Affine Geometry: Schrodinger as an Ardent Player

6 Affine Geometry: Schrödinger as an Ardent Player

6.1 A unitary theory of physical fields

When in peaceful Dublin in the early 1940s E. Schrödinger started to think about UFT, he had in mind a theory which eventually would give a unitary description of the gravitational, electromagnetic and mesonic field. Mesons formed a fashionable subject of research at the time; they were thought to mediate nuclear interactions. They constituted the only other field of integral spin then known besides the gravitational and electromagnetic fields. Schrödinger had written about their matrix representations [546 *]. In his new paper, he deemed it “probable that the fields of the Dirac-type can also be accounted for. […] It is pretty obvious that they must result from the self-dual and self-antidual constituents into which the anti-symmetric part of $Rkl$ can be split.” ([545 *], p. 44, 57.) This is quickly constrained by another remark: “I do not mean that the new affine connection will be needed to account for the well-known Dirac fields”. ([545 *], p. 58.) He followed the tradition of H. Weyl and A. Eddington who had made the concept of affine connection play an essential role in their geometries – beside the metric or without any. He laid out his theory in close contact with Einstein’s papers of 1923 on affine geometry (cf. Section 4.3.2 of Part I) and the nonlinear electrodynamics of M. Born & L. Infeld [42 *] (cf. Section 5). On 10 May 1943, M. Born reported to Einstein about Schrödinger’s work: “[…] He has taken up an old paper of yours, from 1923, and filled it with new life, developing a unified field theory for gravitation, electrodynamics and mesons, which seems promising to me. […]” ([168 *], p. 194.) Einstein’s answer, on 2 June 1943, was less than excited:

“Schrödinger was as kind as to write to me himself about his work. At the time I was quite enthusiastic about this way of thinking. Its weakness lies in the fact that its construction from the point of view of affine space is rather artificial and forced. Also, the link between skew symmetric curvature and the electromagnetic states of space leads to a linear relation between electrical fields and charge densities. […]” ([168 *], p. 196.).⁷⁵

As I suppose, the “At the time” refers to 1923. With “skew symmetric curvature”, the antisymmetric part of the Ricci-tensor is meant. Schrödinger believed that Einstein had left affine theory because of “aesthetic displeasure” resulting from a mistake in his interpretation of the theory.

6.1.1 Symmetric affine connection

In his first papers on affine geometry, Schrödinger kept to a symmetric connection.⁷⁶ There is thus no need to distinguish between $+ ∇$ and $− ∇$ in this context. Within purely affine theory there are fewer ways to form tensor densities than in metric-affine or mixed geometry. By contraction of the curvature tensor, second-rank tensors $Kij$ and $Vij$ are available (cf. Section 2.3.1) from which tensor densities of weight $− 1$ (scalar densities) (cf. Section 2.1.5 of Part I) like $∘ --------- det(Kij)$ or $∘ -------- det(Vij)$ can be built. Such scalar densities are needed in order to set up a variational principle.

In his paper, Schrödinger took as such a variational principle:

thus neglecting homothetic curvature as a further possible ingredient.⁷⁷ $Kij (Γ )$ is the Ricci-tensor introduced in (55 *) or (56 *) due to $Γ$ being symmetric. Enthusiastically, he started at the point at which Einstein had given up and defined the symmetric and skew-symmetric quantities:

The variation of (153 *) with respect to the components of the connection $Γ k ij$ now can be written as:

Note that (155 *) is formally the same equation which Einstein had found in his paper in 1923 when taking up Eddington’s affine geometry [141 *]. A vector density $k ˆj$ is introduced via

with $ˆjk$ being interpreted as the (electric) current density. By help of $gˆik$ a (symmetric) metric tensor $gik$ is introduced by the usual relations:

(155 *) can be formally solved for the components of the symmetric connection to give:

with $j = (− g)− 12 g ˆjs, g = det(g ) k ks lm$ .⁷⁸ This expression is similar but unequal to the connection in Weyl’s theory (cf. Section 4.1.1 of Part I, Eq. (100)). The intention is to express $Γ ijk$ as a functional of the components of $Kik$ , insert the expression into (56 *) (for $L = Γ$ ), and finally solve for $Kik$ . What functions here as a metric tensor, is only an auxiliary quantity and depends on the connection (cf. (154 *), (157 *)).

