ELEMENTARY PARTICLES AND FIELD

D. I. Blokhintsev

§ 1. INTRODUCTION

Physicists of the last century regarded elementary, simplest particles*) as material points moving along trajectories according to the laws of classical mechanics.

At the same time, in that same century a new physical concept arose—the concept of a field as a certain continuous entity by means of which the interaction of particles is brought about.

The most characteristic feature of any physical field is the circumstance that, in order to specify the state of a physical field, one must specify the values of the field at all points of space, i.e., an infinitely large number of quantities. In other words, unlike systems of material points, which possess a finite number of degrees of freedom, a field is a system possessing an infinitely large number of degrees of freedom.

The very concept of the field, still within the framework of pre-quantum physics, underwent a significant evolution from the mechanical ether to the Einstein electromagnetic field, containing a complete renunciation of the application of mechanics to the field—the electromagnetic ether.

As is known, this evolution in the conception of the field led to the downfall of mechanistic physics, and idealist philosophers wanted to interpret this situation as proof of the “disappearance of matter.” V. I. Lenin showed the philosophical untenability of this attack on materialism, explaining that electromagnetic mass is just as material as “mechanical” mass.

*) By elementary particles we shall mean those particles which, at the given stage of development of physical knowledge, are regarded as the simplest.

In this connection, the fact is very interesting that, according to the modern conception, the field acquires a number of features characteristic of a medium, of matter (for example, such phenomena as the polarization of the “vacuum,” as zero-point oscillations, are essentially phenomena well known in solid bodies). On the other hand, particles acquire features that are inherent in the field in its classical understanding.

The opposition between field and particle which was characteristic of the last century and which was used by idealists for the “refutation” of materialism is becoming less and less well founded. The seemingly impassable boundary between field and particles, as our knowledge develops, becomes less and less perceptible.

Such is the dialectic of development.

The problem of the interrelation of the field and elementary particles in the light of modern quantum physics will be the main subject of the present article; the range of questions belonging here has long been the subject of the most difficult and profound investigations in the field of theoretical physics.

Even in the classical electron theory there was an effort to achieve unity in the understanding of particles and the field (we have in mind Lorentz’s doctrine of the electromagnetic origin of the mass of electrons), but nevertheless this problem has not yet been resolved completely satisfactorily to this day. Nonetheless, modern theory and experiment make it possible to consider this problem in a new light, which could not have been foreseen by classical physics.

§ 2. WHAT DOES QUANTUM MECHANICS SUGGEST ABOUT THE NATURE OF PARTICLES?

We shall begin our excursion into the just outlined circle of problems by considering those peculiarities in the motion of particles which were discovered by nonrelativistic quantum mechanics.

Usually the exposition of quantum mechanics begins with the introduction of the wave function $\Psi$, which is a function of the coordinates of the particles $x_1, x_2,\ldots, x_n$ (for a system of $n$ particles) and of time $t$. In doing so, no statements are usually made about the very nature of the particles, so that involuntarily there arises the (incorrect) idea that the microparticles*) considered in quantum theory are essentially the same particles as in classical mechanics, and that only the law of their motion proves to be different (wave-like instead of ray-like).

*) This term was adopted in our course of quantum mechanics in order to emphasize the distinction between the particles considered in quantum mechanics and the material points of classical mechanics.

In general, quantum mechanics is still so closely connected with the classical mechanics of a system of material points that even now many physicists analyze its conclusions from the standpoint of classical atomism and, along this path, constantly encounter one apparent paradox after another.

At the same time, a very important feature of nonrelativistic quantum mechanics remains unnoticed: by rejecting the corpuscular laws of motion of particles, it takes only the first step toward establishing a connection between field and particle in a direction wholly unforeseen by classical physics, and thereby prepares the ground for a new understanding of particles.

In this respect the most important thing is the proof of the fact that any strengthening of the localization of a microparticle is connected with a substantial increase in momentum.

This fact is completely foreign to the corpuscular conception of the motion of a particle. In quantum theory a particle acquires the features of a nonlocalized object.

Fig. 1.

This nonlocalization of the particle can be illustrated especially clearly by the example of the behavior of a particle when the volume in which it is enclosed is compressed.

In Fig. 1 a volume is shown whose linear dimension is equal to \(L_0\). Let a particle be inside this volume. We shall compress this volume. Let us first consider this process from the point of view of classical mechanics. Suppose that, before the volume is compressed, the particle was at rest at its center. We assume no forces between the particle and the walls. Then it is obvious that no work will be done in compressing the box from \(L_0\) to \(L\).

