What I am going to talk was already spoken already by Professor Heisenberg.
But unfortunately I couldn’t follow exactly what he had spoken in German.
So please excuse me, if there is too much overlapping in what I am going to say
with what was spoken already by Professor Heisenberg.
Now three years ago I had the opportunity to talk to you here on a little attempt
at the unified theory of elementary particles.
Since then it has become more likely that the new quantum numbers or new degrees of freedom
which were introduced in order to distinguish between various kinds of new and old particles,
could hardly be reconciled with the notion of quantum mechanical motion of a particle,
which was confined within three dimensional ordinary space.
Or four dimensional spacetime of spatial flare activity.
In fact, those theories, which have been successful in classifying new and old particles
and in deriving a number of selection rules related to elementary processes,
have the following general assumptions in common:
In addition to orbital and spin quantum numbers there are extra quantum numbers,
which characterise the properties and the interactions of common as well as strange particles.
You may say old and new particles.
For instance the concept of iso-spin,
which was already discussed by Professor Heisenberg and Professor Pauli and Professor Born,
the concept of isospin, which had been convenient for distinguishing between different charged states of the pi meson,
pi plus, pi zero, pi minus one, as well as between the neutron and the proton.
This concept of isospin was extended by pi’s so as to include the newly discovered unstable particles.
Such as lambda particle, theta meson and so on.
Thus the isospin quantum number was connected with the state of motion in a space, which was called omega space.
A new space, which has apparently no connection with the ordinary three-dimensional space.
And the distinction between strong and weak interactions was attributed to the even-odd rule
for the parity of states of particles in this space.
According to a further definement due to Gell-Mann and Nishijima,
it is necessary to introduce another quantum number, which is called the strangeness.
Which again does not have any relation as yet to the ordinary space.
In any case the point is as follows:
Whereas the old quantum numbers were directly connected with the state of motion in ordinary space,
the new quantum numbers are introduced, apparently at least, with no such connection.
Granted that such theories have something essentially correct in it,
which reveals a new aspect of the world of elementary particles.
Our further step is to have a deeper insight into the significance of new degrees of freedom.
So that we would be able to approach nearer to a unified theory of elementary particles.
At this point one is tempted to raise an old question once more, are these common and strange particles all elementary?
One may well be expecting the answer, namely only very few of them are really elementary.
All others being composite.
In fact, there have been various attempts at reducing the number of really elementary particles.
The earliest of such attempts was perhaps the neutrino theory of light by de Broglie.
More recently there appeared the nuclear pair theory of pi meson by Fermi and Young.
However now that we are informed of the existence of a variety of new particles,
which seems to necessitate the introduction of such a concept as strangeness,
we have to count among really elementary particles some of the strange particles for the following reason:
We expect that the strangeness quantum number of a composite particle
is the sum of strangeness quantum numbers of the constituent elementary particles.
Now all the common particles
such as the nucleon anti-nucleon and pi meson are assumed to have the strangeness quantum number zero.
While the new unstable particles such as the lambda particle,
theta meson are assumed to have the strangeness quantum number which are positive or negative integers different from zero.
Thus we have to count among elementary particles at least two kinds of strange particles
with the strangeness quantum number +1 and -1 respectively.
For instance: According to the recent proposal by Markov and Sakata,
the lambda particle and the anti-lambda particle with a strangeness quantum number -1 and +1 respectively,
are on the list of really elementary particles.
In addition to the neutron, proton, anti-neutron and anti-proton.
All of which have the strangeness zero.
Obviously such an answer immediately gives rise to another question.
What would be the primary interaction between these really elementary particles?
It is clear, that a very strong attraction between a nucleon and anti-nucleon must exist at very short distances
in order that they form a composite particle of a mass as small as that of the pi meson.
In such a case the effect of the interaction might be drastically different
from what we could infer from the usual quantum theory of fields,
which after all is based on the assumption of weak interaction.
Thus the conventional quantum theory of non-linear fields becomes a matter of more urgent needs
as emphasised by Professor Heisenberg in the preceding lecture.
Now the properties of the stationary solutions
of classical non-linear field equations are being investigated by a number of authors.
In spite of the formidable mathematical complications.
In particular just take the case, which Finkelstein has been dealing with.
He concentrated his attention to the so-called particle-like solutions.
Whereas a linear field equation, which is associated with free particles,
had brane-wave solutions extending over the whole space.
Non-linear field equations, which correspond to particles with self- or mutual interaction turned out to have,
in certain cases, stationary solutions that are concentrated in a small region of space.
