On the Question of a Unified Field Theory*)

Remarks on the Quantum-Field Theory of Matter

Ya. I. Frenkel

§ 1. Quantum Mechanics and the Corpuscular Nature of Matter

In modern quantum mechanics, individual particles, as well as a system of interacting particles, are described statistically in exactly the same way (by a wave function in the configuration space of the system).

Since, in this treatment, all particles, which in principle are localizable in space and time, are regarded as unchanging, such a statistical description in fact means a rejection of mechanical determinism in the determination of their motion.

Foreign physicists, developing idealistic views and basing themselves on an analysis of the role of the observer in the action of the observer upon the system under investigation, elevated this indeterminism into a fundamental principle of quantum mechanics. In recent years, in the works of Soviet theoretical physicists, attempts have been made, proceeding from materialist positions, to overcome indeterminism (in the course of events) and to reduce it to the uncertainty inherent in the quantum-mechanical determination of the initial states.

*) The study of the field as one of the forms of matter is becoming ever more topical in contemporary physics. A great deal of scientific research has been devoted to the development of field theory. Soviet theoretical physicists are also persistently working on field theory.

The editors are publishing the articles devoted to field theory by Ya. I. Frenkel, “Remarks on the Quantum-Field Theory of Matter,” and D. I. Blokhintsev, “Elementary Particles and the Field.” In them the authors, from different positions, substantiate one and the same idea of a “unified field theory of matter.” Of course, this is not yet the theory itself, but rather only an attempt to justify its necessity. Far from all even the basic facts have been considered in the light of the new ideas, and not all tendencies can be regarded as indisputable. The editors will provide the pages of the journal to those wishing to express themselves both on the problem as a whole and on individual ideas put forward by the authors of these articles. (Ed.)

I believe that the essence of the question lies not in this, but in the fact that the purely corpuscular model of the structure of matter, which we continue to use to this day, is not adequate to reality when it is a question of the “micro-world”*), since it reflects only the corpuscular aspect of matter, without taking into account its field nature.

§ 2. THE LIMITED DIVISIBILITY OF MATTER

In analyzing the question of the nature of matter, the materialist philosophers of antiquity held that macroscopic solid bodies consist, like gases, of discrete particles—atoms—which may be regarded as very small solid bodies, differing from ordinary ones only by their “continuity.” This latter circumstance did not in the least prevent one, however, from treating these small solid bodies as consisting of still smaller particles, firmly cemented to one another, i.e., from subdividing them, at least mentally, into still smaller elements, proceeding in this direction to infinity.

The physicists of the beginning of our century did not go far beyond the ancients in this question, having merely somewhat refined their ideas about “atoms.” The place of the latter was taken by electrons and nucleons, and both the former and the latter are ascribed definite dimensions and even shape; i.e., these particles (just as among the ancient Greek philosophers) are regarded as miniature solid bodies, practically incapable of further subdivision, but capable in principle of being subdivided into still smaller elements. In the nonuniform motion of the electron, and with an unchanging relative arrangement of its elements, the interaction of the latter with one another gives a resultant force of the electron’s “self-action,” i.e., of its action upon itself, which reduces, in the main, to the force of inertia. How seriously physicists took the principled divisibility of the electron is evident from the large number of attempts to ensure the stability of the electron against disintegration under the action of the electric forces of mutual repulsion between its “elements,” by means of forces of some other—non-electrical—nature capable of balancing them (Poincaré, Mie, and others).

All these attempts proceed from naïve mechanical conceptions of the nature of the basic elements of matter and of the character of their motion—conceptions copied from macroscopic objects.

*) This point of view was first set forth by the author in a public lecture at the Academy of Sciences of the USSR in 1947.

§ 3. THE LIMITEDNESS OF PURELY CORPUSCULAR REPRESENTATIONS AND THE FIELD THEORY OF MATTER

The development of quantum theory has brought to the fore the question of a decisive rejection of these mechanistic views.

First of all, experiment has shown that the fundamental, essentially metaphysical, conception of the theory of matter in classical physics—the conception of the absolute “hardness” or, more precisely, of the immutability of its basic elements (electrons and nucleons)—does not correspond to reality.

We now know that electrons are not “eternal,” but can, with conservation of the total energy, arise and disappear together with their antipodes—positrons, just as, in the absorption of light, light quanta (photons) disappear, and in its emission they arise.

We have very convincing grounds for thinking that the heavier particles—nucleons—will prove to be just as changeable when we are able to command energies above a billion volts, necessary for creating a nucleon–antinucleon pair*).