Quantum mechanics arrives at an entirely different result: the particle inside the box has the smallest (“zero”) energy, equal to \(\dfrac{h^2\pi^2}{2mL_0^2}\) (where \(m\) is the mass of the particle), and in compressing the box work will be done

\[ A=\frac{h^2\pi^2}{2m}\left(\frac{1}{L^2}-\frac{1}{L_0^2}\right)>0. \]

In Fig. 1 the dashed line shows the graph of the particle’s wave function for both cases (the uncompressed and compressed boxes). From the point of view of quantum mechanics, the particle “feels” the presence of the walls of the box—it is nonlocalized, whereas for a classical particle at rest the position of the walls is immaterial.

This same nonlocalization can be illustrated by the so-called “Ehrenfest theorem.”

Let \(x\) characterize the position of the center of the wave packet, and \(\Delta x^2\), \(\Delta x^3\), etc.—the quadratic width of this packet, the cubic width, etc. Then, if the particle moves in a force field characterized by the potential energy \(U(x)\), then according to Ehrenfest’s theorem:

\[ m \frac{d^2 \bar{x}}{dt^2} = -\frac{\partial U(\bar{x})}{\partial x} - \frac{1}{3!} \frac{\partial^3 U(\bar{x})}{\partial x^3} \Delta x^2 -\ldots \]

The first two terms of this equation coincide with Newton’s equation for the motion of a material point along a trajectory. The following terms, containing higher derivatives, indicate that the motion of a quantum particle is determined by the entire form of the force field. In classical mechanics, however, only the field in those places through which the particle’s trajectory passes is important. This behavior of a quantum particle is explained by Fig. 2. In this figure two potential curves are shown: \(abc\) and \(abd\). A classical, localized particle having energy \(E\) will move on the segment \(x_1x_2\), and for it it is completely irrelevant which of the potential curves, \(abc\) or \(abd\), is actually realized (since these curves diverge outside the segment \(x_1x_2\)). On the contrary, the behavior of a quantum particle will be completely different in the two cases: for it, owing to its nonlocalization, the behavior of \(U(x)\) throughout all space is important. In the case of the curve \(abd\), the particle will be localized in the region around the segment \(x_1x_2\); in the case of the curve \(abc\), it will “spread out” through all space, passing through the barrier existing between \(b\) and \(c\).

Fig. 2.

Fig. 3.

Thus, the nonlocalization of particles in quantum mechanics manifests itself in the influence of the entire field in all space on the motion of a quantum particle*).

The second important feature of quantum particles is their identity (sometimes it is infelicitously called the “indistinguishability” of particles). Thus, if one depicts schematically some state of a particle by a square, and the particle itself by a letter, then the two states (\(I\) and \(II\)) shown in Fig. 3 are physically identical (if both particles \(a\) and \(b\) are of the same “kind”).

*) When we say “all space,” what is actually meant is physical infinity, i.e. “all space” may be very small, for example, the space around an atom \(\simeq 10^{-8}\) cm.

This property is also paradoxical from the corpuscular point of view, but is quite comprehensible and natural from the positions that will be set forth below.

For the moment we shall confine ourselves to pointing out only that this identity is closely connected with nonlocalizability. For example, if the two states in question are two different localizations of particles \(a\) and \(b\), then in classical mechanics, by virtue of the motion of particles along trajectories, both attributes ascribed to the particles (their initial positions \(1\) and \(2\)) are preserved at all times (Fig. 4).

Fig. 4.          Fig. 5.

In quantum mechanics, however, the packets initially constructed around \(1\) and \(2\) will spread out; the particles become confused as to their attributes. This spreading of the packets is shown in Fig. 4 by the hatched region.

These two most important properties of particles established by quantum mechanics—nonlocalizability and the identity of particles—become completely clear from the point of view of the concept of a quantized field, to the exposition of which we shall now turn.

§ 3. PARTICLES AS QUANTUM EXCITATIONS OF HARMONIC OSCILLATIONS OF A FIELD

Whereas classical mechanics, as well as nonrelativistic quantum mechanics, deals with systems of a finite number of degrees of freedom, in processes occurring at high energies the number of degrees of freedom becomes variable and unlimited.

A well-studied example of such processes is the soft component of cosmic rays.

Figure 5 shows a diagram of a developing shower consisting of electrons, positrons, and \(\gamma\)-quanta. As the primary particle an electron \(e^{-}\) is assumed. When this electron is decelerated, a \(\gamma\)-quantum arises (and a scattered electron). This quantum then turns into a positron–electron pair \((e^{+})\)—electron \((e^{-})\). Each

a particle of this pair in turn gives rise to a γ-quantum, the latter turn into pairs, etc.

In the process shown in Fig. 5 the number of degrees of freedom has increased from 4 (for the initial electron) to \(4 \times 5 = 20\). The number of particles arising in this process is limited, in the final analysis, only by the energy of the primary particle, and can be made arbitrarily large if the energy of the primary particle is sufficiently great.

