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Indeterminism in Physics
Schrödinger presented this paper 16th June, 1931 to the Congress of A Society for Philosophical Instruction in Berlin. (Source, Schrödinger, 1935, Science and the Human Temperament, Chapter III.)
The profound changes which the picture of the world as presented by Physics has undergone in recent years has brought it about that the so-called problem of causality has come into the limelight; and discussion of this problem, far from being confined to technical and scientific publications, has found a place in the daily press. I do not here wish to pre-judge the question whether the problem discussed is in fact the problem of causality in the philosophical sense merely by using the label of causality. This name has come to be applied to these matters, they sail under this flag, and that is why I employ the expression.
The question at issue is this: given any physical system, is it possible, at any rate in theory, to make an exact prediction of its future behavior, provided that its nature and condition at one given point of time are exactly known? It is assumed, of course, that no external and unforeseen influences act upon the system from without; but such influences can always be eliminated, at least theoretically, if all bodies, fields of force and the like capable of acting upon the system are included within it. It is assumed, in other words, that the condition of these external elements, too, is exactly known at the initial moment of time. It is possible, and indeed if we argue rigorously it is certain, that in order to do so, the system under consideration has to be extended to comprehend the entire universe. Yet it is possible to imagine a finite, self-contained system, and in practice this abstraction is invariably made use of whenever a law of physics is enunciated. The question therefore is whether it is possible exactly to predict the behavior of such a system provided its initial condition be exactly known.
Some fifteen years ago this was never doubted: absolute determinism was, in a manner, the fundamental dogma of practical physics. The clearest example, which had given this direction to physics, was classical mechanics: given a system of mass points, their masses, positions and velocities at an initial point of time, and given the laws of force in accordance with which they act upon one another, it was possible to calculate in advance their movements for all future time. And when applied to the celestial bodies, this theory had been triumphantly confirmed.
Today many physicists assert that such a strictly determinist view cannot do justice to nature, and that this applies equally whether mass points, fields of force or waves are used as the bricks from which we build our system. They make this assertion on the strength of the experimental results obtained in physics during the last thirty years — results which relate to measurements of every kind; on the strength of the long-continued failure of all attempts at comprehending satisfactorily the totality of these experiments through the medium of a deterministic model; and finally on the strength of the very creditable success which has been reached by a departure from a strict determinism.
Evidently such success and failure cannot in itself determine so grave a question. However firmly we may be convinced that it was determinism which was the stumbling block in all the attempts that had been made hitherto, and however strongly we may believe that it is the obstacle preventing a completely satisfactory explanation of all the observed phenomena; however considerable finally the successes achieved by the employment of an undeterministic picture may be, it is unlikely that we shall ever be able to demonstrate the impossibility of finding any deterministic model of nature capable of doing justice to the facts.
The modem attempts to relinquish determinism are rendered particularly interesting by the fact that their claims with regard to the absence of determinism, far from being vague and inaccurate, are quantitatively quite definite and can be expressed in centimeters, grams and seconds. As a simple example, we may take a mass point in motion either in a state of isolation from others or as a member of a system of many mass points exerting force upon each other. The claim which is made is that its movement cannot be foretold with complete accuracy because, among other things, it would be necessary to know its position and velocity at the initial point of time; and it is claimed that it is impossible in principle to determine both of these exactly.
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According to classical physics, and especially mechanics, it would be necessary to undertake certain operations in order to take a mass point to a given place at the initial point of time and in order to impress upon it a given velocity. Thus we might take it between nippers, carry it to the place in question and push it in an appropriate direction. Quantum mechanics teaches us that if such an operation is undertaken with a mass point a great number of times, the same result does not invariably come about even if the operation is always exactly the same. But it further teaches that the result obtained is not entirely a matter of chance. What is claimed is that if you repeat the same experiment a million times and register the frequency with which the different possible results occur, they will in a second million experiments repeat themselves with exactly the same frequency. It is assumed, of course, that all the experiments are exactly identical.
It will be seen that this claim approximates closely to the so-called law of trial and error governing actual measurements. What is peculiar in this theoretical assertion is the fact that there is a rigid limit to the accuracy of observation, a limit which in its turn is determined by a constant of Nature. Hitherto in all our theoretical considerations we had quite unconsciously assumed, that at any rate in principle, observations could be carried out with any degree of accuracy; nobody had dreamed that a correlation of the kind mentioned between the accuracies of the different measurements (in the present instance position and velocity) did in fact exist.
The other assertions made by modern physics in support of indeterminism are essentially of a similar kind, although they are less easily comprehensible, especially to non-physicists; and a discussion of them would not promote our present argument.
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Presumably I may take it as known that some fifty years ago it was grasped that a very large number of so-called natural laws were statistical laws which were fulfilled with extreme accuracy only because the number of individual entities concerned was extremely great. Thus, for example, the pressure exerted by a gas on the walls of the container is taken to be the resultant of a very large number of individual impulses exerted by molecules striking against the container and rebounding from it. Now the kinetic energy of an individual molecule at a given temperature is far from being exactly determined; all that is determined is its average value, while the individual values vary somewhat considerably (their law of distribution being exactly known both theoretically and experimentally). The direction in which the molecules strike the container is wholly contingent and the number of molecules striking it in any unit of time is also, of course, subject to variations.
Nevertheless, the average value of the pressure is a well-defined physical quantity. Its casual fluctuations
are far beyond the limit of experimental accuracy, provided that the surface of the body, which experiences the pressure, and the time which is physically involved in the
"process of averaging" is not too small. If, however, a very light and small body is subjected to pressure these conditions are not fulfilled and, as might have been expected, the purely contingent variations in pressure cause it to execute a trembling motion known as Brownian movement.
But not only the laws governing the stationary equilibria have disclosed their statistical nature: the same holds, in most cases, for the dynamic evolution of physical happening. To put it briefly, all the laws relating to irreversible natural processes are now known definitely to be of a statistical kind; and this means, of course, the great majority of laws, since in the main the course of events in nature is irreversible. As an example I may quote the conduction of heat in a gas.
![]() 001011110...0110100001It is possible that a physical system might be so simple that this meager information would suffice to settle its fate: in that case nature would not be more complicated than a game of chess. To determine the positions in a game of chess, thirty-three facts suffice. If I know of every piece where it is or whether it has been taken, and if I know whose move it is, then I know the position of the game, and a super-player would be able to state definitely whether White could force a win by playing correctly or whether he could only force a draw or whether, if Black plays correctly, White must lose. If nature is more complicated than a game of chess, a belief to which one tends to incline, then a physical system cannot be determined by a finite number of observations. But in practice a finite number of observations is all that we can make. All that is left to determinism is to believe that an infinite accumulation of observations would in principle enable it completely to determine the system. Such was the standpoint and view of classical physics, which latter certainly had a right to see what it could make of it. But the opposite standpoint has an equal justification: we are not compelled to assume that an infinite number of observations, which cannot in any case be carried out in practice, would suffice to give us a complete determination. This is the direction in which modern physics has led us without really intending it. Normal | Teacher | Scholar |