Each of such solutions could be regarded as representing a particular form of the elementary particle with a definite mass,
spin, charge etc.
However, it depends on the procedure of quantisation of non-linear fields
whether such interpretation in the framework of classical field theory survives all changes from classical to quantum theory.
Now that the relativistically invariant methods of quantisation of fields has been greatly refined and developed
in connection with quantum electrodynamics and meson theory during the last decade.
Which Professor Heisenberg already discussed in detail.
Although the method was inseparably connected with the assumption of weak coupling between linear fields,
one may as well expect that some part of it could be taken over by non-linear field theories.
Actually Professor Heisenberg recently developed an ingenious method of quantisation on this line.
Thus he tried to deduce various types of particles as stationary solutions of quantized
non-linear equations for a single spinor field or maybe two such fields.
This is a further step along the lines of de Broglie, Fermi-Young and others as I have mentioned.
However in his formulation the usual Hilbert-space, with a positive-definite metric was to be so generalised,
so that the metric was no longer positive-definite.
This means, that the concept of probability in quantum mechanics cannot be applied straightforwardly.
But the introduction of the strange concept of negative probability is necessitated.
Thus it is still an open question
whether a mathematically consistent theory could be constructed in this way as discussed by Professor Pauli.
In this connection one may look for another formulation,
in which the particle-like solutions of classical field equation could remain after quantisation as something,
which characterises the shape of the particles.
There is some attempt on this line, although it is still far from being accomplished.
In any case such a method would be intimately related to field theories with so-called non-local interactions.
By a non-local interaction we mean an interaction,
which is related to the simultaneous appearance and disappearance of particles not at the same point,
but at nearby points in the four dimensional world.
This is because the form factor in the case of non-local interaction
could be determined from the particle like solutions if you quantise non-linear field equations.
It should be noted further, that these two methods may not be entirely different
from each other but the negative probability and nonlocality may turn out to be two sides of the same thing.
We know over very simple and peculiar example in quantum electrodynamics,
namely the elimination of time-like photons with which the notion of negative probability could be associated,
result in Coulomb-interaction between charged particles.
Which is a kind of nonlocal interaction.
However we do not know exactly, what would be the situation in non-linear field theories.
Now let us turn to the problem of nonlocality of the interaction of the field itself.
When I talked to you three years ago, I discussed the possibility of introducing extra degrees of freedom of motion
of an elementary particle in relation to the nonlocality of the wave field, which is associated with the particle.
Namely, the concept of the field, which had been represented by a function of a set of spacetime coordinates,
was extended so as to include the nonlocal field,
which was represented by a function of two sets of spacetime coordinates.
The ordinary local field being a limiting case, where the defendants on the relative coordinates tend to be a delta function.
The particle associated with such a field has the degrees of freedom of internal motion in addition to those of the motion of its centre of mass.
Obviously, the internal motion, as well as the motion of the centre of mass,
takes place in the same spacetime world of special relativity, namely Minkowski space.
This is an unavoidable restriction, which has its own disadvantage as well as advantage for the following reason:
On the one hand the inherent divergence difficulties of the relativistic field theories could be dealt with precisely,
because the internal and external motions are correlated with each other in the same space
in such a way that the internal motion may serve for smoothing out the divergences attached to the external motion.
On the other hand however, this gives rise in turn to a disadvantage:
Namely in the world of special relativity an invariant spacetime region
cannot be confined to a small volume around the origin.
But extends always to infinity along the light cone.
We could overcome the difficulty by considering the internal motion
as something attached rigidly to the motion of the centre of mass,
provided that the mass was different from zero.
However the situation is very complicated, when the nonlocal field in question interacts with other fields.
The internal motion as well as the external motion would be influenced by the interaction.
Or in other words: the particle would be deformed in a complicated manner.
We do not know yet how to deal with such a deformability adequately.
The problem of causality is closely connected with that of nonlocality as discussed already by Professor Heisenberg.
In order to describe the causal relationship between two events in special theory of relativity,
one is obliged to accept the sharp separation of past,
present and future from each other by the light cone in Minkowski space.
The introduction of nonlocality somewhere in the theory, however, tends to obscure this sharp distinction.
Thus it is difficult in general for a nonlocal theory
to reconcile the requirement of causality with that of special relativity.
Although it may not be impossible but certainly it is very difficult in general.
On the other hand, these requirements are both satisfied in local field theories at the cost of regularity of function
such as propagators on the light cone.
