THE CURRENT STATE OF ELEMENTARY PARTICLE THEORY*

W. Heisenberg

In the first decade of the development of quantum theory, elementary particles were regarded as something given. The theory had in view above all electrons, and also atomic nuclei, treating the latter as a whole. The charge, mass, and magnetic moment of these particles were considered to be known from experiment and by no means constituted an object of theoretical consideration. Later, however, when, alongside electrons and protons, a number of other elementary particles were discovered (neutrons, μ-mesons, π-mesons, neutrinos, and many other particles reliably discovered only very recently), this situation changed completely. First, such an abundance of elementary particles indicated to physicists that the elementary particles themselves, as such, require explanation and that, second, one must at some time create a theory that would make it possible to understand and to obtain, by a purely logical route, the distribution of the masses of elementary particles, their charges, spins, and magnetic moments. In what follows we begin with a historical survey of the various attempts undertaken in the last decade to give a mathematical formulation of the regularities inherent either in individual elementary particles or in particular groups of them. The periods from the completion of the construction of quantum mechanics to the beginning of the Second World War and from the end of the war to the present will be considered separately. However, only by comparing all the work of this period can one give a more or less rigorous analysis of the present state of affairs.

It was established quite some time ago that, for the concept of an elementary particle, the starting premise is the quantization of its wave field. Einstein’s investigations of the theory of radiation had already shown that light quanta can be obtained by

* Report delivered on September 23, 1955, at the Congress of Physicists in Wiesbaden. Naturwiss. Heft 24, 637 (1955).

of quantization of the individual Fourier components of the Maxwell field. Further, soon after the emergence of quantum mechanics, Jordan, Klein, and Wigner were able to show that the successive quantization of nonrelativistic de Broglie matter waves, taking into account electrostatic interaction, strictly leads to Schrödinger’s equation for the many-body problem.

This laid the fundamental foundations for the development of the theory. Namely, one had to begin by establishing general mathematical methods for the quantization of fields, test them on known fields—the fields of radiation and matter—and then, moving from the known to the unknown, try to attain an understanding of the structure of other fields and of their elementary particles.

However, the implementation of this program immediately encountered serious difficulties. Although the mathematical apparatus of quantizing wave fields seemed simple and unambiguous, its application to the interaction of the electron field with the electromagnetic Maxwell field led to an infinite self-energy of the electrons, as well as to other divergences, which made a physical interpretation of the complete system of equations impossible. And only when, using the perturbation method, one confined oneself to the lowest approximation was it possible to obtain convergent and, consequently, physically acceptable solutions.

This first form of quantum electrodynamics brought significant successes. Dirac’s theory of radiation explained the scattering and absorption of light by atoms, as well as the spontaneous emission of light and resonance fluorescence. In the course of time it even became possible to predict such phenomena as had not even been considered in the classical theory and which belong wholly to quantum electrodynamics. These are the polarization of the vacuum around an electric charge and the scattering of light by light. If this is translated into the language of mathematics, then it is a question of nonlocal and nonlinear deviations from Maxwell’s equations. These effects were subsequently also observed experimentally. All these successes seemed to give the theory a solid mathematical foundation, although from the very beginning the existence of quantum electrodynamics required justification, since the presence in it of the aforementioned infinities made it, in the final analysis, unacceptable. Naturally, the efforts of many were directed toward clarifying the features of the mathematical apparatus that cause the appearance of infinities in the theory. Especially carefully studied was the question of whether these infinities arise because perturbation theory is used and whether they would disappear in an exact solution. However, for resolving this question, above all, there was a lack of mathematical means.

When later, alongside electrons and light quanta, other elementary particles were discovered in various phenomena,

particles—neutrons and neutrinos, μ-mesons and others—and thus new particles entered the field of view of theoretical physics; the interaction of these new particles with the old ones was considered by the mathematical methods of the quantum theory of wave fields. The theory of β-decay, proposed by Fermi in 1934, is a particularly successful example of this kind of application of the new methods.

