THE FUNDAMENTS OF THEORETICAL PHYSICS
From Science, Washington, D. C. May 24,
1940.
Science is the
attempt to make the chaotic diversity of our sense-experience correspond to a
logically uniform system of thought. In this system single experiences must be
correlated with the theoretic structure in such a way that the resulting
coordination is unique and convincing.
The sense-experiences are the given
subject-matter. But the theory that shall interpret them is man-made. It is the
result of an extremely laborious process of adaptation: hypothetical, never
completely final, always subject to question and doubt.
The scientific way of forming concepts
differs from that which we use in our daily life, not basically, but merely in
the more precise definition of concepts and conclusions; more painstaking and
systematic choice of experimental material; and greater logical economy. By
this last we mean the effort to reduce all concepts and correlations to as few
as possible logically independent basic concepts and axioms.
What we call physics comprises that
group of natural sciences which base their concepts on measurements; and whose
concepts and propositions lend themselves to mathematical formulation. Its
realm is accordingly defined as that part of the sum total of our knowledge
which is capable of being expressed in mathematical terms. With the progress of
science, the realm of physics has so expanded that it seems to be limited only
by the limitations of the method itself.
The larger part of physical research is
devoted to the development of the various branches of physics, in each of which
the object is the theoretical understanding of more or less restricted fields
of experience, and in each of which the laws and concepts remain as closely as
possible related to experience. It is this department of science, with its
ever-growing specialization, which has revolutionized practical life in the
last centuries, and given birth to the possibility that man may at last be
freed from the burden of physical toil.
On the other hand, from the very
beginning there has always been present the attempt to find a unifying
theoretical basis for all these single sciences, consisting of a minimum of
concepts and fundamental relationships, from which all the concepts and
relationships of the single disciplines might be derived by logical process.
This is what we mean by the search for a foundation of the whole of physics.
The confident belief that this ultimate goal may be reached is the chief source
of the passionate devotion which has always animated the researcher. It is in
this sense that the following observations are devoted to the foundations of
physics.
From what has been said it is clear that
the word foundations in this connection does not mean something analogous in
all respects to the foundations of a building. Logically considered of course,
the various single laws of physics rest upon this foundation. But whereas a building
may be seriously damaged by a heavy storm or spring flood, yet its foundations
remain intact to science the logical foundation is always in greater peril from
new experiences or new knowledge than are the branch disciplines with their
closer experimental contacts. In the connection of the foundation with all the
single parts lies its great significance, but likewise its greatest danger in
face of any new factor. When we realize this, we are led to wonder why the so-called
revolutionary epochs of the science of physics have no more often and more
completely changed its foundation than has actually been the case.
The first attempt to lay a uniform
theoretical foundation was the work of Newton. In his system everything is
reduced to the following concepts: (1) Mass points with invariable mass; (2)
action at a distance between any pair of mass points; (3) law of motion for the
mass point. There was not, strictly speaking, any all-embracing foundation,
because an explicit law was formulated only for the actions-at-a-distance of
gravitation; while for other actions-at-a-distance nothing was established a priori except the law of equality of actio and reactio. Moreover, Newton himself fully realized that time and
space were essential elements, as physically effective factors, of his system,
if only by implication.
This Newtonian basis proved eminently
fruitful and was regarded as final up to the end of the nineteenth century. It
not only gave results for the movements of the heavenly bodies, down to the
most minute details, but also furnished a theory of the mechanics of discrete
and continuous masses, a simple explanation of the principle of the
conservation of energy and a complete and brilliant theory of heat. The
explanation of the facts of electrodynamics within the Newtonian system was
more forced; least convincing of all, from the very beginning, was the theory
of light.
It is not surprising that Newton would
not listen to a wave theory of light; for such a theory was most unsuited to
his theoretical foundation. The assumption that space was filled with a medium
consisting of material points that propagated light waves without exhibiting
any other mechanical properties must have seemed to him quite artificial. The
strongest empirical arguments for the wave nature of light, fixed speeds of
propagation, interference, diffraction, polarization were either unknown or
else not known in any well-ordered synthesis. He was justified in sticking to
his corpuscular theory of light.
