A New Theory of Born’s Electromagnetic Field

P. Tartakovsky, Tomsk

§ 1. Introduction

The importance of introducing into the circle of ideas of quantum theory the idea of a “field”—the fundamental idea of classical physics—raises no doubts. However, despite the fact that attempts to accomplish this task have already been made over the course of five years by major theorists and that in this way a special chapter of quantum theory—quantum electrodynamics—has already been created, until recently science still could not boast of very great achievements, and in quantum electrodynamics there remained a whole series of very serious difficulties and shortcomings. Born’s recent works, while not yet constituting a completed theory, represent, it seems to us, a substantial step forward in the matter of providing a new foundation for the doctrine of the field and of creating a quantum form of the theory of the electromagnetic field capable of answering a number of the most difficult and fundamental questions of contemporary physics.

The purpose of the present article is to acquaint the reader with the basic propositions of Born’s theory. Preliminary to this, however, we shall also briefly dwell on certain works on quantum electrodynamics that preceded Born. This, it seems to us, will help in understanding the significance of Born’s work.

§ 2. Brief Review of the Development of Quantum Electrodynamics

The fundamental idea of field theory consists in the finite velocity of propagation of electromagnetic interactions. All classical electromagnetic theory is built upon this idea; precisely because the processes of interaction of material bodies occur not with infinitely great, but with finite velocity, it becomes necessary to introduce the concept of a “field”—a physical space that is the transmitter of interactions. The law of electromagnetic interactions of macroscopic bodies is expressed, with an enormous degree of accuracy, by the fundamental system of field equations given by Maxwell. Only in the region of the microworld is it necessary to abandon the “classical” description of phenomena, which, as is known, leads to incorrect results, and to pass to a new “quantum” description.

However, the initial development of quantum mechanics remained alien to the idea of the field, the idea of a finite speed of propagation of interactions between particles of matter. Indeed, wishing to express in quantum mechanics, for example, the interaction of an electron with a nucleus, we simply put into Schrödinger’s equation the expression for the potential energy in the same form in which it is used by the usual classical theory of action at a distance.

However, already at a fairly early stage in the development of quantum mechanics the inadequacy of such an approach became clear: it is obvious, for example, that in considering the interaction of very fast particles, we must inevitably take into account also the speed of propagation of the interaction. Likewise, a number of special questions in the theory of radiation led to the necessity of introducing the idea of the field, which alone made it possible to give a correct solution of the corresponding questions. Since, however, both in the domain of the interaction of particles of matter and, in particular, in the domain of radiation, we have a whole series of specifically quantum phenomena, the creation of a new field theory, a new electrodynamics, must proceed along the “quantum path”—the new electrodynamics must be quantum. It seems to us fair to compare the first steps of quantum electrodynamics with the first steps of old quantum mechanics: upon the classical laws of the electromagnetic field, as they had been given by Maxwell’s equations, there were imposed “quantum conditions”—the field was quantized. As a result of the process of quantization it proved possible to obtain certain specifically quantum laws, for example to establish that the energy of a continuous field turns out to be quantized—in other words, it consists of definite energy portions \(h\nu\).

The question of the quantization of the field is fundamental for the new theory. In principle, from a correctly carried out quantization of the field one should expect the resolution of the most basic and fundamental questions—this is precisely what Born strives for in his investigation. We shall briefly dwell on the state of the question of field quantization (and on some consequences following from it) as it was before the appearance of Born’s theory. We shall mainly touch upon two directions, which subsequently proved to be identical.

In one of the first works on quantum electrodynamics Heisenberg and Pauli \(^{1,2}\) considered, alongside systems of material particles, the electromagnetic field as a dynamical system, whose behavior can be described by the method of Hamiltonian mechanics through the introduction of the corresponding coordinates and momenta. The “coordinates” of the field (they should not be confused with the ordinary coordinates of space \(x_1,\ x_2,\ x_3\)) \((q_i)\ i=(1,\ 2,\ 3,\ 4)\) are the components of the four-dimensional vector potential, three components of which give the ordinary vector potential \(A\), while the 4th (temporal) component determines the scalar potential \(\varphi\) of the field. Each of the coordinates \(q_i\) has a definite value at all points of space-time. In order to find the momenta corresponding to the coordinates—

momenta \(p_i\), one must write a suitable expression for the Lagrangian function of the field \(L\). Then \(p_i=\dfrac{\partial L}{\partial \dot q_i}\), where \(\dot q_i\) is the derivative of \(q_i\) with respect to time \(t\).

The form of the function \(L\left(q_i,\dfrac{\partial q_i}{\partial x_k},\dot q_i\right)\) must be chosen so that from the variational principle

\[ \delta \int L\left(q_i,\frac{\partial q_i}{\partial x_k},\dot q_i\right)\,d\tau\,dt \tag{1} \]

the necessary field equations—the “equations of motion”—are obtained. In particular, if we wish to remain on the ground of the classical theory, for \(q_i\) one must obtain d’Alembert’s equations for the vacuum,

\[ \nabla^2 q_i-\frac{1}{c^2}\frac{\partial^2 q_i}{\partial t^2}=0, \tag{2} \]

from which (with certain additional restrictions) one can obtain the system of Maxwell equations. It is known, for example, that in classical electrodynamics

\[ L=\frac{1}{2}\left(\mathbf H^2-\mathbf E^2\right), \tag{3} \]

if \(\mathbf H\) is the magnetic field and \(\mathbf E\) the electric field in the corresponding units.

Having found the momenta \(p_i\), we can also write the equations of motion of the field in Hamiltonian form. The next stage is the quantization of the field: we cease to regard \(q_i\) and \(p_i\) as ordinary quantities, and regard them as quantum quantities, i.e. operators, for which the ordinary rules of multiplication are not valid and therefore “commutation rules” must be introduced, by analogy with the usual rules for quantizing mechanical systems.

These rules are written in the form

\[ [q_\alpha q_\beta]=[p_\alpha p_\beta]=0,\quad [p_\alpha q_\beta']=-\delta_{\alpha\beta}\,\delta(r-r'), \tag{4} \]

where the sign \([\ ]\) denotes quantum Poisson brackets,* \(\delta_{\alpha\beta}=0\) for \(\alpha\ne\beta\) and \(\delta_{\alpha\beta}=1\) for \(\alpha=\beta\); \(\delta(x)=1\) for \(x=0\) and equals 0 for \(x\ne0\); \(r\) and \(r'\) are the radius vectors of two points of space for which the coordinates and momenta are denoted by \(q\) and \(q'\), \(p\) and \(p'\). Thus the quantization of the field proceeds in exactly the same way as the quantization of any dynamical system.