Incidentally, in one of the later investigations of the Fermi interaction, an important classification of interactions of various types was introduced, which had great fundamental significance for the entire subsequent development of the quantum theory of wave fields. The interaction between elementary particles which collide once with a definite kinetic energy (measured in the center-of-mass system) may, as this energy increases, either grow, or decrease, or remain constant. Depending on which of these three cases takes place, the conclusions concerning the applicability of the perturbation-theory method and concerning the physical processes themselves turn out to be different. If the interaction is small and, with increasing energy, decreases or remains constant, the application of perturbation theory even at high energies gives convergent expansions. In this case, in collisions, the single production of new particles is considerably more probable than the simultaneous production of many particles—a process to which the higher approximations of perturbation theory correspond. If, however, the interaction increases with increasing energy, then at a certain energy the solution obtained according to perturbation theory begins to diverge. In this case, in the collision of two very energetic particles, several new particles may be produced at once; however, for the description of such processes perturbation theory is in principle inapplicable. The interaction between electrons and light quanta in quantum electrodynamics belongs to the first group of interactions; the interaction between electrons, neutrinos, and nucleons according to Fermi’s theory of β-decay belongs to the second. It also follows from Fermi’s theory that multiple production of electrons and neutrinos through collisions of sufficiently energetic particles is possible. The fact that multiple production of elementary particles can in reality have an appreciable magnitude was first discovered much later, namely in the study of π-meson showers, and most recently in the study of showers formed by γ-rays. However, it was already possible to establish at once that the coupling constant in the Fermi interaction in β-decay is extremely small. To explain photon and electron showers in cosmic rays, it at first proved sufficient to use the cascade theory of showers, created by Bhabha, Heitler, and Oppenheimer and Carlson, in which only the electrodynamic interaction was taken into account.

At this stage in the development of the theory of elementary particles, a many-year interruption in research in this direction occurred in many countries, an interruption caused by the Second World War. All attention was concentrated on applications of nuclear physics. Nevertheless, one should mention certain results obtained during the war years, since these results play a definite role in the theoretical considerations used at the present time. In particular, the divergences arising in the quantum theory of wave fields had already been investigated in detail. From the fact that for more than ten years after the emergence of the theory it had proved impossible to eliminate the divergences, the conclusion was drawn that the foundations of quantum theory, when applied to wave fields, required certain modifications. Meanwhile, nothing definite was known about the nature of these modifications. It was important, however, to establish which features of the earlier theories would most probably have to be preserved when such modifications were introduced. It became clear that the mathematical expression representing the asymptotic behavior of the wave function in collisions at a sufficiently large distance from the point of collision—the so-called \(S\)-matrix—must exist also in the future theory, and with properties very similar to those possessed by the \(S\)-matrix of the earlier theories. In this connection the properties of the \(S\)-matrix were studied very carefully; here special mention should be made of Møller’s work. From these investigations there followed fundamental conclusions concerning the general structure of the quantum theory of elementary particles. At the same time, solid foundations were laid for the further development of the theory. It was possible to think that the future theory of elementary particles would be mathematically formulated in such a way that the local behavior of the wave function would not be determined, but that, despite this, a unitary \(S\)-matrix could be obtained in the theory.

In the postwar period, on the one hand, significant experimental successes were achieved, bringing a large amount of new data on elementary particles: \(\pi\)-mesons, \(V\)-particles, and \(\tau\)-mesons were discovered; their masses and other properties were determined experimentally. But we shall not dwell here on the achievements of the experimentalists.

On the other hand, quite independently of this advance in experimental physics, important theoretical discoveries were made. Already the studies in quantum electrodynamics carried out in the thirties had shown that the mass and charge obtained for the electron from the original equations ultimately differ from the constants \(m_0\) and \(e_0\) contained in the original equations. The application of perturbation theory leads even to infinite values of the mass and charge, although in the original equations finite values \(m_0\) and \(e_0\) were assumed.