During the nineteenth century the
dispute was settled in favor of the wave theory. Yet no serious doubt of the
mechanical foundation of physics arose, in the first place because nobody knew
where to find a foundation of another sort. Only slowly, under the irresistible
pressure of facts, there developed a new foundation of physics, field-physics.
From Newton's time on, the theory of
action-at-a-distance was constantly found artificial. Efforts were not lacking
to explain gravitation by a kinetic theory, that is, on the basis of collision
forces of hypothetical mass particles. But the attempts were superficial and
bore no fruit. The strange part played by space (or the inertial system) within
the mechanical foundation was also clearly recognized, and criticized with
especial clarity by Ernst Mach.
The great change was brought about by
Faraday, Maxwell, and Hertz-as a matter of fact half-unconsciously and against
their will. All three of them, throughout their lives, considered themselves
adherents of the mechanical theory. Hertz had found the simplest form of the
equations of the electro-magnetic field, and declared that any theory leading
to these equations was Maxwellian theory. Yet toward the end of his short life
he wrote a paper in which he presented as the foundation of physics a
mechanical theory freed from the force-concept.
For us, who took in Faraday's ideas so
to speak with our mother's milk, it is hard to appreciate their greatness and
audacity. Faraday must have grasped with unerring instinct the artificial
nature of all attempts to refer electromagnetic phenomena to
actions-at-a-distance between electric particles reacting on each other. How
was each single iron filing among a lot scattered on a piece of paper to know
of the single electric particles running round in a nearby conductor? All these
electric particles together seemed to create in the surrounding space a
condition which in turn produced a certain order in the filings. These spatial
states, today called fields, if their geometrical structure and interdependent
action were once rightly grasped, would, he was convinced, furnish the clue to
the mysterious electromagnetic interactions. He conceived these fields as
states of mechanical stress in a space-filling medium, similar to the states of
stress in an elastically distended body. For at that time this was the only way
one could conceive of
states that were
apparently continuously distributed in space. The peculiar type of mechanical
interpretation of these fields remained in the background-a sort of placation
of the scientific conscience in view of the mechanical tradition of Faraday's
time. With the help of these new field concepts Faraday succeeded in forming a
qualitative concept of the whole complex of electromagnetic effects discovered
by him and his predecessors. The precise formulation of the time-space laws of
those fields was the work of Maxwell. Imagine his feelings when the
differential equations he had formulated proved to him that electromagnetic
fields spread in the form of polarized waves and with the speed of light! To
few men in the world has such an experience been vouchsafed. At that thrilling
moment he surely never guessed that the riddling nature of light, apparently so
completely solved, would continue to baffle succeeding generations. Meantime,
it took physicists some decades to grasp the full significance of Maxwell's
discovery, so bold was the leap that his genius forced upon the conceptions of
his fellow-
workers. Only
after Hertz had demonstrated experimentally the existence of Maxwell's
electromagnetic waves did resistance to the new theory break down.
But if the electromagnetic field could
exist as a wave independent of the material source, then the electrostatic
interaction could no longer be explained as action-at-a-distance. And what was
true for electrical action could not be denied for gravitation. Everywhere Newton's
actions-at-a-distance gave way to fields spreading with finite velocity.
Of Newton's foundation there now
remained only the material mass points subject to the law of motion. But J. J.
