CAUSALITY AND STATISTICS IN MODERN PHYSICS¹

P. Jordan, Göttingen.

The development of physics over recent decades has again and again brought epistemological questions to the fore. In the theory of relativity, the problem of space and time first received a completed explanation. New questions are being raised by quantum theory. Among them belongs, in particular, the question: does causality exist in elementary physical being? Is the fate of an individual atom completely determined, or are there gaps in the causal determinacy of elementary events?

Physicists at the present time do not doubt that the question of the existence of complete causality can be decided only by experiment; that causality is therefore not an a priori necessity of thought. It is true that a certain measure of causality is a necessary prerequisite for the possibility of physical science in general, as well as of orderly human existence. Fortunately, in our macroscopic world causality does in fact exist, apparently knowing no exceptions. But for the domain of atomic dimensions it follows from this only that this domain is subject to causal laws only on the statistical average. The question still remains: is the fate of each individual atom completely prescribed?

Before turning to this question, it is useful to analyze rigorously the concept of causality. A physicist cannot be satisfied with the approximate idea of the meaning of this word that we possess. Equally, he is entirely uninterested in those metaphysical meanings that many philosophers associate with it. To define causality for the physicist means nothing other than to indicate how its existence or nonexistence can be established experimentally. At the same time, it is already clear that the definition of causality must change progressively in connection with the advances of our views, knowledge, and experimental means. Thus let us consider first of all the role of causality in classical field physics.

¹ Introductory lecture delivered at the University of Göttingen. Naturwissenschaften, 15, 105, 1927.

Classical field physics asserts that one can describe physical reality—where, in using the word “describe,” we mean it here, so to speak, in a purely geographical sense—by specifying, for every point of a four-dimensional space-time region, certain measurable quantities: field strengths, gravitational potentials, and so on. In this, causality takes place in the following sense: let us imagine a bounded and finite region of space, for example in the form of a box. We shall not go into detail concerning how our formulations would have to be carried out while taking into account the exact relativistic space-time relations, which, of course, would present no difficulties. At some definite instant of time, say at 11 o’clock, let the physical state inside the whole box be completely known, completely measured. Next, let the physical state over the entire surface of the box between 11 and 12 o’clock be continuously monitored. Under the circumstances thus established, the physical phenomena inside the box from 11 to 12 o’clock are uniquely determined. This means that if one reproduces, at any time and in any place, the initial state of the box and the temporal course of the processes on its surface, then all the processes inside the box will thereby reproduce themselves as well. Within a certain time interval—of the order of the transverse dimension of the box divided by the speed of light—the processes inside the box will even be independent of the processes on its surface.

All these are assertions that are accessible to experimental verification. Of course, it is assumed here that the initial state of the box will not be so complex that its complete physical numerical characterization is impossible. Thus, for example, one must exclude the case in which there is a living being inside the box—the premise that, in this case too, it is possible to carry out an exact numerical characterization of the physical state is too far from practical possibility. For the purposes of biology, the principle of causality and the problem of causality must be formulated in an essentially different and considerably more complex way than for physics.

But let us return to physical causality. It is necessary to emphasize that this causality is something extraordinarily remarkable. It is by no means equivalent to the existence of physical laws in general—to the existence of mathematical relations between physical quantities in some region of the world. There exists, moreover, a peculiar asymmetry between the spatial and temporal world coordinates. Namely, according to the principle of causality, between known regions of the world separated in time there exists a physical dependence; between spatial—

divided regions of such dependence there never exists.