In any case, the law of conservation of substance in its former classical sense, i.e. the law of conservation of the elementary particles of which, allegedly, alone substance consists, may already now be regarded as experimentally refuted.

This circumstance should in no case be regarded as a threat to the law of conservation of matter. It is self-evident that matter is conserved together with its inherent properties—energy, mass, quantity of motion, moment of quantity of motion, etc. What is wrong is only that matter consists of immutable elementary particles. This old, purely corpuscular, metaphysical model of substance must be abandoned and replaced by new representations, based on the capacity of matter to exist not only in corpuscular form, but also in another, non-corpuscular form.

This proposition is not new. In theoretical physics it has been making its way for more than twenty years; moreover, the second form of manifestation of the nature of matter is called the wave form, or, more generally, the field form.

The waves or fields in question resemble ordinary—“classical” (for example, electromagnetic)—ones only in that they are distributed in space in a continuous manner; however, the actions of these wave fields manifest themselves in the form of discrete quantum effects, which in the simplest case we picture to ourselves as separate particles.

*) At the present time we are acquainted only with the fact of the mutual transformability of protons and neutrons in the nucleus with the simultaneous emission or absorption of an electron or positron and a neutrino.

Thus, the new quantum fields constitute a peculiar dialectical unity of space-time continuity with the discreteness or quantumness of action—a unity for which physicists have so far been unable to devise any other model than that of classical waves, on the one hand, and classical particles, on the other.

This conception is inadequate in the sense that the corpuscular and wave aspects of matter are, as it were, equivalent in it. In reality, however, as we know, the particles or quanta associated by us with the idea of a field can, under appropriate energy conditions, arise and disappear (individually or in pairs)*; whereas the field basis of matter, if it changes at all, in any case cannot disappear completely, being, apparently, the principal seat of the dynamical properties of matter, i.e. of its energy, momentum, and so on.

§ 4. ELEMENTARY PARTICLES AS QUANTA OF THE FIELD

The nature of the relation under consideration is to some extent revealed in the classical theory of the electromagnetic field. In this theory, according to Lorentz, the bearer of the dynamical properties of electrified matter is the field itself, and not the charged particles, which are usually interpreted as its “sources.”

If the sum of the “self-action,” i.e. the force that the electron exerts upon itself, and the external force that it experiences from other particles, is set equal to zero, then its motion will take place in accordance with the laws of Newtonian—or, more precisely, Einsteinian relativistic—mechanics. In this case the energy, momentum, and angular momentum that had previously been attributed to the electron prove to be properties of the electromagnetic field and satisfy conservation laws. Under such conditions it seems natural to regard electrons (and positrons) not as sources of the electromagnetic field, but as its products, leaving entirely aside the forces of their interaction or self-action and considering their displacement in space and time as the result of changes in the field, in accordance with the laws of conservation of energy, etc.

This program, outlined by Lorentz about 50 years ago, has remained unfinished to this day because of the problem of the “internal structure of the electron,” regarded as a small solid body.

Today, after half a century of development of quantum theory, we must return on a new basis to this fundamental problem and reconsider it anew.

* Correspondingly, in the case of photons and in the case of electrons (negative or positive).

With this new point of view, electrons and positrons should be regarded not in the former sense, as small solid balls, but as a special kind of “quanta” of the electromagnetic field. These quanta should not, of course, be confused with photons, which correspond to the energy of the field, and only in the wave zone at that.

The identity of all electrons (with respect to the magnitude of their charge and rest mass) appears obvious from the new point of view, since they all correspond to one and the same field. The question of subdividing each electron into infinitely small elements and of the interaction of the latter with one another loses all meaning: the quanta of electrically charged matter (with a nonzero rest mass) are likewise devoid of any internal structure, as are the quanta of light (whose rest mass is zero).

As is known, the nucleons that form complex nuclei are also associated with a certain “nucleon” field, being its quanta; moreover, the energy and momentum of this field apparently correspond to mesons—charged particles that play the same role with respect to it as photons do with respect to the electromagnetic field.

This question requires further development in connection with the existence of mesons of various masses, which apparently correspond to different states of one and the same field system.

Without delving into these still unresolved questions, we can, however, assert with complete confidence that the laws of macrophysics are so different from the laws of microphysics because the objects of the latter are not “ordinary” particles, for which the “classical” laws were formulated, but forms of matter qualitatively different from them—namely, quantized fields, i.e. fields continuous in space and time that manifest themselves in the form of discrete effects, with which the model representations of particles are associated, even if unchanging ones.

