(59)
which is, in the more exact notation, just Eqs. (8).
CAMBRIDGE, LENINGRAD AND KHARKOV.
FOUNDATIONS OF THE NEW FIELD THEORY
BY
MAX BORN AND LEOPOLD INFELD
Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 144, No. 852.
(Mar. 29, 1934), pp. 425–451.
§ 1. INTRODUCTION.
The relation of matter and the electromagnetic field can be interpreted from two opposite standpoints:—ky
The first which may be called the unitarian standpointkz assumes only one physical entity, the electromagnetic field. The particles of matter are considered as singularities of the field and mass is a derived notion to be expressed by field energy (electromagnetic mass).
The second or dualistic standpoint takes field and particle as two essentially different agencies. The particles are the sources of the field, are acted on by the field but are not a part of the field; their characteristic property is inertia measured by a specific constant, the mass.
At the present time nearly all physicists have adopted the dualistic view, which is supported by three facts.
1. The failure of any attempt to develop a unitarian theory.—Such attempts have been made with two essentially different tendencies: (a) The theories started by Heaviside, Searle and J. J. Thomson, and completed by Abraham, Lorentz, and others, make geometrical assumptions about the “shape” and kinematic behaviour of the electron and distribution of charge density (rigid electron of Abraham, contracting electron of Lorentz); they break down because they are compelled to introduce cohesive forces of non-electromagnetic origin; (b) the theory of Miela formally avoids this difficulty by a generalization of Maxwell’s equations making them non-linear; this attempt breaks down because Mie’s field equations have the unacceptable property, that their solutions depend on the absolute value of the potentials.
2. The result of relativity theory, that the observed dependance of mass on velocity is in no way characteristic of electromagnetic mass, but can be derived from the transformation law.
3. Last, but not least, the great success of quantum mechanics which in its present form is essentially based on the dualistic view. It started from the consideration of oscillators and particles moving in a Coulomb field; the methods developed in these cases have then been applied even to the electro-magnetic field, the Fourier coefficients of which behave like harmonic oscillators.
But there are indications that this quantum electrodynamics meets considerable difficulties and is quite insufficient to explain several facts.
The difficulties are chiefly connected with the fact that the self-energy of a point charge is infinite.lb The facts unexplained concern the existence of elementary particles, the construction of the nuclei, the conversion of these particles into other particles or into photons, etc.
In all these cases there is sufficient evidence that the present theory (formulated by Dirac’s wave equation) holds as long as the wavelengths (of the Maxwell or of the de Broglie waves) are long compared with the “radius of the electron” e2 / mc2 , but breaks down for a field containing shorter waves. The non-appearance of Planck’s constant in this expression for the radius indicates that in the first place the electromagnetic laws are to be modified; the quantum laws may then be adapted to the new field equations.
Considerations of this sort together with the conviction of the great philosophical superiority of the unitarian idea have led to the recent attemptlc to construct a new electrodynamics, based on two rather different lines of thought: a new theory of the electromagnetic field and a new method of quantum mechanical treatment.
It seems desirable to keep these two lines separate in the further development. The purpose of this paper is to give a deeper foundation of the new field equations on classical lines, without touching the question of the quantum theory.
In the papers cited above, the new field theory has been introduced rather dogmatically, by assuming that the Lagrangian underlying Maxwell’s theory
(1.1)
(H and E are space-vectors of the electric and magnetic field) has to be replaced by the expressionld
(1.2)
The obvious physical idea of this modification is the following:-
The failure in the present theory may be expressed by the statement that it violates the principle of finiteness which postulates that a satisfactory theory should avoid letting physical quantities become infinite. Applying this principle to the velocity one is led to the assumption of an upper limit of velocity c and to replace the Newtonian action function of a free particle by the relativity expression . Applying the same condition to the space itself one is lead to the idea of closed space as introduced by Einstein’s cosmological theory.le Applying it to the electromagnetic field one is lead immediately to the assumption of an upper limit of the field strength and to the modification of the action function (1.1) into (1.2).
