Mathematical Theories

Modern physics is not satisfied with only qualitative description of the fundamental structure of matter.

It must try on the basis of careful experimental investigations to get a mathematical formulation of those natural laws that determine the `forms’ of matter, the elementary particles and their forces.

A clear distinction between matter and force can no longer be made in this part of physics, since each elementary particle not only is producing some forces and is acted upon by forces, but it is at the same time representing a certain field of force. The quantum-theoretical dualism of waves and particles makes the same entity appear both as matter and as force.

All the attempts to find a mathematical description for the laws concerning the elementary particles have so far started from the quantum theory of wave fields.

Theoretical work on theories of this type started early in the thirties.

But the very first investigations on this line revealed serious difficulties the roots of which lay in the combination of quantum theory and the theory of special relativity. At first sight it would seem that the two theories, quantum theory and the theory of relativity, refer to such different aspects of nature that they should have practically nothing to do with each other, that it should be easy to fulfill the requirements of both theories in the same formalism. A closer inspection, however, shows that the two theories do interfere at one point, and that it is from this point that all the difficulties arise.

The theory of special relativity had revealed a structure of space and time somewhat different from the structure that was generally assumed since Newtonian mechanics. The most characteristic feature of this newly discovered structure is the existence of a maximum velocity that cannot be surpassed by any moving body or any traveling signal, the velocity of light.

As a consequence of this, two events at distant points cannot have any immediate causal connection if they take place at such times that a light signal starting at the instant of the event on one point reaches the other point only after the time the other event has happened there; and vice versa.

In this case the two events may be called simultaneous. Since no action of any kind can reach from the one event at the one point in time to the other event at the other point, the two events are not connected by any causal action.

For this reason any action at a distance of the type, say, of the gravitational forces in Newtonian mechanics was not compatible with the theory of special relativity. The theory had to replace such action by actions from point to point, from one point only to the points in an infinitesimal neighborhood. The most natural mathematical expressions for actions of this type were the differential equations for waves or fields that were invariant for the Lorentz transformation.

Such differential equations exclude any direct action between ‘simultaneous’ events.

Therefore, the structure of space and time expressed in the theory of special relativity implied an infinitely sharp boundary between the region of simultaneousness, in which no action could be transmitted, and the other regions, in which a direct action from event to event could take place.

On the other hand, in quantum theory the uncertainty relations put a definite limit on the accuracy with which positions and momenta, or time and energy, can be measured simultaneously. Since an infinitely sharp boundary means an infinite accuracy with respect to position in space and time, the momenta or energies must be completely undetermined, or in fact arbitrarily high momenta and energies must occur with overwhelming probability.

Therefore, any theory which tries to fulfill the requirements of both special relativity and quantum theory will lead to mathematical inconsistencies, to divergencies in the region of very high energies and momenta.

This sequence of conclusions might not seem strictly binding, since any formalism of the type under consideration is very complicated and could perhaps offer some mathematical possibilities for avoiding the clash between quantum theory and relativity.

But so far all the mathematical schemes that have been tried did in fact lead either to divergencies, i.e., to mathematical contradictions, or did not fulfill all the requirements of the two theories.

The difficulties came from the point that has been discussed.

The way in which the convergent mathematical schemes did not fulfill the requirements of relativity or quantum theory was in itself quite interesting. For instance, one scheme, when interpreted in terms of actual events in space and time, led to a kind of time reversal; it would predict processes in which suddenly at some point in space particles are created, the energy of which is later provided for by some other collision process between elementary particles at some other point.

The physicists are convinced from their experiments that processes of this type do not occur in nature, at least not if the two processes are separated by measurable distances in space and time.

Another mathematical scheme tried to avoid the divergencies through a mathematical process which is called renormalization; it seemed possible to push the infinities to a place in the formalism where they could not interfere with the establishment of the welldefined relations between those quantities that can be directly observed.

This scheme has led to very substantial progress in quantum electrodynamics, since it accounts for some interesting details in the hydrogen spectrum that had not been understood before.

A closer analysis of this mathematical scheme, however, has made it probable that those quantities which in normal quantum theory must be interpreted as probabilities can under certain circumstances become negative in the formalism of renormalization.

This would prevent the consistent use of the formalism for the description of matter.

The final solution of these difficulties has not yet been found. It will emerge someday from the collection of more and more accurate experimental material about the different elementary particles, their creation and annihilation, the forces between them.

In looking for possible solutions of the difficulties one should perhaps remember that such processes with time reversal as have been discussed before could not be excluded experimentally, if they took place only within extremely small regions of space and time outside the range of our present experimental equipment. Of course one would be reluctant to accept such processes with time reversal if there could be at any later stage of physics the possibility of following experimentally such events in the same sense as one follows ordinary atomic events.

But here the analysis of quantum theory and of relativity may again help us to see the problem in a new light.

The theory of relativity is connected with a universal constant in nature, the velocity of light.

This constant determines the relation between space and time and is therefore implicitly contained in any natural law which must fulfill the requirements of Lorentz invariance.

Our natural language and the concepts of classical physics can apply only to phenomena for which the velocity of light can be considered as practically infinite. When we in our experiments approach the velocity of light we must be prepared for results which cannot be interpreted in these concepts.

Quantum theory is connected with another universal constant of nature, Planck’s quantum of action.

An objective description for events in space and time is possible only when we have to deal with objects or processes on a comparatively large scale, where Planck’s constant can be regarded as infinitely small.

When our experiments approach the region where the quantum of action becomes essential we get into all those difficulties with the usual concepts that have been discussed in earlier chapters of this volume.

There must exist a third universal constant in nature. This is obvious for purely dimensional reasons. The universal constants determine the scale of nature, the characteristic quantities that cannot be reduced to other quantities.

One needs at least three fundamental units for a complete set of units. This is most easily seen from such conventions as the use of the c-g-s system (centimeter, gram, second system) by the physicists. A unit of length, one of time, and one of mass is sufficient to form a complete set; but one must have at least three units. One could also replace them by units of length, velocity and mass; or by units of length, velocity and energy, etc. But at least three fundamental units are necessary. Now, the velocity of light and Planck’s constant of action provide only two of these units. There must be a third one, and only a theory which contains this third unit can possibly determine the masses and other properties of the elementary particles. Judging from our present knowledge of these particles the most appropriate way of introducing the third universal constant would be by the assumption of a universal length the value of which should be roughly io-13 cm, that is, somewhat smaller than the radii of the light atomic nuclei. When from such three units one forms an expression which in its dimension corresponds to a mass, its value has the order of magnitude of the masses of the elementary particles.

If we assume that the laws of nature do contain a third universal constant of the dimension of a length and of the order of io3 cm, then we would again expect our usual concepts to apply only to regions in space and time that are large as compared to the universal constant.

We should again be prepared for phenomena of a qualitatively new character when we in our experiments approach regions in space and time smaller than the nuclear radii.

The phenomenon of time reversal, which has been discussed and which so far has only resulted from theoretical considerations as a mathematical possibility, might therefore belong to these smallest regions.

If so, it could probably not be observed in a way that would permit a description in terms of the classical concepts. Such processes would probably, so far as they can be observed and described in classical terms, obey the usual time order.

But all these problems will be a matter of future research in atomic physics. One may hope that the combined effort of experiments in the high energy region and of mathematical analysis will someday lead to a complete understanding of the unity of matter. The term `complete understanding’ would mean that the forms of matter in the sense of Aristotelian philosophy would appear as results, as solutions of a closed mathematical scheme representing the natural laws for matter.