no physical significance can be directly given to flow of heat, and
there are no operations for measuring it. All we can measure are
temperature distributions and rates of rise of temperature. As at
present defined, a heat current is a pure invention, without physical
reality, for any determined heat flow may always be modified by the
addition of a solenoidal vector, with change in no measurable quantity.
If someone states that throughout all space there is a uniform heat
current of 10^6 cal./cm.^2 sec. in the direction of Polaris, no disproof
can be given, for such a stream is solenoidal, and as much heat flows
out of every closed surface in unit time as flows in. Such a solenoidal
flow is meaningless in terms of operations; we could give meaning to
such a flow only in terms of some slight modification of the solenoidal
condition introduced by the measuring instrument. In all ordinary
conditions the flow of heat given by the simple relation q = k Grad t
corresponds exactly to what our atomic pictures lead to expect in those
cases where the details of the picture can be worked out. But there may
be cases where it is advantageous to supplement the ordinary heat flow
(= k Grad t) by the pure fiction of a solenoidal flow, because in this
way it may be possible to account for new phenomena which appear when
the solenoidal conditions are slightly departed from. Thus if in a
conductor at uniform temperature carrying a steady electric current we
say that a heat current is also flowing proportional to the electric
current and therefore solenoidal, we may provide the possibility for a
simple correlation of phenomena found under those more complicated
conditions when an electric current flows in a conductor of non-uniform
temperature in a magnetic field. If it should turn out that the heat
current is _uniquely_ determined by considerations of this character,
then we have taken the first step away from the pure formalism which
this sort of thing otherwise is in the direction of giving physical
reality to the invention of "heat current."
There are other interesting questions of a fundamental thermodynamic
character, such for example, as whether the entropy concept has any
general significance apart from the scale of our measuring instruments,
and what is the operational significance of applying thermodynamic
concepts to radiation, but we shall not consider these questions here.
ELECTRICAL CONCEPTS
We now set ourselves the problem of finding the meaning of the various
concepts in terms of which we describe the behavior of electrical
systems, assuming that we understand what we mean by "electrical." We
start with the simplest electrical systems, namely, those in which we
deal with static phenomena on a large scale. In such systems there are
independent physical operations by which we may find the magnitude of
any charge, provided that it is effectively concentrated in a
geometrical point. The measurements involved in these operations are
measurements of ordinary mechanical forces; we assume that our knowledge
of mechanics has already taught us how to make such measurements. An
electrically charged body experiences forces, which may be measured by
tying a string to it and pulling on the string with a spring balance
hard enough to keep the body in equilibrium. Three charges are
numerically equal if when each is placed at unit distance from another,
in the absence of the third (or other charge), the forces are always the
same. If furthermore the forces are of unit magnitude, the charges are
defined as unit charge. Having obtained unit charge, we define the
magnitude of any other charge as equal to the force which it experiences
when placed at unit distance from unit charge. This of course is all
very trite; the important thing for us is merely that magnitude of
charge, or quantity of electricity, is an independent physical concept,
and that unique operations exist for determining it. These operations
presuppose the ability to perform certain operations of mechanics.
Having now learned how to measure electrical quantities, we discover
experimentally the inverse square law of force, and later arrive at the
concept of the electric field. As we have seen, the field is an
invention; here we shall use this concept only for the purpose for which
the invention was made, and shall not involve ourselves in any of the
implications of ascribing physical reality to the field. Notice that as
long as we deal only with point charges we do not have to define field
strength in terms of the limiting procedure of making the exploring
charge smaller, for the limiting small charge is necessary only to avoid
the reaction of the exploring charge on the positions of the charges
which generate the field. All this again is trite; the important point
is that the operations by which the inverse square law and the concept
of the field are established presuppose that the charge is given as an
independent concept, since the operations involve a knowledge of
charges. The operations also involve the measurement of forces by the
ordinary _static_ procedure of mechanics with spring balances. With the
means now at our command we establish one very important property of
electric charges, namely that the total amount of charge on an isolated
body of finite size is conserved, no matter how the charge is forced to
rearrange itself by the motion of charges on adjacent bodies.
By procedures exactly like those outlined above, we may treat all the
corresponding magnetic quantities; there is formal parallelism between
the two sets of phenomena, but there is the physical difference that we
have to realize a single magnetic pole by the device of using a very
long slender magnet.
We now give our electrical system more freedom, in that we allow the
charges to be in motion with respect to each other. Perhaps the most
immediate question which we now have to ask is whether charge continues
to be conserved when set in motion, or whether the total charge on an
isolated body is a function of its velocity? To answer this question we
must generalize the procedure by which we assigned a numerical value to
a stationary charge. Perhaps the simplest way is to allow two unit
charges each to move with constant velocity, remaining at unit distance
apart, and measure with a spring balance the force required to keep them
at constant distance apart. Now we immediately find that the force is
altered under these conditions, so that our first impulse is to say that
the charge is a function of the velocity. But as we experiment further,
we find that the state of affairs is very complicated; the force between
the two charges at any moment of their motion depends not only on the
charges, their distance apart, and their velocities, but also on the
angle between the line joining them and the direction of motion in the
lines. Further experiment of other kinds yields other information; it
requires a force to maintain a charge in uniform motion in a magnetic
field, or to maintain a magnetic pole in motion in an electric field. A
moving electric charge exerts a force on a stationary magnetic pole, so
that by definition the moving charge is surrounded by a magnetic field,
and similarly a moving magnetic pole is surrounded by an electric field.
Returning to our two moving electric charges, we are impelled to ask
whether, if all these complications are possible, the numerical constant
(unity for static charges) in the inverse square law of force is a
function of velocity as well as the magnitude of the charges themselves?
If we broaden the question in this way, as we apparently must, our
problem becomes indeterminate, for we are trying to answer two different
questions with a single kind of measurement, namely of the force between
moving charges. I have had no better luck on trying other methods of
measurement. Apparently the operations do not exist by which unique
meaning can be given to the question of whether the magnitude of a
charge is a function of its velocity. On realizing this situation, we
are at first embarrassed to know how to proceed, but we reflect that the
embarrassment is not of our own making, but corresponds to a physical
fact. The concept of charge as a unique and independent thing
essentially pertains only to static systems. We may extend the concept
to moving systems if we wish, as a matter of convenience to ourselves,
but must recognize that such an extension is an invention of ours and
not a reality of nature. Now we do make such an extension, and we make
it in the simplest possible way, that is, we define the charge on an
isolated body in motion as that which we should find on it if we reduced
it to rest and made measurements according to the regular static
procedure. That this is a convenient thing to do depends on the
experimental result that the charge so found is independent of the way
in which velocity is imparted to or removed from the body; in other
words, whenever the body is reduced to rest, the same charge is always
found on it.
Although this is pure definition on our part, it turns out to have a
most simple and convenient connection with experimental facts which were
discovered after the decision to treat a moving charge in this way was
made; the discovery is of the atomic structure of electricity. If then
we agree to call each elementary charge a constant independent of the
velocity, the total charge on a body becomes merely proportional to the
count of the total number of atomic charges on the body, which is
certainly highly convenient and suggestive.
Having now fixed what we mean by the magnitude of a moving charge, we
are ready to turn to the general problem of the behavior of any system
of charged bodies in motion. For the present we consider only phenomena
of the scale of everyday experience. The most general problem that has
meaning here is to determine all measurable properties of the system in
terms of those data which experiment shows can be arbitrarily specified.
Now we have already emphasized that the electromagnetic field itself is
an invention, and is never subject to direct observation. What we
observe are material bodies, with or without charges (including
eventually in this category electrons), their positions, motions, and
the forces to which they are subject. The forces are to be measured
according to definition in mechanical terms, either by the strains in
members of a framework if the system is in equilibrium, or in terms of
accelerations and masses if it is not in equilibrium. The
electromagnetic field as such is not the final object of our
calculations, but the calculation of it is only an intermediate
auxiliary step, convenient to make because our mathematical formulation
gives so simple a connection between electromagnetic field, charges, and
mechanical action that the latter can be calculated at once in terms of
the former. In fact the connection is so simple that in many cases we
have come to regard our problem as solved if we can compute the
electromagnetic field, overlooking the fact that the field has no
immediate meaning in terms of experience.
