The logic of modern physics

by P. W. Bridgman

We now ask what significance is to be ascribed to the sort of conservation that energy does have. We restrict ourselves first to mechanical systems. The motions of a mechanical system satisfy certain differential equations of the second order, and the actual motion is to be found by an integration of the equations. In the integral of a differential equation certain constants appear which are determined by the initial conditions, and are therefore the same during all the future motion of the system; obviously these constants of motion correspond to conservative properties. This reasoning can of course be at once extended. Any system, mechanical or not, whose motion is determined by differential equations, will have certain conservative properties. For the systems of mechanics energy is one of the conservative functions; others are momentum and moment of momentum. Energy is particularly simple, in that it is connected with measurable properties of the system by a simple formula (∑ ½ mv²), and is furthermore scalar, which is also a property of quantity of matter. But to go further and ascribe to energy other properties of matter, such as localization in space, is entirely overlooking the essential difference in the character of the operation by which matter and quantity of energy are measured, that is, overlooking the essential difference in their physical character.

The possible extension of the energy concept from mechanics to thermodynamics receives a sufficient physical explanation in terms of our views of the essentially mechanical character of thermal phenomena. That the idea can be extended also to simple electrical or magnetic systems, in which the effect of velocity of propagation is neglected, is a consequence of the fact that in these systems the equations of motion remain of the same general mechanical type, it having been shown by Maxwell that the equations of such systems may be written in the generalized Lagrangean form. When, however, we extend our formulas to systems in which the velocity of propagation is important (that is, when we consider the field equations in their general form) we find that the Lagrangean equations no longer apply to matter taken by itself, and energy is no longer conserved in the original sense. A new function appears, however, which behaves mathematically in the same way that the energy did before. The equations of motion of the system remain Lagrangean in form if the mechanical parts of the system are supplemented by the electric and magnetic fields in space. In this extended form we have, therefore, a conservative function as before, and the energy concept may be retained in this enlarged aspect. The physical operations by which energy is determined are entirely altered, however, and the physical character of the concept is changed. No more than before is there justification for localizing energy in space, or ascribing to it other properties of matter. Yet the materialization of the energy concept, and the consequent desire that energy be localized in space, is one of the strongest arguments in many minds for the existence of a medium.

As far as I can see, therefore, the existence of conservative functions is involved in the possibility of describing natural phenomena with differential equations. That further there is a conservative function of the precise form found in mechanics is a consequence of the particular form of the equations and the nature of the forces. The question of the significance of the fact that the forces of nature appear to be conservative, with respect to this particular function of mechanics, is of much interest, but it is not our immediate concern now. We are interested rather to ask under what general conditions we shall have conservative functions. Quantum theory strongly suggests that when we pass to phenomena on a small enough scale, we may no longer be able to employ differential equations in our descriptions, and hence the previous reason for the existence of constants connected with the motion disappears. Now there is one obvious remark to be made about this more general situation. Whenever the future history of a system is so connected with its present condition that we can retrace our way to the present from any future configuration, we shall always have conservative functions. For any future configuration contains certain fixed (or conservative) features, in that we can reconstruct the unique present from any future state. There is no reason to expect that the operations by which we find the fixed features will always be simple, as in the mechanical case. Now the determination of the future by the present, and conversely the possibility of reconstructing the present from the future (or the past from the present), is, we are convinced, a property which is at least approximately true of phenomena down to a smaller scale of magnitude than we have yet reached, and so we expect to find these conservative functions in systems whose ultimate laws of motion are much more general than any with which we are yet familiar. The particular form of the conservative function depends on the character of the system. That there is a scalar conservative function for ordinary systems depends of course on particular properties of the system, but we are at least prepared to find that a scalar conservative function does not necessarily mean a differential equation of the second order.

The potential energy of a system has a particular significance with respect to this point of view. In an ordinary mechanical system, the potential energy simply measures the work done by the applied forces in being displaced from the initial to the final positions; that is, the potential energy is a measure of the deviation from the initial position, and so measures a certain feature of the history of the system. In the more general system, in which we may not have differential equations, we may look for something analogous to the potential energy which shall measure the displacement of the system from its initial configuration. Such a measure is always possible as long as the past can be reconstructed from the present (or the present from the future). We recall a remark of Poincaré's[20] to the effect that we of necessity must always have conservation, because if we have a system in which conservation apparently fails, we merely have to invent a new form of potential energy. This remark is obviously not of entire generality, but applies only to such systems as those considered here, in which the past may be reconstructed from the present.

[20]Henri Poincaré, Wissenschaft und Hypothese, translated into German by F. and L. Lindemann, Teubner, Leipzig, 1906, Chap. VIII.

Of late there has been much discussion of the advisability, on the basis of certain quantum phenomena, of giving up conservation as a principle applied to the details of the emission and absorption of light, retaining it only in a statistical sense. It seems to me that the question here in the minds of physicists was always merely one of convenience, and that few, if any, doubted the ultimate applicability of the principle of Poincaré, or thought that we were here concerned with a system of such great generality that the past could not be reconstructed from the present. The question was merely whether those variables in terms of which the potential energy is defined make close enough connection with other things of immediate experimental significance, or whether on the whole the retention of a potential energy is not more trouble than is justified by its convenience, making a treatment from the statistical point of view preferable. However, this is all now a matter of more or less past history, because with the recent extension of the experiments of Compton,[21] we seem to have experimental evidence for the validity of the conservation law in detail for elementary quantum processes, with a corresponding simple potential energy.

[21]W. Bothe and H. Geiger, 2S. f. Phys. 32, 639-663, 1925. A. H. Compton, Proc. Nat. Acad. Soc., II, 303-306, 1925.

Going still deeper, however, there are quantum phenomena which still may have to be treated by statistical methods, and this may mean giving up conservation in detail. We have no experimental evidence, for example, of what the electron is doing while jumping from one quantum orbit to another. A situation like this merely means that those details which determine the future in terms of the past may lie so deep in the structure that at present we have no immediate experimental knowledge of them, and we may for the present be compelled to give a treatment from a statistical point of view based on considerations of probability. But I suppose that no one, except perhaps Norman Campbell,[22] will maintain that such a situation is more than temporary, or will cease to search for consequences of these details of structure which may be open to experimental verification.

[22]Norman Campbell, Time and Chance, Phil. Mag. I, 1106-1117, 1926.

Similarly, we cannot permanently be satisfied with a picture of radioactive phenomena which represents radioactive disintegration as a matter of chance.