"On the History of Unified Field Theories. Part II. (ca. 1930 – ca. 1965)"
Hubert F. M. Goenner 
1 Introduction
2 Mathematical Preliminaries
2.1 Metrical structure
2.2 Symmetries
2.3 Affine geometry
2.4 Differential forms
2.5 Classification of geometries
2.6 Number fields
3 Interlude: Meanderings – UFT in the late 1930s and the 1940s
3.1 Projective and conformal relativity theory
3.2 Continued studies of Kaluza–Klein theory in Princeton, and elsewhere
3.3 Non-local fields
4 Unified Field Theory and Quantum Mechanics
4.1 The impact of Schrödinger’s and Dirac’s equations
4.2 Other approaches
4.3 Wave geometry
5 Born–Infeld Theory
6 Affine Geometry: Schrödinger as an Ardent Player
6.1 A unitary theory of physical fields
6.2 Semi-symmetric connection
7 Mixed Geometry: Einstein’s New Attempt
7.1 Formal and physical motivation
7.2 Einstein 1945
7.3 Einstein–Straus 1946 and the weak field equations
8 Schrödinger II: Arbitrary Affine Connection
8.1 Schrödinger’s debacle
8.2 Recovery
8.3 First exact solutions
9 Einstein II: From 1948 on
9.1 A period of undecidedness (1949/50)
9.2 Einstein 1950
9.3 Einstein 1953
9.4 Einstein 1954/55
9.5 Reactions to Einstein–Kaufman
9.6 More exact solutions
9.7 Interpretative problems
9.8 The role of additional symmetries
10 Einstein–Schrödinger Theory in Paris
10.1 Marie-Antoinette Tonnelat and Einstein’s Unified Field Theory
10.2 Tonnelat’s research on UFT in 1946 – 1952
10.3 Some further developments
10.4 Further work on unified field theory around M.-A. Tonnelat
10.5 Research by and around André Lichnerowicz
11 Higher-Dimensional Theories Generalizing Kaluza’s
11.1 5-dimensional theories: Jordan–Thiry theory
11.2 6- and 8-dimensional theories
12 Further Contributions from the United States
12.1 Eisenhart in Princeton
12.2 Hlavatý at Indiana University
12.3 Other contributions
13 Research in other English Speaking Countries
13.1 England and elsewhere
13.2 Australia
13.3 India
14 Additional Contributions from Japan
15 Research in Italy
15.1 Introduction
15.2 Approximative study of field equations
15.3 Equations of motion for point particles
16 The Move Away from Einstein–Schrödinger Theory and UFT
16.1 Theories of gravitation and electricity in Minkowski space
16.2 Linear theory and quantization
16.3 Linear theory and spin-1/2-particles
16.4 Quantization of Einstein–Schrödinger theory?
17 Alternative Geometries
17.1 Lyra geometry
17.2 Finsler geometry and unified field theory
18 Mutual Influence and Interaction of Research Groups
18.1 Sociology of science
18.2 After 1945: an international research effort
19 On the Conceptual and Methodic Structure of Unified Field Theory
19.1 General issues
19.2 Observations on psychological and philosophical positions
20 Concluding Comment

14 Additional Contributions from Japan

We already met Japanese theoreticians with their contributions to non-local field theory in Section 3.3.2, to wave geometry as presented in Section 4.3, and to many exact solutions in Section 9.6.1. The unfortunate T. Hosokawa showed, in the paper mentioned in Section 4.3, that “a group of motions of a Finsler space has at most 10 parameters” [287]. Related to the discussion about exact solutions is a paper by M. Ikeda on boundary conditions [299]. He took up Wyman’s discussion of boundary conditions at spacelike infinity and tried to formulate such conditions covariantly. Thus, both spacelike infinity and the approach to it were to be defined properly. He expressed the (asymmetric) metric by referring it to an orthormal tetrad tetrad gij → aAB = gijξiAξjB, where ξiA are the tetrad vectors, orthonormalized with regard to eAδAB, eA = ±1. The boundary condition then was aAB → eA δAB for ρ → ∞, where ∫ Q ∘ -----i--j ρ (P Q ) := P γijdu du and the integral is taken over a path from P to Q on a spacelike hypersurface k k 1 2 3 x = x (u ,u ,u ,σ ) with parameters i u and metric γ.

Much of the further research in UFT from Japan to be discussed, is concerned with structural features of the theory. For example, S. Abe and M. Ikeda generalized the concept of motions expressed by Killing’s equations to a non-symmetric fundamental metric [1]. Although the Killing equations (43*) remain formally the same, for the irreducible parts of gab = hab + kab, they read as:

h c h c h c h ℒ ξhab = 2∇ (aξˇb) = 0, ℒξkab = ξ ∇ckab + k b ∇aξˇc + ka ∇b ˇξc = 0. (497 )
In (497*), ˇξ = h ξr c cr. All index-movements are done with h ab. The authors derive integrability conditions for (497*); it turns out that for space-time the maximal group of motions is a 6-parameter group.267

Possibly, in order to prepare a shorter way for solving (30*), S. Abe and M. Ikeda engaged in a systematic study of the concomitants of a non-symmetric tensor gab, i.e., tensors which are functionals of gab [301*, 300]. A not unexpected result are theorems 7 and 10 in ([301], p. 66) showing that any concomitant which is a tensor of valence 2 can be expressed by hab,kab,karkr ,karkr ks b s b and scalar functions of g, g h k as factors. In the second paper, pseudo-tensors (e.g., tensor densities) are considered.

A different mathematical interpretation of Hoffmann’s meson field theory as a “unitary field theory” in the framework of what he called “sphere-geometry” was given by T. Takasu [598]. It is based on the re-interpretation of space-time as a 3-dimensional Laguerre geometry. The line element is re-written in the form

2 1 2 22 3 2 k s k 2 ds = (dξ ) + (dξ ) + (dξ ) (ϕs(x )dξ + ϕ4(x )dt) . (498 )
It can be viewed as “the common tangential segment of the oriented sphere with center ξk and radius r = ∫ [ϕsxk )dξs+ ϕ4 (xk )]dt dt …” ([599].

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