncertainty relation,” we obtain for the uncertainty in determining the velocity of the ball the following value:

\[ \Delta v=\frac{h}{\Delta q\cdot m};\qquad \Delta v=\frac{6.55\cdot 10^{-27}}{10^{-5}\cdot 10^{-1}} =6.55\cdot 10^{-20}\ \text{cm/sec}. \]

It is obvious that in practice there is not a single method that would give the velocity with such accuracy. The experimental uncertainty exceeds the fundamental one, and everything proceeds as if the latter did not exist at all.*

The uncertainty relation does not contradict the principle of causality

It is now appropriate here to pose the question: what, properly speaking, is being criticized in the principle of causality by contemporary physicists and philosophers, and what grounds for such “criticism” are provided by the concrete material of the new quantum, wave mechanics?

The basis for criticism of the principle of causality in quantum mechanics, as we know, is the proposition which may be briefly formulated as follows: “in principle no experiment can give the initial \(p_0\) and \(q_0\) for an electron simultaneously so precisely that the law of motion could be applied for an exact prediction of its \(p\) and \(q\) at any moment of time \(t\). These limits, which restrict accuracy, as we have seen, are given by the relation: \(\Delta p\,\Delta q \geq h\), or \(\Delta E\,\Delta t \geq h\).

It is now necessary to preface the discussion with several remarks concerning how, in physics, problems of the unambiguous determination of the course of physical phenomena in time are usually posed.

To solve such problems, as we know, one always proceeds from certain “initial conditions” and from a certain “law of motion.” Moreover, if the law of motion and the initial conditions are exactly the same for two or several problems, then the same result everywhere obtains; that is, we are dealing with the unambiguity of causal connection.

It is clear that both the initial conditions and the law of motion are different for different forms of motion (mechanics, thermodynamics).

If, for example, in Newtonian mechanics or Einsteinian mechanics the initial conditions are certain values of the momentum \(p_0\) and the coordinate \(q_0\) at the moment of time

* See the previously cited work of de Broglie.

$t_0$, and the law of motion is some relation between momenta and coordinates and their derivatives, then for the solution of certain heat-conduction problems one needs the heat-conduction equation and initial conditions in the form of specifying the initial distribution of temperatures; entirely different conditions are required to answer the question of how a wave propagates. Thus, we see that for an unambiguous determination of the course of a phenomenon in time in physics it is necessary to know, first, the characteristic of the “state” of this phenomenon and the so-called law of motion. After all that has been said about the so-called initial conditions, the question now immediately arises: to what extent can momentum and coordinate objectively characterize an electron?

If, according to Schrödinger, the electron as a discrete particle, generally speaking, does not exist, then in the statistical interpretation of wave mechanics the electron is assumed to be a particle, more precisely—a material point with a definite momentum, i.e. the electron is characterized by the same six quantities $(p_i, q_i,\ i = 1, 2, 3)$ as a material point in Newtonian mechanics. At least the examples usually adduced, as we have seen, in defense of the uncertainty relation are calculated in just this way.

The calculations presuppose, as we saw above, a certain “super-observer,” to whom the momentum and coordinates of the electrons are known “exactly” at one and the same time, but the result, owing to the uncertainty relation, turns out to be indeterminate for an ordinary observer.

Now, first of all, it is necessary to note the following: if the question is posed of the principle of causality as of a certain objective category, then it is not difficult to see that the uncertainty relation in no way gives one the right to deny this principle even in the case where one agrees with the assertion that the momentum $p$ and the coordinate $b$ exactly characterize the particle.*

Indeed, if causality is criticized as an objective category, then inevitably in place of objective causality one puts objective acausality, and, first of all, we must demand a definition of what is to be called objectively acausal.

From the standpoint of formal logic, one could speak of a critique of the principle of causality when it had been, for example, “proved” that for one and the same exactly measured (in the classical sense) initial data $p_0$ and $q_0$ in

* If, however, $p$ (momentum) and $q$ (coordinate), already as applied to the electron, lose their objective meaning and must be replaced by some other characteristics, then the uncertainty relation generally ceases to be a problem (see this below).

at the moment \(t_0\), under one and the same law of motion, the theoretical calculation of the state of material particles at some moment \(t\) gives one and the same result for all cases, while a control, absolutely exact (in the classical sense) experimental verification of the predicted result would give the most diverse values, although the conditions of all the experiments were absolutely identical. This would be the definition of “objective acausality” with respect to the motion of a material point. This is precisely what the uncertainty relation does not give.

Thus, if, on the one hand, we accept the uncertainty relation and, on the other hand, interpret the principle of causality as an objective category, then even from the standpoint of formal logic the uncertainty relation neither refutes nor confirms the principle of causality. Moreover, it even excludes an “exact” verification of the principle by the fact that it excludes the possibility of an “exact” measurement of the initial conditions.

The agnostic point of view on this question is developed very vividly in Dirac’s book The Principles of Quantum Mechanics. The last remarks do not resolve, but merely clarify, the question* . It is now necessary to determine whether momentum and coordinate characterize the state of the electron.

Do \(p\) and \(q\) characterize the electron?

Be that as it may, we have found ourselves faced with the fact of the fundamental impossibility of predicting the “future” of the electron by the method customary in mechanics—not because the principle of causality has been violated, but because we cannot simultaneously measure the momentum and the coordinate with sufficient accuracy.

Thus we see that purely physical propositions do indeed provide grounds for certain agnostic statements.

It is now appropriate to pose the question:

How justified is the assertion that the objective concepts of “exact” (definite) momentum and coordinate are indeed applicable to the electron at one and the same moment, i.e., is not the uncertainty relation precisely the limit of applicability of the concepts of momentum and coordinate, beyond which these concepts become indeterminate—in other words, are subjected to the fate, usual in physics, of an “absolutely exact” measurement?

Let us give an example. Suppose we are given a uniformly ac—

* In the sense that, even formally logically, the uncertainty relation does not imply a denial of the principle of causality.

third metal plate, and it is required to calculate the distribution of temperatures on it at some moment \(t\). At our disposal we have the heat-conduction equation and a set of thermometers of every possible accuracy for determining the initial distribution of temperatures. Suppose that, in striving for an absolutely exact measurement, we set ourselves the aim of measuring the temperature in a region of space comparable with the size of a molecule. By this very fact we are still immeasurably far from obtaining absolutely exact initial distributions, for the differential equation requires, strictly speaking, “ideally,” that the temperatures of “all points” of the plate be found; but, after all, even with our imaginary thermometer such a confusion may occur that nowhere, in any single measurement, will even one molecule at the instant of measurement strike our thermometer; i.e., the task of “exact measurement” and, consequently, of predicting the future distribution of temperatures with such accuracy, has no meaning.

