4.1 Weyl’s theory

4.1.1 The geometry

Weyl’s fundamental idea for generalising Riemannian geometry was to note that, unlike for the comparison of vectors at different points of the manifold, for the comparison of scalars the existence of a connection is not required. Thus, while lengths of vectors at different points can be compared without a connection, directions cannot. This seemed too special an assumption to Weyl for a genuine infinitesimal geometry:

In contrast to this, Riemann made the much stronger assumption that line elements may be compared not only at the same place but also at two arbitrary places at a finite distance.

In order to invent a purely “infinitesimal” geometry, Weyl introduced the 1-dimensional, Abelian group of gauge transformations,

Let us now look at what happens to parallel transport of a length, e.g., the norm of a tangent vector along a particular curve with parameter to a different (but infinitesimally neighbouring) point:

Thus, in Weyl’s connection (100), both the gravitational and the electromagnetic fields, represented by the metrical field and the vector field , are intertwined. Perhaps, having in mind Mie’s ideas of an electromagnetic world view and Hilbert’s approach to unification, in the first edition of his book, Weyl remained reserved:

However, a little later, in his paper accepted on 8 June 1918, Weyl boldly claimed:

The adverse circumstances alluded to in the first quotation might be linked to the difficulties of finding a satisfactory Lagrangian from which the field equations of Weyl’s theory can be derived. Due to the additional group of gauge transformations, it is useful to introduce the new concept of gauge-weight within tensor calculus as in Section 2.1.5⁶⁴. As the Lagrangian must have gauge-weight , we are looking for a scalar of gauge-weight . Weitzenböck has shown that the only possibilities quadratic in the curvature tensor and the line curvature are given by the four expressions [391]

Perhaps Bach (alias Förster) was also dissatisfied with Weyl’s calculations: He went through the entire mathematics of Weyl’s theory, curvature tensor, quadratic Lagrangian field equations and all; he even discussed exact solutions. His Lagrangian is given by , where the invariants are defined by

[4]

4.1.2 Physics

While Weyl’s unification of electromagnetism and gravitation looked splendid from the mathematical point of view, its physical consequences were dire: In general relativity, the line element had been identified with space- and time intervals measurable by real clocks and real measuring rods. Now, only the equivalence class was supposed to have a physical meaning: It was as if clocks and rulers could be arbitrarily “regauged” in each event, whereas in Einstein’s theory the same clocks and rulers had to be used everywhere. Einstein, being the first expert who could keep an eye on Weyl’s theory, immediately objected, as we infer from his correspondence with Weyl.

In spring 1918, the first edition of Weyl’s famous book on differential geometry, special and general relativity Raum–Zeit–Materie appeared, based on his course in Zürich during the summer term of 1917 [396]. Weyl had arranged that the page proofs be sent to Einstein. In communicating this on 1 March 1918, he also stated that

He then asked whether Einstein would be willing to communicate a paper on this new unified theory to the Berlin Academy ([321], Volume 8B, Document 472, pp. 663–664). At the end of March, Weyl visited Einstein in Berlin, and finally, on 5 April 1918, he mailed his note to him for the Berlin Academy. Einstein was impressed: In April 1918, he wrote four letters and two postcards to Weyl on his new unified field theory – with a tone varying between praise and criticism. His first response of 6 April 1918 on a postcard was enthusiastic:

Einstein’s “objection” is formulated in his “Addendum” (“Nachtrag”) to Weyl’s paper in the reports of the Academy, because Nernst had insisted on such a postscript. There, Einstein argued that if light rays would be the only available means for the determination of metrical relations near a point, then Weyl’s gauge would make sense. However, as long as measurements are made with (infinitesimally small) rigid rulers and clocks, there is no indeterminacy in the metric (as Weyl would have it): Proper time can be measured. As a consequence follows: If in nature length and time would depend on the pre-history of the measuring instrument, then no uniquely defined frequencies of the spectral lines of a chemical element could exist, i.e., the frequencies would depend on the location of the emitter. He concluded with the words

Einstein’s remark concerning the path-dependence of the frequencies of spectral lines stems from the path-dependency of the integral (102) given above. Only for a vanishing electromagnetic field does this objection not hold.

Weyl answered Einstein’s comment to his paper in a “reply of the author” affixed to it. He doubted that it had been shown that a clock, if violently moved around, measures proper time . Only in a static gravitational field, and in the absence of electromagnetic fields, does this hold:

Einstein saw the problem, then unsolved within his general relativity, that Weyl alluded to, i.e., to give a theory of clocks and rulers within general relativity. Presumably, such a theory would have to include microphysics. In a letter to his former student Walter Dällenbach, he wrote (after 15 June 1918):

Another famous theoretician who could not side with Weyl was H. A. Lorentz; in a paper on the measurement of lengths and time intervals in general relativity and its generalisations, he contradicted Weyl’s statement that the world-lines of light-signals would suffice to determine the gravitational potentials [211].