Dirac \(^{2}\) proposed another method of quantizing the field, which may be called the “method of direct quantization.” Quanti-

* Let us recall that for a conjugate coordinate and momentum \(q\) and \(p\) we have

\[ [qp]=\frac{i}{h}(pq-qp)=1, \]

where \(h\) is Planck’s constant divided by \(2\pi\).

the quantum Dirac field is not regarded as a dynamical system, but plays, in his words, “a simpler and more fundamental role,” being the transmitter of all interactions between particles of matter.

What is characteristic of Dirac’s theory is that, although the field is regarded in the spirit of classical theory as something continuous, nevertheless its states change by jumps, as is generally the case in quantum theory. This removes one of the fundamental difficulties of classical electrodynamics: the point is that the latter, when considering the field created by a moving electron, almost neglects the inverse action of the field on the motion of the electron (the reaction of the field is considered small); in other problems, when considering the influence of an external field on the motion of an electron, one has to neglect the change in the field caused by the electron. If these “inverse actions” are not regarded as small, then the theory reaches an impasse. For the quantum theory of the field, which considers its jump-like changes, these difficulties do not exist.

The quantized field transmits interactions between charges by means of a system of plane waves of different frequencies, propagating with velocity \(c\). Dirac himself considered only the one-dimensional case; Fock and Podolsky \(^{3}\) extended his considerations to the actual three-dimensional electromagnetic field and obtained the ordinary Coulomb law as the result of interaction mediated by waves. Rosenfeld \(^{4}\) proved the formal equivalence of Dirac’s theory and the Heisenberg–Pauli theory.

The idea of Dirac’s method of quantizing the field appears especially clearly in Dirac’s fundamental work (the one-dimensional case). Let the interaction be determined by a potential \(V\) and propagate according to the ordinary wave laws. Then \(V\) is a solution of the equation

\[ \frac{\partial^{2}V}{\partial x^{2}}-\frac{1}{c^{2}}\frac{\partial^{2}V}{\partial t^{2}}=0 \tag{5} \]

and can be formed by the superposition of plane waves:

\[ V=\int_{-\infty}^{+\infty}\left\{a_\nu e^{i\nu\left(t-\frac{x}{c}\right)}+b_\nu e^{i\nu\left(t+\frac{x}{c}\right)}\right\}\,d\nu . \tag{6} \]

(Here amplitudes of the type \(a_{-\nu}\) are complex conjugates of \(a_\nu\).) Substituting \(V\) into the energy expression

\[ H=\frac{1}{8\pi}\int\left\{\left(\frac{\partial V}{\partial x}\right)^{2}+\frac{1}{c^{2}}\left(\frac{\partial V}{\partial t}\right)^{2}\right\}dx, \]

we obtain:

\[ H=\frac{1}{c}\int_{0}^{\infty}\nu^{2}\left(a_\nu a_{-\nu}+b_\nu b_{-\nu}\right)d\nu . \tag{7} \]

Up to now we have remained within the domain of classical theory. If we now “quantize” the field, then according to Dirac this will be equivalent to saying that the amplitudes of the individual harmonic oscillations in the expression \(V\) are regarded as quantum quantities—operators—and we introduce for them definite commutation rules. In doing so we proceed in such a way that each \(a_\nu\) and \(b_\nu\) is regarded directly as a separate harmonic oscillator. Dirac writes these rules in the form:

\[ [a_\nu,b_{\nu'}]=0,\qquad [a_\nu,a_{\nu'}]=\frac{ic}{\nu}\,\delta(\nu+\nu'), \tag{8} \]

which in expanded form gives:

\[ a_\nu a_{-\nu}-a_{-\nu}a_\nu=-\frac{\hbar c}{\nu}. \tag{9} \]

Next Dirac replaces each of the complex quantities \(a_\nu\) and \(a_{-\nu}\) by \(p+iq\) and \(p-iq\). Then, remembering that \(p\) and \(q\) are quantum, noncommuting quantities, and calculating \(a_\nu a_{-\nu}\) and \(a_{-\nu}a_\nu\), as well as using (9), it is easy to find

\[ a_\nu a_{-\nu}=p^2+q^2-\frac{\hbar c}{2\nu}. \tag{10} \]

This expression leads to the fact that in the expression for the energy there appears a term containing \(p^2+q^2\), i.e. representing the energy of a certain oscillator, if \(q\) is taken as the coordinate and \(p\) as the momentum. But according to quantum mechanics the energy of a quantized oscillator is equal to \(\left(n+\frac{1}{2}\right)\hbar\nu\), where \(n\) is an integer. If so, then substitution of (10) into (7) for the part depending on \(a\) gives

\[ H_a=\int_0^\infty\left\{\left(n_\nu+\frac{1}{2}\right)\hbar\nu-\frac{1}{2}\hbar\nu\right\}d\nu =\nu\int_0^\infty n_\nu\hbar\nu d\nu \tag{11} \]

and an analogous formula for \(H_b\). Thus, thanks to quantization, the energy of the continuous field proves to be the sum of separate quanta of different frequencies. The numbers of quanta \(n_\nu\) show how much a given frequency is “excited.” We may have zero-quantum, one-, two-, etc. quantum states of the field, at definite frequencies, between which, according to what was said above, abrupt transitions occur (in interaction with matter). We shall not dwell here on the manner in which Dirac considers the interaction of charges. It will be sufficient to point out the following: the interaction is considered by means of successive approximations: the zeroth corresponds to noninteracting particles and to the zero-quantum state of the field; the first—to a one-quantum state, when at each frequency there is one quantum; the action of the first particle is, so to speak, emitted, but not absorbed by the second; finally, the second approximation, which is precisely the one of interest to us, corresponds again to the zero-quantum state of the field—the action is “absorbed” by the second particle. It is precisely in this approximation that one obtains

the Coulomb law of interaction is obtained. But here one must note one essential defect, inherent, however, in all quantum theories (up to Born) and also present in classical theory. The point is that, alongside the expression of the Coulomb potential, there appears an infinite additive constant, expressing the so-called energy of the electron “on itself,” or “self” energy (Selbstenergie). The appearance of this infinitely large energy, as Born rightly points out, is inadmissible for a satisfactory physical theory. This defect occurs both in quantum and in classical theory, and the root of this defect in the theory should probably be sought not in the rules of quantization, but somewhere deeper.

