Nevertheless, in terms of mathematical and physical concepts, a lot has been learned even from failed attempts at unification, vid. the gauge idea, or dimensional reduction (Kaluza–Klein), and much still might be learned in the future.
In the following I shall sketch, more or less chronologically, and by trailing Einstein's path, the history of attempts at unifying what are now called the fundamental interactions during the period from about 1914 to 1933. Until the end of the thirties, the only accepted fundamental interactions were the electromagnetic and the gravitational, plus, tentatively, something like the “mesonic” or “nuclear” interaction. The physical fields considered in the framework of “unified field theory” including, after the advent of quantum (wave) mechanics, the wave function satisfying either Schrödinger's or Dirac's equation, were all assumed to be
classical fields. The quantum mechanical wave function was taken to represent the field of the electron, i.e., a matter field. In spite of this, the construction of quantum field theory had begun already around 1927 [
50,
173,
177,
174,
178]
.
For the early history and the conceptual development of quantum field theory, cf. Section 1 of Schweber [
321]
, or Section 7.2 of Cao [
26]
; for Dirac's contributions, cf. [
189]
. Nowadays, it seems mandatory to approach unification in the framework of quantum field theory.
General relativity's doing away with
forces in exchange for a richer (and more complicated) geometry of space and time than the Euclidean remained the guiding principle throughout most of the attempts at unification discussed here. In view of this geometrization, Einstein considered the role of the stressenergy tensor
${T}^{ik}$
(the sourceterm of his field equations
${G}^{ik}=\kappa {T}^{ik}$
) a weak spot of the theory because it is a field devoid of any geometrical significance.
Therefore, the various proposals for a unified field theory, in the period considered here, included two different aspects:

∙
An inclusion of matter in the sense of a desired replacement, in Einstein's equations and their generalisation, of the energymomentum tensor of matter by intrinsic geometrical structures, and, likewise, the removal of the electric current density vector as a nongeometrical source term in Maxwell's equations.

∙
The development of a unified field theory more geometrico for electromagnetism and gravitation, and in addition, later, of the “field of the electron” as a classical field of “de Brogliewaves” without explicitly taking into account further matter sources^{
$\text{6}$
}
.
In a very Cartesian spirit, Tonnelat (Tonnelat 1955 [
355]
, p. 5) gives a definition of a unified field theory as

“a theory joining the gravitational and the electromagnetic field into one single hyperfield whose equations represent the conditions imposed on the geometrical structure of the universe.”
No material source terms are taken into account^{
$\text{7}$
}
. If however, in this context, matter terms appear in the field equations of unified field theory, they are treated in the same way as the stressenergy tensor is in Einstein's theory of gravitation: They remain alien elements.
For the theories discussed, the representation of matter oscillated between the pointparticle concept in which particles are considered as
singularities of a field, to particles as everywhere regular field configurations of a solitonic character. In a theory for continuous fields as in general relativity, the concept of pointparticle is somewhat amiss. Nevertheless, geodesics of the Riemannian geometry underlying Einstein's theory of gravitation are identified with the worldlines of freely moving pointparticles. The field at the location of a pointparticle becomes unbounded, or “singular”, such that the derivation of equations of motion from the field equations is a nontrivial affair. The competing paradigm of a particle as a particular field configuration of the electromagnetic and gravitational fields later has been pursued by J. A. Wheeler under the names “geon” and “geometrodynamics” in both the classical and the quantum realm [
412]
. In our time, gravitational solitonic solutions also have been found [
234,
24]
.
Even before the advent of quantum mechanics proper, in 1925–26, Einstein raised his expectations with regard to unified field theory considerably; he wanted to bridge the gap between classical field theory and
quantum theory, preferably by deriving quantum theory as a consequence of unified field theory. He even seemed to have believed that the quantum mechanical properties of particles would follow as a fringe benefit from his unified field theory; in connection with his classical teleparallel theory it is reported that Einstein, in an address at the University of Nottingham, said that he

“is in no way taking notice of the results of quantum calculation because he believes that by dealing with microscopic phenomena these will come out by themselves. Otherwise he would not support the theory.” ([90] , p. 610)
However, in connection with one of his moves, i.e., the 5vector version of Kaluza^{
$\text{8}$
}
's theory (cf. Sections 4.2 , 6.3 ), which for him provided “a logical unity of the gravitational and the electromagnetic fields”, he regretfully acknowledged:

“But one hope did not get fulfilled. I thought that upon succeeding to find this law, it would form a useful theory of quanta and of matter. But, this is not the case. It seems that the problem of matter and quanta makes the construction fall apart.”^{
$\text{9}$
}
“Eine Hoffnung ist aber nicht in Erfüllung gegangen. Ich dachte, wenn es gelingt, dieses Gesetz aufzustellen, dass es eine brauchbare Theorie der Quanten und Materie bilden würde. Aber das ist nicht der Fall. Die Konstruktion scheint am Problem der Materie und der Quanten zu scheitern.” ([95] , p. 442)
Thus, unfortunately, also the hopes of the eminent mathematician Schouten^{
$\text{10}$
}
, who knew some physics, were unfulfilled:

“[...] collections of positive and negative electricity which we are finding in the positive nuclei of hydrogen and in the negative electrons. The older Maxwell theory does not explain these collections, but also by the newer endeavours it has not been possible to recognise these collections as immediate consequences of the fundamental differential equations studied. However, if such an explanation should be found, we may perhaps also hope that new light is shed on the [...] mysterious quantum orbits.”^{
$\text{11}$
}
“[...] Anhäufungen von positiver und negativer Elektrizität, die wir in den positiven Wasserstoffkernen und in den negativen Elektronen antreffen. Die ältere Maxwellsche Theorie erklärt diese Anhäufungen nicht, aber auch den neueren Bestrebungen ist es bisher nicht gelungen, diese Anhäufungen als selbstverständliche Folgen der zugrundeliegenden Differentialgleichungen zu erkennen. Sollte aber eine solche Erklärung gefunden werden, so darf man vielleicht auch hoffen, dass die [...] mysteriösen Quantenbahnen in ein neues Licht gerückt werden.” ([300] , p. 39)
In this context, through all the years, Einstein vainly tried to derive, from the field equations of his successive unified field theories, the existence of elementary particles with opposite though otherwise equal electric charge but unequal mass. In correspondence with the state of empirical knowledge at the time (i.e., before the positron was found in 1932/33), but despite theoretical hints pointing into a different direction to be found in Dirac's papers, he always paired electron and proton^{
$\text{12}$
}
. Of course, by quantum field theory the dichotomy between matter and fields in the sense of a dualism is minimised as every field carries its particlelike quanta. Today's unified field theories appear in the form of gauge theories; matter is represented by operator valued spinhalf quantum fields (fermions) while the “forces” mediated by “exchange particles” are embodied in gauge fields, i.e., quantum fields of integer spin (bosons). The spacetime geometry used is rigidly fixed, and usually taken to be Minkowski space or, within string and membrane theory, some higherdimensional manifold also loosely called “spacetime”, although its signature might not be Lorentzian and its dimension might be 10, 11, 26, or some other number larger than four. A satisfactory inclusion of gravitation into the scheme of quantum field theory still remains to be achieved.
In the period considered, mutual reservations may have existed between the followers of the new quantum mechanics and those joining Einstein in the extension of his general relativity. The latter might have been puzzled by the seeming relapse of quantum mechanics from general covariance to a mere Galileior Lorentzinvariance, and by the statistical interpretation of the Schrödinger wave function. Lanczos
^{
$\text{13}$
}
, in 1929, was well aware of his being out of tune with those adherent to quantum mechanics:

“I therefore believe that between the `reactionary point of view' represented here, aiming at a complete fieldtheoretic description based on the usual spacetime structure and the probabilistic (statistical) point of view, a compromise [...] no longer is possible.”^{
$\text{14}$
}
“Ich glaube darum, dass zwischen dem hier vertretenen `reactionären Standpunkt', der eine vollständige feldtheoretische Beschreibung auf Grund der normalen RaumZeitStruktur erstrebt, und dem wahrscheinlichkeitstheoretischen (statistischen) Standpunkt ein Kompromiss [...] nicht mehr möglich ist.” ([197] , p. 486, footnote)
On the other hand, those working in quantum theory may have frowned upon the wealth of objects within unified field theories uncorrelated to a convincing physical interpretation and thus, in principle, unrelated to observation. In fact, until the 1930s, attempts still were made to “geometrize” wave mechanics while, roughly at the same time, quantisation of the gravitational field had also been tried [
283]
. Einstein belonged to those who regarded the idea of unification as more fundamental than the idea of field quantisation [
94]
. His thinking is reflected very well in a remark made by Lanczos at the end of a paper in which he tried to combine Maxwell's and Dirac's equations:

“If the possibilities anticipated here prove to be viable, quantum mechanics would cease to be an independent discipline. It would melt into a deepened `theory of matter' which would have to be built up from regular solutions of nonlinear differential equations, – in an ultimate relationship it would dissolve in the `world equations' of the Universe. Then, the dualism `matterfield' would have been overcome as well as the dualism `corpusclewave'.”^{
$\text{15}$
}
“Sollten sich die hier vorausgeahnten Möglichkeiten als wirklich lebensfähig erweisen, so würde die Quantenmechanik aufhören, eine selbständige Disziplin zu sein. Sie würde verschmelzen mit einer vertieften `Theorie der Materie', die auf reguläre Lösungen von nichtlinearen Differentialgleichungen aufzubauen hätte, – in letztem Zusammenhang also aufgehen in den `Weltgleichungen' des Universums. Der Dualismus `MaterieFeld' würde dann ebenso überwunden sein, wie der Dualismus `KorpuskelWelle'.” ([197] , p. 493)
Lanczos' work shows that there has been also a smaller subprogram of unification as described before, i.e., the view that somehow the electron and the photon might have to be treated together.
Therefore, a common representation of Maxwell's equations and the Dirac equation was looked for (cf. Section
7.1 ).
During the time span considered here, there also were those whose work did not help the idea of unification,
e.g., van Dantzig^{
$\text{16}$
}
wrote a series of papers in the first of which he stated:

“It is remarkable that not only no fundamental tensor [first fundamental form] or tensordensity, but also no connection, neither Riemannian nor projective, nor conformal, is needed for writing down the [Maxwell] equations. Matter is characterised by a bivectordensity [...].” ([367] , p. 422, and also [363, 364, 365, 366] )
If one of the fields to be united asks for less “geometry”, why to mount all the effort needed for generalising Riemannian geometry?
A methodological weak point in the process of the establishment of field equations for unified field theory was the constructive weakness of alternate physical limits to be taken:

∙
no electromagnetic field
$\to $
Einstein's equations in empty space;

∙
no gravitational field
$\to $
Maxwell's equations;

∙
“weak” gravitational and electromagnetic fields
$\to $
Einstein–Maxwell equations;

