Additional Contributions from Japan

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"On the History of Unified Field Theories. Part II. (ca. 1930 – ca. 1965)"

Hubert F. M. Goenner

	Abstract
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
	Acknowledgements
	References
	Footnotes
	Biographies

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:

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

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 ].