Out of My Later Years Part 3

Cla.s.sical mechanics contains one point which is unsatisfactory in that, in the fundamentals, the same ma.s.s constant is met twice over in two different roles, namely as "inertial ma.s.s" in the law of motion, and as "gravitational ma.s.s" in the law of gravitation. As a result of this, the acceleration of a body in a pure gravitational field is independent of its material; or, in a coordinate system of uniform acceleration (accelerated in relation to an "inertial system") the motions take place as they would in a h.o.m.ogeneous gravitational field (in relation to a "motionless" system of coordinates). If one a.s.sumes that the equivalence of these two cases is complete, then one attains an adaptation of our theoretical thinking to the fact that the gravitational and inertial ma.s.ses are identical.

From this it follows that there is no longer any reason for favoring, as a fundamental principle, the "inertial systems"; and, we must admit as equivalent in their own right, also non-linear transformations of the coordinates (x1, x2, x3, x4). If we make such a transformation of a system of coordinates of the special theory of relativity, then the metric ds2=dx12 + dx22 + dx32 - dx42.

goes over to a general (Riemannian) metric of Bane ds2 = g dx dx (Summed over and ).

where the g, symmetrical in and , are certain functions of x1 ... x4 which describe both the metric property, and the gravitational field in relation to the new system of coordinates.

The foregoing improvement in the interpretation of the mechanical basis must, however, be paid for in that-as becomes evident on closer scrutiny-the new coordinates could no longer be interpreted, as results of measurements by rigid bodies and clocks, as they could in the original system (an inertial system with vanis.h.i.+ng gravitational field).

The pa.s.sage to the general theory of relativity is realized by the a.s.sumption that such a representation of the field properties of s.p.a.ce already mentioned, by functions g (that is to say by a Riemann metric), is also justified in the general case in which there is no system of coordinates in relation to which the metric takes the simple quasi-Euclidian form of the special theory of relativity.

Now the coordinates, by themselves, no longer express metric relations, but only the "neighborliness" of the things described, whose coordinates differ but little from one another. All transformations of the coordinates have to be admitted so long as these transformations are free from singularities. Only such equations as are covariant in relation to arbitrary transformations in this sense have meaning as expressions of general laws of nature (postulate of general covariancy).

The first aim of the general theory of relativity was a preliminary statement which, by giving up the requirement of const.i.tuting a closed thing in itself, could be connected in as simple a manner as possible with the "facts directly observed." Newton's gravitational theory gave an example, by restricting itself to the pure mechanics of gravitation. This preliminary statement may be characterized as follows: (1) The concept of the material point and of its ma.s.s is retained. A law of motion is given for it, this law of motion being the translation of the law of inertia into the language of the general theory of relativity. This law is a system of total differential equations, the system characteristic of the geodetic line.

(2) In place of Newton's law of interaction by gravitation, we shall find the system of the simplest generally covariant differential equations which can be set up for the g,-tensor. It is formed by equating to zero the once contracted Riemannian curvature tensor, (R = 0).

This formulation permits the treatment of the problem of the planets. More accurately speaking, it allows the treatment of the problem of motion of material points of practically negligible ma.s.s in the gravitational field produced by a material point which itself is supposed to have no motion (central symmetry). It does not take into account the reaction of the "moved" material points on the gravitational field, nor does it consider how the central ma.s.s produces this gravitational field.

a.n.a.logy with cla.s.sical mechanics shows that the following is a way to complete the theory. One sets up as field equation Rik - gikR = - Tik.

where R represents the scalar of Riemannian curvature, Tik the energy tensor of the matter in a phenomenological representation. The left side of the equation is chosen in such a manner that its divergence disappears identically. The resulting disappearance of the divergence of the right side produces the "equations of motion" of matter, in the form of partial differential equations for the case where Tik introduces, for the description of the matter, only four further functions independent of each other (for instance, density, pressure, and velocity components, where there is between the latter an ident.i.ty, and between pressure and density an equation of condition).

By this formulation one reduces the whole mechanics of gravitation to the solution of a single system of covariant partial differential equations. The theory avoids all internal discrepancies which we have charged against the basis of cla.s.sical mechanics. It is sufficient-as far as we know-for the representation of the observed facts of celestial mechanics. But, it is similar to a building, one wing of which is made of fine marble (left part of the equation), but the other wing of which is built of low grade wood (right side of equation). The phenomenological representation of matter is, in fact, only a crude subst.i.tute for a representation which would correspond to all known properties of matter.

There is no difficulty in connecting Maxwell's theory of the electromagnetic field with the theory of the gravitational field so long as one restricts himself to s.p.a.ce, free of ponderable matter and free of electric density. All that is necessary is to put on the right hand side of the above equation for Tik, the energy tensor of the electromagnetic field in empty s.p.a.ce and to a.s.sociate with the so modified system of equations the Maxwell field equation for empty s.p.a.ce, written in general covariant form. Under these conditions there will exist, between all these equations, a sufficient number of the differential ident.i.ties to guarantee their consistency. We may add that this necessary formal property of the total system of equations leaves arbitrary the choice of the sign of the member Tik, a fact which was later shown to be important.

The desire to have, for the foundations of the theory, the greatest possible unity has resulted in several attempts to include the gravitational field and the electromagnetic field in one formal but h.o.m.ogeneous picture. Here we must mention particularly the five-dimensional theory of Kaluza and Klein. Having considered this possibility very carefully I feel that it is more desirable to accept the lack of internal uniformity of the original theory, because I do not consider that the totality of the hypothetical basis of the five-dimensional theory contains less of an arbitrary nature than does the original theory. The same statement may be made for the projective variety of the theory, which has been elaborated with great care, in particular, by v. Dantzig and by Pauli.