In principle, as we shall see below, the case just cited is in no way different from what we have in wave mechanics. However, here at present no one connects criticism of the principle of causality with these facts, because it is clear that it is not the principle of causality that loses its meaning here, but the requirements that we have presented to the problem concerning the very essence of heat as molecular motion.

Thus, it is now necessary to clarify to what extent \(p\) and \(q\) characterize the electron and its state, to what extent \(p\) and \(q\) give knowledge of the “present” of this form of motion of matter, knowledge that is necessary, as we have seen, for determining the behavior of the phenomenon in time.

The Uncertainty Relation as a Relation of Interaction

Usually the uncertainty relation is derived, for example, in the form \(\Delta p\,\Delta q \geq h\), and it is said that the errors in the knowledge of the momentum and simultaneously of the coordinate are connected in precisely this way; then physical grounds are sought which lead exactly to such a connection between the accuracy of determining the momentum and the accuracy of determining the coordinate. This connection is found in the character of the particular interaction (on this, it would seem, all physicists agree) which occurs during observation between the observed object and the observing apparatus. It is established that interactions occur precisely of such a kind that, by observing the position of a particle as accurately as desired, we risk changing its momentum as strongly as desired; i.e., in the final analysis, the presence precisely of such a cha-

the character of the interaction of the observing apparatus with the observed object leads to the uncertainty relation. Therefore, in many respects it might be more correct to proceed in the opposite way: to establish certain aspects of the interaction which classical mechanics evidently left in the shadows or, at any rate, represented inaccurately—for example, by the classical concepts of momentum and coordinate—and then, for the interaction of the observing apparatus with the observed object as a particular case of physical interaction in general, to obtain the same uncertainty relations.

Apparently it is in this direction that the development of this question is proceeding. The second relation \(\Delta E \Delta t \ge h\) can no longer now be interpreted as simply as was done in Heisenberg’s 1927 paper: as a simple connection between the corresponding errors of observation and \(\Delta t\). Rather, what corresponds here is the corresponding duration of the experiment, the corresponding duration of that same interaction of the observing apparatus with the observed object. In this direction, it seems to us, goes Bohr’s interpretation of this relation; in the same direction, as it were, goes the work of Landau and Peierls.

True, at present one must rather proceed from the uncertainty relation to the establishment of certain features of those interactions which lead to the uncertainty relation; but the ultimate goal appears to us to be precisely that which we formulated above.

But let us turn to our relations.

The mere assumption that the measurement of a coordinate itself, for example, causes a change in the momentum is wholly insufficient for interpreting the uncertainty relations.

Here there are still no fundamental differences in the posing of the question of changes in the new quantum mechanics from the analogous posing in classical physics.

True, classical physics always tacitly admitted the possibility of such an organization of the experiment that under these conditions this phenomenon (i.e., the influence of the measurement process itself) becomes arbitrarily small; and in this sense the relation of inaccuracies establishes limits which did not exist in classical physics.

But this is not the main point.

Classical physics, in general form, never denied the influence of the measuring apparatus on what is being measured, but in it there was always, at worst, assumed the fundamental possibility of taking this influence into account.

The new quantum mechanics establishes limits to this

possibility. If, for example, we know exactly the momentum of a particle before observing its position, then at the moment of observation we alter, to some extent, the former momentum. And this alteration at the moment of observation cannot be measured exactly. This alteration is said to be uncontrollable.

The principle of uncontrollability, if one may say so, is the soul of the uncertainty relation in its modern interpretation.

This principle is quite foreign as a general principle to classical physics, and here lies precisely what is fundamentally new in the formulation of the question of the measurement of certain physical quantities.

Above we have already mentioned that in the usual derivations of the uncertainty relation (from the dualism of waves and corpuscles directly, or from a certain inequality for a function satisfying certain conditions, etc.) only the existence of a connection between the corresponding errors \((\Delta p, \Delta q)\) is established, but it is not possible to trace the dynamical interpretation of this connection.

The dynamical interpretation is given by separate postulates. (Measurement of position changes the momentum... etc.)

Likewise, the concept of uncontrollability is not organically connected with the usual derivations of the relation; it does not follow from the general propositions underlying those derivations.

Let us look more closely at the use of the concept of uncontrollability.

A consequence of uncontrollability is the formulation: it is impossible simultaneously to know exactly the momentum and the coordinate of a particle. In the sense of the relation, at different moments of time, separately, we can know with arbitrary accuracy only the momentum and only the coordinate. Suppose we are concerned with observing separately the momentum and the coordinates of a particle at the moments \(t_2\) and \(t_1\), where at the outset \(t_2 - t_1 = \Delta t\) is sufficiently large \((t_1, t_2\) being the times indicated by the pointers of the measuring instruments).

It is very important for the further discussion of this question that the influence of the observation of momentum and of coordinate on the object observed is in a certain sense not symmetric: whereas an exact observation of position leads to very serious disturbances in the observed system (the momentum changes sharply), observation of only the momentum can be carried out almost entirely without penalty for the objective course of the process being observed. When observing the momentum, we are said to “lose our former knowledge of the particle’s position,” but the objective course of the process may nevertheless remain undisturbed. For example, by using the Doppler effect to measure momentum, we can illuminate the electron with very long waves and thereby make the influence of the Compton effect as small as desired.

Therefore, if we observe the momentum and the coordinate in time separately, then, depending on whether we observe the momentum before or after observing the coordinate, we shall obtain completely different results for the value of the momentum. The point is that, by measuring the position, we change the momentum, and then the measurement of the momentum gives that value of the changed momentum which the particle has already after the observation of the position.

The reverse order of experiments, however, leaves open the question of how the momentum has changed under the influence of the observation of the coordinate.

In the first case the possibility of control* is obtained only at the expense of the non-simultaneity of the measurement: our experiments are separated by a time interval \(\Delta t\).

In discussing the uncertainty relation, in nonrelativistic quantum mechanics one usually does not touch upon the question of the role of time in the process of measurement. It is assumed that the measurement of physical quantities can be carried out in an arbitrarily small interval of time, so that it makes sense to speak of the exact value of a physical quantity at a given instant.**

It is assumed, therefore, that at any instant we can exactly measure, for example, only the coordinate or only the momentum.

It is easy to show that such an assertion (the possibility of an exact instantaneous measurement), with respect to momentum and energy, contradicts the same relation \(\Delta p \Delta q \geq h\).

Indeed, suppose that we have the inertial motion of a particle and that at the instant \(t_1\) we have exactly measured its position, and after this, at the instant \(t_2\), also exactly its momentum.

The instantaneous measurement at \(t_1\) gave exactly the position of the particle \(q_0\) and instantaneously changed the velocity.

The instantaneous measurement at \(t_2\) exactly determined the momentum \(mv\), a momentum which pertains exactly to the instant \(t_2\).