Theoretically, the principle of causality in field physics is justified by two circumstances, which we shall only briefly indicate here, without entering into a mathematical proof that they do indeed condition the validity of the principle of causality in the form stated. First, physical laws, i.e. the mathematical relations satisfied by the numerical characteristics of the field, are nothing other than differential equations and, moreover, chiefly, in the first approximation, linear partial differential equations of the second order. Secondly, it is known that, if one is to have in the four-dimensional domain of the world the simplest geometry in which the theorem of Pythagoras is valid, then it is necessary to introduce not time itself, but imaginary time as a world coordinate. This circumstance is very essential. If, instead of this, the four-dimensional world had four real dimensions (and at the same time the general physical laws, i.e. the differential equations of electromagnetism and gravitation, remained unchanged), then there would exist, so to speak, something more than causality. Namely, knowing exactly a small region of the world, one could infer the physical state of regions of the world arbitrarily distant in space or in time. If, on the contrary, the world had two real and two imaginary dimensions, then causality would not exist at all. It might happen that motions suddenly arose inside a bounded box, without there being any causes for this either inside the box or having come through its walls from outside. Such is the significance of the principle of causality in field physics. It is not in itself a law of nature—the laws of nature are differential equations to which the physical field is subject. The law of causality is a mathematical consequence of the laws of nature—a theorem of the mathematical theory of hyperbolic differential equations, applied to the laws of nature.

Thus one may indeed expect that the principle of causality will prove untenable when we pass from classical field physics to quantum theory, for in this transition precisely those fundamental physical assumptions which we have indicated as the sources of the validity of the principle of causality undergo a profound change. The very description of physical reality, as we now know, cannot be carried out in the manner adopted in classical physics. Physical quantities are not distributed continuously in the domain of the world; physical motions do not proceed in an entirely continuous manner; there exist elementary discontinuities, there exist quantum jumps. Under these conditions, nothing more than statistical causality remains of causality. If we experiment with very large

with a number of identical atoms, or if we repeat an experiment performed with a small number of atoms an infinite number of times, we always arrive at results that are in agreement with the principle of causality. We have just indicated that physical lawfulness and physical causality are not one and the same. It is therefore not superfluous to emphasize that on this point, with respect to physical lawfulness, one can say the same as with respect to causality: everything that we know up to the present time is, chiefly, statistical lawfulness.

In the knowledge of this lawfulness, as is known, important successes have recently been achieved. At the present time one can, for example, calculate the spectrum connected with the motions of electrons inside the atom, in principle just as well as, in classical mechanics, the motion of the planets. However, despite the fact that the course of the calculation is very similar in both cases, there is also an essential difference in the meaning of the results of the calculation. A classical calculation makes it possible to draw a conclusion about the fate of precisely our definite planetary system. A calculation in quantum mechanics in general does not permit any conclusions to be drawn about a definite single atom, but is applicable to an average value taken over a large number of identical atoms. In fact, let us consider, for example, the behavior of an atom under the influence of some external action, for example, incident light or an electronic impulse; the calculation gives a result which we must in no case understand according to the classical scheme in the sense that, for definite values of the phase constants of the atom, completely definite events will occur. We must interpret the result of the calculation only in this way: there is a definite probability that the atom will do one thing, but there is also a definite probability that it will do something else.

We have an analogous situation in optics. Indeed, the classical theory of optics makes it possible to calculate all interference experiments in flawless agreement with reality. But if a calculation gives a definite intensity of light at a known place, this does not mean that in reality a corresponding quantity of energy will in fact be released there. On the contrary, the classical wave field signifies only a certain probability that light quanta will arrive at the given point. As is known, for every ray of material corpuscles one can construct the corresponding wave ray, which will be related to the corpuscles in exactly the same way as the wave ray of light is to light quanta. And here, too, as everywhere, the purely statistical nature of the laws of quantum mechanics known up to the present time is revealed.

We must therefore concentrate our attention not on discrete, discontinuous elementary states and elementary

processes, but on their probabilities. With these probabilities, continuously varying quantities again enter into the description of physical reality, and at the same time, in one of the most fundamental points, we again approach the classical, continuous method of description. Therefore the supposition suggests itself that for continuously varying quantities there should exist a law of causality for probabilities, analogous to the previously formulated law of causality of classical field physics. This is indeed the case, although in a more abstract form than in the classical theory.

As is known, Schrödinger arrived, by his own independent path, at the formulation of quantum mechanics, which mathematically proved to be equivalent to the matrix theory developed from Heisenberg’s ideas. He discovered the mathematical relations of quantum mechanics which, by virtue of the mathematical equivalence of the two theories, were, to be sure, in essence already contained in matrix theory; nevertheless, their explicit formulation represents a major enrichment of quantum mechanics.