Electromagnetic theory now presents us with a solution of the general
problem; this solution is contained in the four-field equations of
Maxwell, the constitutive equations, and those additional equations
(quite often lost sight of) which give the forces exerted by the field
on electric charges, or currents, or dielectrics. Let us inquire how we
may set about testing the physical correctness of these equations. We
may begin with one of the simplest possible tests, and inquire whether
the equations are correct in stating that the force acting on a charge
moving in an electric field is simply the product of the charge and the
field strength. This, on the face of it, is a surprising statement. The
field itself is affected by the motion of the charges which generate it,
and it is natural to expect a converse effect. If, furthermore, we have
sympathy with the medium point of view, it is easy to think that
whatever it is in the medium that gets hold of a charge and exerts a
force on it will find it harder to take hold when the charge is in
motion.
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]
[Footnote 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 10^6 or 10^7 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.
[Footnote 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.
It is evident, therefore, that the concepts which enter the field
equations have entirely lost their large scale significance; they have
become blurred, fused together, and fewer in number. A precise analysis
of this situation has probably never been attempted and would obviously
be difficult: it would be interesting to know at least how many really
independent concepts there are at this order of phenomena. An attempt at
an analysis would probably be worth while from a physical point of view
in suggesting possible experiments by which the number of physically
independent concepts could be extended.
Since the quantities in the field equations are meaningless in the naked
form in which they enter the equations, it is meaningless to inquire
whether the equations as they stand are true or not. In our present
state of experimental knowledge it is also meaningless to ask whether,
for example, the inverse square law between electric charges continues
to hold, or whether an accelerated charge radiates. These questions have
meaning only when applied to phenomena on a scale large enough to
correspond to possible experiment.
There is a rather interesting obverse to the statement that it is
meaningless to ask whether the field equations are true, namely, that it
may not be meaningless to state that they are false. A statement is not
true unless it is true in every particular, but it is false if it is
false in a single particular. If we can show that a single consequence
of the field equations of Lorentz, when integrated or averaged in such a
way as to correspond to experimental possibilities, is false, then the
equations must be false. It seems that, regarded as a complete
description of physical behavior on a small scale, the equations must be
judged false, because they contain no suggestion of quantum phenomena.
Even if we have to recognize that the equations are false, there can be
no question that they correspond to an important part of reality, and
that they have been of the greatest service to physics. What is the
significance of the success that they have attained? It is to be noticed
that all the phenomena to which the Lorentz equations have been
successfully applied, although not large scale phenomena in the ordinary
sense of the word, are nevertheless phenomena involving the coöperation
of a number of atoms, and that the equations unquestionably fail when
applied to phenomena involving single electrons. It appears from our
best present evidence that on a small scale the behavior of nature is
governed by quantum principles and is therefore quite different from
large scale behavior, Which we have seen is governed by the Maxwell
equations. There must of course be a transition zone in which the
character of phenomena changes from quantum to Maxwell. Now any program
like that of Lorentz is almost inevitably bound to begin to give correct
results when we get up as far as the transition zone, for the simple
reason that the relations of Maxwell have been put into the equations
and are always there ready to appear as soon as the quantum relations
begin to give way. The physical significance of the success of the
Lorentz program seems to be that the transition from Maxwell to quantum
takes place at a stage pretty far down toward the individual atoms. To
find the precise details of the transition from Maxwell to quantum
phenomena constitutes a large part of the program of the immediate
future.
All this skepticism about the classical work of Lorentz is likely to be
rather irritating or depressing, particularly if one attempts to imagine
what other course could have been adopted. Indeed it does seem that we
find ourselves in a real quandary; Lorentz was practically forced,
because of the character of the mathematical tools at his command, to
take the course that he did, in spite of any recognition of the physical
meaninglessness of the mathematical operations. We have already seen
that conventional mathematics does not correspond to the physical
reality; it cannot easily make a qualified statement subject to
limitations, and it recognizes no difference between the physically big
and the physically little and the corresponding change in the
operational meaning of its symbols. It begins by being a most useful
servant when dealing with phenomena of the ordinary scale of magnitude,
but ends by dragging us by the scruff of the neck willy nilly into the
inside of the electron where it forces us to repeat meaningless
gibberish. Larmor recognized this, and in his electron theory, developed
practically contemporaneously with that of Lorentz, endeavored to treat
electrons as wholes, and not to make meaningless statements about their
insides.[25] But he was much less successful than Lorentz in making his
analysis give physical results, and one may suspect that it was at least
in part due to difficulty with his tools.
[Footnote 25: Joseph Larmor, Æther and Matter, Cambridge University,
Press, 1900. In this book the electron is treated as a point singularity
in the ether.]
What we should like to be able to do is easy to see. The things that go
into our equations must have independent physical meaning, and the
character of our mathematical formulation should change to keep pace
with the change in the physical operations which give meaning to the
terms. For example, electrical density has a meaning for large scale
phenomena, but means nothing on a small scale. Our ultimate electric
unit is the electron; when we get down to this scale of magnitude, our
mathematics ought to be making statements about the relative behavior of
discrete electrons, and not mention so much as by implication the
density at points inside an electron. But this sort of thing we
apparently cannot yet do; the proper mathematical language has not been
developed. Such a language, when developed, must not only be able to
resist the temptation to burrow inside the electron, but must also try
to get along without the field concept, which we have seen is liable to
so much physical abuse, and must reduce effects in complicated
electrical systems to the ultimate elements that have physical meaning,
namely, a dual action between pairs of electrical charges, with no
implications about physical action where the charges are not.
THE NATURE OF LIGHT AND THE CONCEPTS OF
RELATIVITY
We have already discussed several aspects of the theory of relativity in
connection with the relation to it of some of our fundamental concepts.
There are still other topics connected with relativity which demand
attention; most of these involve the properties of light. It will now be
convenient to discuss together the properties of light and these
concepts of relativity. We restrict our discussion of light to those
simple properties which bear on the theory of relativity.
Practically all our thinking about optical phenomena is done in terms of
an invention, by means of which these phenomena are assimilated to
those of ordinary mechanical experience, and so made easier to think
about. To realize that invention has been active here, we must think
ourselves back into that naive frame of mind in which experience is
given directly in terms of sensation. The most elementary examination of
what light means in terms of direct experience shows that we never
experience light itself, but our experience deals only with things
lighted. This fundamental fact is never modified by the most complicated
or refined physical experiments that have ever been devised; from the
point of view of operations, light means nothing more than things
_lighted_. Now experience shows that these things lighted may stand to
each other in varied relations; in attempting to reduce these relations
to order and understandability we make a certain invention. This is
prompted by several cardinal experimental facts: in the first place,
things lighted have a simple geometrical relation to each other, in that
screens placed on straight lines between the lighted objects may
suppress the illumination of one or the other and themselves become
illuminated. This leads to the concept of rectilinear beams of light,
which is no more than a description of the geometrical relation between
lighted objects. Then we have the experimental fact of the asymmetrical
relation of the lighted objects, described in terms of sources and
sinks. Finally, we have the discovery made at a much later stage, and
not possible until physical measurements had reached a high refinement,
that light has properties analogous to the velocity of material things.
This was first discovered in connection with astronomical phenomena in
the shift of the time of eclipse of Jupiter's satellites and in
aberration, but was later found to hold for purely terrestrial
phenomena, in that a beam of light reflected from a distant mirror does
not return to the source until after the lapse of a time interval that
can be measured with means sufficiently refined. This property of return
after the lapse of time is precisely like that of material things, such
as a messenger despatched for an answer, or a ball or a water wave
bouncing from a wall. These various properties of light lead quite
naturally and almost inevitably to the invention of light as a thing
that _travels_, "thing" not necessarily connoting material thing.
The question now for us is whether we shall regard this as a mere
invention, made for convenience in thinking, or shall go further and
ascribe physical reality to it, that is, shall we think of light as
capable of independent physical existence in the space between the
matter that constitutes the source and the mirror? Now in spite of the
resemblances pointed out above, there is at least one universal and
fundamental difference between a thing that travels and light. We have
independent physical evidence of the continued existence of the ball,
for example, at all intermediate points of space; we can see it, or hear
it, or feel the wind in the air as it passes, or even touch it. All
these phenomena are independent of the initial and terminal phenomena,
and hence by our criterion for the physical reality of an invention, we
are justified in ascribing physical reality to the ball in transit. But
with the beam of light it is entirely different; the only way by which
we can obtain physical evidence of the intermediate existence of the
beam is by interposing some sort of a screen, and this act destroys just
that part of the beam whose existence we have thereby detected. There is
no physical phenomenon, whatever by which light may be detected apart
from the phenomena of the source and the sink (understanding that a
mirror is included in the idea of, sink); that is, no phenomenon exists
independent of the phenomenon which led us to the invention of a thing
travelling. Hence from the point of view of operations it is meaningless
or trivial to ascribe physical reality to light in intermediate space,
and light as a thing travelling must be recognized to be a pure
invention.