The measurement exactly determined the momentum. This means that it is precisely this value that the particle’s momentum has immediately before and after the measurement in all cases of the particle’s motion. For a free particle, an exact measurement of the momentum at the instant \(t_2\) gives that momentum which the particle possesses, beginning from the instant \(t_1\), all the time before and after the second experiment.

If this is true, then by the instant \(t_2\) we could also know the coordinate of the particle just as exactly:

\[ q_{t_2}=q_0+v(t_2-t_1), \]

* Control of the momentum.
** V. A. Fok, Principles of Quantum Mechanics.

which would contradict the relation:

\[ \Delta p\,\Delta q \gtrsim h. \]

Usually here uncontrollability is invoked for help. The assertion that the measurement of momentum is uncontrollable destroys our previous knowledge of the position. But what does it mean: that an observation destroys our previous knowledge of the position? It means only one thing: that as a result of the second observation, immediately after it the particle may, generally speaking, be at any point of space, however far removed from \(q_0\), from the position which the particle occupied immediately after the first experiment (at the moment \(t_1\)).

But the particle can get from \(q_0\) to any \(q\) as a result of the second observation only by moving with some velocity \(v\) during some interval of time \(\Delta t\).

Consequently, the uncertainty \(\Delta q\) in \(q\) at the moment \(t_2\), \(|q=q_0+v(t_2-t_1)|\), can be obtained, with exact knowledge of the position at the moment \(t_1\) (\(q_0\)), only at the expense of an error in the time \((t_2-t_1)\) or at the expense of an error in the determination of the velocity. There is no other choice.

Either \(\Delta t=t_2-t_1\) is measured with an error \(\sigma\), or the velocity \(v\) does not exactly correspond to the actual velocity.

The first assumption does not withstand any criticism: \(\sigma\) cannot be an error in time. Indeed, this would give an error in the coordinate:

\[ \Delta q=v\sigma, \]

\(\sigma\) must satisfy the relation:

\[ v\sigma\,\Delta p \gtrsim h. \tag{a} \]

Relation \((a)\) must be valid for an exactly measured momentum \((mv)\), for finite \(v\) (with the result of the momentum measurement also known exactly), which leads to the absurd requirement \((\sigma=\infty)\), i.e. if, when measuring the coordinate, the clocks indicated exactly the time when the apparatus finished the measurement, then when measuring the momentum the same clocks for some reason must necessarily make an unimaginable error \((\sigma\to\infty)\), and this error in the clock reading must be the smaller the greater the error the apparatus gives in the momentum, which, of course, cannot be, for the clocks are in no way connected with the apparatus measuring the momentum.

It thus becomes necessary to assume, in order for the inequality \(\Delta p\,\Delta q \gtrsim h\) to be fulfilled, that the very measurement of the momentum changes somewhat the measured momentum. This is, generally speaking, so: whatever long waves we may—

illuminated the electron, observing, by the Doppler effect, its momentum, there will still always be the corresponding Compton effect, by virtue of which the momentum changes somewhat.

Let, during the time \(\Delta t\), the average velocity of the particle be \(v+\Delta v\); \(v\) is the value of the velocity which the apparatus indicated; \(\Delta v\) is the error in the velocity as a result of the observation. Then the error in the position at the moment of completion of the second experiment is:

\[ \Delta q = \Delta v\,\Delta t. \]

But, one asks, what prevents us from making the interval \(\Delta t\) between the measurements of coordinate and momentum arbitrarily small?

Analyzing the physical conditions of observation, we may assert that only in the best case, while preserving the same sequence of observations, can we begin the second observation, i.e. the observation of momentum, immediately after the observation of the coordinate has been completed. Then the interval between the two readings of the observation instruments will, in the limiting case, be equal to, and generally speaking greater than, the time necessary for carrying out the second experiment. Consequently, \(\Delta t\) cannot be less than the duration of the second observation.

The duration of the experiment is composed of the time of interaction of the apparatus with the observed object and the time necessary for registering this interaction by the pointer of the instrument. Assuming that the transmission mechanism works ideally, only in this ideal case can we identify \(\Delta t\)—the duration of the experiment—with the duration of the interaction of the apparatus with the particle.

If \(\Delta t\) is the duration of the interaction of the apparatus with the particle, then \(\Delta v\) is nothing other than the mean change of velocity during this interaction.

But \(\Delta q\) must satisfy the relation:

\[ \Delta p\,\Delta q \geq h, \tag{β} \]

therefore,

\[ \Delta p\,\Delta v\,\Delta t \geq h \tag{γ} \]

or

\[ m\,\Delta v^{2}\,\Delta t \geq h. \tag{δ}* \]

* It is not superfluous to recall how, in determining momentum in thought experiments, “the old knowledge of the particle’s position” is lost, and how the error in position is obtained.

Let us recall the experiment discussed, for example, by Heisenberg in Die Physikalischen Prinzipien der Quantentheorie.

The point is to determine the velocity of an electron from its deflection in a magnetic field. The electrons fly out of the first slit, pass a distance \(a+l\), and are caught by the second slit. Let \(a\) be traversed in the magnetic field, \(l\) in free motion. The error in position is calculated as follows: if the electron flew with velocity \(v\), then the experiment lasts, in order to determine the velocity, for a time equivalent to

Thus we see that the relation:

\[ \Delta p\, \Delta q \gtrless h \]

in its dynamical interpretation (the more precisely the position is measured, the more strongly the momentum changes) does not yet lead to uncertainty.

Relation \((\beta)\) becomes an uncertainty relation from the moment when the condition of uncontrollability is introduced; and the condition of uncontrollability, in turn, leads to the relation:

\[ m(\Delta v)^2 \Delta t \gtrless h, \]

the consequence of which in the present case is uncontrollability. And in general the relations \((\beta)\) and \((\delta)\) only together, as we see, give uncertainty.

Thus, not only does a measurement of position change the momentum, but a measurement of momentum also changes the momentum of the particle; and, most importantly, the shorter the duration of the experiment, the shorter the time of interaction of the apparatus with the particle, the more the momentum changes.

Relation \((\delta)\) shows that an exact observation of momentum is possible only in an infinitely long experiment.

At least,

\[ \Delta t=\frac{a+l}{\Delta v}, \]

and if the observation of the velocity gave an error \(\Delta v\), then the inaccuracy in the value of the position of the particle after the impulse is at least equal to:

\[ \Delta q=\frac{a+l}{v}\,\Delta v;\qquad \Delta q=\Delta t\,\Delta v. \]

Consequently, the more precisely the momentum is determined \((\Delta v \to 0)\), the longer the duration of the experiment \(\Delta t\) must be, other conditions being equal; and conversely. We thus see that for this experiment it is necessary that \(\Delta t\) satisfy the inequality:

\[ \Delta p\, \Delta v\, \Delta t \gtrless h. \]

The same is true in determining the velocity of a particle by means of the Doppler effect (see, for example, L. de Broglie, Introduction à l’étude de la mécanique ondulatoire).

