Ladies and gentlemen, I am very much pleased to have an opportunity to talk here in Linde on my way back from New York to Kyoto, Japan.

My only regret is that I cannot talk in German. My only choice is English, which I happened to learn a few years earlier than German.

But in any case, it is not easy to speak because my native language is entirely different from any of the European languages. Now, I am going to talk about some of my works, which I have been engaged in for these few years.

Modern atomic theory has been trying to draw as complete a picture of the material world as possible

in terms of as few elementary constituents as possible. It seemed that we came closer to the goal than ever before in 1932, when the neutron was discovered.

The electrons, protons, neutrons turned out to be the only constituents of ordinary substances, just while the photons were associated with the electromagnetic field.

The positron was discovered in the same year, but this was welcomed as the confirmation of the already successful theory of the electron by Professor Dirac.

On the other hand, however, it was clear even at that time that the picture was not yet complete. There were two outstanding problems.

The one was the beta decay and the other was the nuclear process. The success of Fermi's theory of beta decay led us to accept the existence of the neutrino,

which had been postulated by Pauli. A relativistic field theory of nuclear forces led us further to another new elementary particle, and the duality of field and particle seemed to presuppose the existence of the particle,

which is nowadays called mesons, which were to be associated with the nuclear force field. One type of meson, which was later called mu meson, was discovered by Anderson and Neddermeyer in 1937,

but later turned out to have very little, if any, to do with the nuclear forces. Instead, the pi meson, which was discovered by Professor Powell in 1947,

is supposed to be the one that is mainly responsible for nuclear forces, or at least a considerable part of nuclear forces.

This appeared already to be a little too complicated to be accepted as something final. On the contrary, it turned out later to be merely the beginning of further complications.

As discussed already by Professor Powell the day before yesterday, since 1947, a great variety of unstable particles were discovered in cosmic rays one after another.

Some of them were created artificially by high-energy accelerators. It seems that more and more new particles are disclosed as we go further and further in search for the high-energy region.

Professor Powell ascribed the responsibility to me, but as a matter of fact, I merely needed one type of meson, whereas Professor Powell discovered a great number of extra particles, which I did not need.

At any rate, it seems that we are in an, so to speak, open world, in the sense that a small number of elementary particles which have been familiar to us

are not the sole elementary constituents of our world. But are merely the more stable members of a large family of elementary particles.

Of course, there is still room for the argument that most of the newly discovered unstable particles are not really elementary, but are compound systems

which consist of two or more elementary particles in the true sense. However, irrespective of whether we took such a rather conservative view

of the present situation of the theory of elementary particles, or a more radical view of accepting a great variety of particles as elementary,

all of them being elementary, we could not help asking ourselves the question, what is the elementary particle? At first sight, there is little difficulty in defining the elementary particle precisely in mathematical terms.

In relativistic quantum mechanics, which was established already by 1930, cheaply by Professor Dirac, Heisenberg, and Bowery,

the duality of wave and corpuscle is best represented by the concept of quantized fields. In classical physics, a field is defined as a function of three space variables and one time variable.

Or, one may say, it is the totality of infinitely many quantities with space-time coordinates as parameters. In quantum theory of fields, we have to replace the function defining a field

by a great number of operators in order to take into account the duality of wave and particle. So, the quantized field is the totality of infinitely many operators,

or it may have a number of components. In order that the field theory is in conformity with the principle of relativity, the field quantities must transform linearly under Lorentz transformations.

Let us call such a field which we have been using a local field in order to distinguish it from a non-local field which will be discussed later on.

Now, an elementary particle could be defined as that which is associated with an irreducible local field. A field is called irreducible if it can no longer be decomposed into parts,

each of which transforms linearly by itself under Lorentz transformations. This is the abstract mathematical representation of the fact that the particle is elementary.

It cannot be decomposed further into more elementary constituents. In this way, the so-called spin of the elementary particle is defined precisely.

For instance, the scalar field or the pseudo-scalar field which we encounter in the theory of mesons, in particular the theory of pi mesons with only one component, is associated with the particles with spin zero,

whereas the field describing the electron is called the spinor field and is associated with the particles with spin one-half.