The status of light is exactly the same as that of an electric field;
there is not the slightest warrant for ascribing physical reality to
either at points of empty space--light and field-at-a-point have no
meaning until we go there and make experiments with some material thing.
Of course the electromagnetic theory of light makes this resemblance
inevitable, provided the theory and our views of the nature of light and
the field are correct.
It cannot be denied that there are some phenomena which uncritically
considered appear to justify thinking of light as a thing that travels;
these will now be discussed. Probably the argument to which most
significance is usually ascribed is derived from the phenomena of
energy. The passing of light from source to sink is accompanied by the
transfer of energy. But energy is conserved, so that we have to ask
where the energy is in the time interval between the emission of light
from the source and its absorption by the sink. There is an obvious
answer: the energy is in transit, of course, somewhere in the
intermediate space between source and sink. If we think of light as
propagated through a medium, then the medium is such that energy may
reside in it, as in the electromagnetic theory of light, or if light is
more material and ballistic in character, the thing that travels has
itself energy. We notice in the first place that the conservation
principle involves the time concept, because what we mean by
conservation is that the total energy of the universe, at a fixed
instant of time, is constant. That is, we have to integrate over all
space the local energy at a definite instant of time, and this involves
spreading the time concept over all space. It is further evident that
unless we spread the time concept over space in the right way we shall
not get conservation. The proof that it is possible to spread the time
concept over space in such a way as to give conservation involves a
knowledge of the properties of light. It would seem, then, that we ought
not to assume conservation in deducing the properties of light, when a
knowledge of the properties of light is necessary to establish
conservation. These considerations cannot be accepted as final, however,
until a detailed analysis has been made, and this would be most
complicated. But there is a more important consideration derived from
our previous critique of the energy concept, namely, that there is no
basis for asserting that energy is localized in space at all; energy is
not a physical thing, but rather what we would call a property of a
system as a whole. If this view of energy be granted, the whole energy
argument for light as a thing travelling, and also for the existence of
a medium, falls. I believe that similar considerations apply to any
arguments from the conservation of momentum.
The possibility of detecting light in apparently empty space by a screen
constitutes perhaps the most immediate reason for considering light as a
thing that travels. This point of view I believe is characteristic of
the entire attitude of Einstein in deducing the theorems of the special
theory of relativity. Einstein's light signal is for the purposes of the
deduction thought of as a simple spherical wave spreading from the
source and capable of being watched as it spreads by an observer outside
the system, in much the same way that a water wave can be watched. Of
course the light signal cannot actually be watched in transit, but we
can come fairly close to this ideal by placing screens at any point we
please to make the wave visible. It is true that the mere act of showing
the existence of the light destroys that part of the beam whose
existence is detected, but the screen needs only an infinitesimal amount
of light to make it visible, and so by the usual physical argument we
may suppose that the detecting screen produces only an infinitesimal
modification of the total original light.
Our satisfaction with this picture evaporates if our present quantum
views of the nature of light are correct. We can no longer think of the
spherical light pulse as of irreducible simplicity, but it is an
exceedingly complicated thing, perhaps more complicated than a gas from
the point of view of kinetic theory, and simulates simplicity by some
sort of averaging of the effects of the elementary quantum processes of
which it is composed. If the principles of relativity are to continue to
be regarded as fundamental, or even if they are to remain intelligible,
we must apply our reasoning, not to spherical wavelets, but to the
elementary process of which these wavelets are composed. Now the
elementary quantum act is essentially a twofold thing: there is a
discrete act of emission at some discrete material particle, and the act
is consummated by another discrete act (absorption or scattering) at
some other discrete particle. We cannot yet fully characterize the
details of this twofold process, but have to connect the place at which
absorption takes place with the place of emission by statistical
considerations. It is evident, however, that to think of emission as
starting some process like a spherical wavelet travelling like a thing
through space presents an entirely incorrect view, because in the wave
there is no hint of the discrete place which is to terminate it. We may
say crudely that there is no way by which the wave can know what
discrete material particle is to complete the emission process. We may
perhaps try to save the situation by remembering that a spherical wave
is polarized and so has a unique direction associated with it; but
further examination shows that this does not help, because the unique
direction is that of no energy flow, and absorption can take place in
any direction _except_ this. It appears then that instead of being a
help, the thing travelling point of view is a positive embarrassment
when we try to picture by means of it the essentially twofold nature of
the elementary quantum act.
Another plausible argument for light as a thing travelling may be
deduced from our principle of connectivity. Imagine a dark lantern with
a shutter that can be opened or closed so as to emit a momentary flash
of light, a distant mirror, and a receiving instrument near the source.
One of the properties of light that we always assume is that no
permanent trace of the act of emission is left behind in the source. The
most minute examination of all the details of the lantern and its
surroundings at some time after the emission of the flash has not yet
shown any phenomenon that betrays a remembrance of the emission of the
flash, unless perhaps we measure the total energy or momentum and have
some way of knowing what the energy or momentum would have been if the
flash had not been emitted, and in any event we cannot specify the
moment in past time when the signal was emitted. In the same way we
cannot tell from an examination of a mirror whether it has at any time
in the past reflected a beam of light. Consider now two systems, each
consisting of a source and a mirror distant 3 x 10^10 cm., identical in
all respects, except that in one a light signal was flashed from the
source 1.5 seconds ago, and in the other only 0.5 seconds ago. According
to our hypothesis, the most complete examination of source and mirror in
either system fails to show the slightest difference, but nevertheless
there is something essentially different about the two systems, for in
one a light signal arrives at the screen in 0.5 seconds, and in the
other not until 1.5 seconds. This violates what we have suggested might
be regarded as the cardinal and most general principle of all physics,
the principle of essential connectivity, which states that differences
between two systems must be associated with other differences. A most
obvious and simple way of maintaining our principle is merely to point
out that the system really included more than we investigated: the
system properly consists of source, mirror, screen, and all intermediate
space, so that if we had examined intermediate space we should have
found light there in transit in different positions in the two systems,
thus correlating with the differences in subsequent history. This
argument appeals to me as perhaps the strongest that can be advanced for
the view of light as a thing travelling. But it seems in no way
conclusive. Our principle of essential connectivity made no mention of
the time concept, but we have somehow smuggled it in making the
application above. We sought to give a complete description of our
system at some one instant of time, and this involved spreading the time
concept over space. This itself is a questionable operation and may be
done in different ways. But, more important, what is the justification
for supposing that the system can be completely described by giving a
complete description of all the measurable parts of it at some one
epoch? We have seen that in the most general case the principle of
essential connectivity must recognize that the concept of "initial
condition" of a system involves all the past history, and we may have
here a case in point. The answer can be given only by experiment. In
dealing with ordinary experience, when we do not have to distinguish
between local and extended time, and are not dealing with optical
phenomena, there can be little question that experience at least
approximately justifies the expectation that future behavior is
determined in terms of the present condition and that present condition
may be specified in terms of the results of present operations performed
in the system. But before extending this principle to phenomena in which
we have to distinguish between local and extended time, we have to
answer just that question which we are now considering; namely, whether
there are physical phenomena taking place in apparently empty space, and
whether therefore empty space has to be included in the system. We find
ourselves again treading the vicious circle. Perhaps experience will
show that the extension of the principle of connectivity to optical
phenomena involves something like this: namely, the future at any point
in a material system is determined by a complete description of the
_present_ state of the system in the immediate vicinity, and by a
_history_ of the behavior at more distant points, this history extending
over longer and longer intervals of time as the point becomes more
remote.