With whatever long waves we illuminate the particle, one may still expect a Compton effect, changing during the observation of the particle’s velocity its velocity itself by \(\Delta v\). The duration of the experiment cannot be less than

\[ \frac{l}{c}, \]

where \(c\) is the speed of light, and \(l\) is the width of the light signal, which cannot be less than the mean wavelength \(\lambda\). Consequently,

\[ \Delta q=\Delta v\,\Delta t=\Delta v\,\frac{l}{c}, \]

and the duration of the experiment cannot be less than the period of oscillation of this \(\lambda\).

But, decreasing the period, we increase the frequency \(\nu\), and consequently the energy by which the particle’s energy and, therefore, its momentum may change.

Thus, it is precisely the requirement that the relation \(\Delta p \Delta q \gtrsim h\) be fulfilled that leads to these results; it is precisely the relation \(\Delta p \Delta q \gtrsim h\), as we have seen, that insistently requires certain conclusions to be drawn concerning the role\(^*\) of time in the process of measuring momentum.

If, in order to solve a problem of mechanics, it is necessary to consider the motion of a wave packet and its change in time, then for the physical interpretation of the relation as such only the space-time dimensions of the packet are essential, i.e. precisely the packet which was obtained immediately after the measurement, and not the one which subsequently changed with time.

If one permits oneself to choose one’s own coordinate system, i.e. a coordinate system in which the electron is at rest before the observation, then in this coordinate system all the quantities entering into relations \((\delta)\) and \((\beta)\) receive a very simple interpretation: if \(\Delta t\) is the duration of the interaction, then \(\Delta E\) is the change of energy during this time; if the average velocity during the time \(\Delta t\) which the electron acquired in this coordinate system is \(\Delta v\) (previously the electron was at rest), then \(\Delta v \Delta t = \Delta q\) is precisely that spatial interval over which the interaction took place.

In other words, in this system the relation \(2\Delta E \Delta t \gtrsim h\) establishes a relation according to which the interactions between particles and the apparatus can only be of such a kind that the product of the time of interaction and the change of kinetic energy as a result of the interaction is always greater than, or in any case of the order of, Planck’s constant \(h\).

The relation \(\Delta p \Delta q \gtrsim h\) establishes the proposition according to which the interactions between the particles and the observational apparatus can only be of such a kind that the product of the change of momentum of the particle during the time of interaction and the interval over which the interaction occurred cannot be smaller than Planck’s constant.

Or, in a unifying formulation: the nature of the interaction is such that the action function as a result of the interaction cannot acquire a value less than \(h\) in this coordinate system.\(^**\)

Such is the nature of the interaction. Therefore, it would be more correct to call the uncertainty relation—

\(^*\) Generally speaking, the role of time in quantum mechanics is still not at all clear. Time, on the one hand, is a “mere number,” and on the other an “operator.” This question requires special discussion; some of the central questions of quantum mechanics are connected with this problem.

\(^**\) If by action one understands \(\int p\,dq\) and \(2\int E_{kin}\,dt\).

to be called a relation of interaction or action; it is precisely interaction that is the objective side of these relations.

Above we have made clear that the uncertainty relation leads formally and logically to agnosticism only when propositions are advanced which assert that the momentum \(p\) has an exact objective meaning for a particle, including an electron, at any point \(q\) at any moment \(t\). And, correspondingly, the kinetic energy \(E\) always has an exact physical meaning for the electron at any instant \(t\).

But of course nowhere, by anyone, has the proposition ever been proved that the concept of kinetic energy, as it has historically taken shape for us, has a strict physical meaning for an electron at every instant.

Yet since the assumption within the limits of measurement was always fulfilled in the macrocosm, the assumption gradually turned into a habit and then into a conviction, while in essence it remained, in the microcosm, an extrapolation not grounded in anything.

The universality of the relation

\[ E = h\nu \]

shows that kinetic energy too has meaning only with respect to an entire period of oscillation \(\tau\), and not to a separate instant.

When the period of oscillation has not ended, it is just as absurd to ask what the energy is as it is absurd to measure the area of a geometrical figure if the curve is not closed.

In classical mechanics it was asserted that the motion of a particle (including an electron) can always be strictly replaced by the motion of a material point, that there is always such a special point whose motion strictly characterizes the motion of this entire particle.

This hypothesis finds no confirmation in quantum mechanics.

Against criticism of this hypothesis, it would seem, very weighty objections can be raised. For by the very meaning of the uncertainty relation, only the position of the electron can be determined with arbitrary precision. But it is difficult to regard as accidental the fact that measurement of the exact position of an electron necessarily requires extremely energetic actions upon the electron, only as a result of which, as it were, it becomes possible to gather into some small region that reality which we call an electron. If we were to construct an apparatus that would automatically determine only the position of the electron at some moment, and if we were to

we knew in advance that our apparatus is arranged precisely in such a way that it cannot change the momentum of the electron by an amount greater than \(\alpha\), then we could assert in advance that our apparatus will never determine the positions of electrons with an accuracy greater than

\[ \Delta q > \frac{h}{\alpha}. \]

It must be supposed that this astonishing connection between the accuracy in observing the position of an electron and the action applied to the electron may serve as a key to studying, from a certain side, the very structure, if one may so express it, of this reality.*

In classical mechanics it was assumed that for every instant of time and for every material point the limit is always real:

\[ v=\lim_{\Delta t\to 0}\frac{\Delta q}{\Delta t}, \]

where \(v\) is the velocity.

Strictly speaking, the concept of velocity and, correspondingly, the concept of momentum historically took shape and in essence refer not to a single point of space-time \((q,t)\), but to certain two points:

\[ (q_1,t_1)\ \text{and}\ (q_2,t_2).\quad v=\frac{q_2-q_1}{t_2-t_1}, \]

and although the reality of the limit as \(t_2-t_1\to 0\), the reality of the notion of “velocity at a point,” of “true velocity,” for example in the case of electrons—which understandably is of special interest to us—was not, so far as we know, disputed before wave mechanics, it was likewise not substantiated.

Wave mechanics here brings something essentially new.

* Of special interest is the case when it is known that the apparatus behaves passively (the apparatus only receives energy, but does not give it away or, in absolute magnitude, does not change the velocity of the electron), and since the velocity of the electron can be known exactly before the observation of position, it can be said in advance that this apparatus can measure the position of the electron only with an accuracy up to

\[ \Delta q=\frac{h}{p}, \]

where \(p\) is the momentum of the electron.

The greater the velocity of the electron, the more accurately this apparatus determines its position.