However, Weyl still believed in the physical value of his theory. As further “extraordinarily strong support for our hypothesis of the essence of electricity” he considered the fact that he had obtained the conservation of electric charge from gauge-invariance in the same way as he had linked with coordinate-invariance earlier, what at the time was considered to be “conservation of energy and momentum”, where a non-tensorial object stood in for the energy-momentum density of the gravitational field ([398], pp. 252–253).

Moreover, Weyl had some doubts about the general validity of Einstein’s theory which he derived from the discrepancy in value by 20 orders of magniture of the classical electron radius and the gravitational radius corresponding to the electron’s mass ([397], p. 476; [152]).

4.1.3 Reactions to Weyl’s theory I: Einstein and Weyl

There exists an intensive correspondence between Einstein and Weyl, now completely available in volume 8 of the Collected Papers of Einstein [321]. We subsume some of the relevant discussions. Even before Weyl’s note was published by the Berlin Academy on 6 June 1918, many exchanges had taken place between him and Einstein.

On a postcard to Weyl on 8 April 1918, Einstein reaffirmed his admiration for Weyl’s theory, but remained firm in denying its applicability to nature. Weyl had given an argument for dimension 4 of space-time that Einstein liked: As the Lagrangian for the electromagnetic field is of gauge-weight and has gauge-weight in an , the integrand in the Hamiltonian principle can have weight zero only for : “Apart from the [lacking] agreement with reality it is in any case a grandiose intellectual performance”⁷¹ ([321], Vol. 8B, Doc. 499, 711). Weyl did not give in:

After Weyl’s next paper on “pure infinitesimal geometry” had been submitted, Einstein put forward further arguments against Weyl’s theory. The first was that Weyl’s theory preserves the similarity of geometric figures under parallel transport, and that this would not be the most general situation (cf. Equation (49)). Einstein then suggested the affine group as the more general setting for a generalisation of Riemannian geometry ([321], Vol. 8B, Doc. 551, 777). He repeated this argument in a letter to his friend Michele Besso from his vacations at the Baltic Sea on 20 August 1918, in which he summed up his position with regard to Weyl’s theory:

Einstein’s remark concerning “affine geometry” is referring to the affine geometry in the sense it was introduced by Weyl in the 1st and 2nd edition of his book [396], i.e., through the affine group and not as a suggestion of an affine connexion.

From Einstein’s viewpoint, in Weyl’s theory the line element is no longer a measurable quantity – the electromagnetical 4-potential never had been one. Writing from his vacations on 18 September 1918, Weyl presented a new argument in order to circumvent Einstein’s objections. The quadratic form is an absolute invariant, i.e., also with regard to gauge transformations (gauge weight 0). If this expression would be taken as the measurable distance in place of , then

Einstein was unimpressed:

Einstein’s argument is not very convincing: itself is influenced by matter through his field equations; it is only that now is algebraically connected to the matter tensor. In view of the more general quadratic Lagrangian needed in Weyl’s theory, the connection between and the matter tensor again might become less direct. Einstein added:

The last remarks are interesting for the way in which Einstein imagined a successful unified field theory.

4.1.4 Reactions to Weyl’s theory II: Schouten, Pauli, Eddington, and others

Sommerfeld seems to have been convinced by Weyl’s theory, as his letter to Weyl on 3 June 1918 shows:

Schouten, in his attempt in 1919 to replace the presentation of the geometrical objects used in general relativity in local coordinates by a “direct analysis”, also had noticed Weyl’s theory. In his “addendum concerning the newest theory of Weyl”, he came as far as to show that Weyl’s connection is gauge invariant, and to point to the identification of the electromagnetic 4-potential. Understandably, no comments about the physics are given ([295], pp. 89–91).

In the section on Weyl’s theory in his article for the Encyclopedia of Mathematical Sciences, Pauli described the basic elements of the geometry, the loss of the line-element as a physical variable, the convincing derivation of the conservation law for the electric charge, and the too many possibilities for a Lagrangian inherent in a homogeneous function of degree 1 of the invariants (103). As compared to his criticism with respect to Eddington’s and Einstein’s later unified field theories, he is speaking softly, here. Of course, as he noted, no progress had been made with regard to the explanation of the constituents of matter; on the one hand because the differential equations were too complicated to be solved, on the other because the observed mass difference between the elementary particles with positive and negative electrical charge remained unexplained. In his general remarks about this problem at the very end of his article, Pauli points to a link of the asymmetry with time-reflection symmetry (see [246], pp. 774–775; [244]). For Einstein, this criticism was not only directed against Weyl’s theory

In a letter to Besso on 26 July 1920, Einstein repeated an argument against Weyl’s theory which had been removed by Weyl – if only by a trick to be described below; Einstein thus said:

In his book “Space, Time, and Gravitation”, Eddington gave a non-technical introduction into Weyl’s “welding together of electricity and gravitation into one geometry”. The idea of gauging lengths independently at different events was the central theme. He pointed out that while the fourfold freedom in the choice of coordinates had led to the conservation laws for energy and momentum, “in the new geometry is a fifth arbitrariness, namely that of the selected gauge-system. This must also give rise to an identity; and it is found that the new identity expresses the law of conservation of electric charge.” One natural gauge was formed by the “radius of curvature of the world”; “the electron could not know how large it ought to be, unless it had something to measure itself against” ([57], pp. 174, 173, 177).