Thomson pointed out that an electrically charged body in motion must, according
to Maxwell's theory, possess a magnetic field whose energy acted precisely as
does an increase of kinetic energy to the body. If, then, a part of kinetic
energy consists of field energy, might that not then be true of the whole of
the kinetic energy? Perhaps the basic property of matter, its inertia, could be
explained within the field theory? The question led to the problem of an
interpretation of matter in terms of field theory, the solution of which would
furnish an explanation of the atomic structure of matter. It was soon realized
that Maxwell's theory could not accomplish such a program. Since then many
scientists have zealously sought to complete the field theory by some
generalization that should comprise a theory of matter; but so far such efforts
have not been crowned with success. In order to construct a theory, it is not
enough to have a clear conception of the goal. One must also have a formal
point of view which will sufficiently restrict the unlimited variety of
possibilities. So far this has not been found; accordingly the field theory has
not succeeded in furnishing a foundation for the whole of physics.
For several decades most physicists
clung to the conviction that a mechanical substructure would be found for
Maxwell's theory. But the unsatisfactory results of their efforts led to
gradual acceptance of the new field concepts as irreducible fundamentals-in
other words, physicists resigned themselves to giving up the idea of a
mechanical foundation.
Thus physicists held to a field-theory
program. But it could not be called a foundation, since nobody could tell
whether a consistent field theory could ever explain on the one hand
gravitation, on the other hand the elementary components of matter. In this
state of affairs it was necessary to think of material particles as mass points
subject to Newton's laws of motion. This was the procedure of Lorentz in
creating his electron theory and the theory of the electromagnetic phenomena of
moving bodies.
Such was the point at which fundamental
conceptions had arrived at the turn of the century. Immense progress was made in
the theoretical penetration and understanding of whole groups of new phenomena;
but the establishment of a unified foundation for physics seemed remote indeed.
And this state of things has even been aggravated by subsequent developments.
The development during the present century is characterized by two theoretical
systems essentially independent of each other: the theory of relativity and the
quantum theory. The two systems do not directly contradict each other; but they
seem little adapted to fusion into one unified theory. We must briefly discuss
the basic idea of these two systems.
The theory of relativity arose out of
efforts to improve, with reference to logical economy, the foundation of
physics as it existed at the turn of the century. The so-called special or
restricted relativity theory is based on the fact that Maxwell's equations (and
thus the law of propagation of light in empty space) are converted into
equations of the same form, when they undergo Lorentz transformation. This
formal property of the Maxwell equations is supplemented by our fairly secure
empirical knowledge that the laws of physics are the same with respect to all
inertial systems. This leads to the result that the Lorentz transformation-applied
to space and time coordinates-must govern the transition from one inertial
system to any other. The content of the restricted relativity theory can
accordingly be summarized in one sentence: all natural laws must be so
conditioned that they are covariant with respect to Lorentz transformations.
From this it follows that the simultaneity of two distant events is not an
invariant concept and that the dimensions of rigid bodies and the speed of clocks
depend upon their state of motion. A further consequence was a modification of
Newton's law of motion in cases where the speed of a given body was not small
compared with the speed of light. There followed also the principle of the
equivalence of mass and energy, with the laws of conservation of mass and
energy becoming one and the same. Once it was shown that simultaneity was
relative and depended on the frame of reference, every possibility of retaining
actions-at-a-distance within the foundation of physics disappeared, since that
concept presupposed the absolute character of simultaneity (it must be possible
to state the location of the two interacting mass points "at
the same
time").
The general theory of relativity owes
its origin to the attempt to explain a fact known since Galileo's and Newton's
time but hitherto eluding all theoretical interpretation: the inertia and the
weight of a body, in themselves two entirely distinct things, are measured by
one and the same constant, the mass. From this correspondence follows that it is
impossible to discover by experiment whether a given system of coordinates is
accelerated, or whether its motion is straight and uniform and the observed
effects are due to a gravitational field (this is the equivalence principle of
the general relativity theory). It shatters the concepts of the inertial
system, as soon as gravitation enters in. It may be remarked here that the
inertial system is a weak point of the Galilean-Newtonian mechanics. For there
is presupposed a mysterious property of physical space, conditioning the kind
of coordinate.-systems for which the law of inertia and the Newtonian law of
motion hold good.