Now, since a quantized field is a totality of operators which do not obey the ordinary laws of algebra or ordinary numbers, but they are operators which are not commutative with each other in general,

the commutation relations between these quantities are of utmost importance in quantized field theories, and the commutation relations actually determine the statistics of the corresponding assembly of particles.

One of the most attractive features of quantum theory of fields was that it enabled us to deduce the well-known relation between spin and statistics of elementary particles.

The particles with zero or integer spin obey both Einstein statistics and integer spin obey Permah-Dirac statistics. This is a well-known empirical rule which has no exception, and it could be deduced from the mathematical theory of quantized fields.

We take for granted, furthermore, that each type of elementary particle has its own unique mass. The difficulty of the present field theories arises in connection with the apparently very simple problem of the mass.

It is customary to start from this field equation for the field, in our case a quantized field,

which has a form of the well-known wave equation of the second order, so that we assume from the beginning a unique mass for the particles associated with the field.

However, in the present field theories, in the framework of the present field theories, there is no a priori reason why we choose a certain definite value for the mass.

The value of the mass is entirely arbitrary. This is certainly the drawback of the present field theory. So, what one does actually is just to equate the mass which appears in the field equation with the observed mass of the particle in question.

However, this again is objectionable simply because the particle in question is observable for the very reason that it is not free.

We have started from the field equation for a completely free particle, but if it is completely free, it will never be observed. It is observed because it interacts with other particles. So, the problem of the mass of an elementary particle cannot be separated from the

problem of interaction between particles or between quantized fields in the quantum theory of elementary particles. In the usual field theory, we assume the interaction between the fields or between

the particles to be the local interaction, or you may call it the point interaction. Since we have in mind the picture of point particles, we think that elementary particles are points without any size or internal structure,

so that they interact with each other only when they come very close to each other.

The effect of another field on the field, the second field to the first field, will be felt only at the same point. So, in the interaction, the mathematical expression for the interaction in the local field theory with the local interaction is that there is a

certain additional term in the field equation which depends on the product of a number of field quantities at the same space-time point.

This we call, let us call this the local interaction. Now, if we introduce such an interaction, the mass of the particle, which is associated with the field, say field 1, first field, is altered by a certain amount because of the interaction.

In classical electrodynamics, for instance, the energy of the electromagnetic field around a point charge was known to be infinite because the electromagnetic field very close to the point charge is very large,

so the total amount of field energy turned out to be infinitely large, but the total energy of the system, including the point charge and the surrounding field added together,

must be equal to c squared times the total mass of the system. So, already in classical electrodynamics, we encounter the serious difficulty of the infinite mass, or you may call the infinite self-energy, the energy of the field created by the particle.

The self-energies of particles turned out to be infinite again in the quantum theory of fields in general. In the case of the electron, the infinity of the self-energy, the energy of the field, electromagnetic

field surrounding the electron is still infinite, although the degree of infinity is greatly reduced than in classical electrodynamics. This difficulty was known already in 1930 when the quantum theory of fields, in particular quantum electrodynamics, was established.

So, from the viewpoint of the interaction between fields or interaction between particles,

it is not possible to define consistently the mass of the elementary particle. You could start from a certain prescribed value for the mass of a particle, then you introduce the interaction, then it will,

in general, the change in the mass will be infinitely large, so the first definition of the mass at the beginning is meaningless. So, one must admit that the precise definition of the mass of an elementary particle is impossible,

unless one may be able to get rid of the infinite self-energy some way or other. There have been great many attempts to overcome this very serious difficulty in the field theory of elementary particles.

One was the so-called mixed field theory, which was proposed during the last war by Pais and Sakata, independent. Let us take the familiar case of the electron interacting with the electromagnetic field again.

As I said, the self-energy of the electron due to the electromagnetic field produced by the electron itself is infinite. However, if we assume further that the electron interacts at the same time with some other field of appropriate kind in an appropriate manner,

we may hope that the self-energy due to the latter interaction may just counterbalance the electromagnetic self-energy of the electron so that the resultant self-energy may become finite.