The foregoing considerations concern, exclusively, the theory of the field, free of matter. How are we to proceed from this point in order to obtain a complete theory of atomically constructed matter? In such a theory, singularities must certainly be excluded, since without such exclusion the differential equations do not completely determine the total field. Here, in the field theory of general relativity, we meet the same problem of a theoretical field-representation of matter as was met originally in connection with the pure Maxwell theory.

Here again the attempt to construct particles out of the field theory, leads apparently to singularities. Here also the endeavor has been made to overcome this defect by the introduction of new field variables and by elaborating and extending the system of field equations. Recently, however, I discovered, in collaboration with Dr. Rosen, that the above mentioned simplest combination of the field equations of gravitation and electricity produces centrally symmetrical solutions which can be represented as free of singularity (the well known centrally symmetrical solutions of Schwarzschild for the pure gravitational field, and those of Reissner for the electric field with consideration of its gravitational action). We shall refer to this shortly in the paragraph next but one. In this way it seems possible to get for matter and its interactions a pure field theory free of additional hypotheses, one moreover whose test by submission to facts of experience does not result in difficulties other than purely mathematical ones, which difficulties, however, are very serious.

5. Quantum Theory and the Fundamentals of Physics.

The theoretical physicists of our generation are expecting the erection of a new theoretical basis for physics which would make use of fundamental concepts greatly different from those of the field theory considered up to now. The reason is that it has been found necessary to use-for the mathematical representation of the so-called quantum phenomena-new sorts of methods of consideration.

While the failure of cla.s.sical mechanics, as revealed by the theory of relativity, is connected with the finite speed of light (its avoidance of being ), it was discovered at the beginning of our century that there were other kinds of inconsistencies between deductions from mechanics and experimental facts, which inconsistencies are connected with the finite magnitude (the avoidance of being zero) of Planck's constant h. In particular, while molecular mechanics requires that both, heat content and (monochromatic) radiation density, of solid bodies should decrease in proportion to the decreasing absolute temperature, experience has shown that they decrease much more rapidly than the absolute temperature. For a theoretical explanation of this behavior it was necessary to a.s.sume that the energy of a mechanical system cannot a.s.sume any sort of value, but only certain discrete values whose mathematical expressions were always dependent upon Planck's constant h. Moreover, this conception was essential for the theory of the atom (Bohr's theory). For the transitions of these states into one another-with or without emission or absorption of radiation-no causal laws could be given, but only statistical ones; and, a similar conclusion holds for the radioactive decomposition of atoms, which decomposition was carefully investigated about the same time. For more than two decades physicists tried vainly to find a uniform interpretation of this "quantum character" of systems and phenomena. Such an attempt was successful about ten years ago, through the agency of two entirely different theoretical methods of attack. We owe one of these to Heisenberg and Dirac, and the other to de Broglie and Schrodinger. The mathematical equivalence of the two methods was soon recognized by Schrodinger. I shall try here to sketch the line of thought of de Broglie and Schrodinger, which lies closer to the physicist's method of thinking, and shall accompany the description with certain general considerations.

The question is first: How can one a.s.sign a discrete succession of energy value H to a system specified in the sense of cla.s.sical mechanics (the energy function is a given function of the coordinates qr and the corresponding momenta pr)? Planck's constant h relates the frequency H/h to the energy values H. It is therefore sufficient to give to the system a succession of discrete frequency values. This reminds us of the fact that in acoustics, a series of discrete frequency values is coordinated to a linear partial differential equation (if boundary values are given) namely the sinusoidal periodic solutions. In corresponding manner, Schrodinger set himself the task of coordinating a partial differential equation for a scalar function to the given energy function (qr, pr), where the qr and the time t are independent variables. In this he succeeded (for a complex function ) in such a manner that the theoretical values of the energy H , as required by the statistical theory, actually resulted in a satisfactory manner from the periodic solution of the equation.

To be sure, it did not happen to be possible to a.s.sociate a definite movement, in the sense of mechanics of material points, with a definite solution (qr, t) of the Schrodinger equation. This means that the function does not determine, at any rate exactly, the story of the qr as functions of the time t. According to Born, however, an interpretation of the physical meaning of the functions was shown to be possible in the following manner: (the square of the absolute value of the complex function ) is the probability density at the point under consideration in the configuration-s.p.a.ce of the qr, at the time t. It is therefore possible to characterize the content of the Schrodinger equation in a manner, easy to be understood, but not quite accurate, as follows: it determines how the probability density of a statistical ensemble of systems varies in the configuration-s.p.a.ce with the time. Briefly: the Schrodinger equation determines the alteration of the function of the with the time.

It must be mentioned that the result of this theory contains-as limiting values-the result of the particle mechanics if the wave-length encountered during the solution of the Schrodinger problem is everywhere so small that the potential energy varies by a practically infinitely small amount for a change of one wave-length in the configuration-s.p.a.ce. Under these conditions the following can in fact be shown: We choose a region G0 in the configuration-s.p.a.ce which, although large (in every dimension) in relation to the wave length, is small in relation to the practical dimensions of the configuration-s.p.a.ce. Under these conditions it is possible to choose a function of for an initial time t0 in such a manner that it vanishes outside of the region G0, and behaves, according to the Schrodinger equation, in such a manner that it retains this property-approximately at least-also for a later time, but with the region G0 having pa.s.sed at that time t into another region G. In this manner one can, with a certain degree of approximation, speak of the motion of the region G as a whole, and one can approximate this motion by the motion of a point in the configuration-s.p.a.ce. This motion then coincides with the motion which is required by the equations of cla.s.sical mechanics.