This argument seems to be quite convincing. But we believe that a deeper foundation of such an important law is necessary, just as in Einstein’s mechanics the deeper foundation is provided by the postulate of relativity. Assuming that the expression has been found by the idea of a velocity limit it is seen that it can be written in the formmc2 (1 − dτ/dt) ,
wherec2dτ2 = c2dt2 − dx2 − dy2 − dz2 ,
and therefore it has the property that the time integral of mc2 dτ/dt is invariant for all transformations for which dτ 2 is invariant. This four-dimensional group of transformations is larger than the three-dimensional group of transformations for which the time integral of the Newtonian function
is invariant.
So we believe that we ought to search for a group of transformations for which the new Lagrangian expression has an invariant space-time integral and which is larger than that for the old expression (1.1). This latter group is the known group of special relativity but not the group of general space-time transformations.lf Now it is very satisfying that the new Lagrangian belongs to this group of general relativity; we shall show that it can be derived from the postulate of general invariance with a few obvious additional assumptions. Therefore the new field theory seems to be a consequence of this very general principle, and the old one not more than a useful practical approximation, just in the same way as for the mechanics of Newton and Einstein.
In this paper we develop the whole theory from this general standpoint. We shall be obliged to repeat some of the formulæ published in the previous paper. The connection with the problems of gravitation and of quantum theory will be treated later.
§ 2. POSTULATE OF INVARIANT ACTION.
We start from the general principle that all laws of nature have to be expressed by equations covariant for all space-time transformations. This, however, should not be taken to mean that the gravitational forces play an essential part in the constitution of the physical world; therefore we neglect the gravitational field so that there exist coordinate systems in which the metrical tensor gkl has the value assumed in special relativity even in the centre of an electron. But we postulate that the natural laws are independent of the choice of the space-time co-ordinate system.
We denote space-time co-ordinates byx2, x2, x3, x4 = x, y, z, ct.
The differential dxk is, as usual, considered to be a contravariant vector. One can pull the indices up and down with help of the metrical tensor which in any cartesian co-ordinate system (as used in special relativity) has the form
(2.1)
It is not the unit matrix, because of the different signs in the diagonal. Therefore we have to distinguish between covariant and contravariant tensors even in the co-ordinate systems of special relativity. In this case, however, the rule of pulling up and down of indices is very simple. This operation on the index 4 does not change the value of the tensor component, that on one of the indices 1, 2, 3 changes only the sign.
We use the well-known convention that one has
to sum over any index which appears twice.
To obtain the laws of nature we use a variational principle of least action of the form
(2.2)
We postulate: the action integral has to be an invariant. We have to find the form of satisfying this condition.
We consider a covariant tensor field akl ; we do not assume any symmetry property of akl. The question is to define to be such a function of akl that (2.2) is invariant. The well-known answer is that must have the formlg
(2.3)
If the field is determined by several tensors of the second order, can be any homogeneous function of the determinants of the covariant tensors of the order .
Each arbitrary tensor akl can be split up into a symmetrical and anti-symmetrical part:
(2.4)
The simplest simultaneous description of the metrical and electromagnetic field is the introduction of one arbitrary (unsymmetrical) tensor akl; we identify its symmetrical part gkl with the metrical field, its antisymmetrical part with the electromagnetic field.lh
We have then three expressions which multiplied by dτ are invariant
(2.5)
where the minus sign is added in order to get real values of the square roots; for (2.1) shows that |δkl | = −1, therefore always |gkl |< 0.
The simplest assumption for is any linear function of (2.5):
(2.6)
But the last term can be omitted. For if fkl is the rotation of a potential vector, as we shall assume, its space-time integral can be changed into a surface integral and has no influence on the variational equation of the field.li Therefore we can take
(2.7)
We need another condition for the determination of A. Its choice is obvious. In the limiting case of the cartesian co-ordinate system and of small values of fkl, has to give the classical expression
(2.8)
We now leave the general co-ordinate system which has guided us to the expression (2.6) for and calculate in cartesian co-ordinates. Then we have with gkl = δkl (see (2.1))
For small values of fkl the last determinant can be neglected and (2.6) becomes equal to (2.8) only if
(2.9)
We have therefore the result:—
The action function of the electromagnetic field is in general co-ordinates
(2.10)
and in cartesian co-ordinates
(2.11)
where
(2.12)
(2.13)
Let us go back to the expression for in a general co-ordinate system. We denote as usual|gkl | = g,
and we develop the determinant |gkl + fkl | into a power series in fkl. We have then|gkl + fkl| = g + Φ (gkl, fkl)+|fkl|.