This is actually the case if we choose a second field, a scalar field, with which

neutral particles with spin zero and with the mass of the order of meson masses are associated, and which interact with the electron as strongly as the electromagnetic field interacts with the electron.

Moreover, if we extend the same idea to the case of the proton, we obtain the attractive result that the mass of the proton

will be smaller by a small amount of the order of the electron mass than the neutral counterpart, which is supposed to be the neutron. In the case of the electron, the mass of the electron is undoubtedly larger

than the mass of the neutral counterpart, which is supposed to be the neutrino. This is because the mass of the proton or the neutron to start with is assumed to be larger than the mass of the particle associated with the scalar field.

This seems to give rise to a new hope of constructing a consistent field theory which

was free from the, so to speak, pathological difficulties of the appearance of the infinities, divergence difficulties, by assuming the coexistence of a number of fields, known and unknown, in such a way that the

self-energies of all those particles which were associated with these fields became finite on account of mutual compensation,

just as in the case of the combination of the electromagnetic field and the scalar fields. Such an attempt was successful to some extent, but there is little hope in arriving at the complete removal of all infinities, all divergences,

as long as we stick to the local field theories with local interactions. Namely, among the divergences which appear in the field theories, there is one which is called the

vacuum polarization, which is a little bit different from the simple divergence of the self-energy type. We do not want to go into details of the definition of the vacuum polarization and the related matter.

I can just say that this type of divergence cannot be removed by the assumption of coexistence of various types of particles. So we cannot hope that the combination, we don't think there is a happy combination of

a number of fields which will get rid of all the divergence difficulties in field theory. But in spite of this defect, however, the idea of mutual compensation is significant in indicating that the coexistence

of various fields and associated particles is not accidental, but one may be able to find cogent reason for it.

In connection with this, I'd like to mention that recent development in quantum electrodynamics by Tomo Nagashvinga and many others was really remarkable in reproducing all experimental results of unknown in connection with quantum electrodynamics.

And ambiguously and with great accuracy, as already pointed out by Professor Taylor yesterday, however, this was possible

only after replacing this theoretically infinite masses and theoretically infinite electric charge by the observed finite masses and charge. Complete justification for this replacement, we call it usually the renormalization of the mass

and the charge. The complete justification cannot be found in the theoretical framework itself. Now, in connection with this procedure of replacing the theoretically infinite mass or some

other quantities such as electric charge and maybe other physical quantities, which are to be finite but theoretically infinite, replacing this theoretically infinite quantities by the observed finite quantity.

In connection with this procedure of renormalization, various types of interactions which appear commonly in field theories such as quantum electrodynamics or method theories can be divided into two classes.

The first class includes all interactions which are called renormalizable. It is difficult to define precisely within the limit of the time of my talk to define precisely what is renormalizable interaction and what is not.

Roughly speaking, one may just say that in the case of the electrodynamics in which, for example, the electron is just regarded as a particle described by Dirac's wave equation in the particle with a spin one half without no extra interaction,

then such an interaction between the electron, the Dirac's electron and the electromagnetic field is called renormalizable because we have already seen that if you replace the masses and the electric charge of the electron,

which turned out to be infinite theoretically by the finite observed mass and electric charge, then if you start anew from that point, then you arrive at the satisfactory result, results which agree very accurately with all known experiments.

So in that sense, you may say, the interactions appearing in ordinary quantum electrodynamics are renormalizable.

However, there are other kinds of interactions which are more complicated so that the renormalization procedure fails. Or in other words, if you repeat the renormalization procedure a finite number of

times, you cannot arrive at the same results. Still some of the infinities remain. If you go further and further, if you repeat to infinity many times, this amounts to the same thing to introduce the interactions one after another more and more singular.

Or in mathematical terms, one may say that the interactions with higher and higher derivatives of the field quantities must appear one after another without any. In that case, the interaction is no longer the logarithm in general because you can simply think of

the Taylor expansion of a field quantity at a point which is at a finite distance from the origin.

Then you can expand the field quantity at a certain point which is close to a certain origin point in terms of the field quantity at origin. Then derivatives of higher degrees will appear so that if you take the interaction between two fields not just at the

same point but two points at finite distance, then this interaction can be reduced to the interaction at the same point. However, with infinity many times, with derivatives of an arbitrary degree.