∙
no gravitational field but a “strong” electromagnetic field
$\to $
some sort of nonlinear electrodynamics.
A similar weakness occurred for the equations of motion; about the only limiting equation to be reproduced was Newton's equation augmented by the Lorentz force. Later, attempts were made to replace the relationship “geodesics
$\to $
freely falling point particles” by more general assumptions for charged or electrically neutral point particles – depending on the more general (nonRiemannian) connections introduced^{
$\text{17}$
}
. A main hindrance for an eventual empirical check of unified field theory was the persistent lack of a worked out example leading to a new gravitoelectromagnetic effect.
In the following Section
2 , a multitude of geometrical concepts (affine, conformal, projective spaces, etc.) available for unified field theories, on the one side, and their use as tools for a description of the dynamics of the electromagnetic and gravitational field on the other will be sketched. Then, we look at the very first steps towards a unified field theory taken by Reichenbächer^{
$\text{18}$
}
, Förster (alias Bach), Weyl^{
$\text{19}$
}
, Eddington^{
$\text{20}$
}
, and Einstein (see Section 3.1 ). In Section 4 , the main ideas are developed. They include Weyl's generalization of Riemannian geometry by the addition of a linear form (see Section 4.1 ) and the reaction to this approach. To this, Kaluza's idea concerning a geometrization of the electromagnetic and gravitational fields within a fivedimensional space will be added (see Section 4.2 ) as well as the subsequent extensions of Riemannian to affine geometry by Schouten, Eddington, Einstein, and others (see Section 4.3 ). After a short excursion to the world of mathematicians working on differential geometry (see Section 5 ), the research of Einstein and his assistants is studied (see Section 6 ). Kaluza's theory received a great deal of attention after O. Klein^{
$\text{21}$
}
intervention and extension of Kaluza's paper (see Section 6.3.2 ). Einstein's treatment of a special case of a metricaffine geometry, i.e., “distant parallelism”, set off an avalanche of research papers (see Section 6.4.4 ), the more so as, at the same time, the covariant formulation of Dirac's equation was a hot topic. The appearance of spinors in a geometrical setting, and endeavours to link quantum physics and geometry (in particular, the attempt to geometrize wave mechanics) are also discussed (see Section 7 ). We have included this topic although, strictly speaking, it only touches the fringes of unified field theory.
In Section
9 , particular attention is given to the mutual influence exerted on each other by the Princeton (Eisenhart^{
$\text{22}$
}
, Veblen^{
$\text{23}$
}
), French (Cartan^{
$\text{24}$
}
), and the Dutch (Schouten, Struik^{
$\text{25}$
}
) schools of mathematicians, and the work of physicists such as Eddington, Einstein, their collaborators, and others. In section 10 , the reception of unified field theory at the time is briefly discussed.
2 The Possibilities of Generalizing General Relativity: A Brief Overview
As a rule, the point of departure for unified field theory was general relativity. The additional task then was to “geometrize” the electromagnetic field. In this review, we will encounter essentially five different ways to include the electromagnetic field into a geometric setting:

∙
by connecting an additional linear form to the metric through the concept of “gauging” (Weyl);

∙
by introducing an additional space dimension (Kaluza);

∙
by choosing an asymmetric Ricci tensor (Eddington);

∙
by adding an antisymmetric tensor to the metric (Bach, Einstein);

∙
by replacing the metric by a 4bein field (Einstein).
In order to bring some order into the wealth of these attempts towards “unified field theory,” I shall distinguish four main avenues extending general relativity, according to their mathematical direction: generalisation of

∙
geometry,

∙
dynamics (Lagrangians, field equations),

∙
number field, and

∙
dimension of space,
as well as their possible combinations. In the period considered, all four directions were followed as well as combinations between them like e.g., fivedimensional theories with quadratic curvature terms in the Lagrangian. Nevertheless, we will almost exclusively be dealing with the extension of geometry and of the number of space dimensions.
2.1 Geometry
It is very easy to get lost in the many constructive possibilities underlying the geometry of unified field theories. We briefly describe the mathematical objects occurring in an order that goes from the less structured to the more structured cases. In the following, only local differential geometry is taken into account^{
$\text{26}$
}
.
The space of physical events will be described by a real, smooth manifold
${M}_{D}$
of dimension
$D$
coordinatised by local coordinates
${x}^{i}$
, and provided with smooth vector fields
$X,Y,\dots $
with components
${X}^{i},{Y}^{i},\dots $
and linear forms
$\omega ,\nu ,\dots $
, (
${\omega}_{i},{\nu}_{i}$
) in the local coordinate system, as well as further geometrical objects such as tensors, spinors, connections^{
$\text{27}$
}
. At each point,
$D$
linearly independent vectors (linear forms) form a linear space, the tangent space (cotangent space) of
${M}_{D}$
. We will assume that the manifold
${M}_{D}$
is spaceand timeorientable. On it, two independent fundamental structural objects will now be introduced.
2.1.1 Metrical structure
The first is a prescription for the definition of the distance
$ds$
between two infinitesimally close points on
${M}_{D}$
, eventually corresponding to temporal and spatial distances in the external world.
For
$ds$
, we need positivity, symmetry in the two points, and the validity of the triangle equation.
We know that
$ds$
must be homogeneous of degree one in the coordinate differentials
$d{x}^{i}$
connecting the points. This condition is not very restrictive; it still includes Finsler geometry [
280,
125,
223]
to be briefly touched, below.
In the following,
$ds$
is linked to a nondegenerate bilinear form
$g(X,Y)$
, called the first fundamental form ; the corresponding quadratic form defines a tensor field, the metrical tensor, with
${D}^{2}$
components
${g}_{ij}$
such that
$$\begin{array}{c}ds=\sqrt{{g}_{ij}d{x}^{i}d{x}^{j}},\end{array}$$ 
(1)

where the neighbouring points are labeled by
${x}^{i}$
and
${x}^{i}+d{x}^{i}$
, respectively^{
$\text{28}$
}
. Besides the norm of a vector
$\leftX\right:=\sqrt{{g}_{ij}{X}^{i}{X}^{j}}$
, the “angle” between directions
$X$
,
$Y$
can be defined by help of the metric:
$$cos((\u0338X,Y)):=\frac{{g}_{ij}{X}^{i}{Y}^{j}}{\leftX\right\leftY\right}.$$
From this we note that an antisymmetric part of the metrical tensor does not influence distances and norms but angles.
With the metric tensor having full rank, its inverse
${g}^{ik}$
is defined through^{
$\text{29}$
}
$$\begin{array}{c}{g}_{mi}{g}^{mj}={\delta}_{i}^{j}\end{array}$$ 
(2)

We are used to
$g$
being a symmetric tensor field, i.e., with
${g}_{ik}={g}_{\left(ik\right)}$
and with only
$D(D+1)/2$
components; in this case the metric is called Riemannian if its eigenvalues are positive (negative) definite and Lorentzian if its signature is
$\pm (D2)$
^{
$\text{30}$
}
. In the following this need not hold, so that the decomposition obtains^{
$\text{31}$
}
:
$$\begin{array}{c}{g}_{ik}={\gamma}_{\left(ik\right)}+{\phi}_{\left[ik\right]}.\end{array}$$ 
(3)

An asymmetric metric was considered in one of the first attempts at unifying gravitation and electromagnetism after the advent of general relativity.
For an asymmetric metric, the inverse
$$\begin{array}{c}{g}^{ik}={h}^{\left(ik\right)}+{f}^{\left[ik\right]}={h}^{ik}+{f}^{ik}\end{array}$$ 
(4)

is determined by the relations
$$\begin{array}{c}{\gamma}_{ij}{\gamma}^{ik}={\delta}_{j}^{k},{\phi}_{ij}{\phi}^{ik}={\delta}_{j}^{k},{h}_{ij}{h}^{ik}={\delta}_{j}^{k},{f}_{ij}{f}^{ik}={\delta}_{j}^{k},\end{array}$$ 
(5)

and turns out to be [
355]
$$\begin{array}{ccc}{h}^{\left(ik\right)}& =& \frac{\gamma}{g}{\gamma}^{ik}+\frac{\phi}{g}{\phi}^{im}{\phi}^{kn}{\gamma}_{mn},\end{array}$$ 
(6)

$$\begin{array}{ccc}{f}^{\left(ik\right)}& =& \frac{\phi}{g}{\phi}^{ik}+\frac{\gamma}{g}{\gamma}^{im}{\gamma}^{kn}{\phi}_{mn},\end{array}$$ 
(7)

where
$g$
,
$\phi $
, and
$\gamma $
are the determinants of the corresponding tensors
${g}_{ik}$
,
${\phi}_{ik}$
, and
${\gamma}_{ik}$
. We also note that
$$\begin{array}{c}g=\gamma +\phi +\frac{\gamma}{2}{\gamma}^{kl}{\gamma}^{mn}{\phi}_{km}{\phi}_{ln},\end{array}$$ 
(8)

where
$g:=det{g}_{ik}$
,
$\phi :=det{\phi}_{ik},$
$\gamma :=det{\gamma}_{ik}$
. The results ( 6 , 7 , 8 ) were obtained already by Reichenbächer ([
272]
, pp. 223–224)^{
$\text{32}$
}
and also by Schrödinger [
319]
. Eddington also calculated Equation ( 8 ); in his expression the term
$\sim {{\phi}_{ik}}^{\u25c6}{\phi}^{ik}$
is missing (cf. [
57]
, p. 233).
The manifold is called
spacetime if
$D=4$
and the metric is symmetric and Lorentzian, i.e., symmetric and with signature
$sigg=\pm 2$
. Nevertheless, sloppy contemporaneaous usage of the term “spacetime” includes arbitrary dimension, and sometimes is applied even to metrics with arbitrary signature.
In a manifold with Lorentzian metric, a nontrivial real
conformal structure always exists; from the equation
$$\begin{array}{c}g(X,X)=0\end{array}$$ 
(9)

results an equivalence class of metrics
$\left\{\lambda g\right\}$
with
$\lambda $
being an arbitrary smooth function. In view of the physical interpretation of the light cone as the locus of light signals, a causal structure is provided by the equivalence class of metrics [
65]
. For an asymmetric metric, this structure can exist as well; it then is determined by the symmetric part
${\gamma}_{ik}={\gamma}_{\left(ik\right)}$
of the metric alone taken to be Lorentzian. A special case of a space with a Lorentzian metric is Minkowski space, whose metrical components, in Cartesian coordinates, are given by
$$\begin{array}{c}{\eta}_{ik}={\delta}_{i}^{0}{\delta}_{i}^{0}{\delta}_{i}^{1}{\delta}_{i}^{1}{\delta}_{i}^{2}{\delta}_{i}^{2}{\delta}_{i}^{3}{\delta}_{i}^{3}.\end{array}$$ 
(10)

A geometrical characterization of Minkowski space as an uncurved, flat space is given below.
Let
${\mathcal{\mathcal{L}}}_{X}$
be the Lie derivative with respect to the tangent vector X^{
$\text{33}$
}
; then
${\mathcal{\mathcal{L}}}_{{X}_{p}}{\eta}_{ik}=0$
holds for the Lorentz group of generators
${X}_{p}$
.
The metric tensor
$g$
may also be defined indirectly through
$D$
vector fields forming an orthonormal
$D$
leg (bein)
${h}_{\hat{\text{\u0131}}}^{k}$
with
$$\begin{array}{c}{g}_{lm}={h}_{l\hat{\text{j}}}{h}_{m\hat{k}}{\eta}^{\hat{\text{j}}\hat{k}},\end{array}$$ 
(11)

where the hatted indices (“beinindices”) count the number of legs spanning the tangent space at each point
$(\hat{\text{j}}=1,2,\dots ,D)$
and are moved with the Minkowski metric^{
$\text{34}$
}
. From the geometrical point of view, this can always be done (cf. theories with distant parallelism ). By introducing 1forms
${\theta}^{\hat{k}}:={h}_{l}^{\hat{k}}d{x}^{l}$
, Equation ( 11 ) may be brought into the form
$d{s}^{2}={\theta}^{\hat{\text{\u0131}}}{\theta}^{\hat{k}}{\eta}_{\hat{\text{\u0131}}\hat{k}}$
.
A new
physical aspect will come in if the
${h}_{\hat{\text{\u0131}}}^{k}$
are considered to be the basic geometric variables satisfying field equations, not the metric. Such tetradtheories (for the case
$D=4$
) are described well by the concept of fibre bundle. The fibre at each point of the manifold contains, in the case of an orthonormal
$D$
bein (tetrad), all
$D$
beins (tetrads) related to each other by transformations of the group
$O\left(D\right)$
, or the Lorentz group, and so on.
In
Finsler geometry, the line element depends not only on the coordinates
${x}^{i}$
of a point on the manifold, but also on the infinitesimal elements of direction between neighbouring points
$d{x}^{i}$
:
$$\begin{array}{c}d{s}^{2}={g}_{ij}({x}^{n},d{x}^{m})d{x}^{i}d{x}^{j}.\end{array}$$ 
(12)

