On Two Early Unified Field Theories

In this article we shall compare two unified field theories namely, Einstein's first presentation of a four dimensional theory (refs. 1, 2) and Levi-Civita's reformulation of it (ref. 3). It is well known that the problem of a unified field theory (of macroscopic world) is to find a suitable entity within a proposed geometry to represent the electromagnetic field. Gravitation is usually built into the theory since most of theories utilising a four dimensional space time as a model of the physical world consist in a direct extension of General Relativity. Although Einstein himself abandoned later his original choice of an electromagnetic vector potential, our remarks are prompted by two considerations. First, no unified field theory has ever been 'disproved'. In view of the numerous attempts to formulate one it is likely that if one should eventually be accepted, it will in some sense contain the ideas previously discussed. Secondly, we wish to consider the assertion that Einstein's and Levi Civita's theories are equivalent. This has been made by the latter Author and we may find a similar statement in a recent work of Mme Tonnelat (ref. 4, pp. 280—81). Two theories may be regarded as equivalent either if they lead to the same field equations or if the basic geometry of the space time is the same in both. It is clear of course, that the latter sense does not apply in our case. Levi Ci vita's theory employs an ordinary Riemann geometry while Einstein's is, from the outset, non riemannian. We shall show below that the first equivalence does not hold either. At least the electromagnetic field is basically distinct in the two theories. Thus, any similarity between them derives only from the first approximation in which a wave equation for the field quantities is obtained in both.