As far as I know, rather long ago Kramers, in passing, expressed the idea that one should start from indeterminate values \(m_0\) and \(e_0\) and then require that the final values \(m\) and \(e\) correspond to those observed experimentally. This program of renormalizing constants was first consistently carried out in Bethe’s work, completed in 1947; in that work the influence of radiation intensity on the energy levels of the hydrogen atom was studied. It was then established that, after renormalization of the electron charge and mass, the divergences mentioned above, which arise in the application of perturbation theory, no longer occur; moreover, it was shown that the back reaction of the radiation field on electrons leads directly to a displacement of the electron energy levels relative to the levels determined by Sommerfeld’s formula, something long felt by experimentalists and reliably demonstrated by Lamb and Retherford by the method of ultrashort waves. The high accuracy with which the theory predicted the shift of electron levels left no room for doubt as to the plausibility of the proposed explanation. This was a great achievement, since it also showed that quantum electrodynamics contains a considerably larger share of truth than could previously have been thought. Hopes now became especially lively that, with the aid of the renormalization process, the theory could be brought to a mathematically closed form in which no divergences at all would arise. It was soon also shown that all theories belonging to the first group admit the renormalization process, in which the interaction either decreases with increasing energy or remains constant, whereas in the other group of theories, in which the interaction grows with increasing particle energy, the renormalization process cannot be carried out. From this arose the supposition that there exist two fundamentally different forms of quantum field theory: those admitting renormalization and nonrenormalizable ones. In renormalizable theories all divergences can be avoided; in nonrenormalizable theories all divergences remain. Only renormalizable theories turn out to be mathematically noncontradictory, and only they can be used in the quantum theory of elementary particles, which is what we are discussing throughout. Therefore the theory of \(\pi\)-mesons, considering the interaction of \(\pi\)-mesons with nucleons, belongs to the first group. Indeed, experiment found that the spin of \(\pi\)-mesons is zero, and in this case renormalization is possible. However, at this point I must emphasize that the assumptions just stated, however tempting they may have seemed, are undoubtedly incorrect. Renormalizable theories cannot, generally speaking, be mathematically formulated in closed form so as to satisfy the requirements of quantum mechanics on the one hand, and, on the other, the observations of electromag-

V. HEISENBERG

... phenomena of high energies occurring in cosmic rays show that quantum electrodynamics in the region of high energies gives incorrect results, and precisely in the sense of an increasing influence of the nonrenormalizable interaction. This circumstance was discovered only very recently, but nevertheless it is natural to conduct the discussion of the further development of the theory with these latest data taken into account.

Already in the first postwar years the mathematical apparatus of quantum field theory was advanced very successfully by a series of important works by Tomonaga, Schwinger, Feynman, Dyson, and others. In this new representation the relativistic invariance of the theory could be fully used, and this made it possible to reveal the structure of the theory very clearly. The new methods also made it possible to consider rigorously the questions of the conditions under which quantum field theories of this type satisfy the causality requirements of the special theory of relativity. In this direction considerable progress was achieved by Stueckelberg and Fierz. They were able to show that a theory in which the \(S\)-matrix of the scattering wave is defined and which satisfies, in a certain sense, the conditions of quantum theory, generally speaking does not satisfy the causality requirements in the sense of the special theory of relativity. Moreover, it turned out that some additional conditions must also be observed; we shall not go into their details here. Attempts also continued to develop the so-called nonlocal theory, in which a relativistically invariant action-at-a-distance is assumed; however, up to the present time these investigations have not yielded encouraging results.

The theory of \(\pi\)-mesons and the interaction of \(\pi\)-mesons with nucleons were also considered with the aid of the new methods. True, here the results were considerably less satisfactory than in quantum electrodynamics. Whereas in quantum electrodynamics the expansion in perturbation theory is carried out essentially in powers of \(e^{2}/\hbar c\), and the expansion converges well, in the theory of \(\pi\)-mesons the corresponding coupling constant is a large quantity. Although the theory permits renormalization, the first approximation of perturbation theory is unacceptable, which points to the necessity of applying other approximate methods. I recall Wentzel’s work on strong coupling and Tomonaga’s work on intermediate coupling. The results obtained there have a very remote relation to experiment. In any case, quantitative agreement, such as there was for the displacement of the fine-structure levels in quantum electrodynamics, is out of the question here.