As the principal objections to the forms of quantum electrodynamics that have existed up to now, Born\(^5\) indicates the following, concerning both the process of quantization of the field and the choice of the basic “classical” field equations, which are then quantized:

1. The difference in the understanding of time and coordinates (time is an ordinary quantity, coordinates are quantum ones). 2. From this follow difficulties with relativistic invariance, which requires complete symmetry of time and space. 3. The not very intelligible treatment of the amplitudes \(a_\nu\) of plane waves as quantum quantities. 4. The non-closed character of the theory (the imposition of different principles—classical electrodynamics and quantum mechanics). 5. The absence in the theory of an organic concept of the radius of the electron, as well as of a natural explanation, on the basis of the field equations themselves, of such properties of the electron as charge, spin, the ratio of the masses of the electron and proton, etc. In general, the electron in quantum electrodynamics, just as in the classical one, is something “given from outside,” not entering into the theory as an organic part. In his demand for the “inclusion” of the electron in the foundations of field theory, Born comes rather close to the old unitary theories. 6. Finally, the last remark is the already mentioned question of the infinite self-energy.

Since, in the main, the deviations of the theory from agreement with experimental data are found at distances of the classical radius of the electron \(\dfrac{e^2}{m_0 c^2}\), according to Born the construction of a new theory first of all requires a revision of the “classical” foundations of the theory—the field equations, which should not unconditionally be taken, as was done earlier, from Maxwellian theory. Only the elimination of the defects of the classical theory itself will give the required result. The second stage in the construction of the theory is the introduction of suitable quantization rules, which would not be a simple transfer, but a reasonable generalization to the case of the electromagnetic field of the quantization rules of ordinary quantum mechanics. Only quantization rules selected in this way can in the future reveal the “riddles of the electron.” Two works by Born are devoted to these questions.

§ 3. Born’s field theory

In our exposition of the foundations of Born’s theory we shall depart from the order adopted by the author himself, and first dwell on a generalization of the equations of the electromagnetic field, i.e. on the “classical” part of Born’s works. In doing so we shall base ourselves, chiefly, on the first paper, since it is simpler and requires knowledge only in the domain of the ordinary theory of the electromagnetic field. The exposition of the same questions in the second article, however, has a more general and rigorous character; in individual cases we shall also refer to the second work. The methods of quantization proposed by Born we shall consider in the following paragraph.

In considering the electromagnetic field we must make use of four independent variables (spatial coordinates and time)

\[ x_1=x,\quad x_2=y,\quad x_3=z,\quad x_4=ix_0=ict \tag{12} \]

and four dependent variables (the components of the four-dimensional vector potential)

\[ \Phi_1,\ \Phi_2,\ \Phi_3,\ \Phi_4=i\Phi_0 . \tag{13} \]

The components of the electric and magnetic field strengths, as is known, are given by means of the quantities

\[ f_{kl}=\frac{\partial \Phi_l}{\partial x_k}-\frac{\partial \Phi_k}{\partial x_l}=-f_{lk}, \tag{14} \]

which form an antisymmetric tensor of rank 2.*

\[ \mathbf{H}=(f_{23},\ f_{31},\ f_{12}),\quad \mathbf{E}=i(f_{14},\ f_{24},\ f_{34}), \tag{15} \]

i.e. the magnetic field is expressed through the “spatial” components of the electromagnetic tensor (indices 1, 2, 3), the electric field through the “temporal” ones (index 4).

Wishing to find the equations of the electromagnetic field, i.e. differential equations connecting \(f_{kl}\), we may make use of the same device that mechanics uses: compose a suitable expression for the Lagrangian function \(L\) and then apply the variational principle. The Euler equations for the corresponding variational problem will give the field equations, which, in this way, are “equations of motion.” Thus the construction of any electromagnetic theory is reduced to the choice of the form of the Lagrangian function. The classical theory gives \(L\) in the form:

\[ L=\frac{1}{2}\sum_{k>l} f_{kl}^{2}=\frac{1}{4}\sum_{k,l} f_{kl}^{2}. \tag{16} \]

* These formulas are nothing other than the ordinary formulas of electrodynamics:

\[ \mathbf{H}=\operatorname{rot}\mathbf{A},\quad \mathbf{E}=\operatorname{grad}\varphi-\frac{1}{c}\frac{\partial \mathbf{A}}{\partial t}, \]

if \(\mathbf{A}\) \((\Phi_1,\Phi_2,\Phi_3)\), \(\varphi=\Phi_0\) and the coordinates and time are expressed by (12).

Introducing the momenta,

\[ p_{kl}=\frac{\partial L}{\partial \Phi_{kl}}=\frac{\partial L}{\partial f_{kl}}, \tag{17} \]

where, for brevity, we have put

\[ \frac{\partial \Phi_k}{\partial x_l}=\Phi_{kl}, \]

we find, in the classical case,

\[ p_{kl}=f_{kl}. \tag{18} \]

From the variational principle \(\delta\int Ld\tau=0\) (\(d\tau\) is a four-dimensional element of volume) we obtain the field equations in Maxwellian form, if we recall (15)*. But we have already seen that the quantization of these equations does not lead to satisfactory results, no matter by what method it is carried out, including the one proposed by Born. Therefore it is necessary to change the form of the Lagrange function, i.e., to pass to a new field theory. This change, however, must be such that the classical function is obtained from the new one as a limiting expression.

Born proposes to choose \(L\) in the form

\[ \left. \begin{gathered} L=\frac{1}{a^2}\left(\sqrt{1+a^2F}-1\right) \quad \text{or} \quad L=b^2\left(\sqrt{1+\frac{1}{b^2}F}-1\right);\\ F=\sum_{k,l} f_{kl}^{\,2},\quad a=\frac{1}{b}. \end{gathered} \right\} \tag{19} \]

From dimensional considerations it is immediately clear that \(b\) represents a certain field strength, which we may call critical. If it is regarded as large (and we shall see that this is indeed the case), then, expanding \(L\) in a series in powers of the small quantity \(a\), in the first approximation we find \(L=\frac{1}{2}F\), i.e., the classical expression (16). Obviously, expression (19) could also be written without the coefficients \(a\) or \(b\), if the units of field strength are chosen in the corresponding way**.

Starting from this expression for \(L\), we find for the momenta

\[ p_{kl}=\frac{\partial L}{\partial f_{kl}}=\frac{f_{kl}}{\sqrt{1+a^2F}}. \tag{20} \]

We see that \(p_{kl}\) forms a new tensor, not coinciding with \(f_{kl}\) and passing into it only in the limit if one sets \(a=0\).

* With these notations \(L=\frac{1}{2}(\mathbf{H}^2+\mathbf{E}^2)\), which is the classical form of the Lagrange function of the field.

** In the article by Born and Infeld such a choice of the Lagrange function is substantiated. As the basis one takes the expression for the Lagrange function of the general theory of relativity \(L=\sqrt{-|g_{kl}|}\), where \(|g_{kl}|\) is the determinant formed from the so-called fundamental tensor. We put \(L=\sqrt{-|a_{kl}|}\), where \(|a_{kl}|=g_{kl}+f_{kl}\). On passing to the Galilean coordinate system and taking into account the limiting value of \(L\), expression (19) is obtained from this.