Mechanical materialism, in contrast to dialectical materialism, considers of all forms of motion (in the broad sense of the word) only one, the simplest form—mechanical motion. Until recently it seemed to many physicists that in the realm of mechanics, and in particular of quantum mechanics, this simplest form of mechanical motion and the mechanical determinism associated with it were the only possible ones. But here too the mechanistic picture of the world suffered a fiasco: microphysical processes lost their mechanical character and acquired a more general and complex character, which might be called quantum-field (or corpuscular-field).

Foreign physicists see the way out of the difficulties connected with the old purely mechanistic conception of matter only in rela-

…rejection of determinism and causality, i.e., ultimately, in the rejection of materialism. In contrast to them, Soviet physicists must seek the solution of the question in the further development of theoretical physics on the basis of the philosophy of dialectical materialism. The principal direction for further work, in the author’s opinion, is the construction of a monistic field theory of matter.

§ 5. FUNDAMENTALS OF A MONISTIC FIELD THEORY OF MATTER

In the presently existing variants of the field theory of matter, the latter is regarded as a combination of quantized fields, on the one hand, and particles, on the other, with the energy of the entire system being associated partly with the field, partly with the particles, and, finally, with the interaction of the field and the particles. This dualistic interpretation leads to the preservation of all those fundamental difficulties which are characteristic of a purely corpuscular theory, giving rise, in particular, to divergent or indeterminate expressions for energy, mass, etc. I believe that these difficulties are rooted in the purely corpuscular image attributed to substance, considered only as an aggregate of particles, and that they can be eliminated only by a complete and decisive rejection of limited corpuscular conceptions. The conception of particles as quanta of the field does not require an independent foundation: its foundation must be the conception of a quantized field.

At the same time, the field itself is considered in two forms of its manifestation, whereas the concept of particles in the classical sense of the word does not appear at all. This means that all the dynamical properties of matter which were formerly attributed to particles and are still associated with particles, such as, for example, mass, energy, quantity of motion, and so on, pass wholly and completely to the field.

Under such conditions, the very name of the physical doctrine of matter and motion must be changed in such a way that the word “mechanics” is entirely driven out of it and the currently accepted name “quantum mechanics” is replaced by the name quantum-field theory.

I believe that the outlines of this theory have been correctly sketched by me, and that the knowledge of the true nature of matter lies along this nonmechanical path.

Along this same path the question posed above concerning the overcoming of indeterminism is also resolved: quantum field theory is strictly deterministic. Apparent indeterminism appears in quantum theory only when and insofar as we attempt to reconcile with one another the field and corpuscular descriptions of phenomena. In quantum-field theory such a comparison proves possible only on a purely statistical…

basis, i.e., on the basis of probability theory. This circumstance, however, in no way affects the determinism of physical phenomena considered from the quantum-field point of view.

§ 6. Radicalization of the Corpuscular Theory in Accordance with the Quantum-Field Theory

In concluding this article I should like to give a brief summary of two papers that I published earlier and that were devoted to certain features of the corpuscular treatment of matter in connection with the theory of relativity, but without taking field theory into account*).

From the purely corpuscular (and deliberately “incorrect”) point of view, it seems natural to radicalize still further the departures of quantum theory from the classical conceptions of the motion of unchanging material particles.

Since elementary particles can disappear and reappear, it seems natural to consider processes in which they are, apparently, conserved as “regenerative,” in the sense that the motion of a particle in space can be described as its disappearance at the initial place and its appearance at another place, more or less close to the initial one. Such an interpretation follows immediately from the consideration of the motion of the electron in Dirac’s relativistic theory.

As Schrödinger showed, when considering the velocity of an electron moving by inertia, alongside the constant component of the velocity there is found a component with frequency \(2m_0c^2/h\), which Schrödinger connects with the transition of the electron from states of positive energy to states of negative energy.

If the nonexistent states of negative energy are replaced by positrons, then this result can be interpreted as the appearance, next to the electron under consideration, of an electron–positron pair, with the positron mutually annihilating with the original electron, which is thereby replaced by a new electron in a new position. In the case of particles with finite rest mass, such “regeneration” of the particle must be observed at an infinitely nearby position, in accordance with the classical conception of the continuity of motion; in the case of photons, however, which have no rest mass, the concept of the particle’s “trajectory” loses all meaning.

*) Relativistic Quantum Mechanics of Composite Particles, ZhETF, 1947; — Theory of the Motion of Particles in Relativistic Quantum Mechanics. DAN, 1943.