CORPUSCULAR ASPECT OF MATTER

Ya. I. Frenkel

1. INTRODUCTION

In a previous article on the “quantum-field theory of matter,” recently published in this journal[^1], I attempted to develop a new program for the development of quantum theory—not as a quantum mechanics of particles, but as a quantum field theory.

The purpose of the present note is to recall the circumstance that the corpuscular aspect of matter, which is revealed especially clearly in the existence of tracks of charged particles in a Wilson chamber, can be interpreted from a purely wave (or field) point of view when one considers the interaction of the principal object (a rapidly moving charged “particle”) with a multitude of other objects (the atoms of the “medium,” which are excited or ionized by this particle). In this connection, as Mott showed as early as 1929[^2], it proves necessary to describe all these “particles,” taking account of their interaction with the charged particle, in the many-dimensional configuration space corresponding to the totality of what we call the “charged particle” and the “medium.”

Among some authors there has until now been current the erroneous opinion that taking into account the interaction of the “particle” with the “medium” leads to the establishment of an uncertainty relation between the coordinates and momenta of this “particle.” In reality, as we shall now see, this relation is connected with the field aspect of that which we incorrectly call the “particles” of matter, whereas taking into account the interaction of one such “particle” with the surrounding “medium” leads to an erroneous corpuscular conception of the micro-objects of nature as particles.

2. APPARENT DIFFICULTIES OF THE FIELD REPRESENTATION OF MATTER

In considering various questions of quantum mechanics relating to the motion of one particle in a given external field, the wave function \(\psi(x, y, z, t)\) is interpreted as the amplitude of the probability that the particle is at the moment of time \(t\) at the point with coordinates \(x, y, z\); more precisely, the square of the modulus of this function, \(|\psi|^2\), multiplied by the volume element, \(dx\,dy\,dz\), represents a measure of the probability that the particle described by the \(\psi\)-waves is at the moment \(t\) in the volume element \(dx\,dy\,dz\).

This interpretation is connected with the corpuscular conception of matter: it presupposes that matter consists of separate particles and that the motion of one of these particles, in the absence of others, is controlled by the de Broglie wave, according to purely statistical laws establishing a connection between the intensity of the wave at different points of space and the probability of finding the particle at one point or another.

In contrast to this statistical point of view, which was advanced by M. Born in 1926, de Broglie and Schrödinger at first (1924–1925) tried to interpret the quantity \(|\psi|^2\) as the density of matter itself, i.e., to regard the latter as a purely wave formation. This—at first sight very attractive—point of view, which promised to reduce the corpuscular-wave dualism of matter to a monistic wave conception of it, was soon rejected in favor of the statistical or dualistic interpretation proposed by Born. The reasons for this are well known. Nevertheless, I consider it necessary to dwell here briefly on the most important of these reasons:

1) A wave packet constructed by de Broglie and Schrödinger to represent a material particle spreads out with the passage of time. This spreading, however, does not pose any threat to the existence of separate immutable particles from Born’s point of view, since it pertains not to the particle itself, but only to the spatial distribution of the probability of its localization.

2) The conception of plane sinusoidal waves of “matter” makes it possible to explain the phenomena of interference and diffraction of these waves when they pass through a crystal, whereas an analogous explanation becomes impossible if these waves are replaced by a multitude of wave packets which, under such conditions, merely break apart and split up.

In connection with the second of these considerations, one more point must be noted.

3) From the phenomena of diffraction of cathode rays it follows that the ability to interfere with one another is possessed not

“resultant” waves, but waves controlling the motion of one and the same individual electron. Thus the diffraction pattern produced by a monochromatic beam of electrons passing through a crystal differs only in its greater intensity from that which corresponds to one of these electrons.