The components of these two tensors make it possible to determine the four electromagnetic vectors \(\mathbf{E}, \mathbf{D}, \mathbf{H}, \mathbf{B}\), with \(f_{kl}\) determining the vectors \(\mathbf{B}\) and \(\mathbf{E}\), and \(p_{kl}\) the vectors \(\mathbf{H}\) and \(\mathbf{D}\). In the first paper this connection is expressed by the formulas

\[ \mathbf{B}=(f_{23}, f_{31}, f_{12}),\quad \mathbf{E}=i(f_{14}, f_{24}, f_{34}), \]
\[ \mathbf{H}=(p_{23}, p_{31}, p_{12}),\quad \mathbf{D}=-i(p_{14}, p_{24}, p_{34}). \tag{21} \]

Applying the variational principle \(\delta \int L d\tau=0\), it is easy to obtain the system of equations

\[ \sum_{l=1}^{4} \frac{\partial p_{kl}}{\partial x_l}=0,\quad (k=1,2,3,4), \tag{22} \]

and from the definition of \(f_{kl}\) there follow the identical relations

\[ \sum_{l=1}^{4} \frac{\partial f_{kl}^{*}}{\partial x_l}=0,\quad (k=1,2,3,4), \tag{23} \]

where \(f_{kl}^{*}\) represents the tensor “dual” with respect to \(f_{kl}\)—it is obtained from the components of the tensor \(f_{kl}\) by replacing the indices \(23, 31, 12\) respectively by \(14, 23, 34\), and conversely. Equalities (22) and (23) constitute the equations of the electromagnetic field. In outward form they coincide completely with Maxwell’s equations for the vacuum, as is easily verified if, in place of \(f_{kl}^{*}\) and \(p_{kl}\), one substitutes their values according to (20). However, this analogy is only external, since according to (20) the relation between \(p_{kl}\) and \(f_{kl}\) (i.e., \(\mathbf{H}, \mathbf{D}\) with \(\mathbf{B}, \mathbf{E}\)) is by no means the same as in Maxwell’s theory: substituting the values of \(p_{kl}\) from (20) into (22) gives nonlinear differential equations, whereby Born’s theory differs very sharply from Maxwell’s; the equations of Maxwell’s theory turn out to be only an approximate form of the field equations. It should be noted that the very fact of introducing four vectors instead of two, which in Maxwell’s theory has meaning only for a medium with dielectric constant and magnetic permeability different from unity*.

Born indicates that the same field equations can be obtained by introducing the Hamiltonian function \(H=-L+\sum p_{kl}f_{kl}\) and regarding \(f_{kl}\) as functions of \(p_{kl}\) (and not conversely, as we have done up to now), for which one must use the variational principle \(\delta \int H d\tau=0\), choo-

* In the second article, where the calculations are carried out in a more perfect approximation, the relation between the vectors is obtained in the form:

\[ \mathbf{H}=\frac{\mathbf{B}-G\mathbf{E}}{\sqrt{1+F-G^2}},\quad \mathbf{D}=\frac{\mathbf{E}+G\mathbf{B}}{\sqrt{1+F-G^2}}, \]

where \(F\) is specified by condition (19), while \(G\) also represents a function of \(f_{kl}\). For \(G=0\) (approximately) we obtain the same relations as those obtained directly from equation (20).

having expressed \(H\) in terms of \(p_{kl}\). We have complete symmetry with respect to the variables \(f_{kl}\) and \(p_{kl}\).

In the new theory it is easy to derive “conservation laws,” representing a generalization of the classical laws of conservation of energy and momentum for the electromagnetic field.

As in classical electrodynamics, these laws are expressed by the assertion that the four-dimensional divergence of a certain tensor is equal to zero. Only in the limiting case does this tensor pass into the tensor of electromagnetic impulse-work of ordinary electrodynamics. If one starts from the first way of writing the field equations (starting from the Lagrange function), then the conservation equations are easily obtained in the form:

\[ \sum_l \frac{\partial}{\partial x_l}\left(L\delta_{lj}-\sum_k p_{lk} f_{kj}\right)=0. \tag{24} \]

The second method of calculation would give:

\[ \sum_l \frac{\partial}{\partial x_l}\left(H\delta_{lj}-\sum_k f^*_{lk}p^*_{kj}\right)=0. \tag{25} \]

The author proposes to regard as the generalization of the classical tensor a certain linear combination which, in the limit, gives the classical value; for example, the component \(T_{44}\) passes into \(\frac{1}{2}(\mathbf H^2+\mathbf E^2)\), i.e., represents the classical value of the field energy, as it should.

The difference between the new theory and the ordinary one appears most clearly when considering the problem of an electron at rest (the static problem). In this case the vector potential is equal to zero, so that:

\[ \mathbf B=0,\quad \mathbf E=-\operatorname{grad}\varphi, \]

\[ L=\frac{1}{a^2}\left(\sqrt{1-a^2\mathbf E^2}-1\right) =\frac{1}{a^2}\left(\sqrt{1-a^2(\operatorname{grad}\varphi)^2}-1\right), \tag{26} \]

and the “equations of motion” give:

\[ \operatorname{div}\mathbf D=0,\quad D_x=\frac{E_x}{\sqrt{1-a^2\mathbf E^2}} = \frac{-\dfrac{\partial\varphi}{\partial x}} {\sqrt{1-a^2(\operatorname{grad}\varphi)^2}}. \tag{27} \]

If \(\mathbf D\) is not continuous everywhere, then the integral over a surface enclosing a singular point is not equal to zero; one may set

\[ \int \mathbf D_r\,ds=4\pi e \tag{28} \]

and regard (28) as the definition of charge.

