This paper presents Chapter X (section 3) from the book THE NATURE OF THE PHYSICAL WORLD by A. S. EDDINGTON. The contents of this book are based on the lectures that Eddington delivered at the University of Edinburgh in January to March 1927.
The paragraphs of original material are accompanied by brief comments in color, based on the present understanding. Feedback on these comments is appreciated.
The heading below links to the original materials.
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Development of the New Quantum Theory
The “New Quantum Theory” originated in a remarkable paper by Heisenberg in the autumn of 1925. I am writing the first draft of this lecture just twelve months after the appearance of the paper. That does not give long for development; nevertheless the theory has already gone through three distinct phases associated with the names of Born and Jordan, Dirac, Schrodinger. My chief anxiety at the moment is lest another phase of reinterpretation should be reached before the lecture can be delivered. In an ordinary way we should describe the three phases as three distinct theories. The pioneer work of Heisenberg governs the whole, but the three theories show wide differences of thought. The first entered on ‘the new road in a rather matter-of-fact way; the second was highly transcendental, almost mystical; the third seemed at first to contain a reaction towards classical ideas, but that was probably a false impression. You will realise the anarchy of this branch of physics when three successive pretenders seize the throne in twelve months; but you will not realise the steady progress made in that time unless you turn to the mathematics of the subject. As regards philosophical ideas the three theories are poles apart; as regards mathematical content they are one and the same. Unfortunately the mathematical content is just what I am forbidden to treat of in these lectures.
Heisenberg’s paper was about the Uncertainty Principle where two related quantities could not be determined accurately even on a theoretical basis. This uncertainty was entering the picture because space and time were being treated as absolute, and the variability in the “substantialness of substance” was not fully understood.
As the substance became less substantial, the nature of space and time also became more diffused inside the atom. This phenomenon of quantization was simply bypassed when material units of space and time were used. This amounted to treating space and time as absolute.
I am, however, going to transgress to the extent of writing down one mathematical formula for you to contemplate; I shall not be so unreasonable as to expect you to understand it. All authorities seem to be agreed that at, or nearly at, the root of everything in the physical world lies the mystic formula
qp—pq = ih/2π
We do not yet understand that; probably if we could understand it we should not think it so fundamental. Where the trained mathematician has the advantage is that he can use it, and in the past year or two it has been used in physics with very great advantage indeed. It leads not only to those phenomena described by the older quantum laws such as the h rule, but to many related phenomena which the older formulation could not treat.
On the right-hand side, besides h (the atom of action) and the merely numerical factor 2π, there appears i (the square root of -1) which may seem rather mystical. But this is only a well-known subterfuge; and far back in the last century physicists and engineers were well aware that √-1i in their formulae was a kind of signal to look out for waves or oscillations. The right-hand side contains nothing unusual, but the left-hand side baffles imagination. We call q and p co-ordinates and momenta, borrowing our vocabulary from the world of space and time and other coarse-grained experience; but that gives no real light on their nature, nor does it explain why qp is so ill-behaved as to be unequal to pq.
It is here that the three theories differ most essentially. Obviously q and p cannot represent simple numerical measures, for then qp—pq would be zero. For Schrodinger p is an operator. His “momentum” is not a quantity but a signal to us to perform a certain mathematical operation on any quantities which may follow. For Born and Jordan p is a matrix—not one quantity, nor several quantities, but an infinite number of quantities arranged in systematic array. For Dirac p is a symbol without any kind of numerical interpretation; he calls it a q-number, which is a way of saying that it is not a number at all.
I venture to think that there is an idea implied in Dirac’s treatment which may have great philosophical significance, independently of any question of success in this particular application. The idea is that in digging deeper and deeper into that which lies at the base of physical phenomena we must be prepared to come to entities which, like many things in our conscious experience, are not measurable by numbers in any way; and further it suggests how exact science, that is to say the science of phenomena correlated to measure-numbers, can be founded on such a basis.
One of the greatest changes in physics between the nineteenth century and the present day has been the change in our ideal of scientific explanation. It was the boast of the Victorian physicist that he would not claim to understand a thing until he could make a model of it; and by a model he meant something constructed of levers, geared wheels, squirts, or other appliances familiar to an engineer. Nature in building the universe was supposed to be dependent on just the same kind of resources as any human mechanic; and when the physicist sought an explanation of phenomena his ear was straining to catch the hum of machinery. The man who could make gravitation out of cog-wheels would have been a hero in the Victorian age.
Nowadays we do not encourage the engineer to build the world for us out of his material, but we turn to the mathematician to build it out of his material. Doubtless the mathematician is a loftier being than the engineer, but perhaps even he ought not to be entrusted with the Creation unreservedly. We are dealing in physics with a symbolic world, and we can scarcely avoid employing the mathematician who is the professional wielder of symbols; but he must rise to the full opportunities of the responsible task entrusted to him and not indulge too freely his own bias for symbols with an arithmetical interpretation. If we are to discern controlling laws of Nature not dictated by the mind it would seem necessary to escape as far as possible from the cut-and-dried framework into which the mind is so ready to force everything that it experiences.
I think that in principle Dirac’s method asserts this kind of emancipation. He starts with basal entities inexpressible by numbers or number-systems and his basal laws are symbolic expressions unconnected with arithmetical operations. The fascinating point is that as the development proceeds actual numbers are exuded from the symbols. Thus although p and q individually have no arithmetical interpretation, the combination qp—pq has the arithmetical interpretation expressed by the formula above quoted. By furnishing numbers, though itself non-numerical, such a theory can well be the basis for the measure-numbers studied in exact science. The measure-numbers, which are all that we glean from a physical survey of the world, cannot be the whole world; they may not even be so much of it as to constitute a self-governing unit. This seems the natural interpretation of Dirac’s procedure in seeking the governing laws of exact science in a non-arithmetical calculus.
I am afraid it is a long shot to predict anything like this emerging from Dirac’s beginning; and for the moment Schrodinger has rent much of the mystery from the p’s and q’s by showing that a less transcendental interpretation is adequate for present applications. But I like to think that we may have not yet heard the last of the idea.
Schrodinger’s theory is now enjoying the full tide of popularity, partly because of intrinsic merit, but also, I suspect, partly because it is the only one of the three that is simple enough to be misunderstood. Rather against my better judgment I will try to give a rough impression of the theory. It would probably be wiser to nail up over the door of the new quantum theory a notice, “Structural alterations in progress—No admittance except on business”, and particularly to warn the doorkeeper to keep out prying philosophers. I will, however, content myself with the protest that, whilst Schrodinger’s theory is guiding us to sound and rapid progress in many of the mathematical problems confronting us and is indispensable in its practical utility, I do not see the least likelihood that his ideas will survive long in their present form.
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