These difficulties can be avoided by the
following postulate: natural laws are to be formulated in such a way that their
form is identical for coordinate systems of any kind of states of motion. To
accomplish this is the task of the general theory of relativity. On the other
hand, we deduce from the restricted theory the existence of a Riemannian metric
within the time-space continuum, which, according to the equivalence principle,
describes both the gravitational field and the metric properties of space.
Assuming that the field equations of gravitation are of the second differential
order, the field law is clearly determined.
Aside from this result, the theory frees
field physics from the disability it suffered from, in common with the
Newtonian mechanics, of ascribing to space those independent physical
properties which heretofore had been concealed by the use of an inertial
system. But it cannot be claimed that those parts of the general relativity
theory which can today be regarded as final have furnished physics with a
complete and satisfactory foundation. In the first place, the total field appears
in it to be composed of two logically unconnected parts, the gravitational and
the electromagnetic. And in the second place, this theory, like the earlier
field theories, has not up till now supplied an explanation of the atomistic
structure of matter. This failure has probably some connection with the fact
that so far it has contributed nothing to the understanding of quantum phenomena.
To take in these phenomena, physicists have been driven to the adoption of
entirely new methods, the basic characteristics of which we shall now discuss.
In the year nineteen hundred, in the
course of a purely theoretic investigation, Max Planck made a very remarkable
discovery: the law of radiation of bodies as a function of temperature could
not be derived solely from the laws of Maxwellian electrodynamics. To arrive at
results consistent with the relevant experiments, radiation of a given
frequency had to be treated as though it consisted of energy atoms of the
individual energy hν, where h is Planck's universal constant.
During the years following, it was shown that light was everywhere produced and
absorbed in such energy quanta. In particular Niels Bohr was able largely to
understand the structure of the atom, on the assumption that atoms can have
only discrete energy values, and that the discontinuous transitions between
them are connected with the emission or absorption of such an energy quantum.
This threw some light on the fact that in their gaseous state elements and
their compounds radiate and absorb only light of certain sharply defined
frequencies. All this was quite inexplicable within the frame of the hitherto
existing theories. It was clear that at least in the field of atomistic phenomena
the character of everything that happens is determined by discrete states and
by apparently discontinuous transitions between them, Planck's constant h
playing a decisive role.
The next step was taken by de Broglie. He
asked himself how the discrete states could be understood by the aid of the
current concepts, and hit on a parallel with stationary waves, as for instance
in the case of the proper frequencies of organ pipes and strings in acoustics.
True, wave actions of the kind here required were unknown; but they could be
constructed, and their mathematical laws formulated, employing Planck's
constant h. De Broglie conceived an electron revolving about the atomic nucleus
as being connected with such a hypothetical wave train, and made intelligible
to some extent the discrete character of Bohr's "permitted" paths by
the stationary character of the corresponding waves.
Now in mechanics the motion of material
points is determined by the forces or fields of force acting upon them. Hence
it was to be expected that those fields of force would also influence de
Broglie's wave fields in an analogous way. Erwin Schrodinger showed how this influence
was to be taken into account, re-interpreting by an ingenious method certain
formulations of classical mechanics. He even succeeded in expanding the wave
mechanical theory to a point where without the introduction of any additional
hypotheses, it became applicable to any mechanical system consisting of an
arbitrary number of mass points, that is to say possessing an arbitrary number
of degrees of freedom. This was possible because a mechanical system consisting
of n mass points is mathematically equivalent to a considerable degree to one
single mass point moving in a space of 3 n dimensions.
On the basis of this theory there was
obtained a surprisingly good representation of an immense variety of facts
which otherwise appeared entirely incomprehensible. But on one point, curiously
enough, there was failure: it proved impossible to associate with these
Schrodinger waves definite motions of the mass points-and that, after all, had
been the original purpose of the whole construction.
The difficulty appeared insurmountable,
until it was overcome by Born in a way as simple as it was unexpected. The de
Broglie-Schrodinger wave fields were not to be interpreted as a mathematical
description of how an event actually takes place in time and space, though, of
course, they have reference to such an event. Rather they are a mathematical
description of what `we can actually know about the system. They serve only to
make statistical statements and predictions of the results of all measurements
which we can carry out upon the system.