In the case of radial symmetry (a singular point at the origin of coordinates), \(\varphi=\varphi(r)\), and from (27) we easily obtain

\[ \frac{d}{dr}\left(r^2D_r\right)=0,\quad D_r=\frac{E_r}{\sqrt{1-a^2\mathbf E^2}} = \frac{\varphi'}{\sqrt{1-a^2\varphi'^2}}; \tag{29} \]

hence:

\[ r^{2}\mathbf D=e. \]

Here \(e\) is a constant of integration, which, in accordance with (29), must be assigned the value of the charge. The displacement \(\mathbf D\) is therefore expressed as in the ordinary theory \(\left(\mathbf D=\dfrac{e}{r^{2}}\right)\) and for \(r=0\) becomes infinite. Substituting the value of \(\mathbf D\) into the second of equations (29), we obtain the expression for the potential \(\varphi\):

\[ \varphi=\frac{e}{r_0}\int\limits_{r/r_0}^{\infty}\frac{dx}{\sqrt{1+x^{4}}} \tag{30} \]

(here we have put \(\dfrac{r}{r_0}=x\)), where \(r_0\) is determined from the condition

\[ r_0=\sqrt{ae}, \]

i.e. the critical field strength

\[ b=\frac{1}{a}=\frac{e}{r_0^{2}}. \tag{31} \]

For large values of \(r\), (30) gives approximately \(\varphi=\dfrac{e}{r}\), i.e. it represents the usual Coulomb potential. For \(r\) of the order of \(r_0\), and especially smaller than \(r_0\), we obtain deviations from Coulomb’s law, and at the point \(r=0\) we get:

\[ \varphi(0)=\frac{e}{r_0}\int\limits_0^\infty\frac{dx}{\sqrt{1+x^{4}}} =1.85407\cdot\frac{e}{r_0^{2}}, \tag{32} \]

and not infinity. This means that the energy of our point charge “of itself” remains finite. The field strength is directed along the radius and is equal to

\[ \mathbf E_r=\frac{e}{r_0^{2}}\frac{1}{\sqrt{1+\left(\dfrac{r}{r_0}\right)^{4}}}. \tag{33} \]

For large \(r\) we again obtain Coulomb’s law \(\mathbf E=\dfrac{e}{r^{2}}\). For \(r=0\) the field strength in absolute value reaches the critical value \(|\mathbf E|=b=\dfrac{e}{r_0^{2}}\); at this point the field strength evidently undergoes a discontinuity equal to \(2\dfrac{e}{r_0^{2}}\).

Thus, although we introduce into consideration the electron (charge) as a point inhomogeneity of the field (a singular point), nevertheless the critical radius \(r_0\) organically enters the theory, which we may rightly regard as the radius of the “electron.” In a certain sense the electron proves to possess finite dimensions. But from

From all that has been said earlier it follows that such a consideration does not lead to the difficulties arising in the classical theory in connection with the assumption of a finite electron (holding its parts together), since the finiteness of the dimensions is a consequence of a special kind of field equations and is, so to speak, apparent.

Considering the motion of an electron in a weak external field (in comparison with the critical quantity \(b\)), Born finds an expression for the Lagrangian function of the electron; for this one must first introduce a coordinate system in which the electron is at rest at the given instant (the system moves together with the electron). In this system one may represent the Lagrangian function of the field as consisting of two terms, of which the first, by virtue of relativistic invariance, can be expressed in the form of (26), while the second term reduces simply to the product of the charge and the potential of the external field \(\varphi_0\).

If the first term is calculated by the formula

\[ \int L' \, dx_0\,dy_0\,dz_0 = \frac{4\pi}{a^2} \int_0^\infty \left(\sqrt{1-a^2\varphi'^2}-1\right) r^2\,dr \]

(here \(x_0, y_0, z_0\) are the coordinate system just mentioned), then one obtains the expression \(-4\pi \cdot 0.619\,\dfrac{e^2}{r_0}\), which we may set equal to \(-4\pi m_0 c^2\). Thus for the rest mass one obtains the equation:

\[ m_0 c^2 = 0.619\,\frac{e^2}{r_0}, \tag{34} \]

and the Lagrangian function in this coordinate system is expressed as follows:

\[ \Lambda = m_0 c^2 + e\varphi_0 . \]

Transition to a coordinate system with respect to which the electron is in motion gives the usual expression for the Lagrangian function:

\[ \Lambda = m_0 c^2 \sqrt{1-\frac{\mathbf v^2}{c^2}} + e\left[\varphi_0-\left(\frac{\mathbf v}{c},\,\mathbf A\right)\right], \tag{35} \]

where \(\mathbf v\) is the velocity of the electron, and \(\mathbf A\) is the vector potential of the external field.

From (34) we find for the radius of the electron \(r_0\) \(0.619\,\dfrac{e^2}{m_0c^2}\), which, upon substituting the values \(e, m_0, c\), gives a quantity of the order of \(10^{-13}\) cm\(*\). For the critical field \(b\) one obtains approximately \(10^{16}\) absolute units.

The expression for the Lagrangian function (35) shows that, at least in the first approximation, Born’s electron obeys the usual classical equations of motion. In connection with the question of the mass of the electron it is interesting to note the following circumstance,

* In the second paper a different numerical coefficient is obtained, approximately twice as large.

which was recently pointed out by Ya. I. Frenkel[^6]. If, as the initial expression, one takes the value of the density of electromagnetic momentum, which in Born’s theory should be written in the form:

\[ \mathbf{G}=\frac{1}{c}\frac{\mathbf{E}\times\mathbf{H}}{\sqrt{1-a^2(\mathbf{E}^2-\mathbf{H}^2)}} \tag{36} \]

(here \(\times\) is the sign of vector multiplication; the value of \(\mathbf{H}\) corresponds to Born’s \(\mathbf{B}\)), then, as in ordinary electrodynamics (where \(\mathbf{G}=\frac{\mathbf{E}\times\mathbf{H}}{c}\)), for the total momentum of the electron one obtains

\[ \mathbf{G}=-\frac{m_0\mathbf{v}}{\sqrt{1-\frac{\mathbf{v}^2}{c^2}}}, \]

where

\[ m_0=\frac{1}{c}\cdot\frac{2}{3}\int\frac{\mathbf{E}^2}{\sqrt{1-a^2\mathbf{E}^2}}\,d\tau \tag{37} \]

instead of the expression of the usual theory

\[ m_0=\frac{1}{c^2}\cdot\frac{2}{3}\int \mathbf{E}^2 d\tau. \tag{38} \]

If by \(U\) we denote the electrostatic energy, then (38), as is known, leads to the relation \(m_0=\frac{4}{3}\frac{U}{c^2}\), whereas calculation shows that for the case of a point electron (37) gives

\[ m_0=\frac{U}{c^2} \]

in complete agreement with the theory of relativity. With this remark we shall conclude the consideration of Born’s “classical” field theory. The “riddle of the electron” must receive its final solution in the process of quantizing the field equations. As a result of a correctly carried out field quantization, Born hopes to obtain the value of the electron charge (the multiplicity of every charge with respect to the elementary one), its spin, etc. However, this question still cannot be considered solved: in Born’s first work only the general path toward solving the problem of generalizing the quantization rules is outlined.

