Yano and M. Ohgane generalized the non-holonomic UFT to arbitray dimension: -dimensional space is a non-holonomic hypersurface of -dimensional Riemannian space. Hoffmann, J. The one who really made progress, although unintentionally and unnoticed at the time, was O.
Klein who extended Abelian gauge theory for a particular non-Abelian group, which almost corresponds to SU 2 gauge theory [ ]. In a letter to his friend H. Zangger in Zurich of 27 February , he wrote:.
We learn this from the letter of 8 August to his friend Besso:. He wanted to keep the four-dimensional space-time continuum and the diffeomorphism group as the covariance group, but wished to replace the Riemannian metric by a generalized concept. Unlike the concept of a bi-vector used by Schouten in [ ] , p. In the latter case, the coordinate transformations for each point are independent. In place of the Riemannian metric, a contravariant bi-vector is defined via.
Here, a symmetry condition is demanded:. Now, the tensorial p-point equations admit the special case that the two points coordinates coincide. By a suitable rimming operation was reached where is the matrix diag.
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The rimming operations were performed with representations of the Lorentz group. Also, mixed bi-vectors were introduced and a tensorial three-point equation had to be satisfied:. The mixed bi-vector is interpreted as the non-infinitesimal affine connection because the relation. In his paper, he set out to. In this note in Nature, Born added a Lorentzian metric in momentum space satisfying as well the corresponding Einstein field equations cf.
About a decade later, in , H.
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It seems that he saw his non-local field theory as introducing new degrees of freedom and leading a step away from the point-particle concept: a possibility for avoiding infinite self-energies. At this time, both Yukawa and Einstein lived in Princeton and got to know each other. In , Yukawa possibly had not the least inkling of the Einstein— Bargmann paper of Of course, Bargmann in Princeton then could have told him. He made only a cryptic remark about his non-local field theory when expressing the then prevailing ignorance about elementary particle theory:.
He then gave references to his papers on non-local field theory.
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In a letter of 23 January , Born informed Einstein also privately about his theory of elementary particles:. The laws of nature are invariant not only with regard to the relativistic transformations but also with regard to the substitutions […]. All amounts to replace your fundamental invariant by the symmetrical quantity where. Although some of those involved were in direct personal contact, no concrete evidence for a conscious transfer of ideas could be established. RSS News Contacts. Part II. Hide Subsections.
Abstract 1 Introduction 2 Mathematical Preliminaries 2.
Tonnelat Hubert F. Relativity , 17 , 5, doi This work is licensed under a Creative Commons License. Mathematical Preliminaries. Metrical structure. Affine geometry. Differential forms. Classification of geometries. Their form was artificially narrowed, the formulas became incorrect. This led to a logical contradiction of the theory, to unsolvable paradoxes. Your email address will not be published. Albert and Mileva Einstein, Related Posts. Carl von Ossietzky and Political Reason.
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Tweets by SciHiBlog. October 6. December December 5. Imprint Privacy Statement. Email Subscription. Relation Browser. Tonnelat of the Sorbonne reported on some mathematical results she had obtained on this theory independently of Einstein and Kaufman. In a later approach by M. It amounted to a mathematical discussion of all logically possible cases and subcases without furthering UFT as a physical theory; cf.
Unified field theory
Sections As a functional of the metric, its first and second derivatives, the Ricci tensor becomes a rather complicated expression. To then find exact solutions of the remaining field equation in is a difficult task. In a paper dealing with approximations of the field equations, M. Tonnelat tried to show the superiority of her method by applying a scheme of approximations to her weak field equations [ ]. However, the resulting equations of 4th order for weak electromagnetic fields k ij and of 1st order for weak gravitational fields h ij are still as complicated as to not allow a physical interpretation.
In fact, the solution of the problem to remove the connection from the field equations neither helped the search for exact solutions nor contributed to a convincing physical interpretation of the theory. Nonetheless, it was of crucial importance for the proofs given by A.
Lichnerowicz that the initial value problem could be well posed in UFT. For physics, no new insights were gained. In the latter, the total field itself defines its own sources. The expression of this tensor then would be imposed by the very geometric principles, and not by conclusions from an alien theory as interesting as it might be. Another approach for the introduction of the energy-momentum tensor of matter T ik is the following. First a symmetric metric must be be chosen, e.
Then in the symmetrical part of the Ricci tensor K ij L a term of the form of the Einstein tensor G ik h mn is separated out. The method is applicable because mixed geometry can always be re-interpreted as Riemannian geometry with many extra fields geometric objects. Its ambiguity lies in the choice of the Riemannian metric. By the formulation within a Riemannian geometry, the unifying strength of a more general geometry is given up, however.
Also, according to a remark by M. An advantage is that the variational principle ensures the existence of an identity [ ]. A disadvantage is that the Lagrangian density for the matter part depends not only on the metric but also on its derivatives of 1st and 2nd order. For the metric Treder took the symmetric part h ij of the asymmetric fundamental tensor g ij. In order to avoid infinitely many degrees of freedom for extended bodies, some limit procedure had to be introduced.
Clauser showed in great detail that the method is applicable there for charged particles [ 83 ]; cf. However, we have mentioned already in Section 9.