### 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:

“If we make no further assumption, the points of a manifold remain totally isolated from each other with regard to metrical structure. A metrical relationship from point to point will only then be infused into [the manifold] if a principle for carrying the unit of length from one point to its infinitesimal neighbours is given.”

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.

However, the possibility of such a comparison ‘at a distance’ in no way can be admitted in a pure infinitesimal geometry.” ([397], p. 397)

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

besides the diffeomorphism group (coordinate transformations). At a point, Equation (98) induces a local recalibration of lengths while preserving angles, i.e., . If the non-metricity tensor is assumed to have the special form , with an arbitrary vector field , then as we know from Equation (57), with regard to these gauge transformations
We see a striking resemblance with the electromagnetic gauge transformations for the vector potential in Maxwell’s theory. If, as Weyl does, the connection is assumed to be symmetric (i.e., with vanishing torsion), then from Equation (42) we get
Thus, unlike in Riemannian geometry, the connection is not fully determined by the metric but depends also on the arbitrary vector function , which Weyl wrote as a linear form . With regard to the gauge transformations (98), remains invariant. From the 1-form , by exterior derivation a gauge-invariant 2-form with follows. It is named “Streckenkrümmung” (“line curvature”) by Weyl, and, by identifying with the electromagnetic 4-potential, he arrived at the electromagnetic field tensor .

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:

By a proper choice of the curve’s parameter, we may write (101) in the form and integrate along to obtain . If is taken to be tangent to , i.e., , then
i.e., the length of a vector is not integrable; its value generally depends on the curve along which it is parallely transported. The same holds for the angle between two tangent vectors in a point (cf. Equation (44)). For a vanishing electromagnetic field, the 4-potential becomes a gradient (“pure gauge”), such that , and the integral becomes independent of the curve.

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:

“Again physics, now the physics of fields, is on the way to reduce the whole of natural phenomena to one single law of nature, a goal to which physics already once seemed close when the mechanics of mass-points based on Newton’s Principia did triumph. Yet, also today, the circumstances are such that our trees do not grow into the sky.” ([396], p. 170; preface dated “Easter 1918”)

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

“I am bold enough to believe that the whole of physical phenomena may be derived from one single universal world-law of greatest mathematical simplicity.” ([397], p. 385, footnote 4)

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 2.1.5. As the Lagrangian must have gauge-weight , we are looking for a scalar of gauge-weight . has shown that the only possibilities quadratic in the curvature tensor and the line curvature are given by the four expressions [391]

While the last invariant would lead to Maxwell’s equations, from the invariants quadratic in curvature, in general field equations of fourth order result. Weyl did calculate the curvature tensor formed from his connection (100) but did not get the correct result; it is given by Schouten ([310], p. 142) and follows from Equation (51):
If the metric field and the 4-potential are varied independently, from each of the curvature-dependent scalar invariants we do get contributions to Maxwell’s equations.

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

with
where is the Riemannian curvature tensor, , and is the electromagnetic 4-potential [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

“As I believe, during these days I succeeded in deriving electricity and gravitation from the same source. There is a fully determined action principle, which, in the case of vanishing electricity, leads to your gravitational equations while, without gravity, it coincides with Maxwell’s equations in first order. In the most general case, the equations will be of 4th order, though.”

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:

“Your note has arrived. It is a stroke of genious of first rank. Nevertheless, up to now I was not able to do away with my objection concerning the scale.” ([321], Volume 8B, Document 498, 710)

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

“Regrettably, the basic hypothesis of the theory seems unacceptable to me, [of a theory] the depth and audacity of which must fill every reader with admiration.” ([395], Addendum, p. 478)

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:

“The most plausible assumption that can be made for a clock resting in a static field is this: that it measure the integral of the normed in this way [i.e., as in Einstein’s theory]; the task remains, in my theory as well as in Einstein’s, to derive this fact by a dynamics carried through explicitly.” ([395], p. 479)

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):

“[Weyl] would say that clocks and rulers must appear as solutions; they do not occur in the foundation of the theory. But I find: If the , as measured by a clock (or a ruler), is something independent of pre-history, construction and the material, then this invariant as such must also play a fundamental role in theory. Yet, if the manner in which nature really behaves would be otherwise, then spectral lines and well-defined chemical elements would not exist. [...] In any case, I am as convinced as Weyl that gravitation and electricity must let themselves be bound together to one and the same; I only believe that the right union has not yet been found.” ([321], Volume 8B, Document 565, 803)

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” ([321], Vol. 8B, Doc. 499, 711). Weyl did not give in:

“Your rejection of the theory for me is weighty; [...] But my own brain still keeps believing in it. And as a mathematician I must by all means hold to [the fact] that my geometry is the true geometry ‘in the near’, that Riemann happened to come to the special case is due only to historical reasons (its origin is the theory of surfaces), not to such that matter.” ([321], Volume 8B, Document 544, 767)

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:

“[Weyl’s] theoretical attempt does not fit to the fact that two originally congruent rigid bodies remain congruent independent of their respective histories. In particular, it is unimportant which value of the integral is assigned to their world line. Otherwise, sodium atoms and electrons of all sizes would exist. But if the relative size of rigid bodies does not depend on past history, then a measurable distance between two (neighbouring) world-points exists. Then, Weyl’s fundamental hypothesis is incorrect on the molecular level, anyway. As far as I can see, there is not a single physical reason for it being valid for the gravitational field. The gravitational field equations will be of fourth order, against which speaks all experience until now [...].” ([99], p. 133)

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

“[...] by the prefixing of this factor, so to speak, the absolute norming of the unit of length is accomplished after all” ([321], Volume 8B, Document 619, 877–879)

Einstein was unimpressed:

“But the expression for the measured length is not at all acceptable in my opinion because is very dependent on the matter density. A very small change of the measuring path would strongly influence the integral of the square root of this quantity.”

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:

“Of course I know that the state of the theory as I presented it is not satisfactory, not to speak of the fact that matter remains unexplained. The unconnected juxtaposition of the gravitational terms, the electromagnetic terms, and the -terms undeniably is a result of resignation.[...] In the end, things must arrange themselves such that action-densities need not be glued together additively.” ([321], Volume 8B, Document 626, 893–894)

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:

“What you say here is really marvelous. In the same way in which Mie glued to his consequential electrodynamics a gravitation which was not organically linked to it, Einstein glued to his consequential gravitation an electrodynamics (i.e., the usual electrodynamics) which had not much to do with it. You establish a real unity.” [327]

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

“but also against every continuum-theory, also one which treats the electron as a singularity. Now as before I believe that one must look for such an overdetermination by differential equations that the solutions no longer have the character of a continuum. But how?” ([103], p. 43)

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:

“One must pass to tensors of fourth order rather than only to those of second order, which carries with it a vast indeterminacy, because, first, there exist many more equations to be taken into account, second, because the solutions contain more arbitrary constants.” ([99], p. 153)

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”

“from the wrong end – as its author might consider; but I trust that my treatment has not unduly obscured the brilliance of what is unquestionably the greatest advance in the relativity theory after Einstein’s work.” ([59], p. 198)

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

“that his non-Riemannian geometry is not to be applied to actual space-time; it refers to a graphical representation of that relation-structure which is the basis of all physics, and both electromagnetic and metrical variables appear in it as interrelated.” ([59], p. 197)

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:

“He admits the physical comparison of length by optical methods [...]; but he does not recognise physical comparison of length by material transfer, and consequently he takes to be a function fixed by arbitrary convention and not necessarily a constant.” ([59], pp. 220–221)

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 [407408]. 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:

“While it was not difficult to adapt also Maxwell’s equations of the electromagnetic field to this principle [of general relativity], it proved insufficient to reach the goal at which classical field physics is aiming: a unified field theory deriving all forces of nature from one common structure of the world and one uniquely determined law of action.[...] My book describes an attempt to attain this goal by a new principle which I called gauge invariance. (Eichinvarianz). This attempt has failed.” ([410], p. V)

#### 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 [244243].

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 divergence

with the 4-potential and the electromagnetic field . Due to his constraint, Weyl had navigated around another problem, i.e., the formulation of the Cauchy initial value problem for field equations of fourth order: Now he had arrived at second order field equations. In the paper in 1921, he changed his Lagrangian slightly into
with a factor in Weyl’s connection (100),
In both presentations, he considered as an advantage of his theory:

“Moreover, this theory leads to the cosmological term in a uniform and forceful manner, [a term] which in Einstein’s theory was introduced ad hoc” ([402], p. 474)

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:

“At the most we can only regard the assumed form of action [...] as a step towards some more natural combination of electromagnetic and gravitational variables.” ([59], p. 212)

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

“to reconciliate Reichenbächer’s idea: matter causes a ‘deformation’ of the metrical field and Einstein’s idea: inertia and gravitation are one.” ([400], p. 561, footnote)

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, , had helped him in this. 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

would have to be solved. Eisenhart wished to partially reinterpret Weyl’s theory: In place of putting the vector potential equal to Weyl’s gauge vector, he suggested to identify it with , where is the electrical 4-current vector (-density) and the mass density. He referred to Weyl, Eddington’s book, and to Pauli’s article in the Encyclopedia of Mathematical Sciences [116].

Einstein’s rejection of the physical value of Weyl’s theory was seconded by , 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:

“The part independent of the ‘electrical’ vector is found to be , a tensor which has been considered by Einstein from time to time in connection with the theory of gravitation.” ([237], p. 623)

After the Second World War, research following Weyl’s classical geometrical approach with his original 1-dimensional Abelian gauge-group was resumed. The more important development, however, was the extension to non-Abelian gauge-groups and the combination with Kaluza’s idea. We shall discuss these topics in Part II of this article. The shift in Weyl’s interpretation of the role of the gauging from the link between gravitation and electromagnetism to a link between the quantum mechanical state function and electromagnetism is touched on in Section 7.