Again,
${g}_{ij}$
is required to be homogeneous of rank 1.
2.1.2 Affine structure
The second structure to be introduced is a linear connection
$L$
with
${D}^{3}$
components
${L}_{ij}^{k}$
; it is a geometrical object but not a tensor field and its components change inhomogeneously under local coordinate transformations^{
$\text{35}$
}
. The connection is a device introduced for establishing a comparison of vectors in different points of the manifold. By its help, a tensorial derivative
$\nabla $
, called covariant derivative is constructed. For each vector field and each tangent vector it provides another unique vector field. On the components of vector fields
$X$
and linear forms
$\omega $
it is defined by
$$\begin{array}{c}{\stackrel{+}{\nabla}}_{k}{X}^{i}=\frac{\partial {X}^{i}}{\partial {x}^{k}}+{L}_{kj}^{i}{X}^{j},{\stackrel{+}{\nabla}}_{k}{\omega}_{i}=\frac{\partial {\omega}_{i}}{\partial {x}^{k}}{{L}_{ki}}^{j}{\omega}_{j}.\end{array}$$ 
(13)

The expressions
${\stackrel{+}{\nabla}}_{k}{X}^{i}$
and
$\frac{\partial {X}^{i}}{\partial {x}^{k}}$
are abbreviated by
${X}_{\parallel k}^{i}$
and
${X}_{,k}^{i}$
, respectively, while for a scalar
$f$
covariant and partial derivative coincide:
${\nabla}_{i}f=\frac{\partial f}{\partial {x}_{i}}\equiv {\partial}_{i}f\equiv {f}_{,i}$
.
We have adopted the notational convention used by Schouten [
299,
309,
389]
. Eisenhart and others [
120,
233]
change the order of indices of the components of the connection:
$$\begin{array}{c}{\stackrel{}{\nabla}}_{k}{X}^{i}=\frac{\partial {X}^{i}}{\partial {x}^{k}}+{{L}_{jk}}^{i}{X}^{j},{\stackrel{}{\nabla}}_{k}{\omega}_{i}=\frac{\partial {\omega}_{i}}{\partial {x}^{k}}{L}_{ik}^{j}{\omega}_{j}.\end{array}$$ 
(14)

As long as the connection is symmetric, this does not make any difference as
${\stackrel{+}{\nabla}}_{k}{X}^{i}{\stackrel{}{\nabla}}_{k}{X}^{i}=2{{L}_{\left[kj\right]}}^{i}{X}^{j}.$
For both kinds of derivatives we have:
$$\begin{array}{c}{\stackrel{+}{\nabla}}_{k}\left({v}^{l}{w}_{l}\right)=\frac{\partial \left({v}^{l}{w}_{l}\right)}{\partial {x}^{k}},{\stackrel{}{\nabla}}_{k}\left({v}^{l}{w}_{l}\right)=\frac{\partial \left({v}^{l}{w}_{l}\right)}{\partial {x}^{k}}\end{array}$$ 
(15)

Both derivatives are used in versions of unified field theory by Einstein and others^{
$\text{36}$
}
.
A manifold provided with only a linear connection L is called
affine space. From the point of view of group theory, the affine group (linear inhomogeneous coordinate transformations) plays a special role: With regard to it the connection transforms as a tensor (cf. Section 2.1.5 ).
For a vector density (cf. Section
2.1.5 ), the covariant derivative of
$\hat{X}$
contains one more term:
$$\begin{array}{c}{\stackrel{+}{\nabla}}_{k}{\hat{X}}^{i}=\frac{\partial {\hat{X}}^{i}}{\partial {x}^{k}}+{{L}_{kj}}^{i}{\hat{X}}^{j}{{L}_{kr}}^{r}{\hat{X}}^{i},{\stackrel{}{\nabla}}_{k}{\hat{X}}^{i}=\frac{\partial {X}^{i}}{\partial {x}^{k}}+{{L}_{jk}}^{i}{\hat{X}}^{j}{{L}_{rk}}^{r}{\hat{X}}^{i}.\end{array}$$ 
(16)

A smooth vector field
$Y$
is said to be parallely transported along a parametrised curve
$\lambda \left(u\right)$
with tangent vector
$X$
if for its components
${Y}_{\parallel k}^{i}{X}^{k}\left(u\right)=0$
holds along the curve. A curve is called an autoparallel if its tangent vector is parallely transported along it at each point^{
$\text{37}$
}
:
$$\begin{array}{c}{X}_{\parallel k}^{i}{X}^{k}\left(u\right)=\sigma \left(u\right){X}^{i}.\end{array}$$ 
(17)

By a particular choice of the curve's parameter,
$\sigma =0$
may be imposed.
A transformation mapping autoparallels to autoparallels is given by:
$$\begin{array}{c}{L}_{ik}^{j}\to {{L}_{ik}}^{j}+{\delta}_{(i}^{j}{\omega}_{k)}.\end{array}$$ 
(18)

The equivalence class of autoparallels defined by Equation ( 18 ) defines a projective structure on
${M}_{D}$
[
404,
403]
. The particular set of connections
$$\begin{array}{c}{}_{\left(p\right)}{{L}_{ij}}^{k}:={{L}_{ij}}^{k}\frac{2}{D+1}{\delta}_{(i}^{k}{L}_{j)}\end{array}$$ 
(19)

with
${L}_{j}:={L}_{im}^{m}$
is mapped into itself by the transformation ( 18 ) [
347]
.
In Part II of this article, we shall find the set of transformations
${{L}_{ik}}^{j}\to {L}_{ik}^{j}+{\delta}_{i}^{j}\frac{\partial \omega}{\partial {x}^{k}}$
playing a role in versions of Einstein's unified field theory.
From the connection
${L}_{ij}^{k}$
further connections may be constructed by adding an arbitrary tensor field
$T$
to its symmetrised part^{
$\text{38}$
}
:
$$\begin{array}{c}{{\overline{L}}_{ij}}^{k}={{L}_{\left(ij\right)}}^{k}+{{T}_{ij}}^{k}={{\Gamma}_{ij}}^{k}+{{T}_{ij}}^{k}.\end{array}$$ 
(20)

By special choice of
$T$
we can regain all connections used in work on unified field theories. We will encounter examples in later sections. The antisymmetric part of the connection, i.e.,
$$\begin{array}{c}{{S}_{ij}}^{k}={{L}_{\left[ij\right]}}^{k}={{T}_{\left[ij\right]}}^{k}\end{array}$$ 
(21)

is called torsion ; it is a tensor field. The trace of the torsion tensor
${S}_{i}:={S}_{il}^{l}$
is called torsion vector ; it connects to the two traces of the affine connection
${L}_{i}:={{L}_{il}}^{l}$
;
${\stackrel{~}{L}}_{j}:={{L}_{lj}}^{l}$
, as
${S}_{i}=\frac{1}{2}({L}_{i}{\stackrel{~}{L}}_{i}).$
2.1.3 Different types of geometry
Affine geometry
Various subcases of affine spaces will occur, dependent on whether the connection is asymmetric or symmetric, i.e., with
${{L}_{ij}}^{k}={{\Gamma}_{ij}}^{k}$
. In physical applications, a metric always seems to be needed; hence in affine geometry it must be derived solely by help of the connection or, rather, by tensorial objects constructed from it. This is in stark contrast to Riemannian geometry where, vice versa, the connection is derived from the metric. Such tensorial objects are the two affine curvature tensors defined by^{
$\text{39}$
}
$$\begin{array}{ccc}{\stackrel{+}{K}}_{jkl}^{i}& =& {\partial}_{k}{L}_{lj}^{i}{\partial}_{l}{L}_{kj}^{i}+{L}_{km}^{i}{L}_{lj}^{m}{L}_{lm}^{i}{L}_{kj}^{m},\end{array}$$ 
(22)

$$\begin{array}{ccc}{\stackrel{}{K}}_{jkl}^{i}& =& {\partial}_{k}{L}_{jl}^{i}{\partial}_{l}{L}_{jk}^{i}+{L}_{mk}^{i}{L}_{jl}^{m}{L}_{ml}^{i}{L}_{jk}^{m},\end{array}$$ 
(23)

respectively. In a geometry with symmetric affine connection both tensors coincide because of
$$\begin{array}{c}\frac{1}{2}({\stackrel{+}{K}}_{jkl}^{i}{\stackrel{}{K}}_{jkl}^{i})={\partial}_{[k}{S}_{]lj}^{i}+2{S}_{j[k}^{m}{S}_{l]m}^{i}+{L}_{m[k}^{i}{S}_{l]j}^{m}{L}_{j[k}^{m}{S}_{l]m}^{i}.\end{array}$$ 
(24)

In particular, in Riemannian geometry, both affine curvature tensors reduce to the one and only Riemann curvature tensor.
The curvature tensors arise because the covariant derivative is
not commutative and obeys the Ricci identity :
$$\begin{array}{ccc}{\stackrel{+}{\nabla}}_{[j}{\stackrel{+}{\nabla}}_{k]}{A}^{i}& =& \frac{1}{2}{\stackrel{+}{K}}_{rjk}^{i}{A}^{r}{S}_{jk}^{r}{\stackrel{+}{\nabla}}_{r}{A}^{i}\end{array}$$ 
(25)

$$\begin{array}{ccc}{\stackrel{}{\nabla}}_{[j}{\stackrel{}{\nabla}}_{k]}{A}^{i}& =& \frac{1}{2}{\stackrel{}{K}}_{rjk}^{i}{A}^{r}+{S}_{jk}^{r}{\stackrel{}{\nabla}}_{r}{A}^{i}\end{array}$$ 
(26)

For a vector density, the identity is given by
$$\begin{array}{c}{\stackrel{+}{\nabla}}_{[j}{\stackrel{+}{\nabla}}_{k]}{\hat{A}}^{i}=\frac{1}{2}{\stackrel{+}{K}}_{rjk}^{i}{\hat{A}}^{r}{S}_{jk}^{r}{\stackrel{+}{\nabla}}_{r}{\hat{A}}^{i}+\frac{1}{2}{V}_{jk}{\hat{A}}^{i},\end{array}$$ 
(27)

with the homothetic curvature
${V}_{jk}$
to be defined below in Equation ( 31 ).
The curvature tensor (
22 ) satisfies two algebraic identities:
$$\begin{array}{ccc}{\stackrel{+}{K}}_{j\left[kl\right]}^{i}& =& 0,\end{array}$$ 
(28)

$$\begin{array}{ccc}{\stackrel{+}{K}}_{\left\{jkl\right\}}^{i}& =& 2{\nabla}_{\{j}{S}_{kl\}}^{i}+4{S}_{m\{j}^{i}{S}_{kl\}}^{m},\end{array}$$ 
(29)

where the curly bracket denotes cyclic permutation:
$${K}_{\left\{jkl\right\}}^{i}:={K}_{jkl}^{i}+{K}_{ljk}^{i}+{K}_{klj}^{i}.$$
These identities can be found in Schouten's book of 1924 ([
299]
, p. 88, 91) as well as the additional single integrability condition, called Bianchi identity :
$$\begin{array}{c}{\stackrel{+}{K}}_{j\{kl\parallel m\}}^{i}=2{K}_{r\{kl}^{i}{S}_{m\}j}^{r}.\end{array}$$ 
(30)