The transformation properties of |gkl + fkl|,g, |fkl| and therefore also of Φ (gkl , fkl) are the same. They transform in the same way as g. If we write
(2.14)
we see, that all expressions in the bracket on the right side of (2.14) are invariant. We have calculated their value in a geodetic co-ordinate system and have found:
Φ/g is an invariant. We have therefore in an arbitrary co-ordinate system:
(2.15)
(2.16)
Both F and G are invariant. We shall bring G into such a form, that its invariance will be evident. For this purpose let us define an antisymmetrical tensor jsklm for any pair of indices, that islj
(2.17)
We can write now G in the following form:
(2.18)
From the last equation we can deduce the tensor character of jsklm . We can also write G in the form
(2.19)
where fljsk is the dual tensor denned by
(2.20)
that is
(2.21)
or also
(2.22)
because
(2.23)
We shall need later the following formulæ:
(2.24)
(2.25)
(2.26)
(2.24)–(2.26) follow from the definition of , and G given above.
The function represented by (2.15) is the simplest Lagrangian satisfying the principle of general invariance. But it differs from that considered in I by the term G2. This is of the fourth order in the fkl and can, therefore, be neglected except in the immediate neighbourhood of singularities (i.e., electrons, see § 6). But the Lagrangian used in I can also be expressed in a general covariant form; for G2 is a determinant, namely, |fkl|, therefore
(2.27)
is also invariant; in cartesian co-ordinates it has exactly the form
(2.28)
Which of these action principles is the right one can only be decided by their consequences. We take the expression given by (2.15) and can then easily return to the other (2.27) or (2.28) by putting G = 0. In any case the solution of the statical problem is identical for both action functions because one has G = 0 in this special case.
§ 3. ACTION PRINCIPLE, FIELD EQUATION AND CONSERVATION LAW.
We write (2.15) in the general form
We shall see that all considerations hold if L is an invariant function of these arguments. As usual we assume the existence of a potential vector φk, so that
(3.1)
Then we have the identity
(3.2)
which can with the help of (2.20) be written:
(3.2A)
We introduce a second kind of antisymmetrical field tensor pkl, which has to fkl a relation similar to that which, in Maxwell’s theory of macrospic bodies, the dielectric displacement and magnetic induction have to the field strengths:
(3.3)
The variation principle (2.2) gives the Eulerian equations
(3.4)
The equation (3.2) (or (3.2A)) and (3.4) are the complete set of field equations.
We prove the validity of the conservation law as in Maxwell’s theory. Assuming a geodetic co-ordinate system, we multiply (3.2) by plm :
(3.5)
In the second and third term we can take plm under the differentiation symbol because of (3.4); in the first term we use the definition (3.3) of plm :
or
If we introduce the tensor
(3.6)
where
(3.7)
we have
(3.8)
In an arbitrary co-ordinate system we have
(3.9)
or, with the usual notation of covariant differentiation
(3.9A)
It follows from (3.3) and (2.25) that we can write also in the form
(3.6A)
§ 4. LAGRANGIAN AND HAMILTONIAN.
can be considered as a function of gkl and fkl. We shall show that is the energy-impulse tensor. We find
(4.1)
(4.2)
(4.3)
Therefore
(4.4)
It follows from (3.6A) and (3.6)
(4.5)
Now it is very easy to generalize our action principle in such a way that it contains Einstein’s gravitation laws; one has only to add to the action integral the term ∫ R√−gdτ , where R is the scalar of curvature. But we do not discuss problems connected with gravitation in this paper.
was regarded as a function of gkl and fkl. We can, however, express also as a function of gkl and pkl. It can be shown that it is possible to solve the equations
(3.3)
with respect to fkl. For this purpose we have to calculate
(4.6)
(4.7)
The Dreams That Stuff is Made of Page 59