I believe, however, that these possibilities will not seem very
satisfactory, and that most physicists will discover in themselves a
very strong disposition to feel that the future is determined in terms
of a complete description of some sort of instantaneous configuration,
time being extended in some suitable way over space. This instinctive
demand that the future be determined in terms of the present may easily
be consistent with the optical phenomena in our two systems consisting
of source, mirror, and screen, without involving the material existence
of light in empty space, provided that our assumption that the emission
of light leaves behind it no permanent record in the source was
incorrect. It may be that detailed examination of a source after
emission will disclose permanent traces, from which the instant of
emission may be found by extrapolation. If the conviction of determinism
of the future is strong, the physicist may well be impelled to search
here for new phenomena indicating such a memory of emission.
Let us now inquire how our physical structure might be affected if we
should give up the identification of light with a thing travelling. One
consequence is that light need no longer be thought of as having the
property of velocity, since velocity, in terms of immediate experience,
is a property of things moving from place to place. Giving up the
concept of light as a thing travelling would enable us, then, to adopt
an alternative method of describing nature with a different concept of
velocity; we have seen that it is possible to define velocity in terms
of operations different from the usual ones, in such a way as to give
the usual numerical results at small velocities, but different results
at high velocities, and in particular to give an infinite velocity for
light.[26]
[Footnote 26: No difficulty arises from the asymmetric character of
light in assigning an infinite velocity to light because those physical
operations by which we discover which is the source and which the sink
are entirely distinct from the operations by which a velocity is
measured; or in other words, even in the limit, it still has meaning to
say that an infinite velocity has a direction associated with it.]
There is now no objection to an infinite number associated with light,
if we no longer think of light as having physical velocity. We may, if
we like, continue for the sake of convenience to talk of the velocity of
light, clearly understanding that the infinite value which must be
ascribed to this velocity corresponds to the fact that the physical
concept of velocity does not apply in all respects to light. We should
now have to revise our process for extending the time concept over
space, because this was formerly so done as to give light a finite
velocity. We are now to make this velocity infinite, which is obviously
to be done simply by setting a distant clock on zero at the instant it
receives a light signal flashed from our clock at its zero. The behavior
of material things now takes on a simple aspect--there is no longer a
finite upper limit to the velocity that can be given a material thing,
and light has no longer the paradoxical property of bearing the same
finite relation to each of two material systems which differ from each
other by a finite amount (that is, the first postulate of relativity
that the velocity of light is 3 x 10^10 in all reference systems). Light
instead now bears the relation of infinitude to each of two systems
which differ from each other by only a finite amount, and this is
natural from the mathematical point of view.
However, all is not simplified by this change in the method of setting
clocks, but a price has to be paid. The price is that we have to give up
the simple connection between the velocity of a thing and its "go and
come" time. Our changes have not affected local time; the time of
passage of light to a distant mirror and return to the source is not
changed, and is therefore still finite, although we describe the
velocity of light as infinite. Now examination shows at once that there
is no immediate connection between the concepts of "velocity" and of "go
and come" time, because the operations involved are different. A
measurement of a linear velocity according to our definition involves
two clocks at two different places or else a clock travelling with the
object, while "go and come" time demands only a single clock at a single
place, and also involves necessarily a reversal of direction of motion
in the object under measurement. We see then that, according to the
definition adopted for velocity, we have the choice either of doing as
Einstein did in the restricted theory of relativity and making "go and
come" time very simply related to velocity; or we may say that refined
physical measurements show that something of significance happens when
the direction of motion is reversed, and that phenomena are not
symmetrical with respect to a reversal of direction. The asymmetry which
results from reversing the direction of motion we may visualize as a
sort of curvature in space and time, as of a small piece of an arc of a
circle bent back on itself, with the two ends diverging. This
alternative way of treating velocity would mean that velocity can be
measured simply only by a specially situated observer; this need not be
considered disturbing, because in fact the operations have been defined
only with respect to such an observer.
Which of these two possible treatments of velocity shall be adopted is
to a certain extent a matter of convenience, determined by the sort of
phenomena in which we are most interested and wish most to simplify.
Einstein's chief concern was with optical phenomena, so that the motive
for his choice is evident. In this choice of Einstein it is not very
evident that the desire to make "go and come" time simply connected with
velocity played a very prominent part, but it seems rather that the
desire to think of light as a thing travelling, with a finite velocity,
was much more influential. This way of thinking of light is fundamental
to all the treatment of restricted relativity; without this sort of
picture all the mathematical deductions would lose their simplicity and
convincingness, for in all the deductions we inevitably think of
ourselves as an observer from outside, watching a thing that we call
light travelling back and forth like any physical thing.
Now there can be no doubt that, when choice is possible, convenience and
simplicity are important considerations; but I believe that there is
another much more important consideration, namely, the most perfect
reproduction possible of the physical situation. It seems to me that it
is very questionable whether Einstein, and all the rest of modern
physics, for that matter, have not paid too high a price for simplicity
and mathematical tractability in choosing to treat light as a thing that
travels. Physically it is the essence of light that it is not a thing
that travels, and in choosing to treat it as a thing that does, I do not
see how we can expect to avoid the most serious difficulties. Of course
the whole problem of the nature of light is now giving the most acute
difficulty. The thing-travelling point of view, even as treated by
Einstein, does not land us in a situation which is at all satisfying
logically. We are familiar with only two kinds of thing travelling, a
disturbance in a medium, and a ballistic thing like a projectile. But
light is not like a disturbance in a medium, for otherwise we should
find a different velocity when we move with respect to the medium, and
no such phenomenon exists; neither is light like a projectile, because
the velocity of light with respect to the observer is independent of the
velocity of the source. On the other hand, in aberration we have a
phenomenon similar to that shown by projectiles. The properties of light
are more like those of a projectile than is perhaps commonly realized,
as is shown in the papers of La Rosa[27] on the ballistic theory of
light. The properties of light remain incongruous and inconsistent when
we try to think of them in terms of material things.
[Footnote 27: M. La Rosa, Scientia, July-August, 1924.]
Einstein's restricted relativity has made a great contribution in so
grouping and coordinating the phenomena that they can all be embraced in
a simple mathematical formula, but he does not seem to have presented
them in such a light that they are simple or easy to grasp physically.
The explanatory aspect is completely absent from Einstein's work.
In view of all our present difficulties it would seem that we ought at
least to try to start over again from the beginning and devise concepts
for the treatment of all optical phenomena which come closer to physical
reality. No one realizes more vividly than I that this is a most
difficult thing to do. If we are ever successful in carrying through
such a modified treatment, it is evident that not only will the
structure of most of our physics be altered, but in particular the
formal approach to those phenomena now treated by relativity theory must
be changed, and therefore the appearance of the entire theory altered. I
believe that it is a very serious question whether we shall not
ultimately see such a change, and whether Einstein's whole formal
structure is not a more or less temporary affair.
Although it is exceedingly difficult to forsee what the treatment of the
future will be like, it is easy to surmise certain of its features. In
essence the elementary process of all radiation perceived as radiation
is twofold. There is some process at the source and some accompanying
process at the sink, and nothing else, as far as we have any physical
evidence; furthermore, the elementary act is unsymmetrical, in that the
source and the sink are physically differentiated from each other. This
is the most complete expression of the physical facts; there is nowhere
any physical evidence for the inclusion of a third element (the ether).
Therefore all the phenomena apprehended by an observer (and this
embraces all physical phenomena) can be determined only by the source
and the sink and the relation to each other of source and sink, for
there is nothing else that has physical meaning in terms of operations.
This formula covers not only the possibility of such first order
phenomena as aberration and the Döppler effect, but also shows that
such second order effects as that looked for by Michelson and Morley
must be non-existent. It will thus be seen that some of the consequences
of relativity theory are implicitly contained in certain very broad
points of view. One interesting question that must be answered before we
can get very far with a new treatment is whether the elementary optical
process is of _necessity_ twofold, or whether we may have emission
without absorption, that is, radiation into empty space. Lewis seems to
imply in recent papers that this is not possible.[28] The astronomers
have already pointed out difficulties in explaining phenomena like the
temperature equilibrium of the planets if we suppose this is the case.
[Footnote 28: For example, in the book: G. N. Lewis, The Anatomy of
Science, Yale University Press, 1926, p. 129.]
OTHER RELATIVITY CONCEPTS
We now turn to some of the other concepts of relativity. One of the most
important of these is the "event"; in fact this concept is made
fundamental by Whitehead.[29] We have already discussed the concept of
"event" under the "identity" concept with which it is closely involved.