3. THE REALITY OF THE DE BROGLIE FIELD FOR A SINGLE ISOLATED MICRO-OBJECT

Despite these facts, the question of the field nature of matter cannot, in my opinion, be considered settled in the negative sense. First of all, there are no grounds for seeking—as de Broglie himself did—the field prototype of matter in the form of a wave packet resembling a particle. Such a representation can apply only to “short-wave” macroscopic objects, but by no means to microscopic ones (to which relatively long waves correspond). If one rejects such a “quasi-corpuscular” representation, then one can quite well reconcile oneself to any constant or time-varying field structure of a micro-object that is not connected with likening it to a particle. In considering, for example, the electromagnetic field of light waves, we by no means try to reduce it to wave packets that would depict individual photons. If one rejects the corpuscularity of matter, then there are absolutely no grounds for trying to represent it in a “quasi-corpuscular” form. The quantization of the field forming this matter consists only in its correspondence to a whole number of micro-objects (quanta); the “form” of these micro-objects cannot have any essential significance, or even any physical meaning at all.*)

We thus see that we ourselves have confused the comparatively simple physical meaning of the concept of a quantized field of matter by trying to impose on it our primitive and naive notion of particles, i.e. by trying to combine in the concept of “matter” two aspects—the field and the corpuscular. This duality can appear only in the case where we are dealing with a collection of some (whole) number of micro-objects of the same kind, and only to the extent to which it is revealed in the fact of field quantization. So long, however, as we are dealing with one micro-object of a given kind, or with an indefinite number of them, grafting the concept of “particle” onto the concept of field has no rationale. In other words, corpuscular representations can appear only in the interpretation of a system

*) Thus, for example, this is the case with the form of light quanta, which may correspond to plane or spherical waves, etc.

micro-objects; so long as we operate with what is called “one micro-object,” or with an indeterminate number of them, the question of corpuscular representations should not arise in our consciousness when describing natural phenomena.

Thus, what we have said above about the superfluity of corpuscular representations in the “interpretation” of quantum “mechanics” and about the possibility of directly treating the field \(\psi\) as physical reality, using the uncertainty relation \(\Delta x \cdot \Delta k > 1\) as an immediate consequence of field representations, remains valid, notwithstanding the considerations set forth in the preceding paragraph.

A certain difficulty, from this point of view, is presented by the question of choosing a rational expression for the potential energy in one case or another; for example, the choice of the potential energy in the form \(\frac{Ze^2}{r}\) when considering the process that we call “the motion of an electron in the Coulomb field of a fixed center.” In setting about the determination of the wave function \(\psi\), which supposedly “controls” the motion of the electron in this case, we must, first of all, forget about the existence of the electron as a point charge, and solve the problem under consideration in roughly the same way as the problem of the oscillations of an electromagnetic field in a spherical endovibrator is solved, with a proper distribution of the “refractive index of the waves.” The choice of precisely this, and not some other, expression for computing this index can then be connected with the macroscopic data on the Coulomb field, which, by a happy accident, give the correct (or almost correct) result.

In exactly the same way, the tunnel effect and, in general, the penetration of the wave function \(\psi\) into a region forbidden by classical mechanics (i.e. corresponding to a negative sign of the kinetic energy) should be regarded simply as a particular case of the total internal reflection of waves when the component of the wave vector in the direction toward the corresponding surface is imaginary, and not as an occasion to embark on complicated and unconvincing arguments about the uncertainty of the velocity of a particle considered in a half-space, as is usually done.

So long as we are considering one “particle,” or an indeterminate number of particles, without taking account of their interaction with one another, our considerations should refer only to the de Broglie field. At the same time, the latter may be treated in a purely classical manner, without, however, considering it scalar, and its oscillatory character may be connected with the presence of a nonzero rest energy (or mass) in the corresponding “particles.”

4. THE DISCOVERY OF THE CORPUSCULAR ASPECT OF MATTER WHEN THE INTERACTION OF A “PARTICLE” WITH THE “MEDIUM” IS TAKEN INTO ACCOUNT

In 1929 Mott showed that, in order to interpret the tracks of rapidly moving charged particles (for example, α-particles in a Wilson chamber), it is necessary to consider these “particles” together with other neutral “particles” forming the material medium in which the motion of the ionizing particle takes place, i.e. to consider the latter together with this medium in the multidimensional configuration space of all particles, and that only under such conditions does the concept of a “track” as the visible trace of an ionizing “particle” acquire physical meaning. If, however, one describes the ionizing “particle” by itself, with the aid of the three-dimensional wave controlling it, then, as was already known earlier, the idea of its corpuscular nature and of the rectilinearity of its “trajectory” loses all meaning.