§ 4. Generalization of the Methods of Quantization

The development of quantum mechanics in general and of the rules of quantization in particular is closely connected with classical mechanics in the Hamilton–Jacobi form. The basis of Hamiltonian mechanics, as is known, is the solution of a definite variational problem. Born connects the introduction and generalization of the principle of quantization with the solution of this variational problem by the method proposed by Hilbert and called the “independence theorem.” In the most pro-

in simple cases (for example, the motion of a material point in a plane) the application of the independence theorem leads to well-known results of mechanics and quantum theory. A logically consistent generalization of the method gives hope of finding correct methods of field quantization.

To clarify the essence of the new method, let us consider the examples analyzed by Born.

In the case of the motion of a point in a plane we have in all three variables \(x, y, z\), and we take the time \(x\) as independent; the coordinates \(y\) and \(z\) are functions of \(x\). The usual method consists in considering the variational principle:

\[ \delta \int L(x,y,z,y',z')\,dx=0, \tag{39} \]

which leads to the Lagrange equations

\[ \frac{d}{dx}\frac{\partial L}{\partial y'}-\frac{\partial L}{\partial y}=0,\quad \frac{d}{dx}\frac{\partial L}{\partial z'}-\frac{\partial L}{\partial z}=0. \]

Instead of the usual transition to the Hamiltonian form of mechanics, we introduce three momenta

\[ p_x=L-y'\frac{\partial L}{\partial y'}-z'\frac{\partial L}{\partial z'},\quad p_y=\frac{\partial L}{\partial y'},\quad p_z=\frac{\partial L}{\partial z'}, \tag{40} \]

which are functions of \(x, y, z, y', z'\), and form a certain integral \(S\) along a curve \(C\) in the space \(x, y, z\):

\[ S=\int_C (p_xdx+p_ydy+p_zdz). \tag{41} \]

If two functions \(\eta, \zeta\) are chosen so that \(y'=\eta(x,y,z)\), \(z'=\zeta(x,y,z)\), and the integral is independent of the form of the curve, depending only on the position of the initial and final point, then

\[ p_x=\frac{\partial S}{\partial x},\quad p_y=\frac{\partial S}{\partial y},\quad p_z=\frac{\partial S}{\partial z}. \tag{42} \]

The choice of \(\eta\) and \(\zeta\) can be made by solving the equations

\[ \frac{\partial p_z}{\partial y}-\frac{\partial p_y}{\partial z}=0 \]

and so on. Then from the second and third equations (42) we find \(\eta\) and \(\zeta\) as functions of \(x, y, z, \dfrac{\partial S}{\partial y}, \dfrac{\partial S}{\partial z}\). Substitution of these values into the first equation, if we denote \(p_x=-H\), gives the Hamilton–Jacobi equation

\[ \frac{\partial S}{\partial x}+H\left(x,y,z,\frac{\partial S}{\partial y},\frac{\partial S}{\partial z}\right)=0. \tag{43} \]

For example, in the case of a free material point with mass equal to unity, we shall have \(L=\dfrac{1}{2}(y'^2+z'^2)\), and it is easy to obtain the Hamilton–Jacobi equation in the form

\[ \frac{\partial S}{\partial x}+\frac{1}{2}\left[\left(\frac{\partial S}{\partial y}\right)^2+\left(\frac{\partial S}{\partial z}\right)^2\right]=0, \]

whose solution can be chosen in the form

\[ S=-\frac{1}{2}(\alpha^2+\beta^2)x+\alpha y+\beta z. \tag{44} \]

This solution, as is not difficult to verify, expresses rectilinear motion.

The transition to quantum mechanics can be carried out either by directly replacing the momenta by the operators* \(\frac{1}{i}\frac{\partial}{\partial x}\), \(\frac{1}{i}\frac{\partial}{\partial y}\), \(\frac{1}{i}\frac{\partial}{\partial z}\), and substituting them into the Hamilton-Jacobi equation, after which the resulting operator is applied to the wave function \(\psi\), or, guided by the limiting transition, we shall regard \(S\) as the phase of a wave process, i.e. put

\[ \psi \longrightarrow e^{iS} \]

and then construct the function \(\psi\) by superposition of plane waves (a wave packet):

\[ \psi=\iint \Phi(\alpha,\beta)e^{i\left[-\frac{1}{2}(\alpha^2+\beta^2)x+\alpha y+\beta z\right]}d\alpha d\beta. \tag{45} \]

By either method we arrive at the Schrödinger equation in the form

\[ \frac{1}{i}\frac{\partial \psi}{\partial x} -\frac{1}{2}\left(\frac{\partial^2\psi}{\partial y^2} +\frac{\partial^2\psi}{\partial z^2}\right)=0. \tag{46} \]

In the general case of arbitrary \(H\) we obtain:

\[ S=-H(\alpha,\beta)+\alpha y+\beta z \tag{44'} \]

\[ \psi=\iint \Phi(\alpha,\beta)e^{i[-H(\alpha,\beta)+\alpha y+\beta z]}d\alpha d\beta \tag{45'} \]

\[ \left\{\frac{1}{i}\frac{\partial}{\partial x} +H\left(\frac{1}{i}\frac{\partial}{\partial y},\frac{1}{i}\frac{\partial}{\partial z}\right)\right\}\psi=0. \tag{46'} \]

The case of a one-dimensional continuous medium (for example, a string) presents an example in which there is one dependent variable \(z(x,y)\) and two independent variables (time \(x\) and coordinate \(y\)). Now the variational principle has the form

\[ \delta\int L(x,y,z,z_x,z_y)\,dxdy=0,\quad \left(z_x=\frac{\partial z}{\partial x}\ldots\right) \]

The Euler equation gives

\[ \frac{\partial}{\partial x}\frac{\partial L}{\partial z_x} +\frac{\partial}{\partial y}\frac{\partial L}{\partial z_y} -\frac{\partial L}{\partial z}=0. \]

Introducing the momenta

\[ p_x=\frac{\partial L}{\partial z_x},\quad p_y=\frac{\partial L}{\partial z_y},\quad p_z=-L+z_x\frac{\partial L}{\partial z_x}+z_y\frac{\partial L}{\partial z_y}, \tag{47} \]

* The units have been chosen so that \(\hbar=1\). With the usual choice of units, it would be \(p_x=\frac{\hbar}{i}\frac{\partial}{\partial x}\), etc.

we choose \(z_x=\xi(x,y,z)\), \(z_y=\eta(x,y,z)\) so that the surface integral

\[ S=\iint (p_x\,dy\,dz+p_y\,dz\,dx+p_z\,dx\,dy) \tag{48} \]

over the surface \(F\) in the space \(x,y,z\) should not depend on the form of \(F\), but only on its bounding curve \(C\). To determine \(\xi\) and \(\eta\) we may use the equations:

\[ \frac{\partial z_x}{\partial y}=\frac{\partial z_y}{\partial x},\quad \frac{\partial p_x}{\partial x}+\frac{\partial p_y}{\partial y}+\frac{\partial p_z}{\partial z}=0. \]