However, more serious objections to the unrestricted application of renormalizable theories appeared only in recent years, and from an entirely different direction. Lee succeeded in developing one case...

renormalizable quantum field theory, which had the advantage over quantum electrodynamics that the equations could be integrated without the use of perturbation theory, while the theory itself was very similar to quantum electrodynamics. The analysis of Lee’s renormalizable field theory, carried out by Källén and Pauli*), showed that, as a consequence of renormalization, the Hamiltonian of the original system is no longer Hermitian. This leads to the fact that, among the stationary states of this Hamiltonian function, there appear states called by Källén and Pauli “ghost states,” which entail that the metric in Hilbert space is no longer positive, and the \(S\)-matrix becomes nonunitary. Physically this means that one must admit negative probabilities, which is logically meaningless. In other words, the renormalizable theory as a quantum theory has no physical interpretation.

I can present this state of affairs somewhat more fully in quantum electrodynamics, although the mathematical proof given may perhaps not be entirely impeccable. If, in the original equations of quantum electrodynamics, we take as the charge the quantity \(e_0\), then after the calculations are carried out the true charge of the electron \(e\) will be some function of \(e_0\). In order to avoid various divergences, convergence in the theory is achieved by introducing a maximum momentum \(P\) (\(P\) is the cutoff momentum; it is assumed that at the end of the calculations the limiting transition \(P \to \infty\) will be performed). In this case the relation between \(e\) and \(e_0\) may be represented in the form

\[ e^2 = \frac{e_0^2}{1 + e_0^2 F(P,e_0^2)} . \]

It can be shown that

\[ F(P,e_0^2) > 0 . \]

According to the arguments of Pauli and Källén, and also according to the recent works of Landau and Tyurin, it has apparently been proved that for the function \(F\) one can indicate a lower bound in the form of the relation

\[ F(P,e_0^2) > \mathrm{const}\cdot \lg P \quad \text{for every } e_0^2 . \]

But this means that for every finite \(e_0\) the quantity \(e^2 \to 0\) as \(P \to \infty\), and this in turn means that every initial charge of finite magnitude corresponds to a true charge equal to zero; in other words, no electromagnetic interaction exists. If one introduces arbitrarily, as the quantity \(e\), the true value of the electron charge, then for finite values of \(P\) exceeding some definite value, the quantity \(e_0^2\)

*) See the following article.

becomes negative, which means that the coupling constant \(e_0\) becomes imaginary, and the Hamiltonian function non-Hermitian. As a consequence of the imaginary coupling constant there appear the already mentioned “ghost states,” which in Hilbert space form an indefinite metric. The energy of these states \(E\), if the assumptions stated are satisfied, is very high; namely \(\log(E/mc^2) \simeq 137\). Practically, at low energies these states play no role, and when a finite and not very large cutoff momentum is adopted, the “ghost states” disappear altogether. Quantum electrodynamics cannot be made a closed, relativistically invariant mathematical theory unless an arbitrary “cutoff” rule is introduced.

These important facts, which deprive the hope for the fundamental significance of renormalized theories of its basis, nevertheless have no very great practical significance for quantum electrodynamics. However, for theories with strong coupling, for example for the interaction of \(\pi\)-mesons with nucleons, they are of fundamental importance. In this case, according to meson theory, the “ghost states” lie in an energy region quite experimentally attainable, say of the order of \(1\ \text{Bev}\). In the theory of \(\pi\)-mesons one can obtain physically interpretable results only when, by introducing a relatively small cutoff momentum, the theory is modified so strongly that at low energies it only very roughly resembles reality, and at high energies is altogether far from it.