So one may say that in general, even the local interaction may give rise to the non-local interaction if you repeat the procedure of renormalization, which was successful in the case of quantum electrodynamics in certain cases of major theories.

In this connection, one may raise the question, is it possible to describe atomic and nuclear phenomena in terms of renormalizable interactions alone?

In that case, then one may to some extent be satisfied with the present formalism of the quantum theory of fields without introducing such things as a non-local interaction. But the answer is very likely to be negative.

The interaction between the electron-neutrino field and the neutron-proton system, which we collectively call the nucleon.

So the nucleon field between the electron-neutrino field and the nucleon field, which appears in Fermi's theory of beta decay, is known to be a linear combination of five independent interactions.

We cannot go into further details of the mathematical formulation of theory of beta decay, but I may only say that there are five types of different interactions. Among them, the so-called tensor interaction is indispensable for accounting for

a great number of experimental results in connection with the beta decay. But unfortunately, this particular type of interaction, the tensor interaction, is not renormalizable. Even if we accept the viewpoint, which I assumed at the very beginning of the Mason's

theory, that the beta decay is not an elementary process, but can be decomposed further into two stages in which creation and annihilation of a virtual Mason, or probably some unknown kind, take place. Even if you accept such a viewpoint, still we need a renormalizable interaction.

Now, if the interaction between fields is not renormalizable in the ordinary sense, as I have said already, this amounts to the same thing, to introduce the interaction which is not just the product of field quantities at the same point.

An interaction which involves derivatives, an interaction which involves the interaction between field quantities at different points, is called non-local interaction.

The introduction of a non-local interaction in field theories can be regarded as a revival of the theory of action at distance, which was thought to be contradictory to the very notion of field in classical physics.

Because the one reason why we stick to field theories is that we could avoid the action at distance. However, in quantum theory of fields, this may not be solved because the quantized field is not

just the field, but contains already fields and particles as two aspects of the same vehicle object. The differentiation may be different in quantum theory of fields, and one may be able to introduce the non-local interaction, which reminds us

a contradiction between the theory of action at distance and field theory, but nevertheless it may be possible to do that in quantum theory. I do not want to go into details of the mathematical formulation of such a theory. One

can just say that if you introduce the non-local interaction, there are a great number of possibilities. Actually, the possibilities of the choice of such general non-local interaction is so great that at the present stage the theory

of non-local interaction is so ugly that there is no definite answer to a great many questions in connection with field theories. But there is one attractive feature which was disclosed recently by Meurer and Christenden in Copenhagen.

Some of the pathological features of the field theories, customary field theories, could be removed by introducing certain type of non-local interaction.

For instance, in the case, simple cases, of meson theory, you can get rid of the infinities in the self-energies or the masses of the nucleons, both for the nucleons and mesons interacting with each other by introducing the non-local interaction between them.

Both of them could be made finite, so you can equate the observed mass with the theoretically reduced finite mass. However, once you introduce the non-local, any kind of non-local interaction, then you cannot just

stop there because there must be some very fundamental modification of the framework of relativistic quantum theory.

This will be necessary because it is not at all clear what will become of the Schrodinger equation in quantum mechanics. Because the Schrodinger equation was necessary in order to determine the wave function or the Schrodinger function for a quantum mechanical

system at a certain instant t, at an arbitrary time instant t, in terms of the wave function at a previous instant. This was possible because the Schrodinger equation as a form of a differential equation with respect to the differentiation with respect to

time of the first order, so that if you give the initial condition of the wave function at a certain time instant, the behavior of the wave function at a later time instant is determined by integrating the Schrodinger equation under the given initial condition.

But this was possible because of the character of the interaction, which was usually assumed to be local. If you once introduced the non-local interaction, then the interaction spread out, so to speak, that the elementary particles themselves to

some extent spread out and no longer be the point particle interacting with each other only when they come to the same point. But they may interact with each other even when they are at a finite distance apart.

This is not easy to say in simple terms, the situation, but it is very likely that we shall no longer have the Schrodinger equation as such. Probably we must find some substitute for the Schrodinger equation.