Schrödinger, moreover, attempted, in connection with his formulas, to develop new physical foundations for the theory of quanta; in doing so he adopted a point of view opposite to the basic conceptions of the theory of quanta developed by Planck, Einstein, and Bohr—stationary states, quantum jumps, and so forth. He attempted to return to quasi-classical conceptions, in which no discontinuities appear at all, and in which, consequently, the principle of causality should have a place in its classical form. But these speculations of Schrödinger’s met with unanimous objection from the other investigators participating in the development of quantum mechanics. For us there is no doubt that Schrödinger’s new concepts must receive a physical interpretation in close connection with the earlier conceptions of stationary states and quantum jumps and with Heisenberg’s ideas—that, consequently, Schrödinger’s laws, like the laws of matrix theory, must be interpreted statistically, as has already been indicated. Such a statistical interpretation of Schrödinger’s theory was given in a very clear and expressive form by Born, on whose considerations the arguments that follow are based.

The essential content of Schrödinger’s discovery, as is known, consists in the following. The laws of quantum mechanics, which in matrix theory are formulated by means of transcendental algebra in the form of a system of an infinitely large number of equations with an infinitely large number of unknowns, can instead be expressed by perfectly ordinary differential equations. In this way, formally, a much closer approach to classical theory is once again achieved. To the question of how—

...whether in the discontinuous entanglement of atomic quantum processes something can be described with the aid of differential equations, we answer, together with Born, as follows: the function that satisfies the differential equation is precisely the probability function.

Let us consider this probability function somewhat more closely, and in doing so elucidate its analogy with classical quantities. To this end let us consider a mechanical system consisting of two material points with rectangular coordinates \(x_1\), \(x_2\) through \(z_1\), \(z_2\), i.e. a system possessing six degrees of freedom. Let us now construct something analogous to the phase space of the system considered in statistical mechanics, namely a coordinate space which, however, has half as many dimensions as the phase space. In our example this will be a six-dimensional space with coordinates from \(x_1\) to \(z_2\). In this coordinate space a system possessing definite coordinates but arbitrary momenta is represented by one definite point, which we shall call the representative point of the system. According to classical mechanics, this representative point of the system would describe a definite path in coordinate space. But if we find this point at a definite moment of time at a known place in coordinate space, then we cannot say in advance how it will move, since from the position of the representative point of the system we can find only the coordinates, but not the momenta, of our two material points. We can determine only the probability that the point will proceed from this place in a definite direction.

Of course, in classical mechanics we can refine this statistical conclusion into an entirely exact prediction if we observe not only the position but also the velocity of our representative point of the system. But this is precisely the point at which quantum mechanics differs from classical mechanics. If, for some quantum-mechanical system, the known coordinates are empirically observable quantities—where we use the word coordinates here in so general a sense that, for example, energy or quantum numbers are also coordinates—then the momenta corresponding to these coordinates are precisely quantities that are in principle inaccessible to observation \(^{1}\). Therefore we can only transfer the statistical question just formulated in classical mechanics into quantum mechanics, and shall probably obtain an answer to it

\(^{1}\) In this connection, with different experimental arrangements it is possible to observe different coordinates; but with one and the same definite arrangement one can, in the best case, observe exactly definite coordinates of the atom, while the corresponding momenta with that very same experimental arrangement will precisely be in principle inaccessible to exact observation.

from Schrödinger’s differential equation. I must say here “probably,” since the corresponding arguments have not yet been completed.

But the following question, closely connected with the one just considered, may be regarded as having received a satisfactory answer thanks to the considerations of Born and Pauli. If, for our system, we know the energy or the quantum numbers—or, more generally, if we know that the system has a certain probability of being in the first quantum state and a certain probability of being in the second quantum state, and so on—then what will be the probability that the point representing the system in coordinate space has rectangular coordinates of quite definite magnitude? This question can now be answered, since Schrödinger’s wave function in coordinate space is known.