A corresponding condition obtains for the curvature tensor
$\stackrel{}{K}$
from Equation ( 23 ).
From both affine curvature tensors we may form two different tensorial traces each. In the first case
${V}_{kl}:={K}_{ikl}^{i}={V}_{\left[kl\right]}$
, and
${K}_{jk}:={K}_{jki}^{i}.{V}_{kl}$
is called homothetic curvature, while
${K}_{jk}$
is the first of the two affine generalisations from
$\stackrel{+}{K}$
and
$\stackrel{}{K}$
of the Ricci tensor in Riemannian geometry. We get^{
$\text{40}$
}
$$\begin{array}{c}{V}_{kl}={\partial}_{k}{L}_{l}{\partial}_{l}{L}_{k},\end{array}$$ 
(31)

and the following identities hold:
$$\begin{array}{ccc}{V}_{kl}+2{K}_{\left[kl\right]}& =& 4{\nabla}_{[k}{S}_{l]}+8{S}_{kl}^{m}{S}_{m}+2{\nabla}_{m}{S}_{kl}^{m},\end{array}$$ 
(32)

$$\begin{array}{ccc}{\stackrel{}{V}}_{kl}+2{\stackrel{}{K}}_{\left[kl\right]}& =& 4{\stackrel{}{\nabla}}_{[k}{S}_{l]}+8{S}_{kl}^{m}{S}_{m}+2{\nabla}_{m}{S}_{kl}^{m},\end{array}$$ 
(33)

where
${S}_{k}:={S}_{kl}^{l}$
. While
${V}_{kl}$
is antisymmetric,
${K}_{jk}$
has both tensorial symmetric and antisymmetric parts:
$$\begin{array}{ccc}{K}_{\left[kl\right]}& =& {\partial}_{[k}{\stackrel{~}{L}}_{l]}+{\nabla}_{m}{S}_{kl}^{m}+{L}_{m}{S}_{kl}^{m}+2{L}_{[lr}^{m}{S}_{mk]}^{r},\end{array}$$ 
(34)

$$\begin{array}{ccc}{K}_{\left(kl\right)}& =& {\partial}_{(k}{\stackrel{~}{L}}_{l)}{\partial}_{m}{L}_{\left(kl\right)}^{m}{\stackrel{~}{L}}_{m}{L}_{\left(kl\right)}^{m}+{L}_{(k\leftm\right}^{n}{L}_{l)n}^{m}.\end{array}$$ 
(35)

We use the notation
${A}_{\left(i\rightk\leftl\right)}$
in order to exclude the index
$k$
from the symmetrisation bracket^{
$\text{41}$
}
.
In order to shorten the presentation of affine geometry, we refrain from listing the corresponding set of equations for the other affine curvature tensor (cf., however, [
355]
).
For a
symmetric affine connection, the preceding results reduce considerably due to
${S}_{kl}^{m}=0$
.
From Equations (
29 , 30 , 32 ) we obtain the identities:
$$\begin{array}{ccc}{K}_{\left\{jkl\right\}}^{i}& =& 0,\end{array}$$ 
(36)

$$\begin{array}{ccc}{K}_{j\{kl\parallel m\}}^{i}& =& 0,\end{array}$$ 
(37)

$$\begin{array}{ccc}{V}_{kl}+2{K}_{\left[kl\right]}& =& 0,\end{array}$$ 
(38)

i.e., only one independent trace tensor of the affine curvature tensor exists. For the antisymmetric part of the Ricci tensor
${K}_{\left[kl\right]}={\partial}_{[k}{\stackrel{~}{L}}_{l]}$
holds. This equation will be important for the physical interpretation of affine geometry.
In affine geometry, the simplest way to define a fundamental tensor is to set
${g}_{ij}:=\alpha {K}_{\left(ij\right)}$
, or
${g}_{ij}:=\alpha {\stackrel{}{K}}_{\left(ij\right)}$
. It may be desirable to derive the metric from a Lagrangian; then the simplest scalar density that could be used as such is given by
$det\left({K}_{ij}\right)$
^{
$\text{42}$
}
.
As a final result in this section, we give the curvature tensor calculated from the connection
${\overline{L}}_{ij}^{k}={\Gamma}_{ij}^{k}+{T}_{ij}^{k}$
(cf. Equation ( 20 )), expressed by the curvature tensor of
${\Gamma}_{ij}^{k}$
and by the tensor
${T}_{ij}^{k}$
:
$$\begin{array}{c}{K}_{jkl}^{i}\left(\overline{L}\right)={K}_{jkl}^{i}(\Gamma )+2{}^{(\Gamma )}{\nabla}_{[k}{T}_{l]j}^{i}2{T}_{[k\leftj\right}^{m}{T}_{l]m}^{i}+2{S}_{kl}^{m}{T}_{mj}^{i},\end{array}$$ 
(39)

where
${}^{(\Gamma )}\nabla $
is the covariant derivative formed with the connection
${\Gamma}_{ij}^{k}$
(cf. also [
309]
, p. 141).
Mixed geometry
A manifold carrying both structural elements, i.e., metric and connection, is called a metricaffine space. If the first fundamental form is taken to be asymmetric, i.e., to contain an antisymmetric part
${g}_{\left[ik\right]}:=\frac{1}{2}({g}_{ij}{g}_{ji})$
, we speak of a mixed geometry. In principle, both metricaffine space and mixed geometry may always be reinterpreted as Riemannian geometry with additional geometric objects: the 2form field
$\phi \left(f\right)$
(symplectic form), the torsion
$S$
, and the nonmetricity
$Q$
(cf.
Equation
41 ). It depends on the physical interpretation, i.e., the assumed relation between mathematical objects and physical observables, which geometry is the most suitable.
From the symmetric part of the first fundamental form
${h}_{ij}={g}_{\left(ij\right)}$
, a connection may be constructed, often called after LeviCivita^{
$\text{43}$
}
[
203]
,
$$\begin{array}{c}{\{}_{ij}^{k}\}:=\frac{1}{2}{\gamma}^{kl}({\gamma}_{li,j}+{\gamma}_{lj,i}{h}_{ij,}),\end{array}$$ 
(40)

and from it the Riemannian curvature tensor defined as in Equation ( 22 ) with
${L}_{ij}^{k}={\{}_{ij}^{k}\}$
(cf.
Section
2.1.3 );
${\{}_{ij}^{k}\}$
is called the Christoffel symbol. Thus, in metricaffine and in mixed geometry, two different connections arise in a natural way. In the remaining part of this section we will deal with a symmetric fundamental form
${\gamma}_{ij}$
only, and denote it by
${g}_{ij}$
.
With the help of the symmetric affine connection, we may define the tensor of
nonmetricity
${Q}_{ij}^{k}$
by^{
$\text{44}$
}
$$\begin{array}{c}{Q}_{ij}^{k}:={g}^{kl}{\nabla}_{l}{g}_{ij}.\end{array}$$ 
(41)

Then the following identity holds:
$$\begin{array}{c}{\Gamma}_{ij}^{k}={\{}_{ij}^{k}\}+{K}_{ij}^{k}+\frac{1}{2}({Q}_{ij}^{k}+{Q}_{ji}^{k}{Q}_{ji}^{k}),\end{array}$$ 
(42)

where the contorsion tensor
${K}_{ij}^{k}$
, a linear combination of torsion
${S}_{ij}^{k}$
, is defined by^{
$\text{45}$
}
$$\begin{array}{c}{K}_{ij}^{k}:={S}_{ji}^{k}+{S}_{ij}^{k}{S}_{ij}^{k}={K}_{ij}^{k}.\end{array}$$ 
(43)

The inner product of two tangent vectors
${A}^{i},{B}^{k}$
is not conserved under parallel transport of the vectors along
${X}^{l}$
if the nonmetricity tensor does not vanish:
$$\begin{array}{c}{X}^{k}{\stackrel{+}{\nabla}}_{k}\left({A}^{n}{B}^{m}{g}_{nm}\right)={Q}_{nml}{A}^{n}{B}^{m}{X}^{l}\ne 0.\end{array}$$ 
(44)

A connection for which the nonmetricity tensor vanishes, i.e.,
$$\begin{array}{c}{\stackrel{+}{\nabla}}_{k}{g}_{ij}=0\end{array}$$ 
(45)

holds, is called metriccompatible^{
$\text{46}$
}
.
J. M. Thomas
^{
$\text{47}$
}
introduced a combination of the terms appearing in
$\stackrel{+}{\nabla}$
and
$\stackrel{}{\nabla}$
to define a covariant derivative for the metric ([
345]
, p. 188),
$$\begin{array}{c}{g}_{ik/l}:={g}_{ik,l}{g}_{rk}{\Gamma}_{il}^{r}{g}_{ir}{\Gamma}_{lk}^{r},\end{array}$$ 
(46)

and extended it for tensors of arbitrary rank
$\ge 3$
.
Einstein later used as a constraint on the metrical tensor
$$\begin{array}{c}0={g}_{\stackrel{ik\parallel l}{+}}:={g}_{ik,l}{g}_{rk}{\Gamma}_{il}^{r}{g}_{ir}{\Gamma}_{lk}^{r},\end{array}$$ 
(47)

a condition that cannot easily be interpreted geometrically [
96]
. We will have to deal with Equation ( 47 ) in Section 6.1 and, more intensively, in Part II of this review.
Connections that are
not metriccompatible have been used in unified field theory right from the beginning. Thus, in Weyl's theory [
397,
395]
we have
$$\begin{array}{c}{Q}_{ijk}={Q}_{k}{g}_{ij}.\end{array}$$ 
(48)

In case of such a relationship, the geometry is called semimetrical [
299,
309]
. According to Equation ( 44 ), in Weyl's theory the inner product multiplies by a scalar factor under parallel transport:
$$\begin{array}{c}{X}^{k}{\stackrel{+}{\nabla}}_{k}\left({A}^{n}{B}^{m}{g}_{nm}\right)=\left({Q}_{l}{X}^{l}\right){A}^{n}{B}^{m}{g}_{nm}.\end{array}$$ 
(49)

This means that the light cone is preserved by parallel transport.
We may also abbreviate the last term in the identity (
42 ) by introducing
$$\begin{array}{c}{X}_{ij}^{k}:={Q}_{ij}^{k}+{Q}_{ji}^{k}{Q}_{ji}^{k}.\end{array}$$ 
(50)