If (48) does not depend on the form of the surface, then one may write

\[ S=\int (Xdx+Ydy+Zdz), \tag{48'} \]

where, evidently, it must be

\[ \frac{\partial Z}{\partial y}-\frac{\partial Y}{\partial z}=p_x. \tag{49} \]

To find in our case the continuum analogue of the Hamilton–Jacobi equation, we regard \(S\) as a functional (a function of functions) and introduce the following generalization of the derivative (surface derivative). Consider integrals over the surface \(F\) bounded by the curve \(C\):

\[ \sigma_{yz}=\int dy\,dz,\quad \sigma_{zx}=\int dz\,dx\ldots \]

and introduce the quantity

\[ \frac{\partial S}{\partial(yz)} = \lim \frac{S}{\sigma_{yz}} \quad (\text{the curve } C \text{ contracts to a point}). \tag{50} \]

It is easy to see that from (50) one obtains:

\[ \frac{\partial S}{\partial(yz)} = \frac{\partial Z}{\partial y}-\frac{\partial Y}{\partial z}, \]

if the integral (48) is written in the form (48′).
By (49) we have:

\[ \frac{\partial S}{\partial(yz)}=p_x,\quad \frac{\partial S}{\partial(zx)}=p_y,\quad \frac{\partial S}{\partial(xy)}=p_z. \tag{51} \]

Hence, by a method analogous to that applied in the case of the motion of a material point, we obtain [putting \(p_z=H(x,y,z,p_x,p_y)\)] the Hamilton–Jacobi equation in the form:

\[ \frac{\partial S}{\partial(xy)} + H\left(x,y,z,\frac{\partial S}{\partial(yz)},\frac{\partial S}{\partial(zx)}\right)=0, \tag{52} \]

which, in contrast to the previous case, is a functional, and not a partial differential equation.

The transition to the quantum theory can be carried out for the case

when \(H\) does not contain \(x,y,z\) explicitly (only this case will be of interest below), i.e., when (52) is written in the form

\[ \frac{\partial S}{\partial(xy)}+ H\left(\frac{\partial S}{\partial(yz)},\frac{\partial S}{\partial(zx)}\right)=0. \tag{52'} \]

For \(S\) we choose a solution in the form

\[ S=\alpha\sigma_{yz}+\beta\sigma_{zx}-H(\alpha,\beta)\sigma_{xy}, \tag{53} \]

with arbitrary constants \(\alpha\) and \(\beta\). In this case it can be shown that

\[ \frac{\partial S}{\partial(yz)}=\alpha,\quad \frac{\partial S}{\partial(zx)}=\beta, \]

i.e., that the functional derivatives are replaced by ordinary ones:

\[ \frac{\partial S}{\partial(yz)}=\frac{\partial S}{\partial\sigma_{yz}}. \]

By analogy with the preceding case, the transition to quantum mechanics is carried out by constructing a wave packet:

\[ \psi(\sigma_{yz},\sigma_{zy},\sigma_{xy}) = \iint \Phi(\alpha,\beta) e^{\,i\left[\alpha\sigma_{yz}+\beta\sigma_{zx}-H(\alpha,\beta)\sigma_{xy}\right]} \,d\alpha\,d\beta . \tag{54} \]

The function \(\psi\) satisfies the differential equation:

\[ \left\{ \frac{1}{i}\frac{\partial}{\partial(xy)} + H\left( \frac{1}{i}\frac{\partial}{\partial(yz)}, \frac{1}{i}\frac{\partial}{\partial(zx)} \right) \right\}\psi=0, \tag{55} \]

obtained from (52) by replacing the momenta by the corresponding operators.

The generalization of the Hamilton–Jacobi equation just considered, and the transition to quantum mechanics, can easily be applied also to a four-dimensional continuum—the electromagnetic field. Now, alongside the four independent variables (12), we have four more dependent variables (13), and the “independent” integral \(S\) must be constructed by integration over a four-dimensional surface in an eight-dimensional space.

To the momenta (17) we further add

\[ p_0=L, \]

which, for the classical expression of the Lagrange function (16), leads to

\[ p_0=\frac{1}{2}\sum_{k>l} f_{kl}^{\,2}. \tag{56} \]

For \(S\), by analogy with (48), we write

\[ S=\int\left\{ p_0\,dx+ \sum_{k>l} p_{kl}\left(d\Phi_k\,dx^{(l)}-d\Phi_l\,dx^{(k)}\right) \right\}, \quad (k,l=1,2,3,4). \tag{57} \]

Here

\[ dx=\frac{1}{i}\,dx_1dx_2dx_3dx_4,\quad dx^{(1)}=\frac{1}{i}\,dx_2dx_3dx_4, \]

Introducing the notation

\[ \sigma_0=\frac{1}{i}\int dx,\quad \sigma_{kl}=\frac{1}{i}\int\left(d\Phi_k\,dx^{(l)}-d\Phi_l\,dx^{(k)}\right)= \]

\[ =\frac{1}{i}\int\left(\frac{\partial\Phi_k}{\partial x_l}-\frac{\partial\Phi_l}{\partial x_k}\right)dx =i\int f_{kl}\,dx \]

(i.e., the \(\sigma_{kl}\) represent, in their way, averaged values of \(f_{kl}\) over the space-time region \(\sigma_0\)), we obtain the Hamilton–Jacobi equation:

\[ \frac{\partial S}{\partial\sigma_0} -\frac{1}{2}\sum_{k>l}\left(\frac{\partial S}{\partial\sigma_{kl}}\right)^2=0. \tag{58} \]

The transition to the quantum theory is again accomplished by constructing a wave packet or simply by introducing the operators

\[ p_0=\frac{1}{i}\frac{\partial}{\partial\sigma_0},\quad p_{kl}=\frac{1}{i}\frac{\partial}{\partial\sigma_{kl}}, \]

which leads to the equation

\[ \left(p_0-\frac{1}{2}\sum_{k,l}p_{kl}^{\,2}\right)\psi =\left\{p_0-\frac{1}{2}\left(\mathbf H^2+\mathbf E^2\right)\right\}\psi=0, \tag{59} \]

since, in accordance with (18) and (15), we must replace the field strengths \(\mathbf H\) and \(\mathbf E\) by operators:

\[ \mathbf H=\frac{1}{i}\left( \frac{\partial}{\partial\sigma_{23}}, \frac{\partial}{\partial\sigma_{31}}, \frac{\partial}{\partial\sigma_{12}} \right),\quad \mathbf E=\frac{1}{i}\left( \frac{\partial}{\partial\sigma_{14}}, \frac{\partial}{\partial\sigma_{24}}, \frac{\partial}{\partial\sigma_{34}} \right). \]

But, as Born points out, equation (59), as one based on the classical value of the field Lagrangian function, cannot be correct.