The event is usually thought of by Einstein as merely an aggregate of
four coordinates, three of space and one of time. The principle of
general relativity, namely, that the laws of nature shall be of
invariant form, when formulated mathematically, involves the assumption
that nature may be analyzed into events, and is expressed by the
requirement that the mathematical relations between the coordinates of a
chain of events shall be invariant. The same idea is also expressed by
Einstein in another form, namely, that nature may be completely
characterized in terms of space-time coincidences. In elaborating this
idea, Einstein assumes that the results of all measurement may be given
in terms of such coincidences.
Now it appears to me most questionable whether the analysis of nature
into events is possible or sufficient. With regard to the coincidence
point of view, it seems perfectly obvious that the world of our
immediate _sensation_ cannot be described in terms of coincidences; how,
for example, shall we describe in terms of space-time coincidence the
photometric comparison of the intensity of two sources of illumination,
or the comparison of the pitch of two sounds, or the location of a sound
by the binaural effect?
[Footnote 29: A. N. Whitehead, An Enquiry Concerning the Principles of
Natural Knowledge, Cambridge University Press, 1919, Chap. V.]
To justify the coincidence point of view we apparently have to analyze
down to the colorless elements beyond our sense perception. It does not
seem unreasonable, perhaps, to expect that the universe is completely
determined in terms of the positions as a function of time of all the
positive and negative electrons; but to introduce such a thesis now
certainly goes beyond present experimental warrant, and is contrary to
the general spirit of relativity, which nowhere else involves any
reference to the small scale structure of things. Even if we were
willing to overlook all these objections, we would still have the fact
that the difference between a positive and negative electron is not
contained in any specification of the mere coordinates.
A further very important doubt in principle as to the possibility of the
analysis of nature into events is afforded by the character of the
concept of event itself. We have seen that the idea of event involves
the existence of discontinuities, and that this can correspond only
approximately to the physical fact, because discontinuities apparently
lose their abruptness as we make our measurements more refined. The
thesis that nature can be described in terms of discontinuities of a
very small scale seems much too special to be made a fundamental part of
a theory of the general pretensions of that of relativity. In fact this,
as well as a consideration to be mentioned later, suggests that the
argument and result of general relativity may be intrinsically
restricted to large scale phenomena.
We now pass from these somewhat special questions to ask why it is that
Einstein was able in the general theory of relativity to obtain new and
physically correct results from general reasoning of an apparently
purely mathematical character. We are convinced that purely mathematical
reasoning never can yield physical results--that if anything physical
comes out of mathematics it must have been put in another form. Our
problem is to find where the physics got into the general theory.
There are two questions to be disentangled here: we have to consider in
the first place the significance of the fact that Einstein has been able
to describe relations in nature in mathematical form, and in the second
place of the fact that he was able to arrive at the mathematical
formulation of these physical relations by reasoning of apparently a
purely mathematical character, from postulates of merely formal
mathematical content (invariance of natural laws in generalized
coordinates). Now the theory of relativity does not seem to differ in
the first respect from any other branch of mathematical physics, such as
the classical mathematical theory of electricity and magnetism, for
instance, and this matter has already been touched in an earlier
chapter. It is a fact that the behavior of nature can in many cases be
expressed to a high degree of precision in mathematical language, and
relativity is not unique in this respect. In any event, we must not
allow this possibility of mathematical formulation to obscure the
essential fact that all physical knowledge is by its nature only
approximate, so that we may expect at any time to find, when we have
carried our measurements to a higher degree of precision, that our
mathematical expression of the laws was not quite exact, as seems now to
be the case with Newton's law of gravitation, for example. I do not
suppose that Einstein would claim that the statements of relativity
differ in this respect from any of our other statements about nature,
although apparently some of his followers see something more here. (From
the operational viewpoint the meaning to be attached to "something more"
is somewhat obscure.)
With respect to the second question, we may stop to notice that the
special theory stands in quite a different position from the general
theory. The special theory is much more physical throughout; its
postulates are physical in character, and it is obvious that the physics
got into the results through the postulates. It seems to me without
question that Einstein showed the intuitive insight of a great genius in
recognizing that there are mutual relations between physical phenomena
which can be described in very much simplified language in terms of
concepts slightly modified from those already in common use. In view of
the remarks made on the nature of light, it is legitimate to wonder,
however, whether the formulations of even the special theory will always
stand. It seems to be true that _all_ the facts of nature, even in the
absence of a gravitational field, cannot be connected by the simple
formulations of the special theory; that the physical relations are
simple only in a sub-group; and that if we wish to deal with _all_
optical phenomena, we have carried our simplifications too far, for the
emission of a light signal is not a simple event, and light is not in
nature like a thing travelling. Just the sorts of physical thing which
are ignored in treating light as the special theory does are coming to
be more and more important in the minds of physicists, and this is a
reason for wondering whether ultimately Einstein's special theory may
not be regarded merely as a very convenient way of tying together a
large group of important physical phenomena, but not as being by any
means a full or complete statement of natural relations.
With respect to the general theory, however, I believe the situation is
quite different. The fundamental postulate that the laws of nature are
of invariant form in all coördinate systems is highly mathematical, and
of an entirely man-made character. Of what concern of nature's is it how
man may choose to describe her phenomena, and how can we expect the
limitations of our descriptive process to limit the thing described?
Furthermore, Einstein's method of connecting his mathematical
formulation and nature by way of coincidences of 4-events (three space,
one time coordinates) seems to be very far removed from reality, since
it entirely leaves out the descriptive background in terms only of which
the 4-event takes on physical significance. Nevertheless, three definite
conclusions about the physical universe have been taken out of the hat
by the conjuror Einstein (shift of the perihelion of Mercury,
displacement of apparent position of stars at the edge of the sun's
disk, and the shift toward the infra-red of spectrum lines from a source
in a gravitational field), and the problem for us as physicists is to
discover by what process these results were obtained.
An examination of what Einstein actually did in deriving his results
will show, I believe, that the situation is really different from that
suggested above. In the first place, the requirement that the laws of
nature be of invariant form actually places no restriction, as any one
can see by setting himself the task of expressing, for example, an
inverse cube law for gravitation in terms of generalized coordinates.
The work of expressing such a law can be attacked in a perfectly routine
way. (The essential difference between the invariability requirements of
the special and general theories is to be noted; the special theory
requires that the velocity of light, for instance, have the same
_numerical_ value in all allowed systems: the general theory merely that
all laws have the same _literal_ form, but with variable numerical
coefficients.) But, as Einstein says, if any one actually attempts to
carry through the work of expressing an inverse cube law in generalized
coordinates, he will find the task prohibitively complicated, and will
seek for some simpler formulation. What Einstein actually did,
therefore, was to require that the laws of nature be _simple_ in
generalized form. Now we know that the law of gravitation as formerly
expressed in ordinary coordinates as an inverse square law was
approximately exact, and was also simple. Any deviations from this law
are small, and all experience leads us to expect that to the first order
of small quantities the deviations can be taken care of mathematically
in the form of small correction terms to this law. This by itself gives
nothing, however, because a small correction term can be added to our
equations in an infinite number of ways. If, however, we know that the
equation must be of a certain type after the correction terms have been
added, the possibilities are so much restricted that the form of the
correction term may be determined. In arguing as to the probable type of
the equation, Einstein advanced the considerations by which physics gets
into the situation.
In the first place, the special theory had prepared us for the
possibility of finding that our measuring instruments might be modified
in a gravitational field, analogously to the shortening of a meter stick
when set into motion. In fact, special theory had indicated that in an
accelerated system the modifications might be too complicated to be
treated by that theory. In the absence, then, of specific information we
must be prepared for the most general possible alteration in space-time
in a gravitational field. In describing space-time we must therefore use
coördinates adapted to handling the most general possible relations,
and these are the generalized coördinates of Riemann, which had been
already discussed by mathematicians. Going back now to Einstein's
criterion that the equations are to be simple, we have the demand that
the equations be simple in generalized coördinates, and of course they
must also reduce to the ordinary equations (that is, the equations of
special relativity) in space where there is no gravitational field. In
deciding the further question as to what the type of equation probably
is, we are influenced by considerations of convenience as well as by
physical considerations. Practically the only type of equation that can
be handled mathematically is linear, so that we shall certainly try
first whether this type of equation may not continue to hold. Now the
Newtonian law of the inverse square may be expressed in terms of a
linear differential equation of the second order in the old Cartesian
coördinates (Poisson's equation), so that our most immediate suggestion
is that the equations remain linear and of the second order in
generalized coördinates. As a matter of fact, this requirement turns
out to be sufficient to determine the small correction term by which the
ordinary equations can be generalized; Einstein's papers must be studied
to see how this works out in detail.