We do not know yet what will be substitute, but one suggestion was made by Professor Heisenberg a number of years ago. He pointed out that it may be that in future theories it may not be possible to define the

Schrodinger function as such, but it seems likely that still we can define the quantity which is called S matrix, which gives us the relation between the initial state and the final state in the limit that by an initial state

we mean the state in a very remote past and by the final state, the state in a very remote future. Since any of the actual experiments, we only have information which refers to a very long period of time.

We have some knowledge at a certain instant and we have some other knowledge at another instant. This time interval is very large from the microscope standpoint, so probably what we need in

connection with actual observation is not the minute change from instant to instant of the physical system, but rather we may be satisfied with the knowledge of the relation, the statistical relation between the state of the

physical system at which are different by the physical system at a certain time instant and another state at another instant. The timing interval between them being very large, very long if you look from the microscope standpoint.

Such a statistical relation can be defined, characterized by the quantity which is called S matrix. The S matrix can be defined and can be calculated theoretically even if we replace the ordinary local interaction by marginal non-local interaction.

Actually, what one says in connection with the finiteness of the mass, finiteness of the self-energy, one refers to one says it in connection with the S matrix.

So, in any case, the introduction of non-local interaction gave rise to the more arbitrariness in field theories, but on the other hand, it gives us a new impetus to proceed further in this direction with the hope that we may have a more unified, more satisfactory theory of elementary particles.

The second step in this direction may be the introduction or the further generalization of the concept of the field.

We have defined the quantized field as the totality of operators characterized by the space-time points as parameters. So, at each point of the space-time, you have one field quantity. The totality of this field, infinity, many field quantities, gives us the notion of the quantized field.

And it turns out that this corresponds to the point particle. Now the question is, we know a great variety of particles with different properties, and we wonder whether we can

cope with this great variety of newly discovered particles if we confine ourselves to the relatively simple notion of field. In the local field theory, the properties of particles, of elementary

particles, are defined in connection with the properties of the quantized field. As I already mentioned, the spin and statistics and mass can be defined in mathematical terms in connection with the quantized field. But it seems that they are not sufficient to characterize the particles, elementary particles, discovered recently.

They are so different from each other. The transformation from one to another is a very complicated procedure. And we don't know yet whether the simple mathematical definition of the elementary

particles in connection with the quantized local field is sufficient for that purpose. We are afraid that probably not. Also the main effect of the present theory is that we cannot have any a priori reason

why in nature there are so many particles with different masses, which are apparently not very regular. So one of the most fundamental problems of theoretical physics today is to derive in some way or other the so-called mass spectrum of particles, which I suppose most of them are supposed to be elementary.

If a great number of particles are really elementary, there's no way to deduce the relations between masses. We just start from the arbitrary values for masses, we just equate them with

the observed masses, or later renormalize them, or a very unsatisfactory process is necessary. So it would be very nice if we have some means of some idea about the masses of the elementary particles.

Certainly the ordinary field theory was incapable of that. Even if you introduce a non-local interaction and you try very hard to obtain the finite mass, still it is not sufficient to understand why there are a number of elementary particles with such and such masses.

Because you may give theoretical interpretation in some more, so to speak, fine structure in the mass spectrum,

but it is very unlikely that, for instance, the striking difference between the mass of the electron and the mass of the mesons and the mass of the proton and the neutron are the heavier particles. It is hard to think that such great differences are only due to the difference caused by the interaction.

Substantial part of the mass of the proton and the neutron at least must be something which is intrinsic to the particle rather than due to the interaction with other fields.

So we want to have a theory which gives from the beginning the substantial part of the masses of the elementary particles. For that purpose, I have introduced the notion of the non-local field several years ago.

This is, so to speak, a mathematical formulation of an object, of an elementary object, which is elementary in the sense that it can no longer be decomposed further into more elementary constituents.