However, the principles of field quantization examined here can be applied to the “classical” field theory based on the form of the Lagrangian function chosen by Born. This application has not at present been carried through to completion, and only general principles are given, containing many hypothetical elements.

First of all, the assumption is made that it is possible to divide the world region under consideration in such a way that it represents the “product” of two dual* regions, i.e.,

\[ dx=d\sigma\cdot d\sigma',\quad V=\int dx=\sigma\cdot\sigma'. \tag{60} \]

* Introducing new variables \(u_1,u_2,u_3,u_4\), we write:

\[ dx=dx_1dx_2dx_3dx_4 =\frac{\partial(x_1x_2x_3x_4)}{\partial(u_1u_2u_3u_4)} \,du_1du_2du_3du_4 =\sum_{kl} \frac{\partial(x_kx_l)}{\partial(u_1u_2)} \cdot \frac{\partial(x_kx_l)^{*}}{\partial(u_3u_4)} \,du_1du_2du_3du_4, \]

where

\[ \frac{\partial(x_2x_3)^{*}}{\partial(u_3u_4)} = \frac{\partial(x_1x_4)}{\partial(u_3u_4)}, \]

and then we introduce the “dual” regions

\[ d\sigma= \frac{\partial(x_kx_l)}{\partial(u_1u_2)}\,du_1du_2, \quad d\sigma'= \frac{\partial(x_kx_l)^{*}}{\partial(u_3u_4)}\,du_3du_4. \]

Then (57) is written in the form:

\[ S=\int\left\{p_0\,d\sigma'\,d\sigma+\sum_{k>l}p_{kl}\,d\sigma'\,f_{kl}\,d\sigma\right\}. \tag{61} \]

Introducing

\[ P_0=\int p_0\,d\sigma',\quad P_{kl}=\int p_{kl}\,d\sigma',\quad F_{kl}=\int f_{kl}\,d\sigma, \tag{62} \]

we obtain, instead of (61),

\[ S=\int\left\{P_0'\,d\sigma+\sum_{k>l}P'_{kl}\,dF_{lk}\right\}, \]

or, in the simplest case,

\[ S=P_0'\sigma+\sum_{k>l}P'_{kl}F_{lk}. \tag{63} \]

Next, with the aid of (63), we construct a wave packet \((\psi\to e^{iS})\), or, what is the same thing, replace (62) by the operators

\[ P_0'=\frac{1}{i}\frac{\partial}{\partial\sigma},\quad P'_{kl}=\frac{1}{i}\frac{\partial}{\partial F_{kl}}. \]

Then from the Hamilton–Jacobi equation

\[ P_0'+H(P'_{kl})=0 \]

we obtain the quantum equation of the field, substituting for \(H\) its value corresponding to the Lagrangian function (19)*:

\[ \left\{P_0'-\frac{1}{a^2}\sqrt{1-a^2\sum P_{kl}^{\prime 2}}\right\}\psi=0. \tag{64} \]

However, this equation too cannot be satisfactory, first of all because the operator acting on \(\psi\) is nonlinear (the meaning of the square of an operator of the type under consideration is quite unclear). Therefore, instead of (64), by analogy with the way in which Dirac introduced his famous equations—linear in the operators—in place of the Schrödinger equation, we shall consider the pair of operators

\[ \begin{array}{l} a^2\gamma_0P_0'+a\displaystyle\sum_{k>l}\gamma_{kl}P'_{lk}-1\\[6pt] a^2\gamma_0P_0'+a\displaystyle\sum_{k>l}\gamma_{kl}P'_{lk}+1 \end{array} \left\}, \tag{65} \]

the multiplication of which gives the square of the operator (64), if the quantities \(\gamma\) satisfy the conditions:

\[ \gamma_0^2=1,\quad \gamma_0\gamma_{kl}+\gamma_{kl}\gamma_0=0,\quad \gamma_{kl}\gamma_{mn}+\gamma_{mn}\gamma_{kl}=\delta_{km}\delta_{ln}, \]

* Born’s expression for the Hamiltonian function was obtained in the form

\[ H=-\frac{1}{a^2}\sqrt{1-a^2P}, \]

where \(P=\sum p_{kl}^{2}\). In the text the small \(p_{kl}\) have been replaced by capital \(P_{kl}\), i.e. by “averaged momenta.”

analogous to the conditions satisfied by Dirac matrices.

Born expresses the supposition that the quantum field equations obtained with the aid of the operators (65) should explain not only the existence of the elementary charge, but also the spin of the electron, as well as the ratio of the masses of the proton and the electron. But carrying out all these calculations is a matter for the future.

Recently Weyl has expressed a number of doubts about the correctness of the path taken by Born in developing methods for quantizing the field. Born himself considers the results obtained so unusual that he is for the moment unable to give them a definite physical interpretation.

It is beyond doubt, however, that the “classical” part of Born’s theory is of very substantial interest and is a generalization of electrodynamics that frees it from a number of substantial difficulties.

Proofreading note. Recently Born and Infeld (Proc. Roy. Soc., (A) 147, 522, 1934) proposed another method for quantizing the field, based on special commutation rules for the components of the vectors $\mathbf{D}$ and $\mathbf{B}$. These latter are now regarded as “independent variables,” while $\mathbf{E}$ and $\mathbf{H}$ are obtained by differentiating the energy $U$, regarded as a function of $\mathbf{D}$ and $\mathbf{B}$.

The field equations turn out to be a consequence of the commutation rules, a special rule for differentiation with respect to time and coordinates, and one further supplementary condition. From these same rules follow the laws of conservation of energy and momentum for the field. There are as yet no concrete applications of the new form of the theory.

LITERATURE

1. W. Heisenberg and W. Pauli, Z. Physik, 56, 1, 1929.
2. P. A. M. Dirac, Proc. Roy. Soc. (A) 136, 453, 1932.
3. V. Fock and B. Podolsky, Sow. Phys., 1, 801, 1932; see also P. Dirac, V. Fock and B. Podolsky, Sow. Phys., 2, 468, 1932.
4. Rosenfeld, Z. Physik, 76, 729, 1932.
5. M. Born, Proc. Roy. Soc. (A) 143, 410, 1934; M. Born and L. Infeld, Proc. Roy. Soc. (A) 144, 425, 1934.
6. J. Frenkel, Proc. Roy. Soc. (A) 146, 930, 1934.
7. H. Weyl, Phys. Rev., 46, 505, 1934.