All this looks pretty mathematical, but as a matter of fact there is
much physical content, because systems which can be described by linear
equations of the second order have definite physical properties. The
requirement that the equations be linear corresponds to one of the most
fundamental properties of our universe--the causality concept would not
be possible or would be much modified in a universe governed by
non-linear equations, for the joint effect of two causes acting together
would not be the sum of their effects acting separately, so that the
analysis of a situation into simple elements would be impossible and the
causality concept probably would not have arisen. Furthermore, an
equation of the type of Poisson of the second order means that there are
propagation phenomena, and equations of mechanics of the second order
involve the existence of a scalar energy function. If, then, the
behavior of the universe can be described by differential equations at
all, these equations must be linear of the second order if the universe
is to have the broadest physical characteristics of our own universe.
What Einstein really did, therefore, was to demand that even when
space-time is warped by the presence of a gravitational field, those
physical phenomena which can be described in terms of differential
equations continue to be described by linear differential equations of
the second order; that is, that nature continues to be describable in
terms of a causality concept, with propagation phenomena, and a simple
energy function. The consequences of a guess like this about the
properties of nature appeal to our physical intuition as being worth
following out, and of course we know the experimental justification.
Several general comments may now be made on the structure reached in
this way. In the first place, the whole structure is only descriptive in
character; we find certain correlations in nature which we describe with
considerable completeness in mathematical equations, without introducing
any new element of explanation or of mechanism. We have seen that as we
increase our range from the realm of ordinary phenomena to phenomena of
different character we arrive at a stage where for a time the process of
explanation apparently halts, and we have to be satisfied with a
statement of mere correlation between elements; later, however, these
elements may be accepted as the ultimates in a broadened scheme of
explanation, and the explanatory process resumed. Are we at such a stage
now with the general relativity theory, and may later a new scheme of
explanation be established based on the correlations of Einstein? This
is of course a matter of individual judgment; I personally question
whether the elements of Einstein's formulation, such as curvature of
space-time, are closely enough connected with immediate physical
experience ever to be accepted as an ultimate in a scheme of
explanation, and I very much feel the need for a formulation in more
intimate physical terms.
In the second place, we must repeat the comment already made in
discussing time, namely, that there is still a very wide gap between the
theory and its physical application, in that we have no way of
identifying our physical clocks and our physical measures of time with
the thing called time in the formulas. This gap must be filled by a
specification of the physical structure of a clock.
It has always been very puzzling to understand why Einstein has so
strenuously insisted that the shift toward the infra-red is an integral
part of the general theory, and that if the shift is not found, the
theory must fall. In other words, Einstein insists that the assumption
that an atom is a clock is an integral part of his theory. I believe
that this attitude may be due to a realization by Einstein of that very
flaw in the logical structure which we are now emphasizing. In the
absence of any method of specifying the details of construction of at
least one clock, relativity becomes a purely academic affair, unless
there exist in nature concrete things which may serve as clocks.
Einstein _must_ either be able to tell how to construct a clock, or else
be able to point to a specific example of a clock. He chose the atom as
the specific thing. Doubtless the reason was the apparent simplicity of
the vibrating mechanism of an atom, as shown by the precise equality of
the frequencies emitted by all atoms of the same element. If the atom is
not a clock, where in nature can one be found? But in the last few years
we have come to appreciate the exceedingly complicated quantum structure
of an atom, and Einstein's thesis loses much of its instinctive appeal.
Since Einstein created the theory of relativity, it is perhaps
ungracious to question his right to stipulate that the assumption that
the atom is a clock is an integral part of the theory. This, however,
degenerates to a mere matter of language, and does not touch the
arbitrary nature of the procedure. It does not prevent us from having a
second brand of relativity theory, that of X instead of Einstein,
exactly like that of Einstein except that perhaps now the "clock" is
constructed in terms of the life period of a radioactively
disintegrating element. The only way to eliminate the arbitrariness
seems to be to postulate that _all_ natural processes, which run
naturally of themselves independently of what we may do, may equally
well serve as clocks and give the same results. But in answering the
question of the operational meaning of "independently of what we may do"
we shall effectively have to answer the question of what is a clock.
This point of view may possibly, however, get us a little nearer to our
goal of finding how to specify the structure of a clock.
Finally, the general theory is not completely general, but applies only
to a certain range of phenomena, just as we saw that the special theory
does not embrace all optical phenomena. The general theory applies only
to those phenomena which can be described in terms of differential
equations, that is, par excellence, to large scale phenomena. If quantum
phenomena cannot be described by differential equations,[30] as
apparently now they cannot, general relativity cannot by its very nature
be applicable. General relativity does not give us a comprehensive
formulation of the behavior of all nature, and as far as we can see, we
are still as far as ever from such a general formulation.
[Footnote 30: This statement now takes on a very questionable aspect in
view of the new quantum wave mechanics (March, 1927).]
ROTATIONAL MOTION AND RELATIVITY
Physically there is a great difference between the behavior of systems
in uniform relative rectilinear motion and those in uniform relative
rotation. The special theory of relativity states that there is a triply
infinite number of systems with all possible uniform rectilinear
velocities with respect to each other, in all of which physical
phenomena have exactly the same mutual relations, that is, natural laws
are the same. Now the mere formulation of the principle suggests the
sense in which "system" is here used. It is obvious that "system" refers
only to a part of the universe; we are not making a hopelessly academic
statement about what would happen if we had an infinite number of
universes to experiment with, but are talking about operations that may
be approximately realized in our own universe. The "system" of the
formulation we may think of as a completely equipped laboratory, out in
empty space, so far from the heavenly bodies that they can have no
effect. The different systems of the formulation are different
laboratories, all built to exactly the same architectural blue prints.
The phenomenon to which the postulates of relativity apply are phenomena
which pertain entirely only to one or another of these laboratories. The
meaning of this restriction is not completely definite and has, in any
special case, to be judged partly by the context. Obviously, to see from
the window of a laboratory another laboratory passing with a certain
velocity cannot be counted as one of the allowed phenomena. Still less
is it one of the allowed phenomena to observe that the center of gravity
of the entire stellar universe has a certain velocity of translation
with respect to the laboratory. The special principle of relativity
contains by implication therefore the statement that certain very large
and important classes of physical phenomena may be isolated and treated
as taking place unaffected by the rest of the universe. Granted now the
possibility of isolation, we have a second statement, which is usually
treated as if it were the entire statement of the restricted principle,
namely, that there is a triply infinite set of systems in which these
phenomena run in the same way independent of the relative motion of the
systems with respect to each other. When once the significance of the
observation is grasped that absolute motion has no meaning in terms of
operations, we see that this last statement takes immediately a most
simple and satisfying aspect, in fact, so simple and inevitable that we
are inclined to see in this the complete essence of the situation and
regard the meaninglessness of absolute motion as affording peremptory
proof of the restricted principle.
With this bias we now turn to examine the facts of rotary motion, and
are disconcerted to find them quite different. No meaning in terms of
measuring operations can be given to absolute rotary motion any more
than to absolute translation, but nevertheless phenomena are obviously
entirely different in different systems in relative rotary motion
(phenomena of rupture, for example), so that apparently there are
physical phenomena by which the concept of absolute rotary motion might
be given a certain physical significance. Given two worlds like our own
in empty space, but surrounded by impenetrable clouds, and each provided
with a Foucault pendulum, then we believe that it is physically possible
that we may find on one of these worlds the plane of rotation of the
pendulum gradually changing in direction, while on the other it remains
stationary. This difference we regard as possible without other
accompanying physical phenomena which are causally related to the
rotation of the pendulum (of course we have to make the two worlds of
infinitely rigid material and eliminate other phenomena which we regard
as purely incidental), so that we apparently have here a contradiction
of our cardinal physical principle of essential connectivity. We are
certainly not inclined to give up our principle, and we believe that as
a physical fact, if the clouds could be evaporated, an observer in one
world would find that he was rotating with respect to the system of the
fixed stars, whereas the corresponding observer on the other world would
find that he was stationary. Our principle of essential connectivity is
therefore maintained, in that the rotation of the plane of the pendulum
is connected with a rotation with respect to the rest of the universe of
the entire world in which the pendulum is mounted. As far as I am aware,
no other way of maintaining our principle has ever been suggested. But
this demands that we give up our physical hypothesis of the possibility
of isolating a system. There is here no question of limiting behavior;
we believe that no matter how far our rotating world gets from the rest
of the universe the Foucault pendulum would always behave in the same
way; the system can never be isolated, but such local phenomena as the
invariance of the plane of the pendulum are always essentially
determined by the rest of the universe.