It is curious that in this case \(\Delta q=\frac{h}{p}=\lambda\), precisely equal in its dimensions to the de Broglie wavelength. Whether the basic role here in narrowing the region belonging to the electron is played merely by the very moment of interaction, or whether in general this region is the smaller the greater the velocity of the electron itself—these are purely speculative questions. Although in our conception of the atom we still assign to such a rapidly moving electron a region very small in comparison with the dimensions of the atom.

Not to mention that the former definition of velocity or momentum as

\[ v=\frac{q_2-q_1}{t_2-t_1} \]

is devoid of any physical meaning, we have in wave mechanics, for a free electron, the relation:

\[ p=\frac{h}{\lambda}, \]

where \(h\) is still Planck’s constant, and \(\lambda\) is the de Broglie wave.

This relation connects the momentum not with a point, nor with two infinitely close points, but with an interval.

Momentum essentially pertains to a certain interval, and not to a point.

If the velocity at a point for an electron has no meaning, as, for example, the temperature of an electron has no meaning, then it is absurd to speak of some inaccuracies or uncertainties in the measurement of the velocity of an electron at a point, just as it is absurd to speak, for example, of an iron electron in an iron atom, or of the fact that a water molecule is wet. From this point of view, the limiting relation:

\[ \Delta p\,\Delta q = h. \]

is a precisely measured limit of applicability to the electron of the concept of velocity at a point.

Thus, in the relation of inaccuracies or uncertainties there is, in this way, no inaccuracy, no uncertainty; and in the very essence of the matter it is more correct to call this relation, as we have already said above, a relation of interaction.

Generally speaking, two ways of posing the question are possible here: either in each elementary case we have an interaction precisely of such a kind (i.e., in each elementary act one that gives uncertainty), or in each elementary case we have no restrictions connecting, for example, the time of interaction and the energy participating in this interaction, respectively \(\Delta p\) and \(\Delta q\), but only statistically, on the average, do these restrictions appear. And in nature, then, in this way, there exist such interactions which can carry out observation in the classical sense of the word; but in quantum mechanics one cannot isolate only these interactions and construct only from them the totality of measurements that gives the value of the measured quantities. In composing such a totality, such interactions will necessarily enter into it which, on the average, again lead to the relation of inaccuracies. If

in reality we are dealing with the second case, then the path along which our article proceeds in the interpretation of “C. N.” (assuming that the electron is such a reality in which objectively exact concepts \(p\) and \(q\) are inapplicable by virtue of its very structure, if one may put it so) is not a path toward resolving the question; and then the question facing physics in its further development will be to find also methods of selecting such elementary acts of interaction, which carry out observation, as would make it possible to perform a measurement without substantially disturbing the process being observed.

This path does not fit within the framework of contemporary quantum mechanics and is impossible without a radical alteration of its foundations.

We are inclined to regard as correct the first proposition, according to which in every elementary act there is an interaction which in each elementary case leads to indeterminacy, which is a direct consequence of the proposition that the electron, by its physical essence, can no longer be characterized by exact \(p\) and \(q\). If this proposition is true in every elementary case, then it may also be statistically true for the mean, as is usually obtained.

We are inclined to regard the first proposition as correct on the grounds that the quantum postulate is valid also for elementary processes, and the uncertainty relation is an immediate consequence precisely of the quantum postulate.*

Of course, in calculations, using corpuscular terminology, one may also speak, in the limiting case of the inequality

\[ \Delta p\, \Delta q \gg h,\ \text{i.e.}\ \Delta p\, \Delta q = h \]

of \(\Delta q\) and \(\Delta p\) as observational errors, supposing that the interaction occurs instantaneously somewhere at a point on the interval \(\Delta q\) and that there exists such a special point of the electron which performs such functions; but one must always remember that this is an abstraction, and from this one cannot draw the well-known conclusions criticizing the principle of causality.

In other words, to the question we have posed: do \(p\) and \(q\) simultaneously characterize the electron, we answer in the negative; and the principle of causality is not to blame if the matter stands precisely so, just as it is difficult to blame the principle of causality for the fact that, for example, the state of society at some moment cannot be described by momentum and coordinate and in this way precisely predict its future, and, it seems,

* Bohr, The Quantum Postulate and the New Development of Atomistics.

no one from this point of view has criticized causality in society.

Strictly speaking, it is here that the main task of our article is resolved. Thus, we posed the question whether the new wave mechanics rejects the principle of causality, and came to the conclusion that the criticism of the principle of causality is connected with the assertion that the electron is characterized simultaneously and exactly by momentum and coordinate. A closer consideration of the question shows the groundlessness of such an assertion and thus the logical groundlessness of the criticism of the principle of causality, i.e., the principle of causality is, even at this stage, criticized after all from the same old philosophical positions, although it is often asserted that the uncertainty relation logically and inevitably leads to the refutation of the principle of causality.

If we insist on the characterization of the process of studying the stages of any new phenomenon which we gave above, then the modern stage in the study of the electron is characterized mainly precisely by the fact that what is being studied, chiefly, in this new reality is that common element which it has with other, already relatively studied forms of motion. This aspect of study has hitherto, in the case of the electron, been dominant. The electron is either represented as a material point and is characterized by \(p, q\), as is a material point in classical mechanics, or else, in addition, a common element is found with phenomena of an entirely different region, also relatively well studied—we speak of the attribution to the electron of certain properties of waves—and as a result of these analogies the electron turns out to possess the most varied and, from the point of view of analogy, mutually exclusive properties (see above on the development of physics).

These analogies (which, as experience shows, correctly reflect certain aspects of objective reality), uncritically extended to details, lead to uncertainty.

When it is said that the electron is both waves and corpuscles, the fact is usually not emphasized that electrons are in essence neither waves nor corpuscles; that before us is in essence a new form of moving matter, which only very schematically, only in certain of its features, can be approximately represented by those analogies which we use in the given case—only within certain boundaries, only within certain limits. In crossing these boundaries, we must inevitably be punished by various kinds of absurdities, i.e., in this case these analogies already begin to play an inhibiting role for the further study of the electron. Finally, some believe that recently we have been approaching a wave picture of the world, but

precisely in the last decade the so-called phase velocity has been losing more and more of its specific physical meaning, both in electromagnetic waves and in waves of matter. Yet phase velocity is fundamentally connected with the concept of a wave*.

Thus, earlier it was believed that the electron is simply a material point of the most ordinary mechanics; now, however, it turns out that, on the path of studying the electron, we have advanced so far that such an abstraction must already be abandoned. The electron appears before us in a more complex form. And the task of science in the nearest future is to understand the electron as this peculiar form of moving matter.