Then, from Equation ( 39 ), the curvature tensor of a torsionless affine space is given by
$$\begin{array}{c}{K}_{jkl}^{i}\left(\overline{\Gamma}\right)={K}_{jkl}^{i}\left({\{}_{nm}^{r}\right\})+2{}^{\left({\{}_{jk}^{i}\right\})}{\nabla}_{[k}{X}_{l]j}^{i}2{X}_{[k\leftj\right}^{m}{X}_{l]m}^{i},\end{array}$$ 
(51)

where
${}^{\left({\{}_{jk}^{i}\right\})}\nabla $
is the covariant derivative formed with the Christoffel symbol.
Riemann–Cartan geometry is the subcase of a metricaffine geometry in which the metriccompatible connection contains torsion, i.e., an antisymmetric part
${L}_{\left[ij\right]}^{k}$
; torsion is a tensor field to be linked to physical observables. A linear connection whose antisymmetric part
${S}_{ij}^{k}$
has the form
$$\begin{array}{c}{S}_{ij}^{k}={S}_{[i}{\delta}_{j]}^{k}\end{array}$$ 
(52)

is called semisymmetric [
299]
.
Riemannian geometry is the further subcase with vanishing torsion of a metricaffine geometry with metriccompatible connection. In this case, the connection is derived from the metric:
${\Gamma}_{ij}^{k}={\{}_{ij}^{k}\}$
, where
${\{}_{ij}^{k}\}$
is the usual Christoffel symbol ( 40 ). The covariant derivative of
$A$
with respect to the LeviCivita connection
$\stackrel{{\{}_{ij}^{k}\}}{\nabla}$
is abbreviated by
${A}_{;k}$
. The Riemann curvature tensor is denoted by
$$\begin{array}{c}{R}_{jkl}^{i}={\partial}_{k}{\{}_{lj}^{i}\}{\partial}_{l}{\{}_{kj}^{i}\}+{\{}_{km}^{i}\left\}{\{}_{lj}^{m}\right\}{\{}_{lm}^{i}\left\}{\{}_{kj}^{m}\right\}.\end{array}$$ 
(53)

An especially simple case of a Riemanian space is Minkowski space, the curvature of which vanishes:
$$\begin{array}{c}{R}_{jkl}^{i}\left(\eta \right)=0.\end{array}$$ 
(54)

This is an invariant characterisation irrespective of whether the Minkowski metric
$\eta $
is given in Cartesian coordinates as in Equation ( 10 ), or in an arbitrary coordinate system. We also have
${\mathcal{\mathcal{L}}}_{X}{R}_{ijkl}=0$
where
$\mathcal{\mathcal{L}}$
is the Liederivative (see below under “symmetries”), and
$X$
stands for the generators of the Lorentz group.
In Riemanian geometry, the socalled geodesic equation,
$$\begin{array}{c}{X}_{;k}^{i}{X}^{k}\left(u\right)=\sigma \left(u\right){X}^{i},\end{array}$$ 
(55)

determines the shortest and the straightest curve between two infinitesimally close points.
However, in metric affine and in mixed geometry
geodesic and autoparallel curves will have to be distinguished.
A
conformal transformation of the metric,
$$\begin{array}{c}{g}_{ik}\to {g}_{ik}^{\prime}=\lambda {g}_{ik},\end{array}$$ 
(56)

with a smooth function
$\lambda $
changes the components of the nonmetricity tensor,
$$\begin{array}{c}{Q}_{ij}^{k}\to {Q}_{ij}^{k}+{g}_{ij}{g}^{kl}{\partial}_{l}\sigma ,\end{array}$$ 
(57)

as well as the LeviCivita connection,
$$\begin{array}{c}{\{}_{ij}^{k}\}\to {\{}_{ij}^{k}\}+\frac{1}{2}({\sigma}_{i}{\delta}_{j}^{k}{\sigma}_{j}{\delta}_{i}^{k}+{g}_{ij}{g}^{kl}{\sigma}_{l}),\end{array}$$ 
(58)

with
${\sigma}_{i}:={\lambda}^{1}{\partial}_{i}\lambda .$
As a consequence, the Riemann curvature tensor
${R}_{jkl}^{i}$
is also changed; if, however,
${{R}^{\prime}}_{jkl}^{i}=0$
can be reached by a conformal transformation, then the corresponding spacetime is called conformally flat. In
${M}_{D}$
, for
$D>3$
, the vanishing of the Weyl curvature tensor
$$\begin{array}{c}{C}_{jkl}^{i}:={R}_{jkl}^{i}+\frac{2}{D2}({\delta}_{[k}^{i}{R}_{l]j}+{g}_{j[l}{R}_{k]}^{i})+\frac{2R}{(D1)(D2)}{\delta}_{[l}^{i}{g}_{k]j}\end{array}$$ 
(59)

is a necessary and sufficient condition for
${M}_{D}$
to be conformally flat ([
397]
, p. 404, [
299]
, p. 170).
Even before Weyl, the question had been asked (and answered) as to what extent the conformal and the projective structures were determining the geometry: According to Kretschmann (and then to Weyl) they fix the metric up to a constant factor ([
195]
; see also [
401]
, Appendix 1; for a modern approach, cf. [
65]
).
The geometry needed for the preand nonrelativistic approaches to unified field theory will have to be dealt with separately. There, the metric tensor of space is Euclidean and not of full rank; time is described by help of a linear form (Newton–Cartan geometry, cf. [
63,
64]
). In the following we shall deal only with relativistic unified field theories.
Together with Ricci, he developed tensor calculus and introduced covariant differentiation. He worked also in the mechanical manybody problem, in hydrodynamics, general relativity theory, and unified field theory. Strongly opposed to Fascism in Italy and dismissed from his professorship in 1938.
Projective geometry
Projective geometry is a generalisation of Riemannian geometry in the following sense: Instead of tangent spaces with the light cone
${\eta}_{ik}d{x}^{i}d{x}^{k}=0$
, where
$\eta $
is the Minkowski metric, in each event now a tangent space with a general, nondegenerate surface of second order
$\gamma $
will be introduced. This leads to a tangential cone
${g}_{ik}d{x}^{i}d{x}^{k}=0$
in the origin (cf. Equation ( 9 )), and to a hyperplane, the polar plane, formed by the contact points of the tangential cone and the surface
$\gamma $
. In place of the
$D$
inhomogeneous coordinates
${x}^{i}$
of
${M}_{D}$
,
$D+1$
homogeneous coordinates
${X}^{\alpha}$
(
$\alpha =0,1,2,\dots ,D$
) are defined^{
$\text{48}$
}
such that they transform as homogeneous functions of first degree:
$$\begin{array}{c}{X}^{\alpha}\frac{\partial {{X}^{\prime}}^{\nu}}{\partial {X}^{\mu}}={{X}^{\prime}}^{\nu}.\end{array}$$ 
(60)

The connection to the inhomogeneous coordinates
${x}^{i}$
is given by homogeneous functions of degree zero, e.g., by
${x}^{i}=\frac{{X}^{i}}{{\phi}_{\alpha}{X}^{\alpha}}$
^{
$\text{49}$
}
. Thus, the
${X}^{\alpha}$
themselves form the components of a tangent vector. Furthermore, the quadratic form
${g}_{\alpha \beta}{X}^{\alpha}{X}^{\beta}=\epsilon =\pm 1$
is adopted with
${g}_{\alpha \beta}$
being a homogeneous function of degree
$2$
. A tensor field
${{{{T}_{{n}_{1}}^{{m}_{1}}}_{{n}_{2}}^{{m}_{2}}}_{{n}_{3}}^{{m}_{3}}}_{\dots}^{\dots}$
(cf. Section 2.1.5 ) depending on the homogeneous coordinates
${X}^{\mu}$
with
$u$
contravariant (upper) and
$l$
covariant (lower) indices is required to be a homogeneous function of degree
$r:=ul$
.
If we define
${\gamma}_{\mu}^{i}:=\frac{\partial {x}^{i}}{\partial {X}^{\mu}}$
, with
${\gamma}_{\mu}^{i}{X}^{\mu}=0$
, then
${\gamma}_{\mu}^{i}$
transforms like a tangent vector under point transformations of the
${x}^{i}$
, and as a covariant vector under homogeneous transformations of the
${X}^{\alpha}$
.
The
${\gamma}_{\mu}^{i}$
may be used to relate covariant vectors
${a}_{i}$
and
${A}_{\mu}$
by
${A}_{\mu}={\gamma}_{\mu}^{i}{a}_{i}.$
Thus, the metric tensor in the space of homogeneous coordinates
${g}_{\alpha \beta}$
and the metric tensor
${g}_{ik}$
of
${M}_{D}$
are related by
${g}_{ik}={\gamma}_{i}^{\alpha}{\gamma}_{k}^{\beta}{g}_{\alpha \beta}$
with
${\gamma}_{\mu}^{i}{\gamma}_{k}^{\mu}={\delta}_{k}^{i}.$
The inverse relationship is given by
${g}_{\alpha \beta}={\gamma}_{\alpha}^{i}{\gamma}_{\beta}^{k}{g}_{ik}+\epsilon {X}_{\alpha}{X}_{\beta}$
with
${X}_{\alpha}={g}_{\alpha \beta}{X}^{\beta}.$
The covariant derivative for tensor fields in the space of homogeneous coordinates is defined as before (cf. Section 2.1.2 ):
$$\begin{array}{c}{\nabla}_{\alpha}{A}^{\beta}\left(X\right)=\frac{\partial {A}^{\beta}\left(X\right)}{\partial {X}^{\alpha}}+{\Gamma}_{\alpha \nu}^{\beta}\left(X\right){A}^{\nu}\left(X\right).\end{array}$$ 
(61)

The covariant derivative of the quantity
${\gamma}_{k}^{\mu}$
interconnecting both spaces is given by
$$\begin{array}{c}{\nabla}_{\rho}{\gamma}_{k}^{\mu}=\frac{\partial {\gamma}_{k}^{\mu}}{\partial {x}^{\rho}}+{\Gamma}_{\rho \sigma}^{\mu}{\gamma}_{k}^{\sigma}{\{}_{kl}^{m}\}{\gamma}_{\rho}^{l}{\gamma}_{m}^{\mu}.\end{array}$$ 
(62)

2.1.4 Cartan's method
In this section, we briefly present Cartan's oneform formalism in order to make understandable part of the literature. Cartan introduces oneforms
${\theta}^{\hat{a}}$
(
$\hat{a}=1,\dots ,4$
) by
${\theta}^{\hat{a}}:={h}_{l}^{\hat{a}}d{x}^{l}.$
The reciprocal basis in tangent space is given by
${e}_{\hat{\text{j}}}={h}_{\hat{\text{j}}}^{l}\frac{\partial}{\partial {x}^{l}}$
. Thus,
${\theta}^{\hat{a}}\left({e}_{\hat{\text{j}}}\right)={\delta}_{\hat{\text{j}}}^{\hat{a}}$
. The metric is then given by
${\eta}_{\hat{\text{\u0131}}\hat{k}}{\theta}^{\hat{\text{\u0131}}}\otimes {\theta}^{\hat{k}}$
. The covariant derivative of a tangent vector with beincomponents
${X}^{\hat{k}}$
is defined via Cartan's first structure equations,
$$\begin{array}{c}{\Theta}^{i}:=D{\theta}^{\hat{\text{\u0131}}}=d{\theta}^{\hat{\text{\u0131}}}+{\omega}_{\mathit{}\hat{l}}^{\hat{\text{\u0131}}}\wedge {\theta}^{\hat{l}},\end{array}$$ 
(63)

where
${\omega}_{\mathit{}\hat{k}}^{\hat{\text{\u0131}}}$
is the connection1form, and
${\Theta}^{\hat{\text{\u0131}}}$
is the torsion2form,
${\Theta}^{\hat{\text{\u0131}}}={S}_{\hat{l}\hat{m}}^{\hat{\text{\u0131}}}{\theta}^{\hat{l}}\wedge {\theta}^{\hat{m}}$
. We have
${\omega}_{\hat{\text{\u0131}}\hat{k}}={\omega}_{\hat{k}\hat{\text{\u0131}}}.$
The link to the components
${L}_{\left[ij\right]}^{k}$
of the affine connection is given by
${\omega}_{\mathit{}\hat{k}}^{\hat{\text{\u0131}}}={h}_{l}^{\hat{\text{\u0131}}}{h}_{\hat{k}}^{m}{L}_{\hat{r}m}^{l}{\theta}^{\hat{r}}$
^{
$\text{50}$
}
. The covariant derivative of a tangent vector with beincomponents
${X}^{\hat{k}}$
then is
$$\begin{array}{c}D{X}^{\hat{k}}:=d{X}^{\hat{k}}+{\omega}_{\hat{l}}^{\hat{k}}{X}^{\hat{l}}.\end{array}$$ 
(64)

