The mathematical form and physical content of unified field theories derived from a variational principle

The Einstein theory of general relativity reduces as a first approximation to the old Newtonian theory and provides a satisfactory amount of such small effects as the notion of the perihelion of Mercury and the gravitational deflection of light. The greatest triumph of the theory, however, is that the field equations themselves determine the notion of an n-particle system - as a first approximation the Newtonian attraction. In this way the separate concepts of matter, field and field - matter interaction are not distinct in general relativity.
Unified field theory seeks to extend general relativity so that other force fields such as electromagnetism are included in the formalism with gravitation. Such a theory, to be successful, must as a first approximation both reduce to the Newtonian - Maxwell field theories and imply the Newtonian - Lorents equations of motion. The exact form of, or higher approximation to, the field equations would be expected to give deviations from this Maxwell theory just as general relativity deviates from the Newtonian theory. Indeed the electromagnetic and gravitational fields may not be separable except as an approximation. It is reasonable to expect that a unified field theory with these characteristics would increase our understanding of matter and provide an improved basis for a new quantum theory.
Taking the so-called affine standpoint one looks for a simple Lagrangian, constructed from the affine connection alone, from which a possable unified field theory can be derived by a variational principle. For a non-symmetric affine connection the simplest Lagrangian seems to be the determinant of a linear combination of the two tensors which result by contracting the curvature tensor. It is found that this Lagrangian gives rise to only two different sets of field equations, one set constituting Schrödinger's theory and the other a new theory, four both sets identify relatlons or conservation laws can be derived and these provide a powerful tool In suggesting possible interpretations of the theories. The next stop is to seee if the approximate form for small fields of either set of field equations contains a subset which could be identified as Maxwell's equations. While Schrödinger's theory is satisfactory on this point the new theory is not and so must be discarded. Schrödinger's a theory contains a factor λ which must be identified as the cosmological constant. From observations of the solar system this constant must be very small. Consequently Schrödinger's theory cannot differ greatly from Einstein's 'weak' theory which may be obtained formally from Schrödinger's by putting λ equal to zero. Also since the conservation law for Schrödinger's theory contains a Lorents term with λ as a multiplicative constant it looks as if the smallness of λ is effectively going to remove electromagnetic effects. This conclusion is supported by the fact that both Einstein's 'weak' and 'strong' theories fail to give equations of motion of a particle system containing force terms other than the gravitational ones. It is difficult to see how Schrödinger's theory with such a small λ could give a significantly different result. All these arguments mean that the new affine theory, Schrödinger's theory, and Einstein's theories each fail to give a satisfactory unified field theory.
For the sake of completeness we derive the detail form for the conservation laws of the new theory. These are found to be of similar form as those found by Schrödinger for his theory. Also the solution of Einstein's 'strong' and 'weak' equations for a static spherically symmetric field are discussed. It is known that in the magnetic case (no electric field) whereas the weak equations have solutions corresponding to a magnetic pole with mass, the strong equations require the mass to be zero. An argument is given to show that in the general combined electric-magnetic case the weak equations may allow the presence of an independent mass parameter, whereas the strong equations, as is known, require the mass to be zero.
All this has been concerned with a non-symmetric affine connection, Part II of this thesis is concerned with a new theory which involves a symmetric affine connection and a non-symmetric metric tensor. Thus the affine approach, according to which the affine connection alone characterises the structure of space, is discarded. The theory developed has a superficial resemblance to a recent theory of Kurgunoglu but it is far simpler and has a quite different interpretation.
The form of the conservation laws not only helps to suggest an interpretation of the theory but also plays a part in the derivation of the field equations themselves. The new theory does in first approximation contoan a Maxwell-like subset. The suggested exact form for Maxwell's equations involves in a complicated way both the gravitational and eleotromagnatic field quantities. The field equations can be solved almost exactly for a static spherically symmetric field and the solutions may be interpreted as a stationary charged particle with mass. A structure parameter is involved and from its influence on the electrostatic field and charge distribution it may be interpreted as the radius of the particle. The gravitational mass of the particle contains a positive contribution which it is reasonable to suppose is the electrostatic energy. For large distances the electrostatic field falls off exponentially and the best models for the elementary particles will depend upon just how slow this fall off must be to fit observation. The theory also requires that in regions where the field is small the currant vector must not vanish since it also acts as the electromagnetic potential. A finite mass for the photon is predicted, the precise value being inversely proportional to the effective range of the electromagnetic field. For ranges of the order of 1000 Km it is approxinately 10-44gms. Finally the field equations are found to require a slowly moving n-particle system to be subject to the usual Newtonian - Lorents force terms.