Let us emphasize still another moment in the development of physics, one very important for our question: a moment connected with the physical essence of phenomena and with the process of their study. The point is that a new form of motion of matter (heat, electricity) is studied from the very beginning not so much from the standpoint of differences as from the standpoint of similarity with already relatively known forms of motion, and such similarity is indeed present. For example, the mathematical apparatus that grew up on mechanical oscillations proves suitable for describing certain phenomena of electromagnetism. These equations really grasp what is common to two entirely different forms of motion, while the pedigree of this apparatus continues to impose a mechanical essence on these new phenomena.

Or it is enough to recall Laplace’s equation and its role in hydrodynamics, electrodynamics, the theory of gravitation, and the theory of heat conduction. Finally, the entire mathematical theory of heat conduction given by Fourier, or the origin of the equations of the electromagnetic field and of hydrodynamics, and, in general, an entire method—the so-called method of analogy.

In the first stages, the study of this common element has always meant in physics a study from the standpoint of already relatively studied forms of motion (electric and magnetic fluids; the fluid that explains elasticity according to Descartes; heat-substance; Lomonosov’s “gravitational” fluid, etc.).

For whatever monstrous “ether” was invented, it

* Dirac, essentially objecting to the term “wave mechanics,” writes that this analogy may lead to serious misunderstandings, for the superposition which occurs in quantum mechanics is profoundly different from that superposition encountered in the classical theory: “If the state of an oscillating membrane is superposed on that same state, the result of such a superposition will be a new state with twice the former amplitude. If, on the other hand, the state of an atomic system is superposed on itself according to the rules of quantum mechanics, then the resulting state will not differ in any way from the original one.”

always proved to be, in essence, a combination of the properties of solid, liquid, and gaseous bodies.*

But when the study of a new form of moving matter advances far enough, then the analogies which previously helped one to orient oneself and to move the process of investigation forward often become a brake on further development; and through the thicket of mutually exclusive analogies it becomes clear that, in essence, we are dealing with fundamentally new things, which cannot be reduced to the “basic” laws, propositions, and “axioms” already known to us.

Gnoseological Conclusions from the “Uncertainty Relation”

Despite the enormous historical experience that persistently points to the development of physical concepts, the idea of development is winning its legitimate place in the physical worldview only with great resistance.

To be sure, the period of natural science has passed when “every change, every development was denied to nature,” when, in contrast to the history of humanity developing in time, only motion in space was ascribed to the history of nature.

But to this day bourgeois physicists cannot admit the idea of development as a methodological principle in its application to the basic physical concepts and categories.

For the ideal of many physicists over the course of many years has been the creation of a logically complete, internally non-contradictory physical system. Often the ideal of a physicist is such a solution of the problem in which, starting from some principle or from the smallest number of “simple” principles, it would be possible to obtain “all” the laws of nature, to explain once and for all “all” phenomena. And once such an initial principle has been found, then everything else must naturally flow from it; the rest is only a question of time.

The history of physics knows quite a few such metaphysical principles upon which similar hopes were placed. It is enough to recall the elevation by Laplace into an absolute of Newtonian mechanics, his all-knowing mind which, knowing exactly the momenta and positions of all particles in the world at a certain moment, could predict both the future and “calculate” the past of the world; or, at the present stage, the formulation

* Einstein’s ether stands somewhat apart.

* Engels, Dialectics of Nature*.

Einstein’s task of a unified field theory—an attempt, in other words (somewhat simplifying the question), to find such a space whose properties could explain both gravitational and electromagnetic phenomena and reduce everything discrete (for example quantum mechanics) to the continuous.

After the principle has been established, everyday work begins: to explain everything on the basis of these principles. Much succeeds. In the general building of science some “details” remain to be finished; assertions circulate in the air and are expressed that all phenomena of the “large-scale” order have been explained and found, that only certain “minor things” remain; they have not yet been fully grasped, are not always clear, but there is hope, and so on.

With the passage of time, out of these minor things “problems” grow; corrections are introduced; and, finally, from the minor things a coffin is put together for the metaphysical ideal of the given moment—to construct a completed physical system.

The metaphysically thinking physicist takes this period as a natural disaster: “the foundations are collapsing,” the supposedly eternal principles are being driven out, the whole “scientific foundation” is “shaking,” and instead of turning to the history of the development of physics and taking account of its experience, there often begins, not a reconsideration of metaphysically ossified physical concepts and definitions, but a revision of man’s cognitive capacity; “world riddles,” “limits of knowledge” are established; even doubt is cast on the possibility of scientific thought in general; the road is opened to faith, religion, and mysticism.

Here is a brief historical note. At the end of the last century the results of science were summed up thus:

“Having summed up this astonishing, complete, and precisely verified and, as it seemed, all-embracing series of laws and principles explaining, as it were, all physical phenomena, the speaker (W. Thomson) came to the possibility of drawing such a conclusion: that all the great discoveries in physics had already been made and that the further process would consist not in the discovery of qualitatively new phenomena, but rather in a more exact quantitative measurement of already known phenomena” *.

Here is how Planck conveys the opinion of another scientist of the same period:

“Of course, in one corner or another one may still notice or remove a speck of dust or a bubble, but the system as a whole stands fairly firmly, and theoretical physics is noticeably approaching that degree of perfection which geometry has already possessed for centuries” **.

This was before the discovery of X-rays, radio-

* Millikan, The Evolution of the Fundamental Concepts of Modern Physics.

* M. Planck, From the Relative to the Absolute*.

activity, electrons, electromagnetic mass. And no more than a decade later Poincaré wrote of the “general rout of principles,” of the “ruins” of the old principles.

Having lived through a great revolution in fundamental physical conceptions in connection with the principle of relativity, many physicists and mathematicians again came forward with the slogan of the axiomatization of physics.

“We see that not only are our conceptions of space and time and motion changing in a radical way according to Einstein’s theory, but I am also convinced that its fundamental equations will make it possible to penetrate into the most hidden processes taking place inside the atom, and, what is especially important, it will become feasible to reduce all physical constants to mathematical constants; and this, in turn, shows that the possibility is approaching of making physics a science of the same kind as geometry; this will be the finest crown for that axiomatic method which, in the questions considered here, makes use of such powerful tools of analysis as the calculus of variations and the theory of invariants.” *

Or the same thing in the book by Friedman and Fredericks:

“The third period (in the development of human thought. The Authors) is the period of the creation of the axiomatization of knowledge, a period that may be characterized as a time of senile skepticism.” **

And there also:

“Fortunately, it is not given to us to see the future, and we do not know whether the epoch, the epoch of axiomatization, the epoch of skepticism, is the deathbed hours of knowledge… But even if this were so, then even then the logical beauty of the end would compel us to welcome the appearance of the principle of relativity.”

And five years later stability is again lost. Here is what Khvolson writes:

“In building, on the basis of the world of sensations, a second world, a provisional one, we counted on approaching the third, the real world.

We hoped thereby to get over from something fluctuating to something stable, from the variable to the constant; but how could we know that we would succeed in this and by what paths?..