In order to arrive at a consistent physical interpretation of his approach, Schrödinger introduced two variables conjugate to $ˆgik,fˆik$ by:

and carried out what he called a contact transformation:

From (160 *) we get:

With the new variables, Eqs. (155 *) and (156 *) may be brought into the form of the Einstein–Maxwell equations:

1- 1- 1- s − (Gik − 2gikG) = Tik + 6(jijk − 2gikj js), (162 ) 1 ∂j ∂j ϕik = -(--ki − --ik), (163 ) 6 ∂x ∂x ∂fˆks ˆk ∂xs = j . (164 )

In (162 *), the (Riemannian) Ricci tensor $Gik$ and Ricci scalar $G$ are formed from the auxiliary metric $gik$ ; the same holds for the tensor $Tik := − (γik − 12gikgrsγrs)$ . Of course, in the end, $gik$ and all quantities formed from it will have to be expresses by the affine connection $Γ$ .

Schrödinger’s assignment of mathematical quantities to physical observables is as follows:
$ϕik$ corresponds to the electromagnetic field tensor $→ → (E, B )$ ,
$fˆik$ corresponds to its conjugate field quantity $(−→D, H→ )$ ,
$k j$ corresponds to the electric 4-current density,
$Tik$ corresponds to the “field-energy-tensor of the electromagnetic field”.

We note from (163 *) that the electric current density is the negative of the electromagnetic 4-potential. The meson field is not yet included in the theory.

Up to here, Schrödinger did not specify the Lagrangian in (153 *). He then assumed:

with a numerical constant $α$ (his Eq. (4,1) on p. 51). According to Schrödinger:

“ $ℒ$ is essentially Born’s Lagrangian, with $ϕkl$ in place of his $→ → (B, E )$ […] $ˆfik$ agrees in form with Born’s contravariant tensor-density $→ → (H, − D )$ […].” ([545 *], p. 52.)

This refers to the paper by Born and Infeld on a non-linear electrodynamics;⁷⁹ cf. Section 5. At the end of the paper, Schrödinger speculated about taking into account a cosmological constant, and about including a meson field of spin $1$ described by a symmetrical rank 2 tensor $ψik$ in a more complicated Lagrangian⁸⁰ :

As field equations, he obtained the following system:

Gik = α-(ϕ rϕkr + gik(w − 1) − 1-jijk), (167 ) w i 6 1-∂jk- -∂ji ϕik = 6(∂xi − ∂xk ), (168 ) [ √ --- ] ∘ -------------------- √-1---∂-- --−-g(ϕkr − I ϕ ∗ kr) = gkrj , w = 1 + 1-ϕrsϕ − (I )2. (169 ) − g∂xr w 2 k 2 rs 2

Here, $∗ ik 1 √1-- ikrs 1 ∗ rs ϕ := 2 −g 𝜖 ϕrs,I2 := 4ϕ ϕrs$ as in [545 *], p. 51, Eq. (4,3). For a physical interpretation, Schrödinger re-defined all quantities $gik,Ak, ϕik,jk$ by multiplying them with constants having physical dimensions. This is to be kept in mind when his papers in which applications were discussed, are compared with this basic publication.

6.1.2 Application: Geomagnetic field

Schrödinger quickly tried to draw empirically testable consequences from his theory. At first he neglected gravity in his UFT and obtained the equations “for not excessively strong electromagnetic fields”:

→ →H = →rot→A, E→ = − →∇V − ∂A-, (170 ) ∂t → r→ot→H − ∂E--= − μ2→A, div→E = − μ2V , (171 ) ∂t

in which the electric current 4-density is replaced by the 4-potential; cf. (163 *). The equations then were applied to the permanent magnetic field of the Earth and the Sun [544 ]. Deviations from the dipole field as described by Maxwell’s theory are predicted by (171 *). Schrödinger’s careful comparison with available data did not show a contradiction between theory and observation, but remained inconclusive. This was confirmed in a paper with the Reverend J. McConnel [419 ] in which they investigated a possible (shielding) influence of the earth’s altered magnetic field on cosmic rays (as in the aurora).
After the second world war, the later Nobel-prize winner Maynard S. Blackett (1897 – 1974) suggested an empirical formula relating magnetic moment $M$ and angular momentum $L$ of large bodies like the Earth, the Sun, and the stars:

with $G$ Newton’s gravitational constant, $c$ the velocity of light in vacuum, and $β$ a numerical constant near $14$ [28 *]. The charge of the bodies was unimportant; the hypothetical effect seemed to depend only on their rotation. Blackett’s idea raised some interest among experimental physicists and workers in UFT eager to get a testable result. One of them, the Portuguese theoretical physicist Antonio Gião, derived a formula generalizing (172 *) from his own unified field theory [225 , 226 ]:

where $e,m0$ are charge and rest mass of the electron, $ξ$ a constant, and $χ$ the “average curvature of space-time”.

Blackett conjectured “that a satisfactory explanation of (172 *) will not be found except within the structure of a unified field theory” [28 *]. M. J. Nye is vague on this point: “What he had in mind was something like Einstein asymmetry or inequality in positive and negative charges.” ([460 ], p. 105.) Schrödinger seconded Blackett; however, he pointed out that it was “not a very simple thing” to explain the magnetic field generated by a rotating body by his affine theory. “At least a general comprehension of the structure of matter” was a necessary prerequisite ([554 ], p. 216). The theoretical physicist A. Papapetrou who had worked with Schrödinger joined Blackett in Manchester between 1948 and 1952. We may assume that the experimental physicist Blackett knew of Schrödinger’s papers on the earth’s magnetism within the framework of UFT and wished to use Papapetrou’s expertise in the field. The conceptional link between Blackett’s idea and UFT is that in this theory the gravitational field is expected to generate an electromagnetic field whereas, in general relativity, the electromagnetic field had been a source of the gravitational field.

Theoreticians outside the circle of those working on unified field theory were not so much attracted by Blackett’s idea. One of them was Pauli who, in a letter to P. Jordan of July 13, 1948, wrote:

“As concerns Blackett’s new material on the magnetism of the earth and stars, I have the following difficulty: In case it is an effect of acceleration the dependency of the angular velocity must be different; in the case of an effect resulting from velocity, a translatory movement ought to also generate a magnetic field. Special relativity then requires that the matter at rest possesses an electric field as well. […] I do not know how to escape from this dilemma.” ⁸¹ ([489 *], p. 543)

Three weeks earlier, in a letter to Leon Rosenfeld, he had added that he “found it very strange that Blackett wrote articles on this problem without even mentioning this simple and important old conclusion.” ([489 *], p. 539) This time, Pauli was not as convincing as usual: Blackett had been aware of the conclusions and discussed them amply in his early paper ([28 ], p. 664).

In 1949, the Royal Astronomical Society of England held a “Geophysical Discussion” on “Rotation and Terrestrial Magnetism”[519 ]. Here, Blackett tried to avoid Pauli’s criticism by retaining his formula in differential form:

For a translation $ω− = 0$ , and no effect obtains. T. Gold questioned Blackett’s formula as being dependent on the inertial system and asked for a radial dependence of the angular velocity $ω$ . A. Papapetrou claimed that Blackett’s postulate “could be reconciled with the relativistic invariance requirements of Maxwell’s equations” and showed this in a publication containing Eq. (174 *), if only forcedly so: he needed a bi-metric gravitational theory to prove it [476 ]. In the end, the empirical data taken from the earth did not support Blackett’s hypothesis and thus also were not backing UFT in its various forms; cf. ([19 ], p. 295).

6.1.3 Application: Point charge

A second application pertains to the field of an electrical point charge at rest [548 *]. Schrödinger introduced two “universal constants” which both appear in the equations for the electric field. The first is his “natural unit” of the electromagnetic field strength $b := e- r20$ called Born’s constant by him, where $e$ is the elementary charge and $2 ∫ ∞ r0 := me0c2 23 0 √1d+xx4-$ the electron radius (mass $m0$ of the electron). The second is the reciprocal length introduced in a previous publication $b √2G- f := c2$ with Newton’s gravitational constant $G$ and the velocity of light $c$ . Interestingly, the affine connection has been removed from the field equations; they are written as generalized Einstein–Maxwell equations as in Born–Infeld theory⁸² (cf. Section 5):

( ) 2 -c2- r w--−-1 μ2c2- Gik = f b2w ϕi ϕkr − gik w − b2 AiAk , (175 ) ϕ = ∂Ak- − ∂Ai-, (176 ) ∘ --------ik----∂xi----∂xk-- 1 ∂ [ √−-g ( c2 ) ] 1 c2 c4 √-------r ----- ϕkr − -2I2 ϕ ∗ kr = μ2Ak, w = 1 + ---2 ϕrsϕrs −-4(I2)2. (177 ) − g ∂x w b 2 b b

If only a static electric field is present, $ϕ∗ ik = 0$ and thus $I2 = ϕ ∗ ikϕik = 0$ .

An ansatz for an uncharged static, spherically symmetric line element is made like the one for Schwarzschild’s solution in general relativity, i.e.,

The solution obtained was:

with $r x := r0-$ . The integral is steadily decreasing from $x = 0$ to $x → ∞$ , where it tends to zero. Like the Schwarzschild solution, (178 *) with (179 *) has a singularity of the Ricci scalar at $x = 0 = r$ . For $r → ∞$ the Schwarzschild (external) solution is reached. According to Schrödinger, “[…] we have here, for the first time, the model of a point source whose gravitational field is accounted for by its electric field energy. The singularity itself contributes nothing” ([548 *], p. 232).

Two weeks later, Schrödinger put out another paper in which he wrote down 16 “conservation identities” following from the fact that his Lagrangian is a scalar density and depends only on the 16 components of the Ricci tensor. He also compared his generalization of general relativity with Weyl’s theory gauging the metric (cf. [689 ]), and also with Eddington’s purely affine theory ([140 ], chapter 7, part 2). From (158 *) it is clear that Schrödinger’s theory is not gauge-invariant.⁸³ He ascribed this weakness to the missing of a third fundamental field in the theory, the meson field. According to Schrödinger the absence of the meson field was due to his restraint to a symmetric connection. Eddington’s theory with his general affine connection would house all the structures necessary to include the third field. It should take fifteen months until Schrödinger decided that he had achieved the union of all three fields.

6.2 Semi-symmetric connection

Schrödinger’s next paper on UFT continued this line of thought: in order to be able to include the mesonic field he dropped the symmetry-condition on the affine connection ([549 *], p. 275). This brings homothetic $V − ik$ curvature into the game (cf. Section 2.3.1, Eq. (65 *)). Although covariant differentiation was introduced through $− i ∇kX$ and $− ∇k ωi$ , in the sequel Schrödinger split the connection according to $k k k Lij = Γij + Sij$ and used the covariant derivative $0 ∇$ (cf. Section 2.1.2) with regard to the symmetric part $Γ$ of the connection.⁸⁴ In his first attempt, Schrödinger restricted torsion to non-vanishing vector torsion by assuming:

with arbitrary 1-form $Y = Yidxi$ . Perhaps this is the reason why he speaks of “weakly asymmetric affinity” without giving a precise definition. Schouten called this type of connection “semi-symmetric” (cf. Section 5 of Part I, Eq. (132)). Obviously, vector torsion $Si = − 3Yi$ . The two contractions of the curvature tensor $i K− jkl$ , i.e., $K− ik$ and $V− ik$ are brought into the form [cf. (63 *)]:

Here, $i∥ k 0$ denotes covariant differentiation with respect to the symmetric connection $Γ$ . Two linear combinations of these tensors are introduced:

A calculation shows that:

with $(Kik )Γ$ being the Ricci-tensor formed with the symmetrized connection, and

In the same way as in (154 *) two tensor densities are introduced:

and a third one according to

Conjugated variables to $ˆgik, ˆfik$ are [cf. (159 *]:

The Lagrangian is to depend only on $Pik$ and $M ik$ . A (symmetric) metric tensor $gik$ is introduced as in (157 *). Variation of the Lagrangian with respect to $Pik$ and $M ik$ leads to field equations now containing terms from homothetic curvature:

and to the same equation (156 *) as before, as well as to an additional equation:

According to Schrödinger, (185 *) and (190 *) “form a self-contained Maxwellian set”. The formal solution for the symmetric part of the connection replacing (158 *) now becomes:

In this paper, Schrödinger changed the relation between mathematical objects and physical observables:
The variables ( $j,ϕ,fˆ$ ) related to the Ricci tensor correspond to the meson field;
whereas ( $Y,M, mˆ$ ) related to torsion describe the electromagnetic field.

His main argument was:

“Now the gravitational field and the mesonic field are actually, to all appearance, universally and jointly produced in the same places, viz. in the heavy nuclear particles. They have at any rate their principal seat in common, while there is absolutely no parallelism between electric charge and mass” ([549 *], p. 282).

In addition, Schrödinger referred to Einstein’s remark concerning the possibility of exchanging the roles of the electromagnetic fields $E→$ by $H→$ and $−→D$ by $−→ B$ ([142 *], p. 418). “Now a preliminary examination of the wholly non-symmetrical case gives me the impression that the exchange of rôles will very probably be imperative, […]” ([549 *], p. 282).

As to the field equations, they still were considered as preliminary because: “the investigation of the fully non-symmetric case is imperative and may have surprises in store.” ([549 *], p. 282.) The application of Weyl’s gauge transformations $gik → λgik$ in combination with

leaves invariant $k Γij ,ϕik$ , and $Mik$ , but changes $γik$ and destroys the vanishing of the divergence $s ˆj,s$ . Schrödinger thought it to be “imperative to distinguish between the potential $jk$ and the current-and-charge $jk$ the two coinciding only in the original gauge.” ([549 *], p. 284.) He also claimed that only the gravitational and mesonic fields had an influence on the auto-parallels (cf. Section 2.1.1, Eq. (22 *)).

These first two papers of Schrödinger were published in the Proceedings of the Royal Irish Academy, a journal only very few people would have had a chance to read, particularly during World War II, although Ireland had stayed neutral. Schrödinger apparently believed that, by then, he had made enough progress in comparison with Eddington’s and Einstein’s publications.⁸⁵ Hence, he wrote a summary in Nature for the wider physics community [547 *]. At the start, he very nicely laid out the conceptual and mathematical foundations of affine geometry and gave a brief historical account of its use within unified field theory. After supporting “the superiority of the affine point of view” he discussed the ambiguities in the relation between mathematical objects and physical observables. An argument most important to him came from the existence of

“a third field […], of equally fundamental standing with gravitation and electromagnetism: the mesonic field responsible for nuclear binding. Today no field-theory which does not embrace at least this triad can be deemed satisfactory at all.” ([549 *], p. 574.)⁸⁶

He believed to have reached “a fully satisfactory unified description of gravitation, electromagnetism and a 6-vectorial meson.”([547 ], p. 575.) Schrödinger claimed a further advantage of his approach from the fact that he needed no “special choice of the Lagrangian” in order to make the connection between geometry and physics, and for deriving the field equations.

As to quantum theory, Schrödinger included a disclaimer (in a footnote): “The present article does not touch on it and has therefore to ignore such features in the conventional description of physical fields as are concerned with their quantum character […].” ([549 *], p. 574.)

In a letter to Einstein of 10 October 1944, in a remark about an essay of his about Eddington and Milne, M. Born made a bow to Einstein:

“My opinion is that you have the right to speculate, other people including myself have not. […] Honestly, when average people want to procure laws of nature by pure thinking, only rubbish can result. Perhaps Schrödinger can do it. I would love to know what you think about his affine field theories. I find all of it beautiful and full of wit; but whether it is true? […]” ([168 *], p. 212–213)⁸⁷

	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