By further external derivation^{
$\text{51}$
}
on
$\Theta $
we arrive at the second structure relation of Cartan,
$$\begin{array}{c}D{\Theta}^{\hat{k}}={\Omega}_{\hat{l}}^{\hat{k}}\wedge {\theta}^{\hat{l}}.\end{array}$$ 
(65)

In Equation ( 65 ) the curvature2form
${\Omega}_{\hat{l}}^{\hat{k}}=\frac{1}{2}{R}_{\hat{l}\hat{m}\hat{n}}^{\hat{k}}{\theta}^{\hat{m}}\wedge {\theta}^{\hat{n}}$
appears, which is given by
$$\begin{array}{c}{\Omega}_{\hat{l}}^{\hat{k}}=d{\omega}_{\hat{l}}^{\hat{k}}+{\omega}_{\hat{l}}^{\hat{k}}\wedge {\omega}_{\hat{l}}^{\hat{k}}.\end{array}$$ 
(66)

${\Omega}_{\hat{k}}^{\hat{k}}$
is the homothetic curvature.

(1)
$d(a\omega +b\mu )=ad\omega +bd\mu $
,

(2)
$d(\omega \wedge \mu )=d\omega \wedge \mu \omega \wedge d\mu $
,

(3)
$dd\omega =0$
.
2.1.5 Tensors, spinors, symmetries
Tensors
Up to here, no definitions of a tensor and a tensor field were given: A tensor
${T}_{p}\left({M}_{D}\right)$
of type
$(r,s)$
at a point
$p$
on the manifold
${M}_{D}$
is a multilinear function on the Cartesian product of
$r$
cotangentand
$s$
tangent spaces in
$p$
. A tensor field is the assignment of a tensor to each point of
${M}_{D}$
. Usually, this definition is stated as a linear, homogeneous transformation law for the tensor components in local coordinates:
$$\begin{array}{c}{T}_{{l}_{1}^{\prime}{l}_{2}^{\prime}{l}_{3}^{\prime}\dots}^{{k}_{1}^{\prime}{k}_{2}^{\prime},{k}_{3}^{\prime}\dots}={T}_{{n}_{1}{n}_{2}{n}_{3}\dots}^{{m}_{1}{m}_{2}{m}_{3}\dots}\frac{\partial {x}^{{n}_{1}}}{\partial {x}^{{l}_{1}^{\prime}}}\frac{\partial {x}^{{n}_{2}}}{\partial {x}^{{l}_{2}^{\prime}}}\frac{\partial {x}^{{n}_{3}}}{\partial {x}^{{l}_{3}^{\prime}}}\frac{\partial {x}^{{k}_{1}^{\prime}}}{\partial {x}^{{m}_{1}}}\frac{\partial {x}^{{k}_{2}^{\prime}}}{\partial {x}^{{m}_{2}}}\frac{\partial {x}^{{k}_{3}^{\prime}}}{\partial {x}^{{m}_{3}}}\cdots \end{array}$$ 
(67)

where
${x}^{{k}^{\prime}}={x}^{{k}^{\prime}}\left({x}^{i}\right)$
with smooth functions on the r.h.s. are taken from the set (“group”) of coordinate transformations (diffeomorphisms). Strictly speaking, tensors are representations of the abstract group at a point on the manifold^{
$\text{52}$
}
.
A
relative tensor
${T}_{p}\left({M}_{D}\right)$
of type
$(r,s)$
and of weight
$\omega $
at a point
$p$
on the manifold
${M}_{D}$
transforms like
$$\begin{array}{c}{T}_{{l}_{1}^{\prime}{l}_{2}^{\prime}{l}_{3}^{\prime}..}^{{k}_{1}^{\prime}{k}_{2}^{\prime},{k}_{3}^{\prime}..}={\left[det\left(\frac{\partial {x}^{s}}{\partial {{x}^{\prime}}^{r}}\right)\right]}^{\omega}{T}_{{n}_{1}{n}_{2}{n}_{3}...}^{{m}_{1}{m}_{2}{m}_{3}...}\frac{\partial {x}^{{n}_{1}}}{\partial {x}^{{l}_{1}^{\prime}}}\frac{\partial {x}^{{n}_{2}}}{\partial {x}^{{l}_{2}^{\prime}}}\frac{\partial {x}^{{n}_{3}}}{\partial {x}^{{l}_{3}^{\prime}}}\frac{\partial {x}^{{k}_{1}^{\prime}}}{\partial {x}^{{m}_{1}}}\frac{\partial {x}^{{k}_{2}^{\prime}}}{\partial {x}^{{m}_{2}}}\frac{\partial {x}^{{k}_{3}^{\prime}}}{\partial {x}^{{m}_{3}}}\cdots \end{array}$$ 
(68)

An example is given by the totally antisymmetric object
${\epsilon}_{ijkl}$
with
${\epsilon}_{ijkl}=\pm 1$
, or
${\epsilon}_{ijkl}=0$
depending on whether
$\left(ijkl\right)$
is an even or odd permutation of (0123), or whether two indices are alike.
$\omega =1$
for
${\epsilon}_{ijkl}$
; in this case, the relative tensor is called tensor density. We can form a tensor from
${\epsilon}_{ijkl}$
by introducing
${\eta}_{ijkl}:=\sqrt{g}{\epsilon}_{ijkl}$
, where
${g}_{ik}$
is a Lorentzmetric. Note that
${\eta}^{ijkl}:=\frac{1}{\sqrt{g}}{\epsilon}^{ijkl}.$
The dual to a 2form (skewsymmetric tensor) then is defined by
${}^{*}{F}^{ij}=\frac{1}{2}{\eta}^{ijkl}{F}_{kl}.$
In connection with conformal transformations
$g\to \lambda g$
, the concept of the gaugeweight of a tensor is introduced. A tensor
${T}_{\dots}^{\dots}$
is said to be of gauge weight
$q$
if it transforms by Equation ( 56 ) as
$$\begin{array}{c}{{T}^{\prime}}_{\dots}^{\dots}={\lambda}^{q}{T}_{\dots}^{\dots}.\end{array}$$ 
(69)

Objects that transform as in Equation ( 67 ) but with respect to a subgroup, e.g., the linear group, affine group
$G\left(D\right)$
, orthonormal group
$O\left(D\right)$
, or the Lorentz group
$\mathcal{\mathcal{L}}$
, are tensors in a restricted sense; sometimes they are named affine or Cartesian tensors. All the subgroups mentioned are Liegroups, i.e., continuous groups with a finite number of parameters. In general relativity, both the “group” of general coordinate transformations and the Lorentz group are present. The concept of tensors used in Special Relativity is restricted to a representation of the Lorentz group; however, as soon as the theory is to be given a coordinateindependent (“generally covariant”) form, then the full tensor concept comes into play.
Spinors
Spinors are representations of the Lorentz group only; as such they are related strictly to the tangent space of the spacetime manifold. To see how spinor representations can be obtained, we must use the 2–1 homomorphism of the group
$SL(2,\mathbf{C})$
and the proper orthochronous Lorentz group, a subgroup of the full Lorentz group^{
$\text{53}$
}
. Let
$\mathbf{A}\in SL(2,\mathbf{C})$
; then
$\mathbf{A}$
is a complex (2by2)matrix with
$det\mathbf{A}=1$
. By picking the special Hermitian matrix
$$\begin{array}{c}\mathbf{S}={x}^{0}1+{\sum}_{p}{\sigma}_{p}{x}^{p},\end{array}$$ 
(70)

where
$1$
is the (2 by 2)unit matrix and
${\sigma}_{p}$
are the Pauli matrices satisfying
$$\begin{array}{c}{\sigma}_{i}{\sigma}_{k}+{\sigma}_{k}{\sigma}_{i}=2{\delta}_{ik}.\end{array}$$ 
(71)

Then, by a transformation
$\mathbf{A}$
from
$SL(2,\mathbf{C})$
,
$$\begin{array}{c}{\mathbf{S}}^{\prime}=\mathbf{A}\mathbf{S}{\mathbf{A}}^{+},\end{array}$$ 
(72)

where
${\mathbf{A}}^{+}$
is the Hermitian conjugate matrix^{
$\text{54}$
}
. Moreover,
$det\mathbf{S}=det{\mathbf{S}}^{\prime}$
which, according to Equation ( 70 ), expresses the invariance of the spacetime distance to the origin:
$$\begin{array}{c}({x}^{{0}^{\prime}}{)}^{2}({x}^{{1}^{\prime}}{)}^{2}({x}^{{2}^{\prime}}{)}^{2}({x}^{{3}^{\prime}}{)}^{2}=({x}^{0}{)}^{2}({x}^{1}{)}^{2}({x}^{2}{)}^{2}({x}^{3}{)}^{2}.\end{array}$$ 
(73)

The link between the representation of a Lorentz transformation
${L}_{ik}$
in spacetime and the unimodular matrix
$\mathbf{A}$
mapping spin space (cf. below) is given by
$$\begin{array}{c}L(\mathbf{A}{)}_{ik}=\frac{1}{2}tr({\sigma}_{i}\mathbf{A}{\sigma}_{k}{\mathbf{A}}^{+}).\end{array}$$ 
(74)

Thus, the map is two to one:
$+\mathbf{A}$
and
$\mathbf{A}$
give the same
${L}_{ik}$
.
Now,
contravariant 2spinors
${\xi}^{A}$
(
$A=1,2$
) are the elements of a complex linear space, spinor space, on which the matrices
$\mathbf{A}$
are acting^{
$\text{55}$
}
.
The spinor is called
elementary if it transforms under a Lorentztransformation as
$$\begin{array}{c}{\xi}^{{A}^{\prime}}=\pm {\mathbf{A}}_{C}^{A}{\xi}^{C}.\end{array}$$ 
(75)

Likewise, contravariant dotted spinors
${\zeta}^{\dot{A}}$
are those transforming with the complexconjugate matrix
$\overline{\mathbf{A}}$
:
$$\begin{array}{c}{\zeta}^{{\dot{B}}^{\prime}}=\pm {\overline{\mathbf{A}}}_{\dot{D}}^{\dot{B}}{\zeta}^{\dot{C}}.\end{array}$$ 
(76)

Covariant and covariant dotted 2spinors correspondingly transform with the inverse matrices,
$$\begin{array}{c}{\xi}_{{B}^{\prime}}=\pm ({\mathbf{A}}^{1}{)}_{B}^{C}{\xi}_{C},\end{array}$$ 
(77)

and
$$\begin{array}{c}{\xi}_{{\dot{B}}^{\prime}}=\pm ({\overline{\mathbf{A}}}^{1}{)}_{\dot{B}}^{\dot{C}}{\xi}_{\dot{C}}.\end{array}$$ 
(78)

