## 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 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}

#### 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 and are available (cf. Section 2.3.1) from which
tensor densities of weight (scalar densities) (cf. Section 2.1.5 of Part I) like
or 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}is the Ricci-tensor introduced in (55*) or (56*) due to 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 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 is introduced via with being interpreted as the (electric) current density. By help of a (symmetric) metric tensor is introduced by the usual relations: (155*) can be formally solved for the components of the symmetric connection to give: with .

^{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 as a functional of the components of , insert the expression into (56*) (for ), and finally solve for . 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 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: In (162*), the (Riemannian) Ricci tensor and Ricci scalar are formed from the auxiliary metric ; the same holds for the tensor . Of course, in the end, and all quantities formed from it will have to be expresses by the affine connection . Schrödinger’s assignment of mathematical quantities to physical observables is as follows:

corresponds to the electromagnetic field tensor ,

corresponds to its conjugate field quantity ,

corresponds to the electric 4-current density,

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 (his Eq. (4,1) on p. 51). According to Schrödinger:“ is essentially Born’s Lagrangian, with in place of his […] agrees in form with Born’s contravariant tensor-density […].” ([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 described by a symmetrical rank 2 tensor in a more complicated
Lagrangian^{80}:

As field equations, he obtained the following system:

Here, as in [545*], p. 51, Eq. (4,3). For a physical interpretation, Schrödinger re-defined all quantities 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”:

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 and angular momentum of large bodies like the Earth, the Sun, and the stars: with Newton’s gravitational constant, the velocity of light in vacuum, and a numerical constant near [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 are charge and rest mass of the electron, a constant, and 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 , 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 . 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 called Born’s constant by him, where is the
elementary charge and the electron radius (mass of the electron). The second
is the reciprocal length introduced in a previous publication with Newton’s gravitational
constant and the velocity of light . Interestingly, the affine connection has been removed from the
field equations; they are written as generalized Einstein–Maxwell equations as in Born–Infeld
theory^{82}
(cf. Section 5):

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 . The integral is steadily decreasing from to , where it tends to zero. Like the Schwarzschild solution, (178*) with (179*) has a singularity of the Ricci scalar at . For 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
curvature into the game (cf. Section 2.3.1, Eq. (65*)). Although covariant differentiation was introduced
through and , in the sequel Schrödinger split the connection according to
and used the covariant derivative (cf. Section 2.1.2) with regard to the symmetric part of the
connection.^{84}
In his first attempt, Schrödinger restricted torsion to non-vanishing vector torsion by assuming:

In this paper, Schrödinger changed the relation between mathematical objects and physical
observables:

The variables () related to the Ricci tensor correspond to the meson field;

whereas () 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 by and by ([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 in combination with

leaves invariant , and , but changes and destroys the vanishing of the divergence . Schrödinger thought it to be “imperative to distinguish between the potential and the current-and-charge 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}