Thus, if one generalizes the results of the latest investigations, one may say that the process of renormalization is in general plainly insufficient to make quantum field theory, with its well-known divergences, a mathematically acceptable theory. Fundamental changes must be introduced into the quantum theory of wave fields in order that it may serve as a reliable basis for theories of elementary particles.

With this state of affairs it is not at all surprising that recently experiment too has been indicating the insufficiency of the old quantum electrodynamics. Thus, in recent years the observations of Schein, Kaplon, and Ritson and of the Turin group have shown that electromagnetic cascade processes of high energies, occurring in cosmic rays, behave quite differently from what would have been expected according to the old quantum electrodynamics. Showers have been observed with an unexpectedly large number of pairs produced by electrons, and multiple production caused by \(\gamma\)-rays has also occurred rather often; the point is that, as a result of a single collision, up to twenty \(\gamma\)-quanta appear. Some particles are also found which, at high energies, radiate considerably more energetically than electrons should according to the old quantum electrodynamics; however, these particles apparently behave in other respects like elec-

...trons. These experiments leave open two possibilities for interpreting the experimental data. Either these strongly radiating particles are simply electrons, and then one must conclude that at high electron energies, of the order of 10 Bev, quantum electrodynamics is invalid and that at such energies an interaction comes into play which decreases with increasing energy and, consequently, is nonrenormalizable. Or else one must assume that these strongly radiating particles are not electrons. Then they may be, for example, particles of spin one, which, as is known, at high energies radiate more energetically than electrons. But this supposition also implies, in equal measure, the existence of a nonrenormalizable electromagnetic interaction. The experimental material at our disposal does not permit a confident choice between these alternatives.

The inadequacy of renormalizable theories and the need for changes in the foundations of quantum field theory follow equally from theoretical considerations and from observations of electromagnetic showers. In this situation, bearing in mind only the difficulties just mentioned, one should once again raise the question of whether it is possible to hope that renormalizable theories will be able to describe elementary particles satisfactorily. Here a fundamental shortcoming of a general nature immediately strikes the eye. Renormalizable theories in their former form apply to a definite group of elementary particles, which are taken as given, for example to electrons and photons or to nucleons and $\pi$-mesons. How can one, starting from such data, arrive at a theory that would explain the distribution of masses and all the other properties of any elementary particles? In this connection quite vague hopes have been expressed that subsequently it would be possible to reduce the masses and interactions of all elementary particles to one gigantic Hamiltonian function, thereby bringing all elementary particles together, and in so doing to prove that the resulting equation can be made renormalizable only for certain values of the masses and coupling constants, and that in these cases convergent solutions can be obtained. In this way the real distribution of masses and the other properties would also be determined. When one recalls the entire extraordinarily rich variety of known elementary particles with their diverse properties, one immediately realizes that such a program can be realized only through a monstrous mathematical apparatus. When, by way of consolation, it is added that perhaps this special mathematics will lead us to certain simple foundations of an entirely different structure, then in any case one must agree only that these simple foundations will turn out to be by no means the renormalizable equations of any definite group of elementary particles known at the present time.

The principal task of research in the field of the theory of elementary particles should be formulated in the following way: such a theory must operate with fundamental equations written for matter in general, and not for individual elementary particles. From these fundamental equations for matter there should follow the existence of all elementary particles, say in the form of proper solutions. Hence, of course, it follows immediately that in such a theory elementary particles play approximately the same role as stationary states for a complex atomic system in quantum mechanics, and not the role that electrons had in the old theory. Further, elementary particles appear, if one may put it so, simultaneously with all their interactions; the conception of free particles without interaction becomes meaningless. All attempts to determine the mass of elementary particles from a system of equations that does not contain the interaction of the particles are doomed from the outset to failure. Whereas earlier in studies of the \(S\)-matrix, in particular in Dyson’s studies, bound states for converging and diverging waves were for the most part not taken into account, in the future, on the contrary, the interests of all researchers must be concentrated precisely on these bound states, since in a correctly formulated theory the elementary particles themselves are, so to speak, bound states. The beginnings of the development of a theory in this direction are found among Japanese authors, in particular in Nishijima. The main difficulty that arises here is to formulate some quantum field theory that contains no divergences or other mathematical contradictions, in order to consider it as an example.