The space of 2spinors can be used as a representation space for the (proper, orthochronous) Lorentz group, with the 2spinors being the elements of the most simple representation
${D}^{(1/2,0)}$
.
Higherorder spinors with dotted and undotted indices
${S}_{C\dots \dot{D}\dots}^{A\dots \dot{B}\dots}$
transform correspondingly.
For the raising and lowering of indices now a real, antisymmetric
$(2\times 2)$
matrix
$\epsilon $
with components
${\epsilon}^{AB}={\delta}_{1}^{A}{\delta}_{B}^{2}{\delta}_{2}^{A}{\delta}_{B}^{1}={\epsilon}_{AB}$
is needed, such that
$$\begin{array}{c}{\xi}^{A}={\epsilon}^{AB}{\xi}_{B},{\xi}_{A}={\xi}^{B}{\epsilon}_{BA}.\end{array}$$ 
(79)

Next to a spinor, bispinors of the form
${\zeta}^{AB},{\xi}^{A\dot{B}}$
, etc. are the simplest quantities (spinors of 2nd order). A vector
${X}^{k}$
can be represented by a bispinor
${X}^{A\dot{B}}$
,
$$\begin{array}{c}{X}^{A\dot{B}}={\sigma}_{k}^{A\dot{B}}{X}^{k},\end{array}$$ 
(80)

where
${\sigma}_{k}^{A\dot{B}}$
(
$k=0,\dots ,3$
) is a quantity linking the tangent space of spacetime and spinor space.
If
$k$
numerates the matrices and
$A$
,
$\dot{B}$
designate rows and columns, then we can chose
${\sigma}_{0}^{A\dot{B}}$
to be the unit matrix while for the other three indices
${\sigma}_{j}^{A\dot{B}}$
are taken to be the Pauli matrices. Often the quantity
${s}_{k}^{A\dot{B}}=\frac{1}{\sqrt{2}}{\sigma}_{k}^{A\dot{B}}$
is introduced. The reciprocal matrix
${s}_{A\dot{B}}^{k}$
is defined by
$$\begin{array}{c}{s}_{j}^{A\dot{B}}{s}_{A\dot{B}}^{k}={\delta}_{j}^{k},\end{array}$$ 
(81)

whereas
$$\begin{array}{c}{s}_{j}^{A\dot{B}}{s}^{jC\dot{D}}={\epsilon}^{AC}{\epsilon}^{\dot{B}\dot{D}}.\end{array}$$ 
(82)

In order to write down spinorial field equations, we need a spinorial derivative,
$$\begin{array}{c}{\partial}_{A\dot{B}}={s}_{A\dot{B}}^{k}{\partial}_{k}\end{array}$$ 
(83)

with
${\partial}_{A\dot{B}}{\partial}^{A\dot{B}}={\partial}_{k}{\partial}^{k}.$
The simplest spinorial equation is the Weyl equation:
$$\begin{array}{c}{\partial}_{A\dot{B}}{\psi}^{A}=0,\dot{B}=1,2.\end{array}$$ 
(84)

The next simplest spinor equation for two spinors
${\chi}^{\dot{B}},{\psi}_{A}$
would be
$$\begin{array}{c}{\partial}_{A\dot{C}}{\chi}^{\dot{C}}=\frac{2\pi}{\sqrt{2}h}m{\psi}_{A};{\partial}^{C\dot{B}}{\psi}_{C}=\frac{2\pi}{\sqrt{2}h}m{\chi}^{\dot{B}},\end{array}$$ 
(85)

where
$m$
is a mass. Equation ( 85 ) is the 2spinor version of Dirac's equation.
Dirac or 4spinors with 4 components
${\psi}^{k}$
,
$k=1,\dots ,4$
, may be constructed from 2spinors as a direct sum of contravariant undotted and covariant dotted spinors
$\psi $
and
$\phi $
: For
$k=1,2$
, we enter
${\psi}^{1}$
and
${\psi}^{2}$
; for
$k=3,4$
, we enter
${\phi}_{\dot{1}}$
and
${\phi}_{\dot{2}}.$
In connection with Dirac spinors, instead of the Paulimatrices the Dirac
$\gamma $
matrices (
$4\times 4$
matrices) appear; they satisfy
$$\begin{array}{c}{\gamma}^{i}{\gamma}^{k}+{\gamma}^{k}{\gamma}^{i}=2{\eta}^{ik}1.\end{array}$$ 
(86)

The Dirac equation is in 4spinor formalism [
51,
52]
:
$$\begin{array}{c}\left(i{\gamma}^{l}\frac{\partial}{\partial {x}^{l}}+\kappa \right)\chi =0,\end{array}$$ 
(87)

with the 4component Dirac spinor
$\chi $
. In the first version of Dirac's equation,
$\alpha $
and
$\beta $
matrices were used, related to the
$\gamma $
's by
$$\begin{array}{c}{\gamma}^{0}=\beta ,{\gamma}^{m}=\beta {\alpha}^{m},m=1,2,3,\end{array}$$ 
(88)

where the matrices
$\beta $
and
${\alpha}^{m}$
are given by
$\left(\begin{array}{cc}0& {\sigma}_{i}\\ {\sigma}_{i}& 0\\ \end{array}\right)$
,
$\left(\begin{array}{cc}0& 1\\ 1& 0\\ \end{array}\right)$
.
The generallycovariant formulation of spinor equations necessitates the use of
$n$
beins
${h}_{\hat{\text{\u0131}}}^{k}$
, whose internal “rotation” group, operating on the “hatted” indices, is the Lorentz group. The group of coordinate transformations acts on the Latin indices. In Cartan's oneform formalism (cf.
Section
2.1.4 ), the covariant derivative of a 4spinor is defined by
$$\begin{array}{c}D\psi =d\psi +\frac{1}{4}{\omega}_{\hat{\text{\u0131}}\hat{k}}{\sigma}^{\hat{\text{\u0131}}\hat{k}}\psi ,\end{array}$$ 
(89)

where
${\sigma}^{\hat{\text{\u0131}}\hat{k}}:=\frac{1}{2}\left[{\gamma}^{\hat{\text{\u0131}}}{\gamma}^{\hat{k}}\right]$
.
Equation (
89 ) is a special case of the general formula for the covariant derivative of a tensorial form
$\psi $
, i.e., a vector in some vector space
$V$
, whose components are differential forms,
$$\begin{array}{c}D\psi =d\psi +{\omega}^{\hat{\text{\u0131}}\hat{k}}{\rho}_{\hat{\text{\u0131}}\hat{k}}\left({e}_{\alpha}\right)\psi ,\end{array}$$ 
(90)

where
$\rho \left({e}_{\alpha}\right)$
is a particular representation of the corresponding Lie algebra in
$V$
with basis vectors
${e}_{\alpha}.$
For the example of the Dirac spinor, the adjoint representation of the Lorentz group must be used^{
$\text{56}$
}
.
Symmetries
In Section 2.1.1 we briefly met the Lie derivative of a vector field
${\mathcal{\mathcal{L}}}_{X}$
with respect to the tangent vector
$X$
defined by
$({L}_{X}Y{)}^{k}:=([X,Y]{)}^{k}={X}^{i}{\partial}_{i}{Y}^{k}{Y}^{i}{\partial}_{i}{X}^{k}$
. With its help we may formulate the concept of isometries of a manifold, i.e., special mappings, also called “motions”, locally generated by vector fields
$X$
satisfying
$$\begin{array}{c}{\mathcal{\mathcal{L}}}_{X}{g}_{ik}:={\partial}_{k}\left({g}_{ij}\right){X}^{k}+{g}_{lj}{\partial}_{i}{X}^{l}+{g}_{il}{\partial}_{j}{X}^{l}=0.\end{array}$$ 
(91)

The generators
$X$
solving Equation ( 91 ) given some metric, form a Lie group
${G}_{r}$
, the group of motions of
${M}_{D}$
. If a group
${G}_{r}$
is prescribed, e.g., the group of spatial rotations
$O\left(3\right)$
, then from Equation ( 91 ) the functional form of the metric tensor having
$O\left(3\right)$
as a symmetry group follows.
A Riemannian space is called (locally)
stationary if it admits a timelike Killing vector; it is called (locally) static if this Killing vector is hypersurface orthogonal. Thus if, in a special coordinate system, we take
${X}^{i}={\delta}_{0}^{i}$
then from Equation ( 91 ) we conclude that stationarity reduces to the condition
${\partial}_{0}{g}_{ik}=0$
. If we take
$X$
to be the tangent vector field to the congruence of curves
${x}^{i}={x}^{i}\left(u\right)$
, i.e., if
${X}^{k}=\frac{d{x}^{k}}{du}$
, then a necessary and sufficient condition for hypersurfaceorthogonality is
${\epsilon}^{ijkl}{X}_{j}{X}_{[k,l]}=0.$
A generalisation of Killing vectors are conformal Killing vectors for which
${\mathcal{\mathcal{L}}}_{X}{g}_{ik}=\Phi {g}_{ik}$
with an arbitrary smooth function
$\Phi $
holds. In purely affine spaces, another type of symmetry may be defined:
${\mathcal{\mathcal{L}}}_{X}{\Gamma}_{ik}^{l}=0$
; they are called affine motions [
425]
.
2.2 Dynamics
Within a particular geometry, usually various options for the dynamics of the fields (field equations, in particular as following from a Lagrangian) exist as well as different possibilities for the identification of physical observables with the mathematical objects of the formalism. Thus, in general relativity, the field equations are derived from the Lagrangian
$$\mathcal{\mathcal{L}}=\sqrt{g}(R+2\Lambda 2\kappa {L}_{M}),$$
where
$R\left({g}_{ik}\right)$
is the Ricci scalar,
$g:=det{g}_{ik},\Lambda $
the cosmological constant, and
${L}_{M}$
the matter Lagrangian depending on the metric, its first derivatives, and the matter variables. This Lagrangian leads to the wellknown field equations of general relativity,
$$\begin{array}{c}{R}^{ik}\frac{1}{2}R{g}^{ik}=\kappa {T}^{ik},\end{array}$$ 
(92)

with the energymomentum(stress) tensor of matter
$$\begin{array}{c}{T}^{ik}:=\frac{2}{\sqrt{g}}\frac{\delta \left(\sqrt{g}{L}_{M}\right)}{\delta {g}_{ik}}\end{array}$$ 
(93)

and
$\kappa =\frac{8\pi G}{{c}^{4}}$
, where
$G$
is Newton's gravitational constant.
${G}^{ik}:={R}^{ik}\frac{1}{2}R{g}^{ik}$
is called the Einstein tensor. In empty space, i.e., for
${T}^{ik}=0$
, Equation ( 92 ) reduces to
$$\begin{array}{c}{R}^{ik}=0.\end{array}$$ 
(94)

If only an electromagnetic field
${F}_{ik}=\frac{\partial {A}_{k}}{\partial {x}^{i}}\frac{\partial {A}_{i}}{\partial {x}^{k}}$
derived from the 4vector potential
${A}_{k}$
is present in the energymomentum tensor, then the Einstein–Maxwell equations follow:
$$\begin{array}{c}{R}^{ik}\frac{1}{2}R{g}^{ik}=\kappa \left({F}_{il}{F}_{k}^{l}+\frac{1}{4}{g}_{ik}{F}_{lm}{F}^{lm}\right),{\nabla}_{l}{F}^{il}=0.\end{array}$$ 
(95)