Let us recall that in the theory of light an analogous situation occurred in the past, when the principal property of light was taken to be the rectilinearity of light rays (in a homogeneous medium). To obtain such a “light ray” it was necessary to use diaphragms or screens capable of selecting and transmitting further only a small portion of the light wave. This circumstance was clarified at the beginning of the last century by Fresnel in his experiments on the diffraction of light and led to the firm establishment of the wave nature of the latter (in connection with the smallness of the wavelength of light waves). In the modern quantum theory of light the matter stands in exactly the same way. The situation has changed only in the respect that the action of light on “electrons” has acquired a quantized character, which it did not have before and which is explained if photons and electrons are considered from the field point of view in their interaction with one another, as well as with other objects.

As Mott showed, the very same thing must be said of the manifestation of the corpuscular aspect of all other micro-objects and, in particular, of the so-called “electrons” or “alpha-particles.” Here the role of screens or diaphragms, which select these “particles” and create the impression of rectilinear or curvilinear trajectories of their motion, is played by those micro-objects of the material medium through which the waves corresponding to the ionizing particles pass.

It is only necessary to remember that the transformation of these waves into “particles” does not in fact take place, but follows from the quantum nature of the waves themselves, or, more precisely, from the quantum character of the effects of their interaction with other waves (or fields), with which we associate representations of other particles.

5. REFINEMENT OF MOTT’S THEORY AND ITS PHILOSOPHICAL COMPLEMENT

Let us briefly recall the essence of Mott’s theory. Alongside the three-dimensional wave \(\Psi(\mathbf{R})\), describing the so-called “alpha-particle,” Mott considers three-dimensional waves \(\psi_s(\mathbf{r})\), describing the atoms of the medium (air) in various states \(s\), from the normal \(s=0\) to the ionized. Further, from these waves he constructs the combined “many-dimensional” wave

\[ \Psi(\mathbf{R}, \mathbf{r}_1, \mathbf{r}_2,\ldots), \]

describing the “\(\alpha\)-particle” and the “atoms” jointly with one another, and, finally, from this combined wave forms a new three-dimensional wave:

\[ f_{s_1,s_2,\ldots}(\mathbf{R}) = \int \Psi(\mathbf{R}, \mathbf{r}_1, \mathbf{r}_2,\ldots)\, \psi^{*}_{s_1}(\mathbf{r}_1)\, \psi^{*}_{s_2}(\mathbf{r}_2)\ldots d\mathbf{r}_1 d\mathbf{r}_2\ldots, \]

which determines the probability that, at the moment of time under consideration, when the “\(\alpha\)-particle” is at the point \(\mathbf{R}\), the atoms are in the quantum states \(s_1, s_2\), etc. If all \(s_i\) \((i=1,2,3,\ldots)\) are equal to zero except, say, \(s_j \ne 0\), then the function \(f\) depends on the distance of atom \(j\) from the point \(O\), at which the \(\alpha\)-ray “begins.” To explain the rectilinearity of the track of the \(\alpha\)-particle, it therefore remains to prove that the function \(f(\mathbf{R})\), for several \(s_j\) different from zero, is very small if these atoms (ionized or excited) are not situated on the straight line passing through \(O\) and \(j\).

We shall not reproduce the proof, which was given by Mott and which the reader may find in his article[^2]. For us it is important to note that the result found by Mott has a very substantial philosophical significance, which escaped the author himself, and which consists in the fact that “particles” are only a certain aspect of purely field processes, appearing when the interaction of waves of one kind with waves of another kind is taken into account (also perceived by us in the form of “particles”); moreover, these wave processes are described in a many-dimensional space connected with the aggregate of the corresponding particles.

This last circumstance appears, at first glance, to present a certain difficulty for the field theory of matter, according to which an aggregate of a large number of identical micro-objects corresponds to highly excited states of the corresponding single quantized field. Mott’s theory, however, is not difficult to transform into a purely field form, considering the atoms of the medium as quanta of the corresponding field, in analogy with how this is done,

for example, in the quantum theory of radiative friction (or of the width of spectral lines).

We cannot dwell here on a more detailed consideration of this question.

References

1. Ya. I. Frenkel, UFN, 42, 69 (1950).
2. N. F. Mott, Wave Mechanics and Nuclear Physics, p. 97, translated by K. V. Nikolsky, ONTI (1936); “The Wave Mechanics of α-Ray Tracks,” Proc. Roy. Soc. A 126, 79 (1929).