If now our system cannot be isolated, we must return to the phenomena of
translational motion. In principle the act of isolation cannot be
performed, the rest of the universe cannot be disregarded, and we should
expect that different states of translational motion as well as
different states of rotational motion with respect to the rest of the
universe would have an effect on phenomena. We set ourselves the problem
of understanding this apparent enormous difference between phenomena of
translation and rotation. We remark that what apparently is a difference
in principle may, in virtue of the approximate character of all
measurement, be only a difference in magnitude, and that translational
effects may exist too small to detect. A physical basis for such a
difference may be found in the enormously different numerical values of
translational and rotational velocities with respect to the rest of the
universe attainable in practice. In describing phenomena of cosmic
magnitude, we may plausibly measure the phenomena in units commensurable
with the scale of the phenomena. Thus in measuring linear distances, we
may perhaps choose as the unit of length the diameter of the stellar
universe, and in measuring rotation, a complete reversal of direction
with respect to the entire universe. This last means a change of angular
orientation of 2 π, the first means a length of the order of 10^6 light
years. Measured in such cosmic units the angular velocities attainable
in practice are incomparably greater than linear velocities. We now see
that it is possible that the real state of affairs is as follows:
namely, phenomena in any system are affected by motion with respect to
the entire universe, whether that motion is of translation or of
rotation, and the magnitude of the effect is connected with the velocity
of the motion by a factor which is of the general order of unity when
velocity is measured in cosmic units. This last is merely an application
of the argument so often made in physics as to the order of magnitude of
unknown numerical factors, and will be found expanded on page 88 of my
book on _Dimensional Analysis_. The linear velocities attainable in
practice are now so exceedingly low that their effect has not yet been
detected experimentally, but angular velocities are high, and the effect
is easily demonstrable. In this light the special principle of
relativity is no different in character from any other physical law; it
is only approximate, and some day our measurements may become refined
enough to detect its limitations.
We have made a hypothesis here, which we may call the hypothesis of the
immanence of the entire universe, namely, that isolation is impossible,
or that the rest of the universe, no matter how distant, always has a
local effect on at least some phenomena. This is essentially the
hypothesis of Mach,[31] and leads to a situation which can, I think, be
contemplated with logical equanimity, although it has always seemed to
many physicists most highly antiphysical in character.
[Footnote 31: E. Mach, The Science of Mechanics, translated by
McCormack, The Open Court Publishing Co., Chicago, 1893. See especially
p. 235.]
It must certainly be admitted that most physical experience justifies us
in thinking that effects may be made as small as we please by getting
far enough away from the cause of the effect. But if we accept the
considerations of the preceding pages, we must be prepared to admit that
as phenomena change in range their character may change, and that in
these new realms we must, at first at least, be satisfied with a mere
statement of correlations. Certainly we have very strong physical
evidence of a formal correlation between the Foucault pendulum and the
rest of the universe. But a correlation of this sort may be without
significance because of its very breadth; we never can prove the
significance of the correlation by performing an experiment with the
rest of the universe absent. Have we really done anything more than
merely get things into such a formal situation that they cannot be
assailed, a possibility which the mere laws of our thinking seem always
to leave open, as has been suggested, or is there any physical content
to what we have done? We have seen that if our correlation is also
suggested by other phenomena, then we may accept it as having physical
content. Now there is just a glimmer of a suggestion that our hypothesis
of the immanence of the universe may be needed in other ways. The
gravitational constant and the velocity of light are always treated as
arbitrary magnitudes thrust on the universe from outside with no
connection with other phenomena. Nevertheless, I suppose that no one
regards this situation as ultimately satisfactory and does not entertain
the hope that some day we may be able to give some sort of account of
the numerical magnitude of these constants. We have not hitherto
succeeded in finding any connection between these constants and small
scale phenomena such as the charge on the electron, its mass, etc., so
that there is some plausibility in expecting that a connection may be
sometime found with cosmic things; indeed general relativity theory
already prepares us for exactly this possibility. Now the velocity of
light and the gravitational constant control small scale experiments,
for of course these two constants can be measured by local experiments,
so that if the cosmic connection is found, we should have a control of
local behavior by cosmic things, and therefore another example of the
immanence of the entire universe. There is no need for me to waste time
in apologizing for the highly speculative character of all this. It is
worth while to emphasize, however, that our general considerations on
the meaning of "explanation" have prepared us to admit as reasonable
just the sort of explanation contained in the hypothesis of the
immanence of the universe, and therefore to reserve a place in our
physical thinking for possibilities of this sort, in spite of the fact
that such considerations are not usually entertained, and may seem to
many opposed to the spirit of physics.
QUANTUM CONCEPTS[32]
The history of quantum theory up to the present is a repetition in many
respects of that of the early theories of electricity, in that all our
thinking has been in mechanical terms. As far as we now know, quantum
phenomena are always associated with atoms. We make for the atom a
mental model with all the properties of the mechanisms of the ordinary
scale of magnitude and with a few impressed properties in addition which
represent the new quantum relations. As we now think of it, the atom has
a massive core about which electrons revolve under an inverse square
law, the connection between the mass of the electron, its acceleration,
and the force acting on it being that usual in Newtonian mechanics.
[Footnote 32: This section was written early in 1926 without access to
recent literature. Our attitude toward quantum phenomena has been so
much changed since then by the "new" quantum mechanics, that a number of
the following statements are superseded as a statement of present
opinion. However it has seemed worth while to let the section stand as
written, because many of the developments actually taken in the new
mechanics follow the lines that it is here urged they ought to take, and
in so far afford interesting confirmation of the point of view of this
essay.]
The space in which the electron circulates is thought of as Euclidean,
and the motion is described in time, which may be measured with clocks
in the usual way. The general equations of electrodynamics do not apply;
there are no propagation effects inside the atom, the motion of the
electrons does not produce a magnetic field, and there is no radiation
when the electron is in one of its possible stable states, in spite of
the acceleration. We may, if we please, in working out the character of
the motion, entirely neglect the electrical origin of the inverse square
law, and treat this merely as an impressed force without further
implications. Superposed on the ordinary spatial, temporal, and
mechanical characteristics of the model are additional quantum
properties, one which determines the particular orbit in which the
electron moves [∫ pdq = nh], and another which determines the
frequency of the radiation emitted when the electron passes from one
allowed orbit to another. No mechanism is suggested to account for these
quantum conditions, although the conditions are formulated in mechanical
terms.
We now have to ask what is the meaning in terms of operations of our
usual concepts of space-time and mechanics when applied to phenomena of
this order. It is of course evident, as has already been emphasized,
that the concepts have entirely changed in character, because we do not
measure an electron orbit, for example, by stepping off the diameter
with meter sticks, or by measuring the time required for light to travel
across the diameter. The particular feature of immediate interest in
this changed situation is the change in number of our concepts on the
atomic level. I shall not attempt to find by an exact analysis the
number of independent concepts at this level; probably such an analysis
is not possible. We may, however, make an approximate suggestion.
Apparently the most important concept in describing relations inside a
quantum system corresponds to that of energy on the ordinary scale.
Changes of energy determine the frequency of emitted radiation, as well
as the relations during collisions of atoms and electrons; these
collisional relations make direct connection with experiment through the
voltages applied to electrons in collision experiments. The analogue of
the momentum concept also seems to have independent significance, as
shown by the Compton effect. The frequency of emitted radiation is also
something with independent experimental significance. I believe that
these three things are all that have direct significance for quantum
experiments made up to the present time. In any event, it is perfectly
evident that on the quantum level the concepts which at present have
operational significance are considerably fewer than on the level of
ordinary experience.