Let me illustrate these general features
of quantum mechanics by means of a simple example: we shall consider a mass
point kept inside a restricted region G by forces of finite strength. If the
kinetic energy of the mass point is below a certain limit, then the mass point,
according to classical mechanics, can never leave the region G. But according
to quantum mechanics, the mass point, after a period not immediately
predictable, is able to leave the region G, in an unpredictable direction, and
escape into surrounding space. This case, according to Gamow, is a simplified
model of radioactive disintegration.
The quantum theoretical treatment of
this case is as follows: at the time to we have a Schrodinger wave system
entirely inside G. But from the time to onwards, the waves leave the interior
of G in all directions, in such a way that the amplitude of the outgoing wave
is small compared to the initial amplitude of the wave system inside G. The
further these outside waves spread, the more the amplitude of the waves inside
G diminishes, and correspondingly the intensity of the later waves issuing from
G. Only after infinite time has passed is the wave supply inside G exhausted,
while the outside wave has spread over an ever-increasing space.
But what has this wave process to do
with the first object of our interest, the particle originally enclosed in G?
To answer this question, we must imagine some arrangement which will permit us
to carry out measurements on the particle. For instance, let us imagine
somewhere in the surrounding space a screen so made that the particle sticks to
it on coming into contact with it. Then, from the intensity of the waves
hitting the screen at some point, we draw conclusions as to the probability of
the particle hitting the screen there at that time. As soon as the particle has
hit any particular point of the screen, the whole wave field loses all its physical
meaning; its only purpose was to make probability predictions as to the place
and time of the particle hitting the screen (or, for instance, its momentum at
the time when it hits the screen).
All other cases are analogous. The aim of
the theory is to determine the probability of the results of measurement upon a
system at a given time. On the other hand, it makes no attempt to give a
mathematical representation of what is actually present or goes on in space and
time. On this point the quantum theory of today differs fundamentally from all
previous theories of physics, mechanistic as well as field theories. Instead of
a model description of actual space-time events, it gives the probability
distributions for possible measurements as functions of time.
It must be admitted that the new
theoretical conception owes its origin not to any flight of fancy but to the
compelling force of the facts of experience. All attempts to represent the
particle and wave features displayed in the phenomena of light and matter, by
direct recourse to a. space-time model, have so far ended in failure. And
Heisenberg has convincingly shown, from an empirical point of view, that any
decision as to a rigorously deterministic structure of nature is definitely
ruled out, because of the atomistic structure of our experimental apparatus.
Thus it is probably out of the question that any future knowledge can compel
physics again to relinquish our present statistical theoretical foundation in
favor of a deterministic one which would deal directly with physical reality.
Logically the problem seems to offer two possibilities, between which we are in
principle given a choice. In the end the choice will be made according to which
kind of description yields the formulation of the simplest foundation,
logically speaking. At the present, we are quite without any deterministic
theory directly describing the events themselves and in consonance with the
facts.
For the time being, we have to admit
that we do not possess any general theoretical basis for physics, which can be
regarded as its logical foundation. The field theory, so far, has failed in the
molecular sphere. It is agreed on all hands that the only principle which could
serve as the basis of quantum theory would be one that constituted a
translation of the field theory into the scheme of quantum statistics. Whether
this will actually come about in a satisfactory manner, nobody can venture to
say.
Some physicists, among them myself,
cannot believe that we must abandon, actually and forever, the idea of direct
representation of physical reality in space and time; or that we must accept
the view that events in nature are analogous to a game of chance. It is open to
every man to choose the direction of his striving; and also every man may draw
comfort from Lessing's fine saying, that the search for truth is more precious
than its possession. (From A. Einstein, Ideas
and Opinions, Bonanza Books, New York, Library of Congress Catalog Card
Number; 54-6644, pp. 323 – 335)