"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

15 Research in Italy

15.1 Introduction

In previous sections, we already have encountered several contributions from Italian researchers. The publication of the 3rd and 4th edition of Einstein’s The Meaning of Relativity in an Italian translation in 1950 and 1953268 seems to have given a boost to research on UFT in Italy. Bruno Finzi went about a fresh derivation of the field equations (287*) to (290*). He started from the Lagrangian Einstein had used in the 4th Princeton edition of his book [156*], i.e., ∗∗ R ik [cf. Section 2.3.2, Eq. (83*)]. However, he did not proceed by varying Einstein’s Lagrangian with regard to the (asymmetric) metric but instead by varying its solenoidal and irrotational parts, separately [201]. While arriving at the correct result, his method is no less arbitrary than what Einstein had tried himself. As one of the major figures in research in the geometry of relativity and unified theories, Finzi became a favoured reviewer of UFT in Italy [473*, 202, 203*]. He made it very clear that the theory did not predict new empirical facts: “Until now, no prevision of verifiable new physical facts have emerged from this unified theory”,269View original Quote but remained a firm believer in Einstein’s unified theory:

“The charm of this theory lies in its generality, its simplicity, and, let’s say it clearly, in its beauty, attributes which the utmost Einsteinian synthesis possesses, more than any other noted today.” ([473], p. 306)270View original Quote

In this spirit, contributions of mathematicians like I. Gasparini Cattaneo (1920 – 2011) [75], or A. Cossu (1922 – 2005) [85, 86], and more or less formal mathematical manipulations by Italian researchers played an important part in the work on UFT. The other leading elder figure Maria Pastori and some of her students present a main example. Already in the 1930s, she had published on anisotropic and “conjugated” skew-symmetric tensors. Hattori’s paper sparked her interest: she intended to reduce his two assumptions concerning skew-symmetric tensors to one [480, 482, 481]. Now in the 1950s, she studied the properties of the new tensorial objects appearing within UFT [483, 484, 485]. Elisa Brinis considered parallel transports conserving the scalar product of two vectors with a non-symmetric fundamental tensor [61]. B. Todeschini arrived at an inhomogeneous d’Alembert’s equation in which the electromagnetic tensor coupled to torsion [609]. F. Graiff derived expressions for the commutation of the ±-derivatives; she also studied alternative forms for the electromagnetic tensor (kij,R [ij], or its duals) in first and second approximation according to the scheme (499*) given below [232, 233]. By adding a term to the Einstein–Straus Lagrangian depending only on the metric tensor, F. de Simoni showed how to trivially derive from a variational principle all the different systems of field equations of UFT including those suggested by Bonnor, or Kursunŏglu (cf. Section 13.1) [115]. Laura Gotusso generalized a theory suggested by Horváth with a Riemannian metric and the connection l l l Lik = L (ik) + F [i jk] where jk is the electrical current vector ji = σ dxi ds. The torsion vector thus is proportional to the Lorenz force S = 1F sj i 2 i s and the autoparallels describe the motion of a charged point particle271 [285*]. Gotusso generalized Horváth’s theory by adding another tensor to the connection: Got l l l l L ik = L (ik) + F [i jk] + U ik satisfying l ik U ik F = 0. With regard to this connection gi+k+∥l = 0 [231]. The Ricci tensor belonging to the connection introduced was not calculated.

15.2 Approximative study of field equations

To P. Udeschini we owe investigations closer to physics.272 In a series of papers, he followed an approximative approach to the field equations of UFT by an expansion of the fundamental tensor starting from flat space:

gij = ηij + bij + cij, (499 )
where b ,c ij ij were assumed to be small of 1st and 2nd order. In linear approximation, the connection then read [652] as:
1 L k= 1ηkr(brj,i + bir,j − bji,r), (500 ) ij 2
whence follow the Eqs. (210*) and (211*) already obtained by Einstein and Straus. The field equations split into two groups related either to the gravitational potential (b(ij)) or to the electromagnetic field Fij ∼ b[ij]. In the identification by Udeschini, i.e., ijrs ij 𝜖 b[rs] =: ψF, where ψ is a constant, the field equations then were:
□b = 0, □b = 0, b ηrs = 0. (501 ) (ij) [ij] [ir],s
The coordinate conditions b(ir),sηrs − brs,iηrs = 0 was used. In this approximation, the current density, Iˆk = 1𝜖klmngmn,l 2 satisfies □ ˆIk = 0. In 2nd approximation ([653, 654, 657]), (500*) is replaced by:
2 1 1 1 1 1 L ijk = L ijk + -ηkr(crj,i + cir,j − cji,r) − ηkr(b[sj]L rsi + b[is]L jrs+ b(rs)Lijs ). (502 ) 2
For cij, the field equations lead to an inhomogeneous wave equation □c(ij) = B (ij) with a lengthy expression for Bij built up from products of bij and 1 Lijk as well as squares of 1 L ijk. Further equations are c[ir],sηrs = Ni and □ 𝜖klmnc [mn ],l = 𝜖klmnB [mn ],l, with Ni being a sum of products of brs and brs,t. For the electrical 4-current density, in 2nd approximation [658] 2 □ Ik = 2𝜖klmnηrsηpqb[rm],p(b(sn),ql − b(nq),sl). As a result, in the 2nd approximation the field equations for the gravitational and electromagnetic fields now intertwine. From his more general approach, Udeschini then reproduced the special case Schrödinger had treated in 1951 [558], i.e., b(ij) = 0,c[ij] = 0: An electromagnetic field, small of first order, generates a gravitational field small of second order. The reciprocal case, i.e., a gravitational field small of first order cannot influence an electromagnetic field of second order [655].