…so, approximately, do many scholars reason nowadays, both physicists and philosophers.” ***

* Hilbert, Die Grundlagen der Physik.

* Friedman and Fredericks, Foundations of the Theory of Relativity, issue 1, Tensor Calculus*, pp. 24 and 27.

* Khvolson, Physics of Our Days, p. 330.

But now, in the opinion of many, Dirac has already begun the axiomatization of the new quantum mechanics.

We are by no means against finding in physics the existing connections between different phenomena; we are not against clear formulations of physical concepts at each given stage of the development of physics; we, too, are for careful, painstaking work that brings physics into “order,” “cleansing” it of “historical accidents.”

But we are against the absolutization of these particular physical concepts for all time, for the establishment of fundamental physical principles in every epoch is at the same time the establishment precisely of those principles to which critical attention must first of all be directed when new facts arise that do not fit into the given physical system built upon these principles. The elevation of given physical concepts into an absolute, as the history of physics shows, is a brake on the further development of science and a source of many “idealistic” vacillations*.

Here is an illustration of what has been said, using the example of the new quantum mechanics.

Dirac has done a very great deal of work on the systematic exposition of quantum mechanics as it is known to us today, and in essence has made the first attempt at its axiomatization.

Dirac, in accordance with historical custom, also attempts to elevate particular physical principles—the principles of quantum mechanics in its modern form—into an absolute.

And since quantum mechanics in its present form, as Dirac explains, makes it possible only to calculate the result of a proposed experiment, and not to provide “a satisfactory description of the whole course of phenomena,” these principles of quantum mechanics, elevated into an absolute, compel him to formulate the aim of theoretical physics in general in the following way:

“The sole aim of theoretical physics consists in calculating results that can be compared with experiment, and there is no need at all for a satisfactory description of the whole course of phenomena.”**

And in the same place:

“Any description of what happens to a photon during the experiment will be merely a mnemonic rule for remembering the final result of the experiment.”***

* On the danger of falling into philosophical relativism, see the page below.

* Dirac, Die Prinzipien der Quantenmechanik*, p. 6.

*** Ibid.

“Description, which quantum mechanics allows us to give, is only a mode of expression, useful for deriving and for retaining in memory the results of experience, and never leading to incorrect consequences.”

That is, at a new stage, on a new basis, the old methodological error is repeated. After all, on the basis of the same Newtonian mechanics, for example, Helmholtz formulated, once and for all, again the same “sole task of theoretical physics.”

“The ultimate goal of the natural sciences consists in finding and studying the motions that underlie all changes, as well as the causes that produce these motions, i.e., in reducing them to mechanics.” *

As Engels, and especially Lenin, showed, this aspect of the development of physics is most intimately connected with the posing of the question of relative and absolute truth**, which in turn is most intimately connected with the central question of philosophy—the understanding of the interrelation of subject and object.

This is one of the points on which metaphysical materialism “broke down.”

“The chief defect of all previous materialism, Feuerbach’s included, is that the object, reality, sensuousness, is conceived only in the form of the object or in the form of contemplation, but not as sensuous human activity, not as practice, not subjectively. Hence the active side, in opposition to materialism, was developed by idealism—but only abstractly, since idealism, of course, does not know real, sensuous activity as such.” ***

Considering the object, reality, only and solely in the form of the object, ignoring “sensuous human activity,” the subjective side of the interaction of subject and object; ignoring human practice as a historical process that clarifies at each stage our knowledge of the objective world, ignoring human practice as the supreme criterion of truth, metaphysical materialism, understandably, sees a firm support in physics only in finding immutable physical, supra-historical, “eternal” truths, generously and uncritically elevating many physical categories into “eternal truths.”

* Helmholtz, The Law of Conservation of Force.

* Lenin, Materialism and Empirio-Criticism; Engels, Anti-Dühring*.

* Marx, Theses on Feuerbach.

But when the development of physics undermines the “eternity” of certain “truths,” the metaphysical materialist reaches an impasse; or, being unable correctly to pose and resolve the question of relative and absolute truth, not seeing in the development of our knowledge a movement “from the relative to the absolute,” not seeing that the denial of a whole series of “fundamental principles” by the development of physics is not the bare negation of the entire preceding stage in the development of knowledge, but its organic growth; not seeing that our knowledge begins ever more accurately to reflect the objective world, the guarantee of which is mankind’s ever increasing capacity actively to intervene in and change this world, the metaphysical materialist is unable, from his own positions, consistently to withstand idealism. Here idealism often comes out very resolutely against tradition and, together with the critique of old physical concepts, declares a campaign against materialism.

“Denying the absolute character of the most important and basic laws, they slid down to the denial of any objective regularity in nature, to declaring the law of nature an empty convention, a ‘limitation of expectation,’ a ‘logical necessity,’ etc. While insisting on the approximate, relative character of our knowledge, they slid down to the denial of the object independent of cognition, which is reflected by this cognition approximately correctly, relatively correctly” *.

Thus the problem is again reduced to the basic question: subject and object; their interrelation is again incorrectly interpreted, but now idealism “inflates hyperbolically” one subjective side of this interrelation, and the character of the development of our knowledge “from the relative to the absolute” is completely distorted.

This question of the relation between relativism and dialectics is perhaps the most important one in explaining Machism’s theoretical conclusions. **

For under the banner of Machism, what is in the main coming forward in contemporary physics is idealism.

In agreement with the majority of physicists, for example, Niels Bohr *** considers that “the consequence of the postulate (the quantum. Authors) is the rejection of a causal spatio-temporal description or coordination of atomic phenomena.”

* Lenin, Collected Works, 3rd ed., vol. XIII, p. 214; Engels, Anti-Dühring, ch. IX. “Morality and Law, Eternal Truths.”

* Lenin, Collected Works*, vol. XIII, p. 252.

* “Quantum Postulate and the New Development of Atomistics,” Russian translation, in Advances in Physical Sciences, vol. VIII, issue 3, p. 307.

THE PRINCIPLE OF INDETERMINACY IN QUANTUM MECHANICS

Mises* considers causality as a concept of pre-scientific thinking: “It turns out that one is inevitably forced to surrender yet another beloved position, having its roots in practical life, in pre-scientific thinking, and elevated by obliging philosophers to the unattainable height of eternal logical categories: the naive concept of causality.”

This author calls the defense of the principle of causality stubbornness—psychologically understandable, it is true. “With a certain psychologically understandable stubbornness, even now people resist the renunciation of a deeply rooted tradition of thought connected with the so-called law of causality.”**

Jordan asserts essentially the same thing: “When modern theory abandoned the idea of causality, it merely came closer to fulfilling its task of describing the present state of our knowledge. Indeed, it is possible to develop a consistent and self-contained physical system of concepts in which the principle of causality has no place. Such a system we have in modern quantum mechanics.”

