"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

17 Alternative Geometries

Although a linear theory of gravitation can be derived as a first approximation of Einstein–Schrödinger-type theories ([641*], pp. 441–446), the results may be interpreted not only in UFT, but also in the framework of “alternative” theories of gravitation. Nevertheless, M.-A. Tonnelat remained defensive with respect to her linear theory:

“ […] this theory does not pursue the hidden aim of substituting general relativity but of exploring in a rather heuristic way some specifically tough and complex domains resulting from the adoption of the principles of a non-Euclidean theory […]”. ([382], p. 327)293View original Quote

We will now describe theories with a different geometrical background than affine or mixed geometry and its linearized versions.

17.1 Lyra geometry

In Lyra geometry, the notion of gauge transformation is different from its use in Weyl geometry. Coordinate and gauge transformations are given the same status: they are defined without a metric or a connection [386, 531]. A reference system now consists of two elements: Besides the usual coordinate transformations xi → xi′ = xi′(x1, x2,...,xn) a gauge transformation x0′ = f0(x1,x2, ...,xn,x0(x1,x2,...,xn )) with gauge function f is introduced. The subgroup of coordinate transformations is given by those transformations for which 0 1 2 n 0 0 f (x ,x ,...,x ;x ) = x. A change of the reference system implies both coordinate and gauge transformations. Tangent vectors ξs under a change of coordinates and gauge transform like:

′ ∂xi′ ξi = λ ----ξs, (546 ) ∂xs
where 0′ λ = xx0- is the gauge factor (Lyra’s “Eichverhältnis”); a basis of tangential space is given by -10 ∂i- x ∂x; a 1-form basis would be x0dxi. The metric then is introduced by ds2 = grs(x0dxr )(x0dxs), and the asymmetric connection i 1 i Γ rs − 2δrϕs is defined via
1 dξi = − (Γ ris − -δirϕs)ξr(x0dxs ),Γ ris = Γsri. (547 ) 2
Here, similar to Weyl’s theory, an arbitrary 1-form ϕk appears and the demand that the length of a transported vector be conserved leads to
k 1 k 1 k k kr Γ ij = --0{ij} + -(δi ϕj + δjϕi − gijg ϕr) (548 ) x 2
with the Christoffel symbol calculated from gij. The curvature tensor is defined by
i 1 0 i 0 i 0 i 0 m 0 i 0 m − K+ jkl = --0-2[∂k(x Γ lj ) − ∂l(x Γ kj ) + x Γ km x Γlj − x Γ lm x Γkj ] (549 ) (x )
Hence the curvature scalar becomes K = --R- + -3ϕ;s + 3ϕsϕ + 20ϕ ϕi (x0)2 x0 ;s 2 s i where the semicolon denotes covariant derivation with regard to gij, and ϕ0 = -1 ∂log(x0)2- i x0 ∂xi.294 In the thesis of D. K. Sen, begun with G. Lyra in Göttingen and finished in Paris with M.-A. Tonnelat, the field equations are derived from the Lagrangian 0 4√ -- ℒ = (x ) gK. In the gauge 0 x = 1, they are given by [571]:
R − 1g R + 3ϕ ϕ − 3g ϕsϕ = − κT . (550 ) ik 2 ik i k 4 ik s ik
Weyl’s field equations in a special gauge are the same – apart from the cosmological term Λg ij. The problem with the non-integrability of length-transfer does not occur here. For further discussion of Lyra geometry cf. [572].

In relying on a weakened criterion for a theory to qualify as UFT suggested by Horváth (cf. Section 19.1.1), after he had added the Lagrangian for the electromagnetic field, Sen could interpret his theory as unitary. In later developments of the theory by him and his coworkers in the 1970s, it was interpreted just as an alternative theory of gravitation (scalar-tensor theory) [573, 574, 310]. In both editions of his book on scalar-tensor theory, Jordan mentioned Lyra’s “modification of Riemannian geometry which is close to Weyl’s geometry but different from it” ([319], p. 133; ([320], p. 154).

17.2 Finsler geometry and unified field theory

Already one year after Einstein’s death, G. Stephenson expressed his view concerning UFT:

“The general feeling today is that in fact the non-symmetric theory is not the correct means for unifying the two fields.”295View original Quote

In pondering how general relativity could be generalized otherwise, he criticized the approach by Moffat [439] and suggested an earlier attempt by Stephenson & Kilmister [591] starting from the line element [cf. (428*)]:

∘ ---------- ds = gjkdxjdxk + Aldxl, (551 )
the geodesics of which correctly describe the Lorentz force.

Since Riemann’s habilitation thesis, the possibility of more general line elements than those expressed by bilinear forms was in the air. One of Riemann’s examples was “the fourth root of a quartic differential expression” (cf. Clifford’s translation in [329], p. 113). Eddington had spoken of the space-time interval depending “on a general quartic function of the dx’s” ([139], p. 11). Thus it was not unnatural that K. Tonooka from Japan looked at Finsler spaces with fundamental form ∘ --------------- 3 aαβγdxαdx βdxγ [648], except that only 3∘ -------α--β---γ | aαβγdx dx dx | is an acceptable distance. With (551*), Stephenson went along the route to Finsler (or even a more general) geometry which had been followed by O. Varga [669*], by Horváth [283] and by Horváth & Moór, [286]. He mentioned the thesis of E. Schaffhauser-Graf [530] to be discussed below. In Finsler geometry, the line element is dependent on the direction of moving from the point with coordinates xi to the point with xi + dxi:

l m j k m dxm- ds = γjk(x ,x˙ ) dx dx , x˙ = du , (552 )
with u an arbitrary parameter. In the approach to Finsler geometry by Cartan [74], the starting point is the line element:296
ds = L (xl, ˙xm )du, (553 )
with L a homogeneous function of the velocity m x˙, and the Finsler metric defined by:
1-∂2(L2-) γjk = 2 ∂x˙j∂ ˙xk. (554 )
Quantities k C ij and k Lij acting as connections are introduced through the change a tangent 4-vector k X is experiencing:
DXk = dXk + L kXsdxt + C kXsd x˙s, (555 ) st st
where the totally symmetric object
s 1-∂3(L2)- Cijk = γksCij = γki,j + γkj,i + γij,k := 4∂ ˙xix˙j ˙xk, (556 )
transforms like a tensor. The asymmetric affine connection is defined by
∂Gp ∂Gp Lijk= {kij}(γ) + γkt(Ctip---- − Ctjp----), (557 ) ∂ ˙xj ∂ ˙xi
with γisγjs = δji. The “geodesic coefficient” 2 2 2 Gp = 14γpr[∂∂˙xr(L∂x)sx˙s − ∂(∂Lxr)] is resulting from the Euler–Lagrange equation d2xk k dt2 + 2G = 0 for L reformulated with the Finsler metric.297 Equation (555*) can also be written as
k k ∗ k s t DX = dX + Lst X dx , (558 )
with Cartan’s connection L∗ijk= Lijk− γktCtjpGpi.

Edith Schaffhauser-Graf at the University of Fribourg in Switzerland hoped that the various curvature and torsion tensors of Cartan’s theory of Finsler spaces would offer enough geometrical structure such as to permit the building of a theory unifying electromagnetism and gravitation. She first introduced the object met before , also known as “Cartan” torsion:

3 2 Sijk = 1-L∂-γik = 1L ∂--(L-)-= LCijk. (559 ) 2 ∂ ˙xj 4 ∂x˙i˙xjx˙k
and, by contraction with γkj, the “torsion” vector Si = 1L ∂-ln|iγ| 2 ∂x˙. With its help and its covariant derivative taken to be:
∂S ∂S ∂Gm Si|k = ---i− ---i----- − (Γikm − CmirΓ skr˙xs)Sm, (560 ) ∂xk ∂ ˙xm ∂ ˙xk
the electromagnetic field tensor is defined by:298
Fij := Sj|i − Si|j. (561 )
The form of the covariant derivative (560*) follows from the supplementary demand that the Finsler spaces considered do allow an absolute parallelism of the line elements. Schaffhauser-Graf used the curvature tensor 0. Varga had introduced in a “Finsler space with absolute parallelism of line elements” ([669], Eq. (37)), excluding terms from torsion. The main physical results of her approach are that a charged particle follows a “geodesic”, i.e., a worldline the tangent vectors of which are parallel. The charge experiences the Lorentz force, yet this force, locally, can be transformed away like an inertial force. Also, charge conservation is guaranteed. For vanishing electromagnetic field, Einstein’s gravitational theory follows. It seems to me that the prize paid, i.e., the introduction of Finsler geometry with its numerous geometric objects, was exorbitantly high.

Stephenson’s paper on equations of motion [590*], from which the quotation above is taken, was discussed at length by V. Hlavatý in Mathematical Reviews [External LinkMR0098611] reproduced below:

The paper consists of three parts. In the first part, the author points out the difference between Einstein–Maxwell field equations of general relativity and Einstein’s latest unified field equations. The first set yields the equations of motion in the form

{ } d2xν ν dxλ dxμ νdxμ ---2-+ λ μ --------+ F μ ----= 0 ds ds ds ds

(ν μ F μdx ∕ds = Lorentz’ force vector). Callaway’s application of the EIH method to the second set does not yield any Lorentz force and therefore the motion of a charged particle and of an uncharged particle would be the same.

In the second part, the author discusses the attempt to describe unified field theory by means of a Finsler metric

∘ ----------- ds = A λdxλ + gλμdx λdxμ,

which leads by means of ∫ δ ds = 0 to (1) with Fμλ = 2∂[λA μ]. However the tensor R μλ does not yield any appropriate scalar term which could be taken as Lagrangian. […]

Remarks of the reviewer: 1) If F μλ is of the second or third class (which is always the case in Callaway’s approximation) we could have Fν (dxμ∕ds ) = 0 μ in (1) for F ⁄= 0 μλ. 2) The clue to Callaway’s result is that four Einstein’s equations ∂{ωRμλ} = 0 do not contribute anything to the equations of motion of the considered singularities. If one replaces these four differential equations of the third order by another set of four differential equations of the third order, then the kinematical description of the motion results in the form

d2xν dx λdx μ ---2-+ Γ λνμ--------= 0. dq dq dq

There is such a coordinate system for which the first approximation of (2) is the classical Newton gravitational law and the second approximation acquires the form (1). […].

Stephenson dropped his plan to calculate the curvature scalar R from (557*) and (551*), and to use it then as a Lagrangian for the field equations, because he saw no possibility to arrive at a term R (g) + FstFst functioning as a Finslerian Lagrangian. His negative conclusion was: “It so seems that this particular generalization of Riemannian geometry is not able to lead to a correct implementation of the electromagnetic field.”299View original Quote

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