Apart from the question of convenience, there may be justification in
continuing to use our old mechanical forms of thought if new
experimental relations are thereby suggested. That a very large number
of such as yet undiscovered relations may be suggested in some such way
is at once evident. Thus we have no present knowledge of any phenomenon
associated with what the electron does when passing from one energy
level to another. How long does it take to make the passage? What is its
path during passage? Is it subject to the ordinary laws of
electrodynamics during passage? When and where is the radiation emitted
that corresponds to passage? When the electron leaves one stable orbit
is the orbit on which it will eventually land already determined? Does
the radiation train emitted during a change from one energy level to
another have a definite length in space, or may it have a variable
length and correspondingly something that corresponds to variable
amplitude? What happens to the radiation when the electron passages are
interfered with before the emission of a quantum has been completed?
What is the mechanism by which the quantum conditions are imposed? Is it
not possible that part of the clew to the riddle of the manner of
transition from purely quantum behavior to the behavior of classical
mechanics may be found in the behavior of the electron during passage
from one energy level to another? Certainly we have a tendency to the
classical behavior under those conditions, such as at high temperature
or in strongly condensed systems, in which the time occupied in passage
might be expected to become a more important part of the total time.
Corresponding to these questions there should be many as yet
undiscovered phenomena, and the mechanical point of view therefore has
its value in suggesting experiments to detect such effects. It is of
course too early to see what the final result will be here; we cannot
tell whether eventually enough new experimental kinds of behavior will
be found to restore the number of independent concepts to that of the
level of ordinary experience or not, or whether indeed it will turn out
that a greater number of concepts is required. It is contrary to our
instincts to expect a greater number, and a smaller number now seems to
us not unnatural, but the considerations of this essay should prepare us
for either possibility.
It is often said that quantum phenomena are inconsistent with ordinary
mechanics, and proofs of this assertion are often offered. I believe
that no such proof, in the spirit in which the attempt is usually made,
can be correct, for it seems to me that the remark of Poincaré applies,
namely, that any sort of behavior can be imitated by a mechanical
system, provided it is only complicated enough. A peremptory proof of
this can be given to any one who is not a believer in vitalism. If a
sentient being can be regarded as a mechanical system, we merely have to
station inside each atom a Maxwell demon, with instructions to make the
atom react according to quantum rules. Opposed to the spirit of this
sort of reduction of quantum phenomena to mechanical terms, we have to
remember that it makes sense to talk about the character of our
conceptual structure only when the number of concepts is reduced to the
number that have independent operational significance, that is, to the
minimum number.
In the meantime let us examine what may be the significance in the light
of present experiment of statements like those ascribed to Bohr that our
usual concepts of space and time may be inapplicable in dealing with
quantum phenomena. This idea is often given the more explicit form that
space and time may be essentially discontinuous at the quantum level.
From the operational point of view, it is most difficult to see exactly
what this more explicit statement means, at least in terms of those
operations by which length and time were originally defined. Thus if
space were discontinuous, it might mean that a point exists which may be
reached by laying off a meter stick fourteen times, for example, and
another point by laying off sixteen times, but that no point can be
found with fifteen applications. Such a state of affairs seems to be
inconsistent with our definition of the counting operation and to have
no concern with any properties of space; for what shall we mean by
laying off a meter stick sixteen times if it cannot be laid off fifteen
times? It is conceivable that space might end, in the sense that beyond
a certain limit there might be some irremovable physical hindrance to
the continued laying off of distances with a meter stick (although I
think that we should be inclined to describe such a state of affairs in
terms of matter enclosing empty space rather than as the end of space),
but to say that space may be discontinuous seems to be meaningless. In
the same way, I believe it meaningless to speak of discontinuous time.
We may have phenomena discontinuous in space and time, but not
discontinuous space or time.
It seems then that we must give up the idea that in the quantum domain
the usual concepts of space and time may fail, in the specific sense
that they may become discontinuous. What may we understand by the
failure of these concepts in a more general sense? No one of course
would expect that even eventually the concepts will have the same
operational significance for the inside of an atom that they have on the
ordinary scale; it must be a modified sort of concept with which we are
concerned, such as we have already seen is given by the field equations
of electrodynamics. If now the number of operationally independent
concepts on the quantum level turns out to be the same as on the level
of ordinary experience, and if there is also the possibility of
continuous transition from the operations of the quantum domain to those
of ordinary experience, then it seems to me that we should say that our
usual concepts of space and time still apply in the quantum domain. But
if the number of operationally independent concepts is either greater or
less than on the ordinary level, then I believe we must say that the
ordinary concepts of space and time cannot apply. One might still look
for the possibility of separating out from the complex of concepts on
the quantum level a group which might change continuously to those of
space and time on the ordinary level, but I think that such a
possibility is very remote when one considers that the total number of
concepts changes, and that in the zone where the number changes the
definitions are not unique by which one extrapolates a concept from one
domain to another.
If Bohr's idea is true that space and time cannot be used in describing
ultimate quantum phenomena, one of the most immediate implications in
terms of experiment might be that phenomena corresponding to
intermediate positions of the electron between stable orbits do not
exist.
Finally, we must comment on the general tactics of the quantum
situation. It would seem that there have already been a sufficient
number of unsuccessful attempts to formulate quantum behavior in terms
of ordinary mechanics to justify the expectation that ultimately
something quite different must evolve. The difficulties of an unmodified
carrying over of ordinary mechanical notions to quantum phenomena may be
illustrated by a simple example. Consider a particle of mass _m_
rotating in a frictionless circular track of radius _r_. Then according
to quantum conditions it can move stably on this track only with certain
definite velocities, such that ∫ pdq = mv 2πr = nh. Suppose now the
particle rotating with one of the allowed velocities, and a tangential
force applied. If the usual mechanical notions of force are still valid,
the particle must respond by moving in its track with continually
increasing velocity. After the velocity has been increased by a small
amount, we remove the force. The motion is now no longer one of the
allowed ones, and the particle must in some way change its velocity; it
must either slow down or speed up. In the first case it must either
radiate energy, which a system of the simple mechanical properties we
have supposed is not capable of doing, or else the law of conservation
of energy fails, and also Newton's first law of motion during the
process of acquiring the steady condition. If, on the other hand, the
particle speeds up, it must increase its energy from nowhere, and again
ordinary mechanics does not apply.
It seems then a mistake to attempt to formulate the quantum conditions
in terms of the notions of ordinary mechanics (momentum, and position
coördinates in either the ordinary or the generalized Lagrangean
sense). It would seem, on the other hand, plausible to expect that
mechanics is not a fundamental thing, but is in some way an effect
produced by the aggregate action of a great many elementary quantum
processes. Amplitude of radiational vibration, for example, may be such
a statistical aspect of a great many processes, in some such way as on
the ordinary level of experience temperature is a statistical aspect of
the average kinetic energy of the atoms. One possibility of this kind
has already been more explicitly indicated; in the elementary process of
emission of radiation, frequency and energy are not two independently
assignable variables, but are connected [E = hν]. That is, on the
quantum level radiation has only a single property, which is properly
neither energy or frequency. [We are now neglecting the polarization
aspect of radiation.] On a higher level, that of ordinary radiation, the
single elementary property has expanded itself into two (energy and
frequency) through the additional variable of the number of elementary
quantum processes in the complex radiation.
The program of the immediate future should be an extension of something
of this sort, namely, to invent new concepts corresponding to the
experimentally independent things on the quantum level (such perhaps as
the resultant of the fusion of the energy and frequency concepts for
radiation), and then to show how the ordinary concepts of mechanics (and
very likely those also of space and time) are generated by statistical
effects in aggregates of great numbers. Perhaps it is yet too early for
an attempt of this sort, because it may seem that there are still too
many possibilities of new experimental discoveries which might upset the
results of elaborate theoretical speculation. If this should really be
felt to be the case, I believe that physics ought for the present to
hold in partial abeyance its theoretical activities in this field, and
devote itself to acquiring as rapidly as possible the necessary
experimental facts. We may emphasize again that the possibility of
carrying out this plausible program can be proved only by experiment; it
may be that more concepts will be required on the quantum level than for
ordinary experience.
The invention of new concepts is certainly not an easy thing, and is
something which physics has always deliberately, and perhaps
justifiably, shirked, as shown by the persistent attempts to carry the
notions of mechanics down into the finest structure of things. This
shirking has not had bad results, but on the contrary good results, as
long as physics has been primarily concerned with phenomena near the
range of ordinary experience, but I believe that as we get farther and
farther away from ordinary experience, the invention of new concepts
will become an increasing necessity.