This Schrödinger function, which is thus a function of six variables and, moreover, depends on time, satisfies Schrödinger’s fundamental differential equation. And with respect to this probability function one may assert that the exact principle of causality is applicable to it. For this, however, it is, of course, necessary to consider not a box in ordinary three-dimensional space, but a six-dimensional box in six-dimensional coordinate space. Then the formulation of the principle of causality will be literally the same as in classical physics; in place of the numerical characteristics of the electric field intensities, and so on, inside and on the surface of the box, there will now appear, however, the numerical characteristics of Schrödinger’s wave function.

We may therefore summarize as follows: classical field physics described the world by means of physical quantities continuously distributed in three-dimensional space and moving continuously in time. Quantum mechanics describes the world by means of an abstract coordinate space that possesses an infinitely large number of dimensions—the number of dimensions is proportional to the number of all particles of matter in the world. In this abstract space there move, in turn, continuously distributed quantities which, however, do not directly describe the elementary events of the atomic world of phenomena, but determine the probabilities of quantum processes. Causality, understood not as the opposite of the metaphysical concept of chance, but as the physical assertion formulated above, formally has a place in exactly the same way in both theories.

We see that, by considering mean values and probabilities, one can eliminate the elementary discontinuities in physical existence and find relations that are mathematically accessible to treatment by methods analogous to those applied to manifestly discontinuous—

values of classical physics. Quantum mechanics here turns out to be a quantitative refinement of Bohr’s correspondence principle, the essence of which, contrary to the widespread view of the exclusive dominion of whole numbers, consists in striving, by considering mean values, to restore a formal analogy with classical laws.

Let us now return from the consideration of continuous mean values back to discontinuous elementary processes. Let us ask what can be said about elementary processes after all the problems concerning mean values have in principle been solved. To answer this is not as simple as it may at first appear, and I would be guilty of very great superficiality if I did not at least point out the difficulties that arise here.

Let us first of all consider the empirical side of the question. It might seem possible to think that experiment in any case gives nothing other than mean quantities. Many (in Göttingen) heard Zernike’s excellent lecture on Brownian motion and on the work of the Swedish physicist Ising. In that lecture, in a vivid and graphic form, the impassable limits were shown which stand in the way of the progressive refinement of measuring technique. It is impossible, for example, to increase the accuracy of a galvanometer measurement beyond a strictly definite limit; it is impossible—because of Brownian motion in all parts of the apparatus. The needle, the suspension thread, the casing, and the surrounding air consist of atoms that are in continuous, disorderly thermal motion, inaccessible to control; the current through the galvanometer consists of individual electrons and therefore likewise exhibits irregular fluctuations of its intensity, accessible only to statistical accounting, which in exactly the same way limit the sensitivity of the apparatus. If one recalls that the same thing occurs in all our instruments, that all our measuring instruments are in thermal motions and tremors, then it would be easy to think that the experimenter can say just as little about the states and processes of an individual atom as quantum mechanics can predict about them. However, there exists a radical way to “tame” the Brownian motion of the apparatus. The simple recipe that a theorist can give the experimenter for this purpose is the following: carry out your experiments at absolute zero temperature! Fortunately, experimental physicists have found yet another recipe, not at all so inconvenient to implement and in principle equivalent to the first. This recipe says: work with a few particles very rich in energy! In comparison with the energy of a fast $\alpha$-particle, the thermal energy of the surrounding atoms is vanishingly small. The thermal jostling of these atoms already

does not exert any action on the flying α-particle. And indeed, as is known, chiefly thanks to the investigations of C. T. R. Wilson, we can empirically trace the fate of an individual α-particle, observe its path, and establish the moment of the quantum jump by which this path comes to an end.

