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This paper presents a review of five papers in which a generalized electrodynamics has been developed. The purpose of the review is to present the results obtained so far, leaving out duplications, false starts, and detailed calculations. The emphasis is on the sequence of ideas, the difficulties encountered, and the methods of procedure.

The components of ρv / c and iρ must transform as components of a four-vector, so that if measured in one coordinate system they are known in all coordinate systems. On the other hand, any operational definition of ρ(r,t) must take account of the positions of all particles at the same time t, that of the s-th particle being r_s(t). Upon performing the Lorentz transformation these will be r_s^′(t_s^′), and the transformed time t_s^′ will be different for each particle. Another observer, in measuring ρ^′, would use r_s^′(t^′), t^′ being the same for all particles. As particles are in motion r_s^′(t_s^′)≠r_s^′(t^′), and there appears to be no necessary relation between ρ(r,t) and ρ^′(r^′,t^′), operationally defined in each coordinate system. It turns out, however, that if in each coordinate system the charge density is defined by ρ(r,t)=Σse_sδ(r-r_s(t)), then relativistic equations of transformation hold.

Paralleling a work of Fock we are able to eliminate the auxiliary conditions in our generalized quantum electrodynamics. As in the work of Fock this leads to a determination of both the electrostatic self-energy and electrostatic particle-particle interaction. Both turn out to be finite and in agreement with results obtained classically.

When the Lagrangian from which the field equations are derived contains second and higher derivatives of the generalized field coordinates, the method of quantizing the field equations developed by Heisenberg and Pauli cannot be immediately applied. By generalizing a method due to Ostrogradsky for converting Lagrange's equations of motion of a particle, when higher derivatives are present, into canonical Hamiltonian form, it becomes possible to perform a similar transformation of the field equations. Applying this method to Podolsky's generalized electrodynamics, we obtain the Hamiltonian of the field and double the usual number of generalized coordinates and momenta. The quantization of the field follows without any special assumptions. The last two sections are devoted to the discussion of the auxiliary conditions and some of their consequences.

If one wishes to derive generalized field equations from a Lagrangian, at the same time preserving the linear character of the equations, one must admit terms involving derivatives of the field quantities. It turns out that the only non-trivial generalization of this kind, leading to differential equations of order below eighth, is obtained by taking L_f=(1 / 8π){1 / 2F_αβ²+a²(∂F_αβ / ∂x_β)²}. This leads to a theory that contains the Landé-Thomas theory and accounts for the choice of sign required when one wishes to consider the total field as consisting of the Maxwell-Lorentz and the Yukawa fields.

The Dirac equation may be written so as to give an eigenvalue problem whose eigenvalues would be values of the electron mass. The equation is solved in several cosmological spaces, none requiring any quantization of m. A space-time suggested by Eddington leads to wave equations that have solutions with quantized m, m depending upon the radius of the universe, constants c and ℏ, and a quantum number. If the radius of the universe is taken as 10²⁸ cm, the lowest mass state in this group of solutions is of the order of 10^-65 g. Conversely if the usual electron mass is considered as the lowest state, the radius of the universe is of the order of 10^-10 cm. Since the only constants occurring in the theory are ℏ, m, and c, one would expect that the radius of the universe would come out in terms of h / mc. Only an occurrence, as the result of quantization, of a large dimensionless number could lead to a reasonable result; but Dirac's equation evidently does not provide such a number, and is therefore unsuited to account for the electron mass in terms of the radius of the universe.

Eddington, in his book Relativity Theory of Protons and Electrons, makes statements to the effect that the invariance of Dirac's equation is an elementary consequence of the tensor form which it acquires in the new wave tensor calculus, in which it is merely an identity derivable on epistemological principles and not on physical hypothesis. It is shown in this article that the invariance of Dirac's equation referred to by Eddington is a purely formal one, while the invariance which is of particular interest in quantum mechanics depends upon the physical interpretation of the wave function. The generalization of physical interpretation suggested by Eddington's work, however, has the desirable quality of leaving Dirac's equation invariant in the physical sense. It is further shown that in the derivation of Dirac's equation Eddington makes use of the usual physical assumptions, but in somewhat disguised form.

In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false. One is thus led to conclude that the description of reality as given by a wave function is not complete.

In this paper a point of view is presented according to which the quantities m and e, the mass and the charge of elementary particles, need not enter into the electrodynamics and the quantum mechanics of electrons, positrons and photons. The only constants entering into the equations, rewritten in this way, are the velocity of light and two independent lengths, namely: a=e² / mc², and b=h / mc. The first of these lengths determines the scale of electrodynamic phenomena, and has no especial relation to electronic radius. The second determines the scale of quantum-mechanical phenomena. In the absence of particles electromagnetic phenomena have no definite scale. This fact, together with the possibility of creation of electron-positron pairs, leads to the belief that a theory of interactions of electrons, positrons, and photons, giving as a by-product a derivation of the ratio a / b=α=e² / hc, could be formulated without introducing the two other pure numbers β=m / M and γ=Gm² / e². This theory is envisaged as a limiting theory, obtainable from the future general theory by putting β=γ=0. It is then considered from the point of view of the necessity of giving up space-time framework for the description of physical phenomena. It is concluded that the first limiting theory should not necessitate abolition of space and time.