As Eddington distinguished natural geometry and actual space from world geometry and conceptual space serving for a graphical representation of relationships among physical observables, he presented Weyl’s theory in his monograph “The mathematical theory of relativity”

Of course, “wrong end” meant that Eddington took Weyl’s theory such

Again, Eddington liked Weyl’s natural gauge encountered in Section 4.1.5, which made the curvature scalar a constant, i.e., ; it became a consequence of Eddington’s own natural gauge in his affine theory, (cf. Section 4.3). For Eddington, Weyl’s theory of gauge-transformation was a hybrid:

In the depth of his heart Weyl must have kept a fondness for his idea of “gauging” a field all during the decade between 1918 and 1928. As he had abandoned the idea of describing matter as a classical field theory since 1920, the linking of the electromagnetic field via the gauge idea could only be done through the matter variables. As soon as the new spinorial wave function (“matter wave”) in Schrödinger’s and Dirac’s equations emerged, he adapted his idea and linked the electromagnetic field to the gauging of the quantum mechanical wave function [407, 408]. In October 1950, in the preface for the first American printing of the English translation of the fourth edition of his book Space, Time, Matter from 1922, Weyl clearly expressed that he had given up only the particular idea of a link between the electromagnetic field and the local calibration of length:

4.1.5 Reactions to Weyl’s theory III: Further research

Pauli, still a student, and with his article for the Encyclopedia in front of him, pragmatically looked into the gravitational effects in the planetary system, which, as a consequence of Einstein’s field equations, had helped Einstein to his fame. He showed that Weyl’s theory had, for the static case, as a possible solution a constant Ricci scalar; thus it also admitted the Schwarzschild solution and could reproduce all desired effects [244, 243].

Weyl himself continued to develop the dynamics of his theory. In the third edition of his Space–Time–Matter [398], at the Naturforscherversammlung in Bad Nauheim in 1920 [399], and in his paper on “the foundations of the extended relativity theory” in 1921 [402], he returned to his new idea of gauging length by setting (cf. Section 4.1.3); he interpreted to be the “radius of curvature” of the world. In 1919, Weyl’s Lagrangian originally was together with the constraint with constant ([398], p. 253). As an equivalent Lagrangian Weyl gave, up to a divergence⁷⁹

Reichenbächer seemingly was unhappy about Weyl’s taking the curvature scalar to be a constant before the variation; in the discussion after Weyl’s talk in 1920, he inquired whether one could not introduce Weyl’s “natural gauge” after the variation of the Lagrangian such that the field equations would show their gauge invariance first ([399], p. 651). Eddington criticised Weyl’s choice of a Lagrangian as speculative:

The changes, which Weyl had introduced in the 4th edition of his book [401], and which, according to him, were of fundamental importance for the understanding of relativity theory, were discussed by him in a further paper [400]. In connection with the question of whether, in general relativity, a formulation might be possible such that “matter whose characteristical traits are charge, mass, and motion generates the field”, a question which was considered as unanswered by Weyl, he also mentioned a publication of Reichenbächer [272]. For Weyl, knowledge of the charge and mass of each particle, and of the extension of their “world-channels” were insufficient to determine the field uniquely. Weyl’s hint at a solution remains dark; nevertheless, for him it meant

Although Einstein could not accept Weyl’s theory as a physical theory, he cherished “its courageous mathematical construction” and thought intensively about its conceptual foundation: This becomes clear from his paper “On a complement at hand of the bases of general relativity” of 1921 [73]. In it, he raised the question whether it would be possible to generate a geometry just from the conformal invariance of Equation (9) without use of the conception “distance”, i.e., without using rulers and clocks. He then embarked on conformal invariants and tensors of gauge-weight 0, and gave the one formed from the square of Weyl’s conformal curvature tensor (59), i.e. His colleague in Vienna, Wirtinger, had helped him in this⁸¹. Einstein’s conclusion was that, by writing down a metric with gauge-weight 0, it was possible to form a theory depending only on the quotient of the metrical components. If has gauge-weight , then is such a metric. In order to reduce the new theory to general relativity, in addition only the differential equation

Einstein’s rejection of the physical value of Weyl’s theory was seconded by Dienes, if only with a not very helpful argument. He demanded that the connection remain metric-compatible from which, trivially, Weyl’s gauge-vector must vanish. Dienes applied the same argument to Eddington’s generalisation of Weyl’s theory [51]. Other mathematicians took Weyl’s theory at its face value and drew consequences; thus M. Juvet calculated Frenet’s formulas for an “-èdre” in Weyl’s geometry by generalising a result of Blaschke for Riemannian geometry [180]. More important, however, for later work was the gauge invariant tensor calculus by a fellow of St. John’s College in Cambridge, M. H. A. Newman [237]. In this calculus, tensor equations preserve their form both under a change of coordinates and a change of gauge. Newman applied his scheme to a variational principle with Lagrangian and concluded: