"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

16 The Move Away from Einstein–Schrödinger Theory and UFT

Toward the end of the 1950s, we note tendencies to simplify the Einstein–Schrödinger theory with its asymmetric metric. Moreover, publications appear which keep mixed geometry but change the interpretation in the sense of a de-unification: now the geometry is to house solely alternative theories of the gravitational field.

Examples for the first class are Israel’s and Trollope’s paper ([308*] and some of Moffat’s papers [440, 441]. In a way, their approach to UFT was a backward move with its use of a geometry Einstein and Schrödinger had abandoned.

In view of the argument demanding irreducibility of the metric, Israel and Trollope returned to a symmetric metric but kept the non-symmetric connection:

“If, then, group-theoretical considerations are accepted as a basic guiding principle in the construction of a unified field theory, it will be logically most economical and satisfactory to retain the symmetry of the fundamental tensor gik, while admitting non-symmetrical Γijk.” ([308*], p. 778)

The Lagrangian was extended to contain terms quadratic in the curvature tensor as well:

--- β„’ = √ − g(a K ijgij + b (K )2 + c K (ij)K (kl)gikgjl + d K [ij]K [kl]gikgjl), (508 ) − − − − − −
where gij = g(ij) and K = K ijgij − −; a,b,.... arbitrary constants. The electromagnetic tensor is identified with K− [ij], and L [iss] = Si “corresponds roughly to the 4-potential”. The field equations, said to follow by varying ij g and k L [ij] independently, are given by:
ij √ --- rs δ(β„’) = ˆs δK− (ij) + − g Wrsδg , (509 )
ˆsij = √ −-g [(a + 2b K )gij + 2c K [ij] + 2dK (ij)], (510 ) − − − 1- 1- Wij = a (K− (ij) − 2K− gij) + 2bK− (K− (ij) − 4K− gij) − 2c Mij(K− [rs]) − 2d Mij(K− (rs)). (511 )
Mij with ij Mijg = 0 is the Maxwell energy-momentum-tensor calculated as if its argument were the electromagnetic field. For b = c = d = 0,Wij reduces to the Einstein tensor. If sij is defined by ∘ ---------- sirˆsjr = δj − det(sˆjr) i, an interpretation of sij as the metric suggests itself. It corresponds to the definition of the metric by a variational derivative in the affine theories of Einstein and Schrödinger.

If a ⁄= 0,b = 0,c = 0 is assumed, and Schrödinger’s star-connection (232*) introduced, the field equations of Israel & Trollope reduce to the system:

∗ is ∗ [is] ∇s ˆs = 0, Si( L) = 0,ˆs ,s = 0, (512 ) K (ij) = 2dMij (K [rs]), K = 0. (513 ) − − −

In the lowest order of an expansion g = η + πœ– γ ij ij ij, it turned out that the 3rd equation of (512*) becomes one of Maxwell’s equations, i.e., is γ ,s = 0, and the first equation of (513*) reduces to K− (ij) = 0. In an approximation up to the 4th order, the Coulomb force and the equations of motion of charged particles in a combined gravitational and electromagnetic field were obtained.

16.1 Theories of gravitation and electricity in Minkowski space

Despite her long-time work on the Einstein–Schrödinger-type unified field theory, M.-A. Tonnelat no longer seems to have put her sole trust in this approach: at the beginning of the 1960s, in her research group a new topic was pursued, the “Euclidean (Minkowskian) theory of gravitation and electricity”, occasionally also named “theory of the graviton” [411*]. In fact, she returned to the beginning of her research carrier: The idea of describing together quanta of spin 0, 1 and 2 in a single theory, like the one of Kaluza–Klein, about which she already had done research in the 1940s [616, 617] (cf. Section 10.1), seems to have been a primary motivation, cf. [638*, 352]; in particular a direct analogy between vector and tensor theories as basis for a theory of gravitation. Other reasons certainly were the quest for an eventual quantization of the gravitational field and the difficulties with the definition of a covariant expression for energy, momentum and stresses of the gravitational field within general relativity [644*]. Tonnelat also may have been influenced by the continuing work concerning a non-standard interpretation of quantum mechanics in the group around de Broglie. In the context of his suggestion to develop a quantum mechanics with non-linear equations, de Broglie wrote Einstein on 8 February 1954:

“Madame Tonnelat, whose papers on the unitary theories you know well, is interested with Mr. Vigier275 and myself in these aspects of the quantum problem, which of course are very difficult.”276

As mentioned by Tonnelat, the idea of developing a theory of gravitation with a scalar or vector potential in Minkowski space went back to the first decade of the 20th century277 [641*]. At the same time, in 1961, when Tonnelat took up the topic again, W. Thirring investigated a theory in which gravitation is described by a tensor potential (symmetric tensor of rank 2) in Minkowski space. The allowed transformation group reduces to linear transformations, i.e., the Poincaré group. He showed that the Minkowski metric no longer is an observable and introduced a (pseudo-)Riemannian metric in order to make contact with physical measurement [603]. This was the situation Tonnelat and her coworkers had to deal with. In any case, her theory was not to be seen as a bi-metric theory like N. Rosen’s [515, 516], re-discovered independently by M. Kohler [335, 336, 337], but as a theory with a metric, the Minkowski-metric, and a tensor field (potential) describing gravitation [638]. Seemingly, without knowing these approaches, Ph. Droz-Vincent suggested a bi-metric theory and called it “Euclidean approach to a metric” in order to describe a photon with non-vanishing mass [127]. In view of the difficulties coming with linear theories of gravitation, Tonnelat was not enthusiastic about her new endeavour ([641*], p. 424):

“[…] a theory of this type is much less natural and, in particular, much less convincing than general relativity. It can only arrive at a more or less efficient formalism with regard to the quantification of the gravitational field.”278View original Quote

A difficulty noted by previous writers was the ambiguity in choosing the Lagrangian for a tensor field. The most general Lagrangian for a massive spin-2 particle built from all possible invariants quadratic and homogeneous in the derivatives of the gravitational potential, can be obtained from a paper of Fierz and Pauli by replacing their scalar field C with the trace of the gravitational tensor potential: C = α ψss, α a proportionality-constant ([196], p. 216).279 Without the mass term, it then contained three free parameters α, a ,a 1 3. After fixing the constants in the Pauli–Fierz Lagrangian, Thirring considered:

1- pq,r rq,p ,p r,q r,q s 1- 2 pq r s L = 2[ψpq,rψ − 2ψpq,rψ + 2ψpq ψ r − ψr ψ s,q] − 2 M [ψpqψ − ψrψ s], (514 )
where M denotes a mass parameter.

Tonnelat began with a simpler Lagrangian:280

--1-- 1- pq,r pq,r M--2 pq √ −-gβ„’ = 4[ψpq,rψ − ψrq,pψ ] − 2 ψpqψ (515 )
where ψpq is the gravitational potential. A more general Lagrangian than (515*) written up in further papers by Tonnelat and Mavridès with constants a,b,c, and the matter tensor M pq [412, 640] corresponds to an alternative to the Pauli–Fierz Lagrangian which is not ghost-free:281
1- pq,r a- ,r ps ,p r,q c- r,q s pq L = 4 ψpq,rψ + 2 ψpr ψ ,s + b ψ pq ψr + 2 ψr ψ s,q] − χM ψpq. (516 )
The field equations of the most general case are easily written down. They are linear wave equations
β–‘ ψpq + linpq(∂r ∂sψmn, a,b,c) = χ Tpq, (517 )
with a tensor-valued linear function lin of its argument ∂ ∂ ψ r s mn also containing the free parameters. Tpq(= Mpq ) is the (symmetric) matter tensor for which, from the Lagrangian approach follows ∂rT kr = 0. However, this would be unacceptable with regard to the conservation law for energy and momentum if matter and gravitational field are interacting; only the sum of the energy of matter and the energy of the tensor field ψmn must be conserved:
∂ (T kr + tkr) = 0. (518 ) r
The so-called canonical energy-momentum tensor of the ψ-field is defined by
i -∂-β„’-- i tk := ∂ψ ψrs,k − δkβ„’, (519 ) rs,i
and is nonlinear in the field variable ψ. For example, if in the general Lagrangian (516*) a = 1,b = − 1,c = 0 are chosen, the canonical tensor describing the energy-momentum of the gravitational field is given by ([73], Eq. (1.3), p. 87)282:
1 1 1 1 tik = -∂iψrs∂kψrs − --ηik∂m ψrs∂m ψrs − -∂iψ ∂kψ + --ηik∂m ψ ∂mψ, (520 ) 2 4 4 8
where ψ := ηrsψrs. As a consequence, (517*) will have to be changed into
β–‘ ψpq + linpq(∂r∂sψmn, a,b,c) = χTpq + tpq, (521 )
which is a nonlinear equation. It is possible to find a new Lagrangian from which (521*) can be derived. This process can be repeated ad infinitum. The result is Einstein’s theory of gravitation as claimed in [238]. This was confirmed in 1968 by a different approach [118] and proved – with varying assumptions and degrees of mathematical rigidity – in several papers, notably [117] and [684]. In view of this situation, the program concerning linear theories of gravitation carried through by Tonnelat, her coworker S. Mavridès, and her PhD student S. Lederer could be of only very limited importance. This program, competing more or less against other linear theories of gravitation proposed, led to thorough investigations of the Lagrangian formalism and the various energy-momentum tensors (e.g., metric versus canonical). The (asymmetrical) canonical tensor does not contain the spin-degrees of freedom of the field; their inclusion leads to a symmetrical, so-called metrical energy-momentum tensor [17]. Which of the two energy-momentum-tensors was to be used in (521*)? The answer arrived at was that the metrical energy-momentum-tensor tensor must be taken [647*, 355, 354*].283

“In an Euclidean theory of the gravitational field, the motion of a test particle can be associated to conservation of mass and energy-momentum only if the latter is defined through the metrical tensor, not the canonical one” [647], p. 373).284View original Quote

Because (518*) is used to derive the equations of motion for particles or continua, this answer is important. In the papers referred to and in further ones, equations of motion of (test-) point particle without or within (perfect-fluid-)matter were studied . Thus, a link of the theory to observations in the planetary system was established [411, 413, 414]. In a paper summing up part of her research on Minkowskian gravity, S. Lederer also presented a section on perihelion advance, but which did not go beyond the results of Mme. Mavridès ([354], pp. 279–280). M.-A. Tonnelat also pointed to a way of making the electromagnetic field influence the propagation of gravitational waves by introducing an induction field ˆpq,r F for gravitation [639*]. In the presence of matter, she defined the Lagrangian

√--- ---−-η-- psqt rv pq β„’mat = 4 πœ–′(μ′)2b b b ψpq,rψst,v + χψ Mpq (522 )
πœ–′μ ′ − 1 bpq := ηpq + (πœ–′μ′ − 1)upuq, bpq = ηpq − ---′-′-upuq. (523 ) πœ–μ
In (523*) up is the 4-velocity of matter and πœ–′,μ′ constants corresponding now to a gravitational dielectric constant and gravitational magnetic permeability. The gravitational induction was ˆpq,r ∂β„’mat F = 2∂ψpq,r and the field equations became:
rp,q pq ∂rF = χM . (524 )

As M.-A. Tonnelat wrote:

“These, obviously formal, conclusions allow in principle to envisage the influence of an electromagnetic field on the propagation of the ‘gravitational rays’, i.e., a phenomenon inverse to the 2nd effect anticipated by general relativity” ([639], p. 227).285View original Quote

Tonnelat’s doctoral student Huyen Dangvu worked formally closer to Rosen’s bi-metric theory [107]. In the special relativistic action principle ∫ 1∫ δ (− mc ds + c β„’ ) = 0, he replaced the metric ηij by a metric gij containing the gravitational field tensor Ο•ij: gij = ηij + Ο•ij. This led to

∫ ∫ ∘ ----i-j 1- 4 δgI = 0;I = − μ0c ds giju u + c d xβ„’ (gij,∂kgij,∂k∂lgij), (525 )
from which one group of field equations followed:
i j L = μ c2√-----uu-------, (526 ) ij 0 1 + 2ψrsurus
with --1- {-∂β„’- (--∂β„’--- ---∂β„’----) } Lij = √ −g ∂Ο•ij − ∂r ∂(∂rΟ•ij) + ∂r∂s(∂(∂r∂sΟ•ij)) and j dxj i j u = ds ,giju u = 1. The second group of field equations is adjoined ad hoc (in analogy with Maxwell’s equations:
r r g girΔ jk + gjrΔik = 2∇k ψij, (527 )
where Δ ijk = 1gks(∂igjs + ∂jgis − ∂sgij) 2. No further consequences were drawn from the field equations of this theory of gravitation in Minkowski space called “semi-Einstein theory of gravitation” after a paper of Painlevé of 1922, an era where such a name still may have been acceptable.

In the mid-1960s, S. Mavridès and M.-A. Tonnelat applied the linear theory of gravity in Minkowski space to the two-body problem and the eventual gravitational radiation sent out by it. Havas & Goldberg [241] had derived as classical equation of motion for point particles with inertial mass mA, A = 1,2, ...,n and 4-velocity i uA:

r 11 2 r i l m ret mA ηiruΛ™A + --Gm Aηir(¨uA − uA ηlmu Au¨A) = fi(∂mg rs ), (528 ) 3
where fi is a functional of the derivatives of the retarded potential. The second term on the left hand side led to self-acceleration. In a calculation by S. Mavridès in the framework of a linear theory in Minkowski space with Lagrangian:
L = ψ ψpq,r + k ψr,qψs , (529 ) pq,r r s,q
the radiation-term was replaced by
2 10-+-42k-χc--κ2M 2η (¨ur − ui η ul ¨um), (530 ) 3(1 + k) 16π A ir A A lm A A
with χ, κ coupling constants and k a numerical constant, MA is connected with gravitational mass [415]. No value of k can satisfy the requirements of leading to the same radiation damping as in the linear approximation of general relativity and to the correct precession of Mercury’s perihelion. By proper choice of k, a loss of energy in the two-body problem can be reached. Thus, in view of the then available approximation and regularization methods, no uncontested results could be obtained; cf. also [416]; [643], pp. 154–158; [644], pp. 86–90).

16.2 Linear theory and quantization

Together with the rapidly increasing number of particles, termed elementary, in the 1950s, an advancement of quantum field theories needed for each of the corresponding fundamental fields was imperative. No wonder then that the quantization of the gravitational field to which particles of spin 2 were assigned also received attention. Seen from another perespective: The occupation with attempts at quantizing the gravitational field in the framework of a theory in Minkowski space reflected clearly the external pressure felt by those busy with research in UFT. Until then, the rules of quantization had been successful only for linear theories (superposition principle). Thus, unitary field theory would have to be linearized and, perhaps, loose its geometrical background: in the resulting scheme gravitational and electromagnetic field become unrelated. The equations for each field can be taken as exact; cf. ([641*], p. 372). For canonical quantization, a problem is that manifest Lorentz-invariance usually is destroyed due to the definition of the canonical variable adjoined to the field ψij:

--∂L--- --∂L-- Πij := ∂(∂ψij), Π := ∂ (∂ψ), ∂t ∂t
where we referred to Tonnelat’s Lagrangian (529*) with k = − 12,ψ = ψrr,t = 1cx0.

A. Lichnerowicz used the development of gravitational theories in Minkowski space during this period for devising a relativistic method of quantizing a tensor field H αβ,λμ(x) simulating the properties of the curvature tensor.286 In particular, the curvature tensor was assumed to describe a gravitational pure radiation field such that

α ρ l H αβ,λμ = 0, l{αH βγ},λμ = 0, H βμ := H β,ρμ = 0. (531 )
l α is a null vector field tangent to the lightcone l dxαdx β = 0 αβ. Indices are moved with l α β and lαβ where βρ β lαρl = δα; cf. (4*) and Section 10.5.3. Let K αβ,λμ(l) be the Fourier transform of Hαβ,λμ(x) and build it up from plane waves:
→ K αβ,λμ(l) = ΣA,B a(A,B, l )(lαnAβ − lβnAα)(lλnBμ − lμnBλ ), (532 )
where A,B = 1,2 and A nα are spacelike orthogonal and normed vectors in the 3-space touching the lightcone along lα. The amplitudes → a(A, B, l ) are then replaced by creation and annihilation operators satisfying the usual commutation relations [373], ([382*], pp. 127–128). Lichnerowicz’ method served as a model for his and M.-A. Tonnelat’s group in Paris. We are interested in this formalism in connection with Kaluza–Klein theory as a special kind of UFT.

The transfer to Kaluza–Klein theory by Ph. Droz-Vincent was a straightforward application of Lichnerowicz’ method: in place of (532*):

→ A A B B K αβ,λμ(l) = ΣA,B a(A,B, l )(lαn β − lβnα)(lλnμ − lμnλ ), (533 )
where now A, B = 0,1,2 and x0 is the 4th spacelike coordinate; the Greek indices are running from 0 to 4. In the tensor H αβ,λμ(x) in (1, 4)-space, through H α0,λμ(x) = β∂ αFλμ, β a constant, also the electromagnetic field tensor Fλμ is contained such that both, commutation relations for curvature and the electromagnetic field, could be obtained [129*]. In a later paper, β2 = 2χ was set with χ being the coupling constant in Einstein’s equations [133*]. The commutation relations for the electromagnetic field Fij were287:
[Fij(x),Flm (x′)] = Σ ηl[i∂j]mπ’Ÿ (x − x′). (534 )

This would have to be compared to the Gupta–Bleuler formalism in quantum electrodynamics.

For linearized JordanThiry theory, Droz-Vincent put [129*, 134]:

√ γγμν = ημν − K α μν μ, ν = 0,1,...,4 (535 )
for the metric density of V 5 and obtained the commutation relations:
′ 2 ′ [α σκ(x),αλμ(x )] = β (PσλPκμ + P σμPκλ)π’Ÿ (x − x ) (536 )
with P σλ = ησλ + 12-∂2- πœ– ∂σ∂λ and the mass parameter πœ–2 introduced into the Klein–Gordon equation but not following from the field equations. In space-time, from (536*):
√ -- ggij = ηij − K αij, βΟ•i = − K α0i (537 )
[Ο•i(x),Ο•j(x′)] = − K2Pij π’Ÿ (x − x′). (538 )
The relation to (534*) is provided by β ∂μFij = KH μ0ij. The tensor 2 2 2 2 Hαβλμ = − (∂ σλα κμ − ∂ σμακλ + ∂ κμα σλ − ∂ σλακμ) corresponds to H αβ,λμ in (531*).

S. Lederer studied linear gravitational theory also in the context of Kaluza–Klein-theory in five dimensions by introducing a symmetric tensor potential Ο• ,A, B = 0,1,...,4 AB comprising massive fields of spin 0, 1, and 2 ([353], pp. 381–283). For the quantization, she started from the linearization of the 5-dimensional metric in isothermal coordinates γAB = ηAB + kAB, ∂5kAB = 0, and the relation kAB = Ο•AB + aμ2∂ABΦ (0) + bηAB Φ(0), where a, b are parameters with 1 + a + 3b = 0, (0) MN Φ = η Ο•MN, and 2 μ ⁄= 0 is connected to the mass of the field.288 The Ο•AB were expressed by creation- and annihilation operators KAB (q),K ∗CD (q) and expanded in terms of an orthonormal tetrad (r) nS with (r) (r) 1- Σrn A n B = μ2qAqB − ηAB tangential to the 4-dimensional surface qRqR − μ2 = 0,q5 = 0, i.e., KAB = ΣrsC (r,s,q)n (rA)n(Bs). The C (r,s,q) were assumed to be self-adjoined operators with commutation relations [C ∗(ij,q),C(lm, q′)] = (δim δjl + δjmδil − &tidle;bδijδlm )δ (qRqR − μ2). &tidle;b is a new numerical parameter. The commutation relations for the fields then were calculated to have the form:

[Ο• (x ),Ο• (x ′)] = (P P + P P − &tidle;bP P )π’Ÿ(x − x ′) (539 ) AB CD AC BD AD BC AB CD
with 2 PAB = ηAB + μ12∂x∂A∂xB- and the Pauli–Jordan distribution π’Ÿ (x − x ′). (539*) translated into
′ [kAB (x),kCD (x )] = 3 − d 2 3 ∂4 ′ (PAC PBD + PADPBC − -----PABPCD − -d (-4---A---B---C---D + ηAB ηCD )π’Ÿ(x − x ) (540 ) 6 9 μ ∂x ∂x ∂x ∂x
and is independent of a if d = (2&tidle;b − 1)(1 + 4a)2 holds. The paper of S. Lederer discussed here in some detail is one midway in a series of contributions to the quantization of the linearized JordanThiry theory begun with the publications of C. Morette-Dewitt & B. Dewitt, [448, 449], continued by Ph. Droz-Vincent [129, 128*],289 and among others by A. Capella [72] and Cl. Roche [511].

These papers differ in their assumptions; e.g., Droz-Vincent worked with the traceless quantity k − 1η ηMN k AB 2 AB MN; for a = 0,&tidle;b = 0, and thus for d = − 1 his results agree with those of S. Lederer. In his earlier paper, A. Capella had taken MN η kMN = 0, and μ = 0. Claude Roche applied the methods of Ph. Droz-Vincent to the case of mass zero fields and quantized the gravitational and the electromagnetic fields simultaneously.

16.3 Linear theory and spin-1/2-particles

With the progress in elementary particle theory, group theory became instrumental for the idea of unification. J.-M. Souriau was one of those whose research followed this line. His unitary field theory started with a relativity principle in 5-dimensional space the underlying group of which he called “the 5-dimensional Lorentz group” but essentially was a product of the 4-dimensional Poincaré group with the group O2 of 2-dimensional real orthogonal matrices. For its infinitesimal generator A, exp(2πA ) = 1 holds wherefrom he introduced the integer n by A2 = − n2. He interpreted − n ≤ 0 as the electric charge of a particle and brought charge conjugation (x5 → − x5) and antiparticles into his formalism [582]. Souriau also asked whether quantum electrodynamics could be treated in the framework of Thiry’s theory, but for obvious reasons only looked at wave equations for spin-0 and spin-1/2 particles. As a result, he claimed to have shown the existence of two neutrinos of opposite chirality and maximum violation of parity in β-decay [583]. By comparing the (inhomogeneous) 5-dimensional wave equation for solutions of the form of the Fourier series +∞ 1 4 5 Ο• = Σ n=− ∞Ο•n (x ,...,x )exp (inx ) with the Klein–Gordon equation in an electromagnetic field, he obtained the spectrum of eigenvalues for charge q and mass m:

∘ ------- n2- q = nk ℏ, m = ℏ ξ2 + a, (541 )
where ∘ --- k := 1ξ 2χπ, χ = 8πcG2-- is the gravitational constant in Einstein’s equations, and a the “mass”-term of the 5-dimensional wave equation, i.e., a free parameter. ξ is the scalar field: ξ2 = − g55. For q = e,n = 1, ξ is of the order of magnitude −32 ≃ 10 cm. Souriau also rewrote Dirac’s equation in flat space-time of five dimensions as an equation in quaternion space for 2 two-component neutrinos. His interpretation was that the electromagnetic interaction of fermions and bosons has a geometrical origin. The charge spectrum is the same as for spin-0 particles except that the constant a in (541*) is replaced by − a2.

O. Costa de Beauregard applied the linear approximation of Souriau’s theory for a field variable Hαβ to describe the equations for a spin-1/2 particle coupled to the photon-graviton system. He obtained the equation α ℏχ¯ ∂ α∂ H5k = − 2ik5 c Ο•γkΟ• (α, β = 1,2,...,5;k = 1,2,...,4), where the wave function Ο• again depends on the coordinate 5 x via 5 exp(k5x ); as before, χ is the coupling constant in Einstein’s field equations. Comparison with electrodynamics led to the identification k5 = ℏe√2χ with e the electric charge. Costa de Beauregard also suggested an experimental test of the theory with macroscopic bodies [91].

16.4 Quantization of Einstein–Schrödinger theory?

Together with efforts at the quantization of the gravitational field as described by general relativity, also attempts at using Einstein–Schrödinger type unified theories instead began. Linearization around Minkowski space was an obvious possibility. But then the argument that the cosmological constant had appeared in some UFTs (Schrödinger) lead to an attempt at quantization in curved space-time. In the course of his research, A. Lichnerowicz developed a method of expanding the field equations around both a metric and a connection which are solutions of equations describing a fixed geometric backgrond [375*]. Quantization then was applied to the quantities varied (semi-classical approximation). The theory was called “theory of the varied field” by Tonnelat [641*], p. 441).290 Lichnerowicz determined the “commutators corresponding to vector meson and to an electromagnetic field (spin 1) on one hand and to a microscopic gravitational field (spin 2, mass 0) on the other hand […] in terms of propagators” [378]. The linearization was obtained by looking at field equations for the varied metric and connection. Let Ψij := δgij be such a variation of the metric gij and X ijk := δLijk a variation of the connection L ijk. It is straightforward to show that the variation of the Ricci tensor is δRij(L ) = ∇sX s− ∇jX s= (− δK ij(L)) ij is −, where the covariant derivative is taken with regard to the connection formed from gij. Also, 2X ijk = ∇j Ψ ik+ ∇i Ψkj − ∇k Ψji. Ph. Droz-Vincent then looked at field equations for a connection with vanishing vector torsion and with Einstein’s compatibility equation (30*) varied, i.e., δg = 0 i+k−βˆ₯l:

2 Rij − λgij =--(∂iΓ j − ∂jΓ i) (542 ) 3
with cosmological constant λ and arbitrary Γ k [131].291 The Riemannian metric which is varied solves Rij − λgij = 0 (Einstein space). Droz-Vincent showed that the variation Ψ ij must satisfy the equations:
∇rΨ = 0 , (543 ) [rj] (Δ + 2λ)Ψ (ij) = ∇ikj + ∇jki, (544 ) 8 (D + 2 λ)Ψ[ij] = -∂ [iδΓ j], (545 ) 3
where D is a differential operator different from the Laplacian Δ for the Riemannian metric introduced by Lichnerowicz ([375], p. 28) such that ∇D ⁄= D ∇. ki = ki(Ψ) is defined by ki(Ψ ) = ∇rΨri − 1∇iΨ s 2 s (indices moved with gij). Equation (543*) follows from the vanishing of vector torsion.

Difficulties arose with the skew-symmetric part of the varied metric. Quantization must be performed such as to be compatible with this condition. The commutators sugested by Lichnerowicz were not compatible with (397*). Droz-Vincent refrained from following up the scheme because:

“The endeavour to establish such a program is, to be sure, a bit premature in view of the missing secure physical interpretation of the objects to be quantized.”292View original Quote

Ph. Droz-Vincent sketched how to write down Poisson brackets and commutation relations for the Einstein–Schrödinger theory also in the framework of the “theory of the varied field” ([130]. In general, the main obstacle for quantization is formed by the constraint equations, once the field equations are split into time-evolution equations and constraint equations. Droz-Vincent distinguished between proper and improper dynamical variables. The system +ij Rij = 0, Dk ˆg − = 0, where D signifies covariant derivation with respect to the star connection (27*), led to 5 constraint equations containing only proper variables arising from general covariance and λ-invariance. By destroying λ-invariance via a term ˆgrsΓ Γ r s, one of the constraints can be eliminated. The Poisson brackets formed from these constraints were well defined but did not vanish. This was incompatible with the field equations. By introducing a non-dynamical timelike vector field and its first derivatives into the Lagrangian, Ph. Droz-Vincent could circumvent this problem. The physical interpretation was left open [133, 135]. In a further paper, he succeeded in finding linear combinations of the constraints whose Poisson brackets are zero modulo the constraints themselves and thus acceptable for quantization [136].

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