There is another point, which one has to keep in mind when one speaks of causality in field theory:
Namely one must ask oneself at the outset what phenomena are the causal laws to be applied.
Such a question has become more significant since the development of field theory in recent years,
by which the procedure of renormalisation turned out to be indispensible.
Generally speaking, the causal relationship is not to be applied directly to the primary quantities in terms
of which the fundamental natural laws are formulated, but to those secondary quantities,
which are so modified from primary quantities as to be more directly connected with observable phenomena.
One may imagine for example, that one would start from an overall spacetime picture in future theory of elementary particles
and then derive causal relationships between phenomena, which are more or less localised.
Under these circumstances it would not be useless to reconsider the whole subject from an entirely different standpoint.
These are two theories of elementary particles started from certain types of classical fields,
which were subject to quantisation afterwards.
In other words: We took up the wave aspect of the elementary particles first.
And the particle aspect subsequently.
Now one may ask the question: “Is it at all possible to reverse the order?”
It is certainly not easy to do so,
because we have thus to leave the well established spacetime structure of special relativity, at least for a moment.
And start anew from the bare fact of appearance and disappearance of various particles in nature.
One may be inclined to think that this is too far departed from the familiar quantum theory of field
to expect to result in anything fruitful.
However, such an approach does not seem very strange, if we recall this bit of Einstein’s theory of general relativity,
namely:
“In the large scale world the spacetime framework is not predetermined independently
of the distribution of matter and energy in it.”
But they are related to each other intimately”
One may well suspect, that such a future regulation of framework and content would exist also in the small-scale world.
Though in a way very much different from the one in the large-scale world.
In the usual field theories the regulation is one-sided
because the spacetime structure is fixed before we consider the wave field.
The opposite side of the mutual regulation would be the inference of the behaviour of elementary particles on the,
so to speak, fine structure of the spacetime world.
Or the small-scale world in a more general sense.
I mean the world including the omega space or some other space,
which may have some relation to the strangeness quantum number.
Such a world may be decomposed asymptotically into the Minkowski space
and the space corresponding to extra degrees of freedom such as isospin and strangeness.
In order to give a little more definitivity to this idea,
let us consider the appearance and disappearance of particles as such.
The structure of the spacetime world in the background being purposely left aside.
These particles can exist in a great variety of different ways.
They can be different from each other, either for the reason of having different masses, spins, charges etc,
or merely for the reason of existing at different places or having different values of energy and momentum.
Let us forget for the moment that such words as spin, energy, momentum etc
are already closely connected with the structure of the spacetime world.
And let us say in broader terms, that the particles can exist in a great number of different ways.
Which can be distinguished between each other by means of a set of certain quantum numbers.
Some of these quantum numbers may be discrete and others may be continuous,
but let us for brevity denote the set of quantum numbers simply by J, which can be zero or an integer.
Then the appearance and disappearance of a particle of type J
could be represented by a creation operator and an annihilation operator,
which satisfy the familiar commutation relation.
There should be some direct or indirect connection between the creation and annihilation operators
with different variants of J.
Otherwise the appearance and disappearance of great variety of particles could be entirely chaotic,
to say nothing of their ordered behaviour in the spacetime world.
Now there are two kinds of ordered behaviour of particles according to our customary way of thinking:
One is the ordered motion of particles in the narrow sense.
In classical mechanics this was represented by a trajectory in space or a world line in spacetime world.
In modern formulation of classical mechanics the corresponding terms appear at the kinetic energy part
or free part of the Lagrangian
The second kind of ordered behaviour is the possibility of transformation between different types of particles.
Such a possibility can be dealt with only in quantum mechanics
and is represented by appropriate terms in the interaction part of the Lagrangian.
Keeping this in mind, we may hope to give order to the chaotic assembly of creation and annihilation operators.
Namely we may assume the existence of a certain function of these operators,
which plays a role similar to the Lagrangian in classical quantum theory of fields.
The invariance or symmetry properties of this function,
with respect to the linear transformations of infinitely many operators,
would be connected with the more familiar symmetry properties in current field theories of elementary particles.
All this is certainly wishful thinking.
I do not at all expect an easy success of such an approach.
I suggest merely that such an approach may not be entirely useless,
since we do not know yet as to what extent the structural world of very small scale would be different
from that of the world more easily accessible to us.
It seems to me,d that the approach from the world picture,
which starts from the classical wave field and then quantises has been so much developed and so widely investigated,
that we are reaching close to the point,
where a complementary approach to the same goal would be of some help for further progress.