Thus, we have here to do with phenomena that cannot be considered in the language of mechanics, whether classical or quantum mechanics: particles multiply and, in principle, multiply without bound.

This compels us to choose an entirely different aspect in understanding the nature of microparticles, one based not on the mechanics of a system of material points, but on field theory. The fact of particle multiplication then no longer, from the very beginning, comes into contradiction with the nature of the field, since the latter by its very essence possesses an unlimited number of degrees of freedom.

From what follows it will be seen that from this “field” point of view particles in different states should be regarded as different kinds of excitation of the field.

Mathematically, a theory of particles proceeding from the field as the fundamental entity may be formulated as follows.

Initially we shall assume the existence of a certain field \(\psi(x,t)\), leaving the question of the existence of particles entirely open.*)

The field \(\psi(x,t)\) can be decomposed into a spectrum, i.e. represented as a superposition of fields (normal oscillations), each of which has a definite frequency of oscillation \(\omega_s\). Let us denote by \(\psi_s(x)\) one of these oscillations, and by \(q_s\) its amplitude. Then the spectral decomposition of the field \(\psi(x,t)\) into normal oscillations may be written in the form:

\[ \psi(x,t)=\sum_s q_s \psi_s(x), \tag{1} \]

where \(s\) is the number of the normal oscillation (if one draws the analogy with the oscillations of a string, then the oscillation with \(s=1\) corresponds to the fundamental tone, \(s=2\) to the first overtone, \(s=3\) to the second overtone, etc.).

*) The field \(\psi(x,t)\) may have a different number of components, transforming linearly under Lorentz transformations. Thus, for the electromagnetic field we have four components of the vector potential \((A_1,A_2,A_3,A_4)\), of which only three are independent (by virtue of the transversality of the field). The positron-electron field \(\psi\) also has four components (the so-called bispinor). A scalar field has only one component, and so on; the field \(\psi(x,t)\), of course, should not be confused with the wave function \(\psi\).

The possibility of the spectral expansion (1) is based on the linearity of the field equations (i.e., it is assumed that the field equations contain \(\psi\) and derivatives of \(\psi\) only to the first degree).

By the very meaning of a normal oscillation, the dynamical variables \(q_s\) (amplitudes) satisfy the equation for an oscillator of frequency \(\omega_s\):

\[ \ddot q_s+\omega_s^2 q_s=0. \tag{2} \]

If plane waves \(\psi_s(\mathbf{x})\sim e^{i\mathbf{k}_s\mathbf{x}}\) are chosen as the normal oscillations \(\left(\mathbf{k}_s\right.\) is the wave vector, \(k_s=\frac{2\pi}{\lambda_s}\), \(\lambda_s\) is the wavelength), then the dependence of \(\omega_s\) on \(k_s\) gives the law of wave dispersion:

\[ \omega_s=\omega(k_s). \tag{3} \]

The phase of the wave will be equal to \(\pm \omega_s t \pm \mathbf{k}_s\mathbf{x}\). It is a number having one and the same value in all reference frames (an invariant), and, consequently, \(\omega_s\) and \(\mathbf{k}_s\) must form a four-dimensional vector, i.e.

\[ \frac{\omega_s^2}{c^2}-k_s^2=\chi^2, \tag{4} \]

where \(\chi^2\) is an invariant (\(\chi\) has the dimension of inverse length). Further, in order that the group velocity of the waves be less than the velocity of light, it is necessary that \(\chi^2\) be greater than zero. Thus the form of the dispersion law is determined from the simplest requirements of invariance of the wave phase.

The energy of each normal oscillation, in accordance with (2), may be written in the form:

\[ E_s=\frac{1}{2}\left(\dot q_s^{\,2}+\omega_s^2 q_s^2\right), \tag{5} \]

and the energy of the whole field \(E\) is equal to the sum of the energies of the individual normal oscillations:

\[ E=\sum_s E_s . \tag{6} \]

Using the field equations (which we do not write out), we can also find the momentum \(\mathbf{G}_s\) of the \(s\)-th normal oscillation. It is evident that it must be directed along the direction of propagation of the wave (i.e., along \(\mathbf{k}_s\)) and must be proportional to the energy of the wave. It can be shown that the momentum of the whole field may be calculated by the formula:

\[ \mathbf{G}=\sum_s \frac{\mathbf{k}_s}{\omega_s} E_s . \tag{7} \]

Such a representation of the energy \(E\) and momentum of the field \(\mathbf{G}\) can be carried out for any linear field. It has turned out that, by means of (1), (6), and (7), any motion of the field \(\psi(\mathbf{x}, t)\) is reduced to an aggregate of oscillators—normal modes.