Haas*** believes that “causality from the point of view of quantum mechanics must be denied for the elementary processes of physics.”

The extreme position on the question of causality is taken by Heisenberg****, who asserts that nature “makes a free choice” in elementary processes. Dirac, for example, in his book***** writes that the process of physics “consists in our equations becoming invariant with respect to ever broader classes of transformations. Such a state of affairs also points to the recognition of the absence of arbitrariness in nature” (emphasis ours).

The idea of freedom of choice in the microcosm is expressed by Jeans, who says: “Apparently, the deadly inevitability of the chain linking cause with effect has disappeared, and we stand before the possibility of a freedom of which until now we had no conception.”

It is important to note that in this connection physicists look upon causality idealistically, as upon a subjective category, and not as upon an objective law-governed regularity of nature. Bohr regards it as a form of describing phenomena, Mises—as a concept, Jordan—as an idea, Schrödinger—as

* Mises, Probability and Statistics, p. 235.
** Ibid., p. 236.
*** Haas, Matter Waves and Quantum Mechanics, p. 115.
**** See the discussion on causality and probability at the 5th Solvay Congress.
***** Dirac, Die Prinzipien der Quantenmechanik, Leipzig, 1930.

setting of our thinking. In his article “Was ist Naturgesetz,” he writes that when causality is discussed, “it is not a matter of a decision about the real structure of nature as it appears before us, but of the expediency or convenience of one or another setting of our thinking with which we approach nature.” This is Machism. For the sake of the “convenience” of practice he preserves causality for macroscopic phenomena. “For practice we shall, of course, preserve causality, since it gives correct results. But for the sake of the ‘convenience’ and ‘expediency’ of quantum mechanics, for the sake of excluding contradictions from it, he denies the principle of causality for the microcosm, considering that phenomena in it are subject to statistical laws, at the basis of which, supposedly, lies acausality: “...it would be an unforgivable logical circle if we were to consider that macroscopic causality should compel us, on account of statistical laws, to accept and postulate absolute causal determinacy.”**

It is interesting to note here that macroscopic causality is interpreted by Schrödinger as the mean, statistical result of acausality in the microcosm. Here we have a formulation of the question directly opposite to that which existed in the classics. There it was considered that statistical regularity could be reduced to dynamic regularity. Here, however, at the basis of dynamic regularity lies statistical regularity. Moreover, causality is “reduced” to acausality.

We have seen that the majority of leading physicists have taken the path of the idealistic denial of causality. And only a few bourgeois physicists are fighting for materialism on this question—true, for its limited, mechanical form. But even some from this latter group of physicists sometimes make concessions to Machism on this question. Planck is an example. He yields to Schrödinger in his assertion that causality does not possess the objectivity of the laws of nature, that the question of the inviolability of the principle of causality is a question of expediency. In reply to Schrödinger’s words: “One of the most burning questions that occupy us now... is the question of the expediency of the inviolability of the postulate of causality,” Planck says: “First of all, I fully agree with you that this question is in essence a question of expediency.”

Among the reasons why a number of physicists have taken idealistic positions in the problem we are considering, not—

* “Die Naturwissenschaften” No. 1, 1929.
** Ibid.

doubt, is the aggravation of the crisis that capitalism is experiencing at the present moment, as well as ignorance of dialectics.

Comrade Lenin put forward, as the most important task of Marxist natural scientists, the task of reworking those achievements in the field of natural science that are made by bourgeois scientists. He repeatedly emphasized that the task consists in the ability “to pursue one’s own line and to struggle against the entire line of forces and classes hostile to us”*. However, this task, in the main, has still not been fulfilled.

The same is true of the critical attitude toward the problem we are considering. Some Marxists reject the “uncertainty relation” only on the grounds that bourgeois physicists draw idealistic conclusions from it (they deny the principle of causality). To be sure, these conclusions, drawn by bourgeois physicists from the theory, are a signal of some kind of trouble within it. But these conclusions alone are certainly not sufficient for rejecting the theory. An analysis of its concrete content must reveal the concrete shortcomings of its premises.

A number of comrades reveal a simplificatory approach to the question, ignorance of science, physical illiteracy; they replace the study of the subject with “searchlight criticism,” with fruitless reasoning “in general.”

Comrade Stalin, in his speech “New Conditions—New Tasks of Economic Construction,” pointed, as one of the most important tasks confronting economic managers under the new conditions, to the task of learning to lead in a new way. He said: “For this it is required, further, that our economic leaders lead enterprises not ‘in general,’ not ‘from the air,’ but concretely, substantively (our emphasis. The Authors), that they approach each question not from the standpoint of general chatter, but in a strictly businesslike manner, that they not confine themselves... to general phrases and slogans, but enter into the technique of the matter, penetrate into the details of the matter (our emphasis. The Authors), penetrate into the ‘trifles,’ for from ‘trifles’ great things are now built”**.

Comrade Stalin’s criticism of the negative “methods” of work “in general,” practiced and sometimes still practiced even now by some economic managers, correctly characterizes the approach of some Marxists to contemporary physical theories. The task of Marxist natural scientists consists in rebuilding all their work on the basis of Comrade Stalin’s slogan about mastering science, in order to—

* Lenin, Materialism and Empirio-Criticism.

** Stalin, “Technique,” p. 15.

at present, seriously to set about reworking physics on the basis of dialectical materialism. The most important condition for solving these tasks is the struggle against all distortions of Marxism-Leninism.

In conclusion, let us note that in this way the crisis of physics at this stage is in essence again connected with the problem of matter, but now on a new basis. If earlier that stage of the crisis, of which Lenin wrote, was connected with the atom of substance, then the present one is connected with the electron and, if one may put it so, with the atom of action. Thus, physics is preparing to go beyond the limits of the electron.

Recalling the statements of bourgeois physicists, we once again note their peculiar orientation. Almost all physicists* connect the uncertainty relation with a critique precisely of those categories of a theoretical-cognitive character under whose banner knowledge of the objective world has in fact developed up to now. And none of the physicists, in essence, considers the question in developed form from the point of view of a change precisely in the physical categories themselves, so profoundly characteristic of the objective course of the development of physics.

In conclusion, the authors express their deep gratitude to B. M. Gessen for systematic assistance in the work and to A. A. Maksimov for a number of valuable suggestions.

* Some questions are formulated very clearly in Fock’s book, for example. But, assuming that every physical quantity can be measured instantaneously (as Fock formulates it), while also asserting that it makes no sense to speak of its value “at a definite moment,” it is very difficult to object to the word “ignorabimus” in the interpretation of the uncertainty relation.

The inapplicability to the electron, for example, of the concept of “impulse at a point,” of impulse referred to an instant, provides a weapon against “ignorabimus.”