The logic of modern physics

by P. W. Bridgman

In attempting to check our statement experimentally, the only additional complication, as compared with the static case which we have already checked, is afforded by the motion of the charge, for we have defined the magnitude of a charge in motion, so there is no difficulty here, and we may furthermore suppose that the field is generated by stationary charges, so that we need not trouble to inquire whether the procedure by which the field was originally defined is here applicable. The task of checking the equation then reduces to the simple physical task of measuring the force on the moving charge. How shall we do this? If the velocity is low, we may tie a string to the charge and measure the force with a spring balance (or its equivalent). But now an examination of the equations shows that in more complicated phenomena perceptible deviations from the static behavior are to be expected only at much higher velocities than can be attained by towing charges with a string and a spring balance, so that it is evidently necessary to check the simple equation for the force on a moving charge also at high velocity. Since at high velocity the spring balance method for measuring forces fails, we are driven to the only procedure that we have, namely a measurement in terms of the resultant acceleration, calculating the force by Newton's first law of mechanics. But this involves a knowledge of the mass of the moving body, which we recognize in general may be a function of the velocity. Now we have already seen, in discussing the concepts of mechanics, that the operations by which mechanical mass is defined cannot be carried out at high velocities, so that either the concept of mechanical mass becomes meaningless at high velocities, or we must adopt another definition. In attempting to give this new definition of mass at high velocities, we are driven to a result of special relativity theory, namely that all mass, mechanical or electrical, must be the same function of velocity. If now electrical mass can be found in terms of velocity, our immediate problem is solved and we shall be in a position to complete the experimental check of the equation. But as a matter of fact, in order to determine electrical mass, we have to use that equation which we are now engaged in trying to establish. Logically we have again the vicious circle, the physical significance of which is that independent operations do not exist for giving unique meaning to the concept of force on a charge at high velocity.

We seemed so close to our goal a minute ago; that we may allow ourselves to jump the logical chasm, and assume that the equation is correct. Electrical mass now becomes a definite function of velocity, mechanical mass the same function, and we are in a position to compare the actual acceleration received by a charge in a field with that calculated by the equation. Our conviction, on the basis of all experience up to the present, is that the two accelerations will be found to agree.

The equation then does somehow make correct connection with experience in that a consequence of the equation can be verified experimentally, in spite of the fact that as the equation stands it is meaningless, because the operations do not exist by which meaning can be given to the individual terms. At low velocities the equation really says what it seems to say, because the individual terms have meaning in terms of operations; and, what is more, what the equation says agrees with experiment. At high velocities the equation does not mean at all what appears on the surface; by itself it has no meaning; it has meaning only when considered as a member of a system of equations, and only in so far as the system of equations makes by implication statements about nature that have meaning in terms of operations that can be carried out physically. The individual terms of the equation of the system do not have meaning at high velocities, and in fact there are more terms than there are independent physical operations.

An exact analysis from the operational point of view of the significance of the equations at high velocities has perhaps never been made, and is not necessary for our immediate purpose. The discussion has brought out, however, that the number of physically independent concepts has been cut down by two at least, in that we have made purely formal definitions of the meaning of quantity of electricity, and of the force exerted by a field on a charge at high velocity. There is no reason to think that there is anything unique about this analysis, or that formal definitions might not have been given to other concepts than charge and force. We can only state that as far as physical content goes the equations have at least two degrees of freedom. It should then be possible to find quite different sorts of equations which agree equally well with experience. In particular, since we have seen that the force on a moving charge has no meaning in terms of independent operations, it should be possible by arbitrary definition to make this force any function of velocity that we please (of course reducing to the proper value at low velocities), and then to determine the other equations so that the entire group of equations is consistent with experiment. So far as I know, no one has tried to give such a modified set of equations, and indeed there is no particular reason why anyone should bother to do this, because the present equations are simple enough, and the modified equations, although perhaps differing greatly in appearance from the present ones, would have no advantage in any greater or different physical content.

But there is no reason to think that the present state of affairs will always continue. We have seen that the decrease in the number of concepts corresponds to our inability to measure as many sorts of physical things at high velocities as at low. Now it is the task of the future experimenter so to refine the possibilities of measurement at high velocities as to restore these two degrees of freedom. In particular, mass should be made measurable in mechanical terms at high velocities. When this restoration has been made, and all the quantities in our equations receive independent physical meaning, the significance of the equations in terms of operations will be quite altered, although the formal appearance will be unchanged. We must then be prepared to find, as always when we change the range of phenomena, that the equations in their present form do not correspond to the facts at all, and that one of the alternative forms allowed by our present two degrees of freedom is the correct form. But until the new experimental facts have been obtained, it seems hardly worth while to attempt to specify the doubly infinite variety of forms which the equations might have consistently with present experiment.[23]

[23]Since this was written, a paper has appeared by V. Bush, Jour. of Math. and Phys., vol. V., No. 3, 1926, in which it is shown that there are advantages in supposing the charge of an electron to change when it is set in motion.