Experiments on interference made with particle rays have given a brilliant proof that the wave character of phenomena of motion as a.s.sumed by the theory does, really, correspond to the facts. In addition to this, the theory succeeded, easily, in demonstrating the statistical laws of the transition of a system from one quantum condition to another under the action of external forces, which, from the standpoint of cla.s.sical mechanics, appears as a miracle. The external forces were here represented by small additions of the potential energy as functions of the time. Now, while in cla.s.sical mechanics, such additions can produce only correspondingly small alterations of the system, in the quantum mechanics they produce alterations of any magnitude however large, but with correspondingly small probability, a consequence in perfect harmony with experience. Even an understanding of the laws of radioactive decomposition, at least in their broad lines, was provided by the theory.

Probably never before has a theory been evolved which has given a key to the interpretation and calculation of such a heterogeneous group of phenomena of experience as has the quantum theory. In spite of this, however, I believe that the theory is apt to beguile us into error in our search for a uniform basis for physics, because, in my belief, it is an incomplete representation of real things, although it is the only one which can be built out of the fundamental concepts of force and material points (quantum corrections to cla.s.sical mechanics). The incompleteness of the representation is the outcome of the statistical nature (incompleteness) of the laws. I will now justify this opinion.

I ask first: How far does the function describe a real condition of a mechanical system? Let us a.s.sume the r to be the periodic solutions (put in the order of increasing energy values) of the Schrodinger equation. I shall leave open, for the time being, the question as to how far the individual r are complete descriptions of physical conditions. A system is first in the condition 1 of lowest energy 1, Then during a finite time a small disturbing force acts upon the system. At a later instant one obtains then from the Schrodinger equation a function of the form = cr,r.

where the cr are (complex) constants. If the r are "normalized," then c1 is nearly equal to 1, c2 etc. is small compared with 1. One may now ask: Does describe a real condition of the system? If the answer is yes, then we can hardly do otherwise than ascribe3 to this condition a definite energy , and, in particular, such an energy as exceeds 1 by a small amount (in any case 1 <>< 2).="" such="" an="" a.s.sumption="" is,="" however,="" at="" variance="" with="" the="" experiments="" on="" electron="" impact="" such="" as="" have="" been="" made="" by="" j.="" franck="" and="" g.="" hertz,="" if,="" in="" addition="" to="" this,="" one="" accepts="" millikan's="" demonstration="" of="" the="" discrete="" nature="" of="" electricity.="" as="" a="" matter="" of="" fact,="" these="" experiments="" lead="" to="" the="" conclusion="" that="" energy="" values="" of="" a="" state="" lying="" between="" the="" quantum="" values="" do="" not="" exist="" from="" this="" it="" follows="" that="" our="" function="" does="" not="" in="" any="" way="" describe="" a="" h.o.m.ogeneous="" condition="" of="" the="" body,="" but="" represents="" rather="" a="" statistical="" description="" in="" which="" the="" cr="" represent="" probabilities="" of="" the="" individual="" energy="" values.="" it="" seems="" to="" be="" clear,="" therefore,="" that="" the="" born="" statistical="" interpretation="" of="" the="" quantum="" theory="" is="" the="" only="" possible="" one.="" the="" function="" does="" not="" in="" any="" way="" describe="" a="" condition="" which="" could="" be="" that="" of="" a="" single="" system;="" it="" relates="" rather="" to="" many="" systems,="" to="" "an="" ensemble="" of="" systems"="" in="" the="" sense="" of="" statistical="" mechanics.="" if,="" except="" for="" certain="" special="" cases,="" the="" function="" furnishes="" only="" statistical="" data="" concerning="" measurable="" magnitudes,="" the="" reason="" lies="" not="" only="" in="" the="" fact="" that="" the="" operation="" of="" measuring="" introduces="" unknown="" elements,="" which="" can="" be="" grasped="" only="" statistically,="" but="" because="" of="" the="" very="" fact="" that="" the="" function="" does="" not,="" in="" any="" sense,="" describe="" the="" condition="" of="" one="" single="" system.="" the="" schrodinger="" equation="" determines="" the="" time="" variations="" which="" are="" experienced="" by="" the="" ensemble="" of="" systems="" which="" may="" exist="" with="" or="" without="" external="" action="" on="" the="" single="">

Such an interpretation eliminates also the paradox recently demonstrated by myself and two collaborators, and which relates to the following problem.

Consider a mechanical system const.i.tuted of two partial systems A and B which have interaction with each other only during limited time. Let the function before their interaction be given. Then the Schrodinger equation will furnish the function after the interaction has taken place. Let us now determine the physical condition of the partial system A as completely as possible by measurements. Then the quantum mechanics allows us to determine the function of the partial system B from the measurements made, and from the ? function of the total system. This determination, however, gives a result which depends upon which of the determining magnitudes specifying the condition of A has been measured (for instance coordinates or momenta). Since there can be only one physical condition of B after the interaction and which can reasonably not be considered as dependent on the particular measurement we perform on the system A separated from B it may be concluded that the function is not unambiguously coordinated with the physical condition. This coordination of several functions with the same physical condition of system B shows again that the function cannot be interpreted as a (complete) description of a physical condition of a unit system. Here also the coordination of the function to an ensemble of systems eliminates every difficulty.4 The fact that quantum mechanics affords, in such a simple manner, statements concerning (apparently) discontinuous transitions from one total condition to another without actually giving a representation of the specific process, this fact is connected with another, namely the fact that the theory, in reality, does not operate with the single system, but with a totality of systems. The coefficients cr of our first example are really altered very little under the action of the external force. With this interpretation of quantum mechanics one can understand why this theory can easily account for the fact that weak disturbing forces are able to produce alterations of any magnitude in the physical condition of a system. Such disturbing forces produce, indeed, only correspondingly small alterations of the statistical density in the ensemble of systems, and hence only infinitely weak alterations of the functions, the mathematical description of which offers far less difficulty than would be involved in the mathematical representation of finite alterations experienced by part of the single systems. What happens to the single system remains, it is true, entirely unclarified by this mode of consideration; this enigmatic happening is entirely eliminated from the representation by the statistical manner of consideration.

But now I ask: Is there really any physicist who believes that we shall never get any inside view of these important alterations in the single systems, in their structure and their causal connections, and this regardless of the fact that these single happenings have been brought so close to us, thanks to the marvelous inventions of the Wilson chamber and the Geiger counter? To believe this is logically possible without contradiction; but, it is so very contrary to my scientific instinct that I cannot forego the search for a more complete conception.

To these considerations we should add those of another kind which also voice their plea against the idea that the methods introduced by quantum mechanics are likely to give a useful basis for the whole of physics. In the Schrodinger equation, absolute time, and also the potential energy, play a decisive role, while these two concepts have been recognized by the theory of relativity as inadmissible in principle. If one wishes to escape from this difficulty he must found the theory upon field and field laws instead of upon forces of interaction. This leads us to transpose the statistical methods of quantum mechanics to fields, that is to systems of infinitely many degrees of freedom. Although the attempts so far made are restricted to linear equations, which, as we know from the results of the general theory of relativity, are insufficient, the complications met up to now by the very ingenious attempts are already terrifying. They certainly will rise sky high if one wishes to obey the requirements of the general theory of relativity, the justification of which in principle n.o.body doubts.

To be sure, it has been pointed out that the introduction of a s.p.a.ce-time continuum may be considered as contrary to nature in view of the molecular structure of everything which happens on a small scale. It is maintained that perhaps the success of the Heisenberg method points to a purely algebraical method of description of nature, that is to the elimination of continuous functions from physics. Then, however, we must also give up, by principle, the s.p.a.ce-time continuum. It is not unimaginable that human ingenuity will some day find methods which will make it possible to proceed along such a path. At the present time, however, such a program looks like an attempt to breathe in empty s.p.a.ce.

There is no doubt that quantum mechanics has seized hold of a beautiful element of truth, and that it will be a test stone for any future theoretical basis, in that it must be deducible as a limiting case from that basis, just as electrostatics is deducible from the Maxwell equations of the electromagnetic field or as thermodynamics is deducible from cla.s.sical mechanics. However, I do not believe that quantum mechanics will be the starting point in the search for this basis, just as, vice versa, one could not go from thermodynamics (resp. statistical mechanics) to the foundations of mechanics.

In view of this situation, it seems to be entirely justifiable seriously to consider the question as to whether the basis of field physics cannot by any means be put into harmony with the facts of the quantum theory. Is this not the only basis which, consistently with today's possibility of mathematical expression, can be adapted to the requirements of the general theory of relativity? The belief, prevailing among the physicists of today, that such an attempt would be hopeless, may have its root in the unjustifiable idea that such a theory should lead, as a first approximation, to the equations of cla.s.sical mechanics for the motion of corpuscles, or at least to total differential equations. As a matter of fact up to now we have never succeeded in representing corpuscles theoretically by fields free of singularities, and we can, a priori, say nothing about the behavior of such ent.i.ties. One thing, however, is certain: if a field theory results in a representation of corpuscles free of singularities, then the behavior of these corpuscles with time is determined solely by the differential equations of the field.

6. Relativity Theory and Corpuscles.

I shall now show that, according to the general theory of relativity, there exist singularity-free solutions of field equations which can be interpreted as representing corpuscles. I restrict myself here to neutral particles because, in another recent publication in collaboration with Dr. Rosen, I have treated this question in a detailed manner, and because the essentials of the problem can be completely shown by this case.

The gravitational field is entirely described by the tensor gv. In the three-index symbols v there appear also the contravariants gv which are defined as the minors of the gv divided by the determinant g( = g). In order that the Rik, shall be defined and finite, it is not sufficient that there shall be, for the environment of every part of the continuum, a system of coordinates in which the gv and their first differential quotients are continuous and differentiable, but it is also necessary that the determinant g shall nowhere vanish. This last restriction is, however, eliminated if one replaces the differential equations Rik = 0 by g2Rik = 0, the left hand sides of which are whole rational functions of the gik and of their derivatives.

These equations have the centrally symmetrical solutions indicated by Schwarzschild.

ds2=- 1/ 1-2m/r dr2 - r2 (d2 + sin2d2) + (1- 2m/r) dt2.

This solution has a singularity at r = 2m, since the coefficient of dr2 (i.e. g11), becomes infinite on this hypersurface. If, however, we replace the variable r by defined by the equation 2 = r - 2m.

we obtain ds2 = - 4 (2m + 2) d2 - (2m + 2)2 (d2 + sin2d2) +2/2m+2 dt2.

This solution behaves regularly for all values of p. The vanis.h.i.+ng of the coefficient of dt2 i.e. (g44) for = 0 results, it is true, in the consequence that the determinant g vanishes for this value; but, with the methods of writing the field equations actually adopted, this does not const.i.tute a singularity. If extends from - to +, then r runs from + to r = 2m and then back to +, while for such values of r as correspond to r < 2m="" there="" are="" no="" corresponding="" real="" values="" of="" p.="" hence="" the="" schwarzschild="" solution="" becomes="" a="" regular="" solution="" by="" representation="" of="" the="" physical="" s.p.a.ce="" as="" consisting="" of="" two="" identical="" "sh.e.l.ls"="" neighboring="" upon="" the="" hypersurface="0," that="" is="" r="2m," while="" for="" this="" hypersurface="" the="" determinant="" g="" vanishes.="" let="" us="" call="" such="" a="" connection="" between="" the="" two="" (identical)="" sh.e.l.ls="" a="" "bridge."="" hence="" the="" existence="" of="" such="" a="" bridge="" between="" the="" two="" sh.e.l.ls="" in="" the="" finite="" realm="" corresponds="" to="" the="" existence="" of="" a="" material="" neutral="" particle="" which="" is="" described="" in="" a="" manner="" free="" from="">

The solution of the problem of the motion of neutral particles evidently amounts to the discovery of such solutions of the gravitational equations (written free of denominators), as contain several bridges.

The conception sketched above corresponds, a priori, to the atomistic structure of matter insofar as the "bridge" is by its nature a discrete element. Moreover, we see that the ma.s.s constant m of the neutral particles must necessarily be positive, since no solution free of singularities can correspond to the Schwarzschild solution for a negative value of m. Only the examination of the several-bridge-problem, can show whether or not this theoretical method furnishes an explanation of the empirically demonstrated equality of the ma.s.ses of the particles found in nature, and whether it takes into account the facts which the quantum mechanics has so wonderfully comprehended.

In an a.n.a.logous manner, it is possible to demonstrate that the combined equations of gravitation and electricity (with appropriate choice of the sign of the electrical member in the gravitational equations) produce a singularity-free bridge-representation of the electric corpuscle. The simplest solution of this kind is that for an electrical particle without gravitational ma.s.s.

So long as the important mathematical difficulties concerned with the solution of the several-bridge-problem, are not overcome, nothing can be said concerning the usefulness of the theory from the physicist's point of view. However, it const.i.tutes, as a matter of fact, the first attempt towards the consistent elaboration of a field theory which presents a possibility of explaining the properties of matter. In favor of this attempt one should also add that it is based on the simplest possible relativistic field equations known today.

Summary.

Physics const.i.tutes a logical system of thought which is in a state of evolution, and whose basis cannot be obtained through distillation by any inductive method from the experiences lived through, but which can only be attained by free invention. The justification (truth content) of the system rests in the proof of usefulness of the resulting theorems on the basis of sense experiences, where the relations of the latter to the former can only be comprehended intuitively. Evolution is going on in the direction of increasing simplicity of the logical basis. In order further to approach this goal, we must make up our mind to accept the fact that the logical basis departs more and more from the facts of experience, and that the path of our thought from the fundamental basis to these resulting theorems, which correlate with sense experiences, becomes continually harder and longer.

Our aim has been to sketch, as briefly as possible, the development of the fundamental concepts in their dependence upon the facts of experience and upon the strife towards the goal of internal perfection of the system. Today's state of affairs had to be illuminated by these considerations, as they appear to me. (It is unavoidable that historic schematic representation is of a personal color.) I try to demonstrate how the concepts of bodily objects, s.p.a.ce, subjective and objective time, are connected with one another and with the nature of the experience. In cla.s.sical mechanics the concepts of s.p.a.ce and time become independent. The concept of the bodily object is replaced in the foundations by the concept of the material point, by which means mechanics becomes fundamentally atomistic. Light and electricity produce insurmountable difficulties when one attempts to make mechanics the basis of all physics. We are thus led to the field theory of electricity, and, later on to the attempt to base physics entirely upon the concept of the field (after an attempted compromise with cla.s.sical mechanics). This attempt leads to the theory of relativity (evolution of the notion of s.p.a.ce and time into that of the continuum with metric structure).

I try to demonstrate, furthermore, why in my opinion the quantum theory does not seem likely to be able to produce a usable foundation for physics: one becomes involved in contradictions if one tries to consider the theoretical quantum description as a complete description of the individual physical system or happening.

On the other hand, up to the present time, the field theory is unable to give an explanation of the molecular structure of matter and of quantum phenomena. It is shown, however, that the conviction to the effect that the field theory is unable to give, by its methods, a solution of these problems rests upon prejudice.

1 It is in the nature of things that we are able to talk about these objects only by means of concepts of our own creation, concepts which themselves are not subject to definition. It is essential, however, that we make use only of such concepts concerning whose coordination to our experience we feel no doubt.

2 This defect of the theory could only be eliminated by such a formulation of mechanics as would command validity for all Bo. This is one of the steps which lead to the general theory of relativity. A second defect, also eliminated only by the introduction of the general theory of relativity, lies in the fact that there is no reason given by mechanics itself for the equality of the gravitational and inertial ma.s.s of the material point.

3 Because, according to a well established consequence of the relativity theory, the energy of a complete system (at rest) is equal to its inertia (as a whole). This, however, must have a well defined value.

4 The operation of measuring A, for example, thus involves a transition to a narrower ensemble of systems. The latter (hence also its function) depends upon the point of view according to which this narrowing of the ensemble of systems is made.

* "I make no hypotheses."

14.

The Fundaments of Theoretical Physics.

SCIENCE IS THE ATTEMPT to make the chaotic diversity of our sense-experience correspond to a logically uniform system of thought. In this system single experiences must be correlated with the theoretic structure in such a way that the resulting coordination is unique and convincing.

The sense-experiences are the given subject-matter. But the theory that shall interpret them is man-made. It is the result of an extremely laborious process of adaptation: hypothetical, never completely final, always subject to question and doubt.

The scientific way of forming concepts differs from that which we use in our daily life, not basically, but merely in the more precise definition of concepts and conclusions; more painstaking and systematic choice of experimental material; and greater logical economy. By this last we mean the effort to reduce all concepts and correlations to as few as possible logically independent basic concepts and axioms.

What we call physics comprises that group of natural sciences which base their concepts on measurements; and whose concepts and propositions lend themselves to mathematical formulation. Its realm is accordingly defined as that part of the sum total of our knowledge which is capable of being expressed in mathematical terms. With the progress of science, the realm of physics has so expanded that it seems to be limited only by the limitations of the method itself.

The larger part of physical research is devoted to the development of the various branches of physics, in each of which the object is the theoretical understanding of more or less restricted fields of experience, and in each of which the laws and concepts remain as closely as possible related to experience. It is this department of science, with its ever-growing specialization, which has revolutionized practical life in the last centuries, and given birth to the possibility that man may at last be freed from the burden of physical toil.

On the other hand, from the very beginning there has always been present the attempt to find a unifying theoretical basis for all these single sciences, consisting of a minimum of concepts and fundamental relations.h.i.+ps, from which all the concepts and relations.h.i.+ps of the single disciplines might be derived by logical process. This is what we mean by the search for a foundation of the whole of physics. The confident belief that this ultimate goal may be reached is the chief source of the pa.s.sionate devotion which has always animated the researcher. It is in this sense that the following observations are devoted to the foundations of physics.

From what has been said it is clear that the word foundations in this connection does not mean something a.n.a.logous in all respects to the foundations of a building. Logically considered, of course, the various single laws of physics rest upon this foundation. But whereas a building may be seriously damaged by a heavy storm or spring flood, yet its foundations remain intact, in science the logical foundation is always in greater peril from new experiences or new knowledge than are the branch disciplines with their closer experimental contacts. In the connection of the foundation with all the single parts lies its great significance, but likewise its greatest danger in face of any new factor. When we realize this, we are led to wonder why the so-called revolutionary epochs of the science of physics have not more often and more completely changed its foundation than has actually been the case.

The first attempt to lay a uniform theoretical foundation was the work of Newton. In his system everything is reduced to the following concepts: (1) Ma.s.s points with invariable ma.s.s; (2) action at a distance between any pair of ma.s.s points; (3) law of motion for the ma.s.s point. There was not, strictly speaking, any all-embracing foundation, because an explicit law was formulated only for the actions-at-a-distance of gravitation; while for other actions-at-a-distance nothing was established a priori except the law of equality of actio and reactio. Moreover, Newton himself fully realized that time and s.p.a.ce were essential elements, as physically effective factors, of his system, if only by implication.

This Newtonian basis proved eminently fruitful and was regarded as final up to the end of the nineteenth century. It not only gave results for the movements of the heavenly bodies, down to the most minute details, but also furnished a theory of the mechanics of discrete and continuous ma.s.ses, a simple explanation of the principle of the conservation of energy and a complete and brilliant theory of heat. The explanation of the facts of electrodynamics within the Newtonian system was more forced; least convincing of all, from the very beginning, was the theory of light.

It is not surprising that Newton would not listen to a wave theory of light; for such a theory was most unsuited to his theoretical foundation. The a.s.sumption that s.p.a.ce was filled with a medium consisting of material points that propagated light waves without exhibiting any other mechanical properties must have seemed to him quite artificial. The strongest empirical arguments for the wave nature of light, fixed speeds of propagation, interference, diffraction, polarization, were either unknown or else not known in any well-ordered synthesis. He was justified in sticking to his corpuscular theory of light.

During the nineteenth century the dispute was settled in favor of the wave theory. Yet no serious doubt of the mechanical foundation of physics arose, in the first place because n.o.body knew where to find a foundation of another sort. Only slowly, under the irresistible pressure of facts, there developed a new foundation of physics, field-physics.

From Newton's time on, the theory of action-at-a-distance was constantly found artificial. Efforts were not lacking to explain gravitation by a kinetic theory, that is, on the basis of collision forces of hypothetical ma.s.s particles. But the attempts were superficial and bore no fruit. The strange part played by s.p.a.ce (or the inertial system) within the mechanical foundation was also clearly recognized, and criticized with especial clarity by Ernst Mach.

The great change was brought about by Faraday, Maxwell and Hertz-as a matter of fact half-unconsciously and against their will. All three of them, throughout their lives, considered themselves adherents of the mechanical theory. Hertz had found the simplest form of the equations of the electromagnetic field, and declared that any theory leading to these equations was Maxwellian theory. Yet toward the end of his short life he wrote a paper in which he presented as the foundation of physics a mechanical theory freed from the force-concept.

For us, who took in Faraday's ideas so to speak with our mother's milk, it is hard to appreciate their greatness and audacity. Faraday must have grasped with unerring instinct the artificial nature of all attempts to refer electromagnetic phenomena to actions-at-a-distance between electric particles reacting on each other. How was each single iron filing among a lot scattered on a piece of paper to know of the single electric particles running round in a nearby conductor? All these electric particles together seemed to create in the surrounding s.p.a.ce a condition which in turn produced a certain order in the filings. These spatial states, to-day called fields, if their geometrical structure and interdependent action were once rightly grasped, would, he was convinced, furnish the clue to the mysterious electromagnetic interactions. He conceived these fields as states of mechanical stress in a s.p.a.ce-filling medium, similar to the states of stress in an elastically distended body. For at that time this was the only way one could conceive of states that were apparently continuously distributed in s.p.a.ce. The peculiar type of mechanical interpretation of these fields remained in the background-a sort of placation of the scientific conscience in view of the mechanical tradition of Faraday's time. With the help of these new field concepts Faraday succeeded in forming a qualitative concept of the whole complex of electromagnetic effects discovered by him and his predecessors. The precise formulation of the time-s.p.a.ce laws of those fields was the work of Maxwell. Imagine his feelings when the differential equations he had formulated proved to him that electromagnetic fields spread in the form of polarized waves and with the speed of light! To few men in the world has such an experience been vouchsafed. At that thrilling moment he surely never guessed that the riddling nature of light, apparently so completely solved, would continue to baffle succeeding generations. Meantime, it took physicists some decades to grasp the full significance of Maxwell's discovery, so bold was the leap that his genius forced upon the conceptions of his fellow-workers. Only after Hertz had demonstrated experimentally the existence of Maxwell's electromagnetic waves, did resistance to the new theory break down.

But if the electromagnetic field could exist as a wave independent of the material source, then the electrostatic interaction could no longer be explained as action-at-a-distance. And what was true for electrical action could not be denied for gravitation. Everywhere Newton's actions-at-a-distance gave way to fields spreading with finite velocity.

Of Newton's foundation there now remained only the material ma.s.s points subject to the law of motion. But J. J. Thomson pointed out that an electrically charged body in motion must, according to Maxwell's theory, possess a magnetic field whose energy acted precisely as does an increase of kinetic energy to the body. If, then, a part of kinetic energy consists of field energy, might that not then be true of the whole of the kinetic energy? Perhaps the basic property of matter, its inertia, could be explained within the field theory? The question led to the problem of an interpretation of matter in terms of field theory, the solution of which would furnish an explanation of the atomic structure of matter. It was soon realized that Maxwell's theory could not accomplish such a program. Since then many scientists have zealously sought to complete the field theory by some generalization that should comprise a theory of matter; but so far such efforts have not been crowned with success. In order to construct a theory, it is not enough to have a clear conception of the goal. One must also have a formal point of view which will sufficiently restrict the unlimited variety of possibilities. So far this has not been found; accordingly the field theory has not succeeded in furnis.h.i.+ng a foundation for the whole of physics.

For several decades most physicists clung to the conviction that a mechanical substructure would be found for Maxwell's theory. But the unsatisfactory results of their efforts led to gradual acceptance of the new field concepts as irreducible fundamentals-in other words, physicists resigned themselves to giving up the idea of a mechanical foundation.

Thus physicists held to a field-theory program. But it could not be called a foundation, since n.o.body could tell whether a consistent field theory could ever explain on the one hand gravitation, on the other hand the elementary components of matter. In this state of affairs it was necessary to think of material particles as ma.s.s points subject to Newton's laws of motion. This was the procedure of Lorentz in creating his electron theory and the theory of the electromagnetic phenomena of moving bodies.

Such was the point at which fundamental conceptions had arrived at the turn of the century. Immense progress was made in the theoretical penetration and understanding of whole groups of new phenomena; but the establishment of a unified foundation for physics seemed remote indeed. And this state of things has even been aggravated by subsequent developments. The development during the present century is characterized by two theoretical systems essentially independent of each other: the theory of relativity and the quantum theory. The two systems do not directly contradict each other; but they seem little adapted to fusion into one unified theory. We must briefly discuss the basic idea of these two systems.

The theory of relativity arose out of efforts to improve, with reference to logical economy, the foundation of physics as it existed at the turn of the century. The so-called special or restricted relativity theory is based on the fact that Maxwell's equations (and thus the law of propagation of light in empty s.p.a.ce) are converted into equations of the same form, when they undergo Lorentz transformation. This formal property of the Maxwell equations is supplemented by our fairly secure empirical knowledge that the laws of physics are the same with respect to all inertial systems. This leads to the result that the Lorentz transformation-applied to s.p.a.ce and time coordinates-must govern the transition from one inertial system to any other. The content of the restricted relativity theory can accordingly be summarized in one sentence: all natural laws must be so conditioned that they are covariant with respect to Lorentz transformations. From this it follows that the simultaneity of two distant events is not an invariant concept and that the dimensions of rigid bodies and the speed of clocks depend upon their state of motion. A further consequence was a modification of Newton's law of motion in cases where the speed of a given body was not small compared with the speed of light. There followed also the principle of the equivalence of ma.s.s and energy, with the laws of conservation of ma.s.s and energy becoming one and the same. Once it was shown that simultaneity was relative and depended on the frame of reference, every possibility of retaining actions-at-a-distance within the foundation of physics disappeared, since that concept presupposed the absolute character of simultaneity (it must be possible to state the location of the two interacting ma.s.s points "at the same time").

The general theory of relativity owes its origin to the attempt to explain a fact known since Galileo's and Newton's time but hitherto eluding all theoretical interpretation: the inertia and the weight of a body, in themselves two entirely distinct things, are measured by one and the same constant, the ma.s.s. From this correspondence follows that it is impossible to discover by experiment whether a given system of coordinates is accelerated, or whether its motion is straight and uniform and the observed effects are due to a gravitational field (this is the equivalence principle of the general relativity theory). It shatters the concepts of the inertial system, as soon as gravitation enters in. It may be remarked here that the inertial system is a weak point of the Galilean-Newtonian mechanics. For there is presupposed a mysterious property of physical s.p.a.ce, conditioning the kind of coordination-systems for which the law of inertia and the Newtonian law of motion hold good.

These difficulties can be avoided by the following postulate: natural laws are to be formulated in such a way that their form is identical for coordinate systems of any kind of states of motion. To accomplish this is the task of the general theory of relativity. On the other hand, we deduce from the restricted theory the existence of a Riemannian metric within the time-s.p.a.ce continuum, which, according to the equivalence principle, describes both the gravitational field and the metric properties of s.p.a.ce. a.s.suming that the field equations of gravitation are of the second differential order, the field law is clearly determined.

Aside from this result, the theory frees field physics from the disability it suffered from, in common with the Newtonian mechanics, of ascribing to s.p.a.ce those independent physical properties which heretofore had been concealed by the use of an inertial system. But it can not be claimed that those parts of the general relativity theory which can to-day be regarded as final have furnished physics with a complete and satisfactory foundation. In the first place, the total field appears in it to be composed of two logically unconnected parts, the gravitational and the electromagnetic. And in the second place, this theory, like the earlier field theories, has not up till now supplied an explanation of the atomistic structure of matter. This failure has probably some connection with the fact that so far it has contributed nothing to the understanding of quantum phenomena. To take in these phenomena, physicists have been driven to the adoption of entirely new methods, the basic characteristics of which we shall now discuss.

In the year nineteen hundred, in the course of a purely theoretic investigation, Max Planck made a very remarkable discovery: the law of radiation of bodies as a function of temperature could not be derived solely from the laws of Maxwellian electrodynamics. To arrive at results consistent with the relevant experiments, radiation of a given frequency had to be treated as though it consisted of energy atoms of the individual energy h.v., where h is Planck's universal constant. During the years following it was shown that light was everywhere produced and absorbed in such energy quanta. In particular Niels Bohr was able largely to understand the structure of the atom, on the a.s.sumption that atoms can have only discrete energy values, and that the discontinuous transitions between them are connected with the emission or absorption of such an energy quantum. This threw some light on the fact that in their gaseous state elements and their compounds radiate and absorb only light of certain sharply defined frequencies. All this was quite inexplicable within the frame of the hitherto existing theories. It was clear that at least in the field of atomistic phenomena the character of everything that happens is determined by discrete states and by apparently discontinuous transitions between them, Planck's constant h playing a decisive role.

The next step was taken by De Broglie. He asked himself how the discrete states could be understood by the aid of the current concepts, and hit on a parallel with stationary waves, as for instance in the case of the proper frequencies of organ pipes and strings in acoustics. True, wave actions of the kind here required were unknown; but they could be constructed, and their mathematical laws formulated, employing Planck's constant h. De Broglie conceived an electron revolving about the atomic nucleus as being connected with such a hypothetical wave train, and made intelligible to some extent the discrete character of Bohr's "permitted" paths by the stationary character of the corresponding waves.

Now in mechanics the motion of material points is determined by the forces or fields of force acting upon them. Hence it was to be expected that those fields of force would also influence De Broglie's wave fields in an a.n.a.logous way. Erwin Schrodinger showed how this influence was to be taken into account, re-interpreting by an ingenious method certain formulations of cla.s.sical mechanics. He even succeeded in expanding the wave mechanical theory to a point where without the introduction of any additional hypotheses, it became applicable to any mechanical system consisting of an arbitrary number of ma.s.s points, that is to say possessing an arbitrary number of degrees of freedom. This was possible because a mechanical system consisting of n ma.s.s points is mathematically equivalent to a considerable degree, to one single ma.s.s point moving in a s.p.a.ce of 3 n dimensions.

On the basis of this theory there was obtained a surprisingly good representation of an immense variety of facts which otherwise appeared entirely incomprehensible. But on one point, curiously enough, there was failure: it proved impossible to a.s.sociate with these Schrodinger waves definite motions of the ma.s.s points-and that, after all, had been the original purpose of the whole construction.

The difficulty appeared insurmountable, until it was overcome by Born in a way as simple as it was unexpected. The De Broglie-Schrodinger wave fields were not to be interpreted as a mathematical description of how an event actually takes place in time and s.p.a.ce, though, of course, they have reference to such an event. Rather they are a mathematical description of what we can actually know about the system. They serve only to make statistical statements and predictions of the results of all measurements which we can carry out upon the system.

Let me ill.u.s.trate these general features of quantum mechanics by means of a simple example: we shall consider a ma.s.s point kept inside a restricted region G by forces of finite strength. If the kinetic energy of the ma.s.s point is below a certain limit, then the ma.s.s point, according to cla.s.sical mechanics, can never leave the region G. But according to quantum mechanics, the ma.s.s point, after a period not immediately predictable, is able to leave the region G, in an unpredictable direction, and escape into surrounding s.p.a.ce. This case, according to Gamow, is a simplified model of radioactive disintegration.

The quantum theoretical treatment of this case is as follows: at the time t0 we have a Schrodinger wave system entirely inside G. But from the time t0 onwards, the waves leave the interior of G in all directions, in such a way that the amplitude of the outgoing wave is small compared to the initial amplitude of the wave system inside G. The further these outside waves spread, the more the amplitude of the waves inside G diminishes, and correspondingly the intensity of the later waves issuing from G. Only after infinite time has pa.s.sed is the wave supply inside G exhausted, while the outside wave has spread over an ever-increasing s.p.a.ce.

But what has this wave process to do with the first object of our interest, the particle originally enclosed in G? To answer this question, we must imagine some arrangement which will permit us to carry out measurements on the particle. For instance, let us imagine somewhere in the surrounding s.p.a.ce a screen so made that the particle sticks to it on coming into contact with it. Then from the intensity of the waves. .h.i.tting the screen at some point, we draw conclusions as to the probability of the particle hitting the screen there at that time. As soon as the particle has. .h.i.t any particular point of the screen, the whole wave field loses all its physical meaning; its only purpose was to make probability predictions as to the place and time of the particle hitting the screen (or, for instance, its momentum at the time when it hits the screen).

All other cases are a.n.a.logous. The aim of the theory is to determine the probability of the results of measurement upon a system at a given time. On the other hand, it makes no attempt to give a mathematical representation of what is actually present or goes on in s.p.a.ce and time. On this point the quantum theory of to-day differs fundamentally from all previous theories of physics, mechanistic as well as field theories. Instead of a model description of actual s.p.a.ce-time events, it gives the probability distributions for possible measurements as functions of time.

It must be admitted that the new theoretical conception owes its origin not to any flight of fancy but to the compelling force of the facts of experience. All attempts to represent the particle and wave features displayed in the phenomena of light and matter, by direct course to a s.p.a.ce-time model, have so far ended in failure. And Heisenberg has convincingly shown, from an empirical point of view, any decision as to a rigorously deterministic structure of nature is definitely ruled out, because of the atomistic structure of our experimental apparatus. Thus it is probably out of the question that any future knowledge can compel physics again to relinquish our present statistical theoretical foundation in favor of a deterministic one which would deal directly with physical reality. Logically the problem seems to offer two possibilities, between which we are in principle given a choice. In the end the choice will be made according to which kind of description yields the formulation of the simplest foundation, logically speaking. At the present, we are quite without any deterministic theory directly describing the events themselves and in consonance with the facts.