Among the previous works in this direction one should mention the rather detailed investigation carried out by the author jointly with Kortel and Mitter. In this work one starts from the simple wave equation

\[ \gamma_\nu \frac{\partial \psi}{\partial x_\nu} + l^2\psi(\bar{\psi}+\psi)=0, \]

where \(\psi\) denotes the wave field of matter, but not the field of elementary particles of any definite kind. The quantization rules are changed in comparison with the quantization rules of the usual quantum field theory, so that renormalization is impossible here; ordinary quantization also leads to absurd mathematical results. These changes lead to the fact that, alongside the usual Hilbert space, there arises a “second Hilbert space” or “Hilbert space of particles” with states that are not physically realized. This second Hilbert space has significance only for virtual intermediate states and leads to the fact that the \(\delta\)-function in the permutation function on the light-

on the cone, which is the cause of all divergences, vanishes. As incoming or outgoing waves, the states of the second Hilbert space do not appear. This second Hilbert space, as was shown somewhat later, is closely connected with the space of the “ghost states” of Chew and Pauli in the renormalized theory. The states of the second Hilbert space likewise make the metric of the total Hilbert space indefinite and lead formally to negative probabilities, which cannot be interpreted from a physical point of view. Further, they are also similar to “ghost states” in that they likewise eliminate the strongest singularities in the perturbation function. However, they are not mathematically identical with the “ghost states” of Chew and Pauli, and differ from them in an essential respect. The states of the second Hilbert space have approximately the same relation to “ghost states” as that between a dipole and a pole. We cannot here go into the details of the theory; let us note, however, that this distinction has as its consequence the fact that, first, in contrast to renormalized theories, transitions from normal states to ghost dipoles do not occur in the \(S\)-matrix, and, second, that for normal states the \(S\)-matrix remains unitary. Negative probabilities have no significance for the \(S\)-matrix and do not affect the physical interpretation. This means, at least in every finite approximation, that the theory is qualitatively consistent with our present knowledge of elementary particles. Of course, it does not at all follow from this that the theory is mathematically fully sound, since it has not yet been proved that the entire set of successive approximations ultimately leads to a finite limiting value. In the lower approximations the spectrum of elementary particles obtained from the so-called simple wave equation exhibits a remarkable resemblance to the experimentally known spectrum of elementary particles. It is especially interesting that these approximations contain a significant part of electrodynamics. But this may be of a purely accidental character.

This theory had to be mentioned here only because it can give an idea of the fundamental bases of the theory of elementary particles. It is also an example of a theory in which, on the one hand, just as in ordinary quantum theory, a unitary \(S\)-matrix can be defined, but, on the other hand, the local behavior of the wave function has no physical interpretation because of the appearance of negative probabilities. The theory also shows how the mass of elementary particles can be obtained as a consequence of the interaction of matter already contained in the basic equations, and how elementary particles can, from the very beginning, be in interaction with one another.

In conclusion, the present state of elementary-particle theory may be characterized as follows. Renormalizable theories of the type of quantum electrodynamics can be very useful as an approximate description of a certain group of elementary particles and their interactions. For the formulation of an all-embracing theory of elementary particles, renormalizable theories are unsuitable. In order to obtain a general, closed theory of elementary particles, a further development of the known methods of field quantization is required. The future mathematical apparatus must yield discrete eigenvalues for the mass, and, for collision processes, a unitary and relativistically invariant \(S\)-matrix. Apparently, an essential role in the development of quantum field theory will be played by an extension of Hilbert space in the sense considered above. Such an extension may open the possibility, while preserving the unitarity of the \(S\)-matrix, of abandoning the local description by a wave function, which is in principle sufficient for the interpretation of experimental data. One may hope that in time it will be possible to obtain from a simple fundamental equation—as the proper solutions of this equation—all elementary particles.