So far we have discussed the extension of ordinary electric phenomena in only one direction, to high velocities. There is another extension which is much more important physically, namely to very small scales of magnitude. This extension is necessary to an understanding of the properties of matter in bulk, the electrical nature of the atom having been once established. Our problem is to show how the statistical average of the behavior of a large number of electrons gives the large scale effects which are within the reach of observation, and which are described by the equations we have just discussed. To get this statistical average we must be able to calculate at least certain features of the behavior of the individual electrons, which means that we must know the form of the equations down to dimensions of the order of those of an electron, or smaller. Now if one contrasts the scale of the supposed dimensions of the electron with the smallest dimensions on which we can make independent experimental verification of those equations, he must admit that there is an enormous chance for change in the type of equation beyond the limit that we can reach by direct experiment, and the chances of guessing the correct extension of the equation to small dimensions are correspondingly almost vanishingly small. (We may perhaps say that experiments on the Brownian motion on a scale a good many atoms in diameter bring us the closest possible directly, which means that we are 10⁶ or 10⁷ fold away from electronic dimensions.) In spite, however, of the apparently enormous chances against it, this program of extending the field equations to small dimensions and following out the consequences was exactly the program which Lorentz set himself.[24] That Lorentz saw that such a program might be carried through must be recognized as a vision of extraordinary genius, and that he was willing to devote to it the years of arduous and detailed calculation that he did is evidence of a pertinacity of purpose of the highest moral order.

[24]See for example, H. A. Lorentz, The Theory of Electrons, B. G. Teubner, 1916.

We now have to examine critically this program and to inquire what is the significance of the measure of success that Lorentz attained. The precise extension of the equations that he made was very simple, for the large scale equations of Maxwell were taken over with as little change as possible. The equations are so familiar that it is not necessary for us to write them in detail; they express relations between the electric and magnetic force vectors (force and induction now becoming the same thing, the difference between them in ponderable bodies being one of the things that is to be explained in terms of the electrons), the space density of electric charge, its velocity, and the force acting on elementary charge. We have to notice that although formally the equations have changed little in appearance, nevertheless the physical content, as judged by the operations, has changed a great deal. Consider, for instance, the meaning of charge density. In the Maxwell equations, ρ was merely the number of discrete elementary charges per unit volume, the distances between these charges being supposed so small compared with the scale of the phenomena involved that their average effect could be fairly represented in terms of their numbers. In the Lorentz equations, on the other hand, ρ has a value different from 0 only inside the electron; everywhere else ρ = 0. Now an examination of the previous discussion, in which we questioned whether the magnitude of the charge might be a function of its velocity, will show that there are no physical operations whatever by which meaning can be given to ρ at individual points inside an electron. There is a single condition on this ρ, namely, that its integral throughout the total volume assigned to the electron shall equal the total static charge of the electron. Obviously a single scalar condition is a pretty blunt tool with which to attempt to determine a point function throughout a volume. Again, the equations talk about the velocity of the charge at interior points of the electron; what possible physical operations are there by which meaning can be assigned to the velocity of an amorphous structureless substance in regions inaccessible to experiment? Here again, the concept as a detailed description of the behavior at a point has become meaningless, and again there is a single integral condition, namely, that the v associated with every ρ must be such that when integrated over the volume of the electron it will give a total transport of charge equal to that carried by the electron in its motion. This again is a single condition on a function distributed through space. Still again, the equations contain the electric and magnetic vectors at points inside the electron. What is the possible meaning of these field vectors in terms of operations? Our procedure for finding the field at a point involves by definition finding the force on an electric charge placed at that point. But there is no charge smaller than an electron, and the procedure degenerates into a fiction. Again there is a single integral condition on the field vectors; the integral of the force on the assumed charge density when taken over the total volume of the electron must give a value corresponding to experiment. Except for this single condition, the concept of the field at points inside the electron is an invention without physical reality. Not only is the field concept meaningless at points inside the electron, but it is meaningless at points outside within a certain distance, because the exploring charge can never be made smaller than the electron itself, and so can never come closer than a certain distance.

The actual state of affairs is much worse than has already appeared. It was shown in the discussion of space and time that no independent physical meaning can be attached to lengths and times as small as have to be assumed in describing the behavior of the individual electrons. The operations Div, Curl, d/dt which enter the field equations are, therefore, physically meaningless as they stand; they have only a mathematical meaning which begins to acquire physical complexion in a most complicated way when the equations are integrated over large enough volumes.