Thus, under certain circumstances, the moments of individual quantum jumps belong to quantities quite accessible to experiment; one asks, then, what can theory say about these moments? The simplest and most immediately suggested answer, evidently, is the following. Theory gives us mean quantities. It says how many quantum jumps must occur in a given interval of time on the average, taken over many separate experiments. Consequently—one may conclude from this—the theory gives, for a single quantum jump, the probability that it will occur in a certain prescribed interval of time. And consequently—one might now conclude further—the true moment of occurrence of the quantum jump is in fact not determined, and there exists only the probability of a quantum jump. But this last conclusion is in reality no longer a necessary consequence of the preceding; it is a hypothesis going beyond it. This is precisely the hypothesis which Bohr, Kramers, and Slater attempted to lay at the foundation of their theory of radiation. But these investigators clearly understood that this hypothesis must lead to a certain definite consequence, namely, that the law of conservation of energy is valid only statistically. This consequence, as is known, has been refuted by the brilliant experiments of Bothe, Geiger, and Compton. At the present time we may definitely assert the following: if an atom, by a quantum jump, emits light, and this light, without encountering on its path obstacles in the form of interference, is absorbed by another atom, then there occurs a quantum jump of the absorbing atom, which is separated in time from the quantum jump of the emitting atom by an interval exactly corresponding to the spatial distance of the atoms. We thus see that the moments of quantum jumps are by no means in any case undetermined.

It would seem that one might try to say this: the moments of quantum jumps are determined insofar as this is required by the exact fulfillment of the law of conservation of energy; in other respects—they are undetermined. However, this somewhat ambiguous explanation is too indefinite for one to be able, using it, to undertake anything when, for example, one has to deal with circumstances complicated by interference. Another way of overcoming these difficulties had already long ago been tried by Wenzel: since the act of absorption in our just-considered example is completely deter-

mined by the preceding act of emission, then the two together could be regarded as a single quantum elementary act, and in that case one might hope that such elementary acts are statistically independent of one another. It appears, however, that even by this route one cannot arrive at simple formulations.

It is highly significant that, in the just-explained formulations of Born—Pauli, nothing is in fact said about the probability of an individual quantum jump; rather, what is spoken of is the probability that the representative point of the system is located at a definite place in coordinate space. Thus one may apparently hope that these considerations will indeed lead us to independent physical elementary probabilities. Namely, although according to quantum mechanics all possible probabilities can in principle be calculated, an essential problem still remains unresolved. For simplicity let us consider a simple example. Imagine that we throw two dice, and let it be empirically found that, on average, one pip comes up together with three just as often as four with five, and twice as often as two twos. If we had a theory which could, in some very complicated and abstract way, predict these empirical facts, then we could be satisfied. In reality, however, we are already satisfied when the theory can be put in the following form: we say that for each die any one of its six positions is equally probable and that the two dice are statistically independent of one another. Only when we state the facts in this way does it seem to us that we have truly understood them.

In the case of two dice, however, the situation is such that we know in advance that there is no possibility of constructing a theory in any way other than the one just described. In quantum mechanics the situation is different: in quantum mechanics at the present time we can calculate all probabilities; but we do not yet understand them! We could assert that we had understood them only if we were able to interpret, in the following way, the mathematical calculations carried out in abstract coordinate space: in known cases there is no prediction as to what nature will do; it may do one thing or another; both are equally probable, and the decision that nature makes in such a case is entirely independent of the decisions it makes in other cases.

In other words: the probabilities given to us by quantum mechanics must be quantitatively reduced to independent elementary probabilities. Only then shall we be able to assert that we have truly understood these laws; only then shall we be able to decide under what conditions and in what way the moment of the quantum-

of the jump and when it is not determined. Only then shall we be able to assess exactly what in physical being is causally determined and what is left to chance.

In conclusion, one circumstance in particular should be emphasized once more. We have just considered as already established that the elementary analysis of the quantum-mechanical laws of probability, which still remains to be carried out, must lead to the conclusion that the known elementary processes are not determined and may occur in different ways with equal probability. But in reality this is by no means obvious. The circumstance that the laws of quantum mechanics are laws of mean values, and can be applied to elementary processes only by means of statistical concepts, is not yet a compelling basis for saying that elementary processes must be laws of probability.

Thus we can pose our question—whether modern physics recognizes determinism, a question which, as we have seen, upon closer examination breaks down into several different questions—for the last time in the following form: are the elementary laws sought for laws of probability or deterministic laws? Is it at all possible for the instant of an individual quantum jump to prove to be undetermined?

In all probability the matter stands such that, in elementary physical processes themselves, incomplete determinations are indeed encountered, i.e. pure probabilities. But, as has already been said, a definite decision is possible only after a further analysis of quantum mechanics in the direction indicated by Born and Pauli.