At the same time, it is so substantial that to be able to contain implicitly a great variety of particles with different masses, spins, and other intrinsic properties, one may say more visually that we have the particles with some internal structure

so that the internal structure may differ from one elementary particle from another, but the totality of all these particles with different internal structures can be described by a certain type of field which is not the ordinary local field

because in the case of the ordinary field, you cannot describe the different internal structures of the different particles. So for that reason, I introduced the notion of the non-local field. I gave up the restriction which was so common in field theories that the field quantity is a point function.

Rather, it may depend on two points in the space-time bar. It looks like to give up completely the classical notion of field.

However, since in quantum theory, the field quantity itself is no longer the ordinary number or ordinary function but rather an operator so that it may be possible to generalize the notion of the field in such a way that the field is a totality of infinite number of quantities,

which may or may not be a point function, defined as a point, or it may be something more general.

Now, I hesitate very much to go further details of the mathematical formulation of the non-local field theories. Instead, I would like to point out a number of characteristic features of non-local field theories in non-mathematical terms.

First of all, if you introduce the non-local field, this amounts to the same thing to introduce some internal degrees of freedom, which was not present in the ordinary field theory.

So you can introduce a certain type of eigenvalue problem characteristic to the internal degrees of freedom. And from that, you can suppose that the masses of the particles may be obtained as eigenvalues of a certain eigenvalue problem characteristic to the internal degrees of freedom.

So if you choose, if you make a very appropriate choice for the internal structure for the elementary particles, characterized by the non-local field,

then you may have a certain discrete spectrum, mass spectrum, as a result of the solution of a certain eigenvalue problem. This is possible, and one type of such example was already given by Max von several years ago, independent of the notion of the non-local field.

But I think that I cannot accept his approach as it stands, because there is some very serious objection to it in connection with the relativistic invariance. Instead, if we give interpretation to his model as relating to the internal structure in connection with the non-local field,

then we can understand that if you adopt a certain model for the internal structure, you can deduce the mass spectrum for the elementary particles.

The model which was adopted by Born may be called the oscillator model. We regard the internal structure as something like the oscillation inside the particle, but the oscillations must be four-dimensional.

We would like to construct a completely relativistic theory, so the vibrations themselves must be four-dimensional, not only the vibrations in space, three-dimensional space, but it must include the oscillation in space, including the time itself.

So we are dealing with a four-dimensional eigenvalue problem. But the characteristic to any four-dimensional problem, we encounter another difficulty of the appearance of degeneration. That means that to a certain eigenvalue for the mass, there are infinitely many, in general, infinitely many states.

So there can be, in general, infinitely many different types of elementary particles with the same mass. It is not easy to get rid of this difficulty.

We do not know yet whether we have to modify the simple oscillator model from the beginning by introducing some other terms, or we just start from this simple model and introduce interaction between non-local fields.

Then, as a result of interaction, this undesired degeneration may be removed. I cannot say definitely, and I have no time to discuss this rather complicated problem further. I can only say that if we go further in this direction, it may happen that a

possible formulation of a unified theory of elementary particles may be found if we are very lucky. In any case, this is just the beginning of one type of theory, and I do not think that this is the only way.

And also this is the notion of the non-local field in connection with the structure, internal structure of the elementary part. If the structure is sufficient for a purpose, probably we need something else, because we know other things which may well be explained in connection with some other notion.

And also, I must say that what we have been discussing was only one side of one feature of the field fields. It was customary to start from the notion of the free field.

At the beginning, we think that the particles are entirely from each other, then introduce interaction and see what will happen due to the interaction. But such an approach presupposes that the interaction between them is rather weak.

The change caused by the interaction between them is rather small, so we stick to the so-called weak coupling approximation. We are well aware of the limitations of such weak coupling approximations in connection with the problem of nuclear forces,

because the nuclear forces are very strong and without validity of the approximation of the weak coupling. However, unfortunately, we do not have yet any such factory relativistic theory which is free from this assumption of weak coupling.

Probably, if we take into account that the interactions are very strong, then some entirely new feature of the field theory may be revealed. But this is a very difficult mathematical problem, and no one knows what will become of that.

So we were obliged to stick to some extent to a great deal to the notion of the weak coupling. So for that very reason, our discussions are greatly limited, and for that very reason, my own work

has only one possibility of a great number of possibilities in constructing a unified theory of elementary particles.