From this approximative approach, Udeschini calculated an additional term for the shift of the frequency ν of a spectral line by the unified field due to g(00) = b00 + c00 with 2 2 2 c00 = ψ μ0H and H the polar magnetic field:

δν- 1- 1- 2 2 2 ν = − c2δU + 2ψ μ 0(δH ). (503 )
U is the Newtonian gravitational potential [656, 659*, 660*]. This result depends crucially on the interpretations for the gravitational and electromagnetic fields. If, in place of g(ij), the quantity lij is chosen to describe the gravitational field (potential), then the 2nd term with the magnetic field drops out of (503*) ([660], p. 446). Due to this and to further ambiguities, it makes no sense to test (503*); at best, the constant ψ eventually needed for other experiments could be determined.

L. Martuscelli studied the assignment of the electromagnetic tensor F ij to the quantity R [ij] [389]. In first approximation, 1 Fij = 2□b [ij], while in second approximation

1 1 Fij = -□b [ij] +--□c[ij] + A [ij] (504 ) 2 2
with a lengthy expression for A [ij] again containing derivatives of products of bij and 1 L k ij and of products of 1 L ijk. A. Zanella wrote down a formal scheme of field equations which in any order n of the approximation looked the same as in the 2nd order field equations. The r.h.s. term in, e.g., n n □ c(ij) = B (ij) contains combination of quantities obtained in all previous orders. Convergence was not shown [723].

15.3 Equations of motion for point particles

While Einstein refused to accept particles as singularities of the unified field, E. Clauser, P. Udeschini, and C. Venini in Italy followed Infeld (cf. Section 9.3.3) by assuming the field equations of UFT to hold only outside the sources of mass and charge treated as singularities:

“In the equations for the unified field, no energy-tensor has been introduced: only the external problem outside the sources of the unified field (masses and charges), assumed to be singularities, exists” ([659], p. 74).273View original Quote

As mentioned in Section 10.3.2, Emilio Clauser (1917 – 1986) used the method of Einstein & Infeld in order to derive equations of motion for point particles. In [81], he had obtained an integral formula for a 2-dimensional surface integral surrounding the singularities. With its help, Clauser was able to show that from Einstein’s weak field equations for two or more “particles” all classical forces in gravitation and electromagnetism (Newton, Coulomb, and Lorentz) could be obtained [82]. In his interpretation, g (ik) stood for the gravitational, g[ik] for the electromagnetic field.

He expanded the fundamental tensor according to:

2k 2k 2k g00 = 1 + Σk=1 λ α g 00, g(mn ) = − δmn + Σk=1 λ2kα2k 2gk(mn), k g[mn ] = Σk=1 λ2kβ2k g[mn], 2k+1 g(0m ) = Σk=1 λ2k+1α2k+1 g (0m), (505 ) 2k+1 2k+1 2k+1 g[0m ] = Σk=1 λ β g [0m ], (506 )
where m, n = 1,2, 3;λ ∼ 1 c with the vacuum-velocity of light c.274 In the n-th step of approximation, the field equations are:
n n n n n r R (ik) = 0, R[ik],l + R [kl],i + R [li],k = 0, L[ir] = 0. (507 )
In the symmetrical part of the fundamental tensor, only terms beginning with 4 λ contribute, in the skew-symmetric part terms from 3 λ on. The Newtonian and Coulomb forces exerted on on a “particle“ from the others, appear in the terms ∼ λ4 together with a force independent of distance and not containing the masses of the “particles“. After laborious calculations, the Lorentz force showed up in the terms ∼ λ6. This result is the very least one would have expected from UFT: to reproduce the effects of general relativity and of electrodynamics.

In a subsequent paper by Clauser, Einstein’s weak system for the field equations of UFT was developed in every order into a recursive Maxwell-type system for six 3-vectors corresponding to electric and magnetic fields and intensities, and to electric and magnetic charge currents [83]. Quasi-stationarity for the fields was assumed.

C. Venini expanded the weak field equations by help of the formalism generated by Clauser and calculated the components of the fundamental tensor gik, or rather of γik := (g(ik) − ηik − 1ηikηrs(g(rs) − ηrs) 2, directly up to 2nd approximation: 4 4 γik,γ00, and 5 γ0m [672]. He applied it to calculate the inertial mass in 2nd approximation and obtained the corrections of special and general relativity; unfortunately the contribution of the electrostatic field energy came with a wrong numerical factor [673]. He also calculated the field of an electrical dipole in 2nd approximation [674]. Moreover, again by use of Clauser’s equation of motion, Venini derived the perihelion precession for a charged point particle in the field of a second one. It depends on both the charges and masses of the particles. However, his formula is not developed as far as that it could have been used for an observational test [675]. In hindsight, it is astonishing how many exhausting calculations Clauser and Venini dedicated to determining the motion of point particles in UFT in view of the ambiguity in the interpretation and formulation of the theory.

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