As we see, so far there has been no mention whatever of particles. Particles turn out to be a quantum phenomenon. Indeed, if we assume that the oscillators of the field obey the laws of quantum mechanics, then we arrive at the existence of particles*).

Indeed, for a quantum oscillator the energy \(E_s\) assumes only discrete values:

\[ E_s = h\omega_s\left(N_s+\frac{1}{2}\right) \tag{8} \]

\(N_s = 0, 1, 2,\ldots\), and \(\dfrac{h\omega_s}{2}\) is the “zero-point” (smallest) energy of the oscillator. Then from (6) and (7) it follows that:

\[ E=\sum_s N_s h\omega_s + E_0, \tag{9} \]

\[ \mathbf{G}=\sum_s N_s h\mathbf{k}_s . \tag{10} \]

Thus, the energy and momentum of the quantized field change discretely (by \(\pm h\omega_s\) and \(\pm h\mathbf{k}_s\), respectively). If all \(N_s=0\), then the field is unexcited. In the classical theory an unexcited field \((q_s=q_s=0)\) would simply mean the absence of any field whatsoever.

In quantum field theory this is not so: if all \(N_s=0\) (there are no light quanta), nevertheless the zero-point energy of the field \(E_0\) is not equal to zero. Modern theory leads to the value \(E_0=\infty\). This is undoubtedly a defect of the theory, and one may think that in a more advanced theory \(E_0\) must turn out to be a finite quantity. However, more important than the numerical value of \(E_0\) is the circumstance that for \(N_s=0\) there exist zero-point oscillations of the field (fluctuations), whose real existence was recently discovered from the displacement of levels in the hydrogen atom**). As it turned out, this dis—

*) The assumption of field quantization was originally substantiated as applied to the electromagnetic field—the well-known Planck law for the distribution of energy in the spectrum of black radiation can be obtained only for a quantized field.

In modern theory the idea of field quantization is extended to any fields, including the positron-electron, meson, and nucleon fields. In this connection, for fields possessing half-integral mechanical angular momentum, the quantization differs somewhat from the quantization of fields with integral angular momentum. Namely, for fields of the first type the possible values of \(N_s\) are limited to two: 0 or 1 (the Pauli principle).

**) See, for example, the review by Ya. A. Smorodinsky \(^{1}\).

is due to the fact that the electron performs Brownian motion in the field of zero oscillations.

An analogous state of affairs also obtains in the case of the positron-electron field. According to modern theory, in the absence of positrons and electrons there exists a “background”—the aggregate of electrons filling all levels of negative energy \((E_s=\hbar\omega_s<0)\). This unexcited state of the positron-electron field corresponds to \(N_s=0\) for \(E_s>0\) and \(N_s=1\) for \(E_s<0\).

However imperfect and preliminary this conception of the “background” may be—which also has infinite (but negative) energy, just as the zero energy of the electromagnetic field does—nevertheless its existence is expressed in the remarkable phenomenon of the scattering of light by light, caused by fluctuations of the electric charge of the background. Although direct experimental proof of the scattering of light by light has still not been given, nevertheless the existence of fluctuations of the electric charge of the background may be regarded as proved by the fact that the spin of the electron must perform Brownian motion in the field of these fluctuations. This motion leads to the experimentally proven change in the ratio of the magnetic and mechanical moments of the electron, in comparison with that predicted from Dirac’s equation \(^{1,2}\).

The existence of zero oscillations of the electromagnetic field and of polarization oscillations (fluctuations) of the positron-electron field leads to the conclusion that the field exists constantly, and in this sense there is no “emptiness.” Therefore this “emptiness” is now justly called by the more cautious word “vacuum.” As we see, the “vacuum” possesses physical properties, and moreover such as are well known to us from phenomena in solid bodies: zero oscillations and polarization.

As for particles, their existence is already connected with an excitation of the field relative to the lowest possible level. Let, for example, all the numbers \(N_{s'}=0\), except for one \(N_s=1\). Then the excitation energy of the field is \(\varepsilon=E-E_0=\hbar\omega_s\), and the excitation momentum is \(\mathbf p=\mathbf G=\hbar\mathbf k_s\).

From the dispersion law (4) it then follows that

\[ \frac{\varepsilon^2}{c^2}-p^2=\hbar^2\varkappa^2=m_0^2 c^2, \tag{11} \]

i.e. the energy \(\varepsilon\) and momentum \(\mathbf p\) of the excitation are related to each other as the energy and momentum of a particle possessing rest mass \(m_0=\dfrac{\hbar\varkappa}{c}\). Thus, excitation of the field is equivalent to the appearance of particles, and the particles themselves are nothing other than excitation of the field.

If the field \(\psi(\mathbf x,t)\) interacts with something, then the character of its excitation may change.

Consider, for example, the interaction of the electron field with an atom. Let the initial state of the electron field correspond to the presence of one electron with energy \(\varepsilon\) and momentum \(\mathbf p\) (from the point of view being set forth, the atom itself should also be regarded as a formation from other fields, but we shall retain the usual terminology for the atom). Then, after interaction with the atom, the state of the electron will change and its energy and momentum will be \(\varepsilon'\), \(\mathbf p'\). Such a process is a process of scattering of an electron; in this case we speak of an inelastic or (when \(\varepsilon'=\varepsilon\)) an elastic collision of the electron; moreover, we are almost ready to ascribe to the electron a trajectory that it traversed in the process of changing \((\varepsilon,\mathbf p)\) into \((\varepsilon',\mathbf p')\). The approximate and incomplete character of such an interpretation is especially clearly visible in those cases when, as a result of the “collision,” pairs of positrons and electrons arise, so that the electron \(\varepsilon_-,\mathbf p\) is transformed into the pairs \((\varepsilon'_-,\mathbf p'_-)\), \((\varepsilon''_+,\mathbf p''_+)\), \((\varepsilon'''_-,\mathbf p'''_-),\ldots\), and so on. (Here the sign \(-\) corresponds to an electron, and the sign \(+\) to a positron.) In this case we are completely unable to distinguish which of the electrons after the impact is “the same one” that was primary.

Meanwhile, from the field point of view we are dealing here simply with a new character of excitation of the electron field, and it is clear that to look here for the primary electron is just as meaningless as, for example, to try to recognize heat by the sign of its origin, if a body has received it from various sources.

If particles are regarded as excited states of a field, then it is clear that the question of the localization of a particle in space cannot have any a priori answer. The answer to this question will depend in an essential way both on the nature of the field whose excitation we are considering and on the form of the excitation itself. These excitations may be localized in a small region of space or, on the contrary, may be distributed over a considerable volume.

Let us examine this aspect of the matter in more detail, first using photons as an example. Let the energy of the field be \(E=\hbar\omega_s+E_0\), so that there is only one photon with frequency \(\omega_s\). The field corresponding to this excitation has the form:

\[ A(\mathbf x)=\sum_s' q_s e^{i\mathbf k_s\mathbf x}, \tag{12} \]

where the sum \(\sum_s'\) extends over all oscillations differing in direction \(\mathbf k_s\), but having one and the same frequency \(\omega_s\) \(\left(k_s=\frac{\omega_s}{c}\right)\). Such a “packet” inevitably has dimensions \(\Delta x\) greater

wavelength \(\left(\lambda_s=\dfrac{2\pi}{k_s}\right)\). It is not difficult to show that for a spherically symmetric field \(A(\mathbf{x})=A(r)\) we obtain

\[ A(r)\simeq \frac{\sin k_s r}{k_s r}. \]

Although such a field is concentrated mainly in the region \(0<r<\dfrac{\lambda_s}{2}\), it nevertheless decreases very slowly with distance from the center of localization \(r=0\). A field of this kind represents the greatest possible localization of one photon. However, one may also consider the case of a strongly localized field, for example \(A(\mathbf{x})=\delta(\mathbf{x})\), \(\delta=\infty\) for \(\mathbf{x}=0\) and \(\delta=0\) for \(\mathbf{x}\ne 0\). Such an extremely concentrated field can no longer be represented by the excitation of a normal oscillation with frequency \(\omega_s\). It necessarily contains oscillations of all frequencies from \(\omega_s=0\) to \(\omega_s=\infty\) and, consequently, corresponds not to one, but to an aggregate of photons.

A similar state of affairs also obtains in the case of particles having a rest mass \(m_0\), for example electrons and positrons. Namely, localization of an electron in a small volume is possible only when the electron momentum \(p\) is appreciable. If \(p\) noticeably exceeds \(m_0c\), then the kinetic energy of the electron will be much greater than \(m_0c^2\) (the energy of a resting electron).

Since an energy \(2m_0c^2\) is needed for the formation of a positron–electron pair, when an electron is localized in a region smaller than \(\dfrac{h}{m_0c}\) (momenta \(p\simeq m_0c\)), pairs will also arise, and it will no longer be possible to speak of the localization of one electron. Therefore the maximum degree of localization of one electron or one positron is determined by the size

\[ \Delta x \simeq \frac{h}{m_0c}. \]

As for the positron-electron field, there is no reason whatever to think that it cannot be concentrated in still smaller regions of space.

Thus, one must distinguish two questions: the question of the localization of a field excitation and the question of the localization of one particle among those proper to the given field. For the localization, concentration, of a field, apparently there are no limitations; for the localization of one particle they do exist.

§ 4. PARTICLES AND THE PRINCIPLE OF SPECTRAL DECOMPOSITION

We see that the notion of an elementary particle is most closely connected with the possibility of a spectral decomposition of a field into harmonic oscillations. The excitation of some harmonic of the field, from the corpuscular point of view, is equivalent to the existence of a particle in a definite state.

However, it is known that one and the same field can be represented in the form of different spectral expansions. Does this mean that one and the same field may have different particles inherent in it?

To answer this question, let us turn to examples.

Let us first consider a transverse (light) electromagnetic field. Such a field can be characterized by the vector potential $\mathbf{A}$ (with $\operatorname{div}\mathbf{A}=0$, so that we are dealing with two independent functions). We can expand $\mathbf{A}$ into linearly polarized oscillations or into circularly polarized ones. Both expansions are completely equivalent and can be written in the form:

\[ \mathbf{A}=\sum_s \sum_{\alpha=1,2} q_{s\alpha}\mathbf{A}_{s\alpha}(x) =\sum_s \sum_{\beta=1,2} Q_{s\beta}\mathbf{B}_{s\beta}(x), \tag{13} \]

where $\mathbf{A}_{s1}$ and $\mathbf{A}_{s2}$ represent two independent linear oscillations (perpendicular to the direction of propagation of the wave), while $\mathbf{B}_{s1}$ and $\mathbf{B}_{s2}$ similarly represent two oscillations with left and right circular motion.

Both expansions are equivalent and equally true. But the first expansion is a spectral expansion of the field with respect to a certain system which interacts with our field and responds to linear oscillations (for example, a simple harmonic dipole); the second expansion is an expansion with respect to a system which responds to circular oscillations. An atom placed in a magnetic field may serve as an example of such systems. Then the atom resonates with linear oscillations parallel to the magnetic field and with circular oscillations in the plane perpendicular to the magnetic field. Such an atom, as it were, combines two types of analyzers: analyzers of linear and analyzers of circular oscillations. With respect to linear analyzers the field consists of linearly polarized particles, and with respect to circular analyzers it consists of circularly polarized particles.

Thus, a field may indeed consist of different particles, depending on what it, the field, interacts with.

However, in the example considered, the differences in the particles associated with the two kinds of spectral expansion are still so insignificant (they are limited to differences in polarizations) that it is more appropriate to speak (as is usually done) of one and the same kind of particles (photons), but existing in two different polarization states. In any case, both ways of formulating the phenomenon (“two kinds of particles” or two kinds of states of one and the same particle) turn out to be completely equivalent.

One can, however, give a more striking example, in which it is shown that differences in the spectral decomposition of a field can lead to the most radical differences in such basic properties of particles as their mass and their mode of interaction.

Let us consider a field described by two scalar functions, \(\psi_1\) and \(\psi_2\), satisfying the equations:

\[ \left. \begin{aligned} -\frac{1}{c^2}\frac{\partial^2 \psi_1}{\partial t^2}+\Delta \psi_1-\chi_1^2 \psi_1 &= g\psi_2,\\ -\frac{1}{c^2}\frac{\partial^2 \psi_2}{\partial t^2}+\Delta \psi_2-\chi_2^2 \psi_2 &= g\psi_1. \end{aligned} \right\} \tag{14} \]

Suppose that we have some system interacting with these fields. Then we may regard it as an analyzer of our field, decomposing the field into \(\psi_1\) and \(\psi_2\). With respect to this system we are dealing with particles of two kinds, whose rest masses will be equal to \(m_1=\dfrac{h\chi_1}{c}\) and \(m_2=\dfrac{h\chi_2}{c}\). At the same time, owing to the presence of coupling between the fields \(\psi_1\) and \(\psi_2\) (the terms on the right-hand sides of the equations), these particles will scatter from one another and transform into one another, i.e. with respect to this analyzer a very lively picture will be observed.

Let us now turn to another analyzer, reacting to the normal oscillations of the combined field \((\psi_1,\psi_2)\). These normal oscillations are described by the functions:

\[ \begin{aligned} \Phi_1 &= \alpha\psi_1+\beta\psi_2,\\ \Phi_2 &= \gamma\psi_1+\delta\psi_2, \end{aligned} \tag{15} \]

where \(\alpha,\beta,\gamma,\delta\) are certain coefficients.

It is not difficult to show that these fields \(\Phi_1\) and \(\Phi_2\) will satisfy the equations:

\[ \left. \begin{aligned} -\frac{1}{c^2}\frac{\partial^2 \Phi_1}{\partial t^2}+\Delta \Phi_1-\chi_1^{\prime 2}\Phi_1 &=0,\\ -\frac{1}{c^2}\frac{\partial^2 \Phi_2}{\partial t^2}+\Delta \Phi_2-\chi_2^{\prime 2}\Phi_2 &=0. \end{aligned} \right\} \tag{16} \]

That is, with respect to a system reacting to the linear combinations \(\Phi_1\) and \(\Phi_2\), the very same field \((\psi_1,\psi_2)\) consists of particles having masses \(M_1=\dfrac{h\chi_1'}{c}\), \(M_2=\dfrac{h\chi_2'}{c}\) and not interacting with one another (the right-hand sides in equations (16) are absent). A simple calculation shows that the masses \(M_1\) and \(M_2\) are found from the formula

\[ M_{1,2}^2=\frac{m_1^2+m_2^2}{2}\pm \sqrt{\frac{(m_1^2-m_2^2)^2}{2}+\frac{h^4g^2}{c^3}}. \tag{17} \]

It is very curious that, for a sufficiently large coupling constant \(g\) between the fields \(\psi_1\) and \(\psi_2\), one of the masses \(M_{1,2}\) becomes imaginary. In this case the equilibrium position of the oscillation corresponds not to a minimum of the potential energy (a “focus”), but to its maximum (more precisely, to a “saddle”).

At the same time, this excitation \(\Phi_2\) does not correspond to any particles at all*), although the field continues to exist and can still depend harmonically on time (the oscillation frequency for \(\Phi_2\) is \(\omega^2 = k^2 - |M_2^2|\)). For this kind of excitation the phase velocity is less than the velocity of light, while the group velocity is greater than the velocity of light. This example shows how close the connection is between the decomposition of a field into a spectrum of harmonic oscillations and the concept of a particle.

In the example considered, the coupling between the fields \(\psi_1\) and \(\psi_2\) was assumed to be linear. This was admitted only for the sake of simplicity. In fact, in modern theory the couplings between fields are nonlinear. For example, the equations for the electron-positron and electromagnetic fields read:

\[ \left. \begin{aligned} \gamma^\mu \frac{\partial \psi}{\partial x_\mu} - \frac{m_0 c}{h}\psi &= \frac{ie}{hc}\,\gamma^\mu \Phi_\mu \psi, \\[6pt] \square^2 \Phi_\mu &= -\frac{4\pi e}{c}\,\psi^{+}\gamma^\mu \psi . \end{aligned} \right\} \tag{18} \]

The first of these equations is the Dirac equation for the electron-positron field \(\psi\), and the second is the d’Alembert equation for the electromagnetic potentials \(\Phi_\mu\). In the right-hand sides of the equations are written the terms expressing the interaction; \(e\) is the coupling constant, in the present case the electric charge. As is evident, these right-hand sides are nonlinear with respect to \(\psi\) and \(\Phi_\mu\)**). Since the electron-positron field always coexists with the electromagnetic field, we may say that the equations for these fields are always nonlinear. But to nonlinear fields the principle of spectral decomposition, based on the linearity of the field equations, is inapplicable.

At the same time, the strict theoretical foundation for the concept of a particle disappears. Indeed, in the case of nonlinear fields it is impossible to represent the energy and momentum of the field in the form of a sum of the energies and momenta of individual excitations, each of which

*) It can be shown that the energy and momentum of the field in this case are not discrete—they are continuous.

**) The interaction terms in the energy give cubic terms of the form
\[ \frac{ie}{hc}\,\Phi_\mu \psi^{+}\gamma^\mu \psi . \]
Therefore here, as in the example considered above, one may expect for the oscillation not only “foci” but also “saddles.” Investigation in this case is much more complicated because of the nonlinearity of the equations.

has a corpuscular character (i.e., \(\dfrac{\varepsilon_s}{c}\) and \(\mathbf{p}_s\) form a four-dimensional vector). In other words, excitations in such fields, generally speaking, cannot be reduced to particles. In the old, classical understanding of fields and particles we could simply speak of a system of particles interacting (through the field). Now, however, we are not entitled to make so definite a division of functions between particles and fields.

In considering the fact of the multiplication of particles, we have already noted the disappearance of the boundary between field and particles. This aspect of the matter can be further emphasized by the interchangeability of the role of particle and field, which also follows from the possibility of the annihilation and creation of particles. The emission and absorption of photons leads to an interaction between electrons. But the converse is also true: the emergence of positron–electron pairs and their subsequent disappearance leads to the scattering of light by light, i.e., to the interaction of photons. Here the roles of field and particles have changed places. In the first case the interaction of particles (electrons, positrons) is due to the electromagnetic field (photons); in the second case the same photons, as particles, interact through the positron–electron field.

In a similar way, nucleons interact by means of the meson field, and, conversely, mesons can interact through the nucleon field. In short, the boundary between field and particles exists only relatively: it is significant only in those cases where the energies and momenta in the system are so small that new particles cannot arise. Only in these cases can particles be opposed to the field. This inseparable and nonlinear connection between different fields has been demonstrated rather widely by modern experiment. Besides the mutual transformations of the electron–positron and electromagnetic fields mentioned above, the transformability and interconnection of these fields with meson fields has also been demonstrated. On the other hand, mesons are produced by nucleons.