In this paper a simple tensor form of Dirac's equation is obtained. This is accomplished by considering ψ_s and γ^α(rs) of the usual equations as being related to a set of n-beins as invariants corresponding to true tensors ψ_μ and Γ^α_β^σ. The results are, however, independant of the choice of the n-beins. It is thus shown that the introduction of the idea of half-vectors in the quantum mechanics, while undoubtedly desirable when dealing exclusively with cartesian coordinates, is unnecessary.

Expressions are obtained, in accordance with Einstein's approximate solution of the equations of general relativity valid in weak fields, for the effect of steady pencils and passing pulses of light on the line element in their neighborhood. The gravitational fields implied by these line elements are then studied by examining the velocity of test rays of light and the acceleration of test particles in such fields. Test rays moving parallel to the pencil or pulse do so with uniform unit velocity the same as that in the pencil or pulse itself. Test rays moving in other directions experience a gravitational action. A test particle placed at a point equally distant from the two ends of a pencil experiences no acceleration parallel to the pencil, but is accelerated towards the pencil by twice the amount which would be calculated from a simple application of the Newtonian theory. The result is satisfactory from the point of view of the conservation of momentum. A test particle placed at a point equally distant from the two ends of the track of a pulse experiences no net integrated acceleration parallel to the track, but experiences a net acceleration towards the track which is satisfactory from the point of view of the conservation of momentum.

It has been shown by Schrödinger and O. Klein that the Kramers-Heisenberg formulas for dispersion and incoherent scattering can both be obtained by an extension of Schrödinger's method for treating dispersion, the terms in the Hamiltonian involving squares of the potentials being neglected. The resulting formulas agree with those of Dirac. They are, however, inconvenient for calculations, as they contain summations with respect to all energy levels combining with the pair of levels under consideration, and thus imply complicated integrations when a continuous spectrum is present. This paper treats the problem of a hydrogen-like atom, acted on by light the wave-length of which is large compared to the size of the atom and the frequency of which is not too near a resonance frequency, and develops a method which obviates the necessity of integrating over the continuous range. It is an extension of the method used by one of us in treating the dispersion by atomic hydrogen. It is applied in detail to the first two levels of H. Formulas are derived for the intensities of the Smekal-Raman lines with a shift corresponding to the first absorption line.

The probability density that an electron have certain momenta is given by the square of the absolute magnitude of a momentum eigenfunction ϒ_nlm (P, Θ, Φ), in which P, Θ, and Φ are spatial polar coordinates of the total momentum vector referred to the same axes as the coordinates r, θ, and φ of the electron. The following general expression for these functions for a hydrogen-like atom is obtained: ϒ_nlm(P, Θ, Φ)={1 / (2π)^{1 / 2}e^±imΦ} {((2l+1)(l-m)! / 2(l+m)!)^{1 / 2}P_l^m(cosΘ)} {π2^2l+4l! / (γh)^{3 / 2}(n(n-l-1)! / (n+l)!)^{1 / 2}ζ^l / (ζ²+1)^l+2C_n-l-1^l+1(ζ²-1 / ζ²+1)} in which ζ=(2π / γh)P, with γ=(4π²μe²Z / nh²)=(Z / na₀). The probability Ξ_nl(P)dP that the electron have a total momentum lying within the limits P and P+dP is also evaluated, and it is shown that the root mean square of the total momentum is equal to the momentum of the electron in a circular Bohr orbit with the same total quantum number.

Dirac showed that, if in the Hamiltonian H momenta η_r conjugate to the co-ordinates ξ_r are replaced by (h / 2πi)∂ / ∂ξ_r, the Schrödinger equation appropriate to the coordinate system ξ_r is (H-E)ψ_ξ=0. Applied to coordinate systems other than cartesian this usually leads to incorrect results. The difficulty is here traced partially to the way in which ψ_ξ is normalized and partly to the choice of H. In H expressions such as qpq^-1p and p² are not equivalent, and the simplified form is generally incorrect. A formula satisfying all the requirements of quantum mechanics for a Hamiltonian of a conservative system, in an arbitrary coordinate system, is therefore developed H=1 / 2μΣr=1r=nΣs=1s=ng^{-1 / 4}p_rg^{1 / 2}g^rsp_sg^{-1 / 4}+U This formula is applied to a case of plane polar coordinates and leads to correct results.