Thus, the field picture extends to all particles, and only nucleons still constitute an exception, since there are as yet no reliable data proving the possibility of the emergence of nucleons at the expense of other fields. However, this exception will hardly retain its force in the future, and therefore we are entitled (provisionally) to speak also of a nucleon field.

The impossibility of decomposing nonlinear fields into particles indicates that the very concept of a particle must be regarded as approximate, valid only insofar as, under special conditions, it is still approximately possible to decompose a nonlinear field into weakly bound linear oscillations.

It should be assumed that in reality such conditions may be realized when such an expansion is possible with a high degree of accuracy.

This will be the case when the concentration of particles is so small that they are at large distances from one another.

The simplest case of this kind will be, for example, a single isolated (free) electron. As modern theory shows, the effect of the nonlinear interaction with the electromagnetic field leads in this case to an infinitely large correction to the mass of the electron, which, however, can be eliminated by an artificial device called the “renormalization” of mass[^1]. The final equation for a free electron is obtained as linear:

\[ \gamma^\mu \frac{\partial \psi}{\partial x_\mu} - \frac{m_0 c}{\hbar}\psi = 0. \]

This indicates that the principle of spectral expansion can be preserved for free particles, and, at the same time, the very concept of a particle can also be preserved.

Another example may be a free meson. Undoubtedly, for it the relation

\[ \frac{\varepsilon^2}{c^2} - p^2 = m_0^2 c^2 \]

is applicable. However, the meson is capable of decaying. For example, the \(\mu\)-meson decays into an electron and, apparently, two neutrinos[^3]; the \(\pi\)-meson into a \(\mu\)-meson and a “neutretto.” Therefore one may consider that the \(\mu\)-meson is formed from the neutrino and electron fields, but it can in no way be considered to consist of particles—an electron and two neutrino particles—since neither the neutrino nor the electron can be localized in a volume of the order of the Compton wavelength of the meson,

\[ \frac{\hbar}{m_0 c}. \]

Only when the meson has decayed and the fields that formed it have “spread out” can we speak of the appearance of such field excitations as represent two free neutrinos and one free electron.

Thus the field concept provides a broader basis for understanding physical phenomena than the particle concept. The particle concept is limited and will remain valid in the future development of the theory only in those cases in which one is dealing with the transformation of one spectral expansion of the field into another, i.e., in particle language, with the case of collision of particles, when some particles come from infinity and are again found moving at infinity in other directions (scattering) or in another form (transformation of particles).

Modern theory is developing a mathematical apparatus[^4][^5][^6][^7] which is intended specifically for the calculation of such

kind of problem is the apparatus of the so-called scattering matrix, by means of which the outgoing waves are calculated from waves incoming from infinity. This apparatus, apparently, possesses a considerable degree of generality*. However, it still includes the fundamental shortcomings of the present-day theory: 1) the necessity of removing, by artificial devices, the infinities that arise when the interaction of fields is taken into account**, and 2) the abundance of different fields (electromagnetic, positron–electron, neutrino, meson, and nucleon fields). Unity in the understanding of these fields has not yet been achieved, but the experimental facts concerning the mutual transformability of these fields may perhaps indicate that the different fields are only different modes of excitation of one single field.

REFERENCES

1. Ya. A. Smorodinskii, UFN XXXIX, 325 (1949).
2. V. Weisskopf, UFN XLI, 165 (1950).
3. G. B. Zhdanov, UFN XXXIX, 512 (1950).
4. W. Heisenberg, Zeits. f. Physik 120, 513, 673 (1943).
5. D. Blokhintsev, ZhETF 16, 480 (1946); 17, 66 (1947); DAN III, 205 (1946).
6. R. Feynman, Phys. Rev. 74, 139, 1430, 1212A (1942).
7. F. Dyson, Phys. Rev. 75, 486, 1737 (1949).
8. M. A. Markov, Journ. Phys. USSR II, 454 (1940).
9. D. Blokhintsev, ZhETF 18, 566 (1948).
10. H. Yukawa, Phys. Rev. 77, 219 (1950).

* Objections that have been raised concerning the impossibility of obtaining discrete levels of systems from this apparatus are based on an incorrect application of it. In this apparatus the discrete levels should appear as maxima in the dispersion curve, if the problem is posed as a problem of particle scattering.

** A very interesting and general path for constructing a theory without infinities was first indicated by M. A. Markov⁸—these are the so-called nonlocalized fields. For a similar conception see also ⁹. Recently, a very interesting work by Yukawa¹⁰ has followed this path.