The components of the metrical tensor are identified with gravitational potentials. Consequently, the components of the (LeviCivita) connection correspond to the gravitational “field strength”, and the components of the curvature tensor to the gradients of the gravitational field. The equations of motion of material particles should follow, in principle, from Equation ( 92 ) through the relation
$$\begin{array}{c}{\nabla}_{l}{T}^{il}=0\end{array}$$ 
(96)

implied by it^{
$\text{57}$
}
. For point particles, due to the singularities appearing, in general this is a tricky task, up to now solved only approximately.
However, the world lines for point particles falling freely in the gravitational field are, by definition, the geodesics of the Riemannian metric. This definition is consistent with the rigourous derivation of the geodesic equation for noninteracting dust particles in a fluid matter description. It is also consistent with all observations.
For most of the unified field theories to be discussed in the following, such identifications were made on internal, structural reasons, as no linkup to empirical data was possible. Due to the inherent wealth of constructive possibilities, unified field theory never would have come off the ground proper as a physical theory even if all the necessary formal requirements could have been
satisfied. As an example, we take the identification of the electromagnetic field tensor with either the skew part of the metric, in a “mixed geometry” with metric compatible connection, or the skew part of the Ricci tensor in metricaffine theory, to list only two possibilities. The latter choice obtains likewise in a purely affine theory in which the metric is a derived secondary concept. In this case, among the many possible choices for the metric, one may take it proportional to the variational derivative of the Lagrangian with respect to the symmetric part of the Ricci tensor. This does neither guarantee the proper signature of the metric nor its full rank. Several identifications for the electromagnetic 4potential and the electric current vector density have also been suggested (cf. below and [
142]
).
For an arbitrary Riemannian manifold this no longer holds true.
2.3 Number field
Complex fields may also be introduced on a real manifold. Such fields have also been used for the construction of unified field theories, although mostly after the period dealt with here (cf.
Part II, in preparation). In particular, manifolds with a complex fundamental form were studied,
e.g., with
${g}_{ik}={s}_{ik}+i{a}_{ik}$
, where
$i=\sqrt{1}$
[
96]
. Also, geometries based on Hermitian forms were studied [
312]
. In later periods, hypercomplex numbers, quaternions, and octonions also were used as basic number fields for gravitational or unified theories (cf. Part II, forthcoming).
In place of the real numbers, by which the concept of manifold has been defined so far, we could take other number fields and thus arrive,
e.g., at complex manifolds and so on. In this part of the article we do not need to take into account this generalisation.
2.4 Dimension
Since the suggestions by Nordström and Kaluza [
237,
180]
, manifolds with
$D>4$
have been used for unified field theories. In most of the cases, the additional dimensions were taken to be spacelike; nevertheless, manifolds with more than one direction of time also have been studied.
3 Early Attempts at a Unified Field Theory
3.1 First steps in the development of unified field theories
Even before (or simultaneously with) the introduction and generalisation of the concept of parallel transport and covariant derivative by Hessenberg (1916/17) [
159]
, LeviCivita (1917), [
203]
, Schouten (1918) [
293]
, Weyl (1918) [
397]
, and König (1919) [
192]
, the introduction of an asymmetric metric was suggested by Rudolf Förster^{
$\text{58}$
}
in 1917. In his letter to Einstein of 11 November 1917, he writes ([
320]
, Doc. 398, p. 552):

“Perhaps, there exists a covariant 6vector by which the appearance of electricity is explained and which springs lightly from the
${g}_{\mu \nu}$
, not forced into it as an alien element.”^{
$\text{59}$
}
“Vielleicht findet sich ein kovarianter Sechservektor der das Auftreten der Elektrizität erklärt und ungezwungen aus den
${g}_{\mu \nu}$
herauskommt, nicht als fremdes Element herangetragen wird.”
Einstein replied:

“The aim of dealing with gravitation and electricity on the same footing by reducing both groups of phenomena to
${g}_{\mu \nu}$
has already caused me many disappointments. Perhaps, you are luckier in the search. I am totally convinced that in the end all field quantities will look alike in essence. But it is easier to suspect something than to discover it.”^{
$\text{60}$
}
“Das Ziel, Gravitation und Elektromagnetismus einheitlich zu behandeln, indem man beide Phänomengruppen auf die
${g}_{\mu \nu}$
zurückführt, hat mir schon viele erfolglose Bemühungen gekostet. Vielleicht sind Sie glücklicher im Suchen. Ich bin fest überzeugt, dass letzten Endes alle Feldgrössen sich als wesensgleich herausstellen werden. Aber leichter ist ahnen als finden.” (16 November 1917 [320] , Vol. 8A, Doc. 400, p. 557)
In his next letter, Förster gave results of his calculations with an asymmetric
${g}_{\mu \nu}={s}_{\mu \nu}+{a}_{\mu \nu}$
, introduced an asymmetric “threeindexsymbol” and a possible generalisation of the Riemannian curvature tensor as well as tentative Maxwell's equations and interpretations for the 4potential
${A}_{\mu}$
, and special solutions (28 December 1917) ([
320]
, Volume 8A, Document 420, pp. 581–587).
Einstein's next letter of 17 January 1918 is skeptical:

“Since long, I also was busy by starting from a nonsymmetric
${g}_{\mu \nu}$
; however, I lost hope to get behind the secret of unity (gravitation, electromagnetism) in this way. Various reasons instilled in me strong reservations: [...] your other remarks are interesting in themselves and new to me.”^{
$\text{61}$
}
“Das Ausgehen von einem nichtsymmetrischen
${g}_{\mu \nu}$
hat mich auch schon lange beschäftigt; ich habe aber die Hoffnung aufgegeben, auf diese Weise hinter das Geheimnis der Einheit (Gravitation–Elektromagnetismus) zu kommen. Verschiedene Gründe flössen da schwere Bedenken ein: [...] Ihre übrigen Bemerkungen sind ebenfalls an sich interessant und mir neu.” ([320] , Volume 8B, Document 439, pp. 610–611)
Einstein's remarks concerning his previous efforts must be seen under the aspect of some attempts at formulating a unified field theory of matter by G. Mie [
228,
229,
230]
^{
$\text{62}$
}
, J. Ishiwara, and G. Nordström, and in view of the unified field theory of gravitation and electromagnetism proposed by David Hilbert.

“According to a general mathematical theorem, the electromagnetic equations (generalized Maxwell equations) appear as a consequence of the gravitational equations, such that gravitation and electrodynamics are not really different.”^{
$\text{63}$
}
“In Folge eines allgem. math. Satzes erscheinen die elektrody. Gl. (verallgemeinerte Maxwellsche) als math. Folge der Gravitationsgl., so dass Gravitation und Elektrodynamik eigentlich garnicht verschiedenes sind.” (letter of Hilbert to Einstein of 13 November 1915 [161] )
The result is contained in (Hilbert 1915, p. 397)^{
$\text{64}$
}
.
Einstein's answer to Hilbert on 15 November 1915 shows that he had also been busy along such lines:

“Your investigation is of great interest to me because I have often tortured my mind in order to bridge the gap between gravitation and electromagnetism. The hints dropped by you on your postcards bring me to expect the greatest.”^{
$\text{65}$
}
“Ihre Untersuchung interessiert mich gewaltig, zumal ich mir oft schon das Gehirn zermartert habe, um eine Brücke zwischen Gravitation und Elektromagnetik zu schlagen. Die Andeutungen, welche Sie auf Ihren Karten geben, lassen das Grösste erwarten.” [100]
Even before Förster alias Bach corresponded with Einstein, a very early bird in the attempt at unifying gravitation and electromagnetism had published two papers in 1917, Reichenbächer [
269,
268]
.
His paper amounts to a scalar theory of gravitation with field equation
$R=0$
instead of Einstein's
${R}_{ab}=0$
outside the electrons. The electron is considered as an extended body in the sense of Lorentz–Poincaré, and described by a metric joined continuously to the outside metric^{
$\text{66}$
}
:
$$\begin{array}{c}d{s}^{2}=d{r}^{2}+{r}^{2}d{\phi}^{2}+{r}^{2}{cos}^{2}\phi d{\psi}^{2}+{\left(1\frac{\alpha}{r}\right)}^{2}d{x}_{0}^{2}.\end{array}$$ 
(97)

Reichenbächer, at this point, seems to have had a limited understanding of general relativity: He thinks in terms of a variable velocity of light; he equates coordinate systems and reference systems, and apparently considers the transition from the Minkowskian to a nonflat metric as achieved by a coordinate rotation, a “Drehung gegen den Normalzustand” (“rotation with respect to the normal state”) ([
269]
, p. 137). According to him, the deviation from the Minkowski metric is due to the electromagnetic field tensor:

“The disturbance, which is generated by the electrons and which forces us to adopt a coordinate system different from the usual one, is interpreted as the electromagnetic sixvector, as is known.”^{
$\text{67}$
}
“Die Störung, die durch die Elektronen erzeugt wird und uns also zur Annahme eines von dem gewöhnlichen abweichenden Weltkoordinatensystem zwingt, wird nun bekanntlich als der elektromagnetische Sechservektor aufgefasst.” ([269] , p. 136)
By his “coordinate rotation”, or, as he calls it in ([
268]
, p. 174), “electromagnetic rotation”, he tries to geometrize the electromagnetic field. As Weyl's remark in Raum–Zeit–Materie ([
398]
, p. 267, footnote 30) shows, he did not grasp Reichenbächer's reasoning; I have not yet understood it either. Apparently, for Reichenbächer the metric deviation from Minkowski space is due solely to the electromagnetic field, whereas gravitation comes in by a single scalar potential connected to the velocity of light. He claims to obtain the same value for the perihelion shift of Mercury as Einstein ([
268]
, p. 177). Reichenbächer was slow to fully accept general relativity; as late as in 1920 he had an exchange with Einstein on the foundations of general relativity [
270,
69]
.
After Reichenbächer had submitted his paper to
Annalen der Physik and seemingly referred to Einstein,

“Planck was uncertain to which of Einstein's papers Reichenbächer appealed. He urged that Reichenbächer speak with Einstein and so dissolve their differences. The meeting was amicable. Reichenbächer's paper appeared in 1917 as the first attempt at a unified field theory in the wake of Einstein's covariant field equations.” ([261] , p. 208)
In this context, we must also keep in mind that the generalisation of the metric tensor toward asymmetry or complex values was more or less synchronous with the development of Finsler geometry [
125]
. Although Finsler himself did not apply his geometry to physics it soon became used in attempts at the unification of gravitation and electromagnetism [
273]
.
3.2 Early disagreement about how to explain elementary particles by field theory
In his book on Einstein's relativity, Max Born, in 1920, had asked about the forces hindering “an electron or an atom” to disintegrate.

“Now, these objects are tremendous concentrations of energy in the smallest place; therefore, they will house huge curvatures of space or, in other words, gravitational fields. The idea that they keep together the dispersing electrical charges lies close at hand.”^{
$\text{68}$
}
“Nun sind diese Gebilde ungeheure Anhäufungen von Energie auf kleinsten Räumen; daher werden sie gewaltige Raumkrümmungen, oder mit anderen Worten, Gravitationsfelder, in sich bergen. Der Gedanke liegt nahe, dass diese es sind, die die auseinanderstrebenden elektrischen Ladungen zusammenhalten.” ([17] , p. 235)
Thus, the idea of a program for building the extended constituents of matter from the fields the source of which they are, was very much alive around 1920. However, Pauli's remark after Weyl's lecture in Bad Nauheim (86. Naturforscherversammlung, 19–25 September 1920) [
244]
showed that not everybody was a believer in it. He claimed that in bodies smaller than those carrying the elementary charge (electrons), an electric field could not be measured. There was no point of creating the “interior” of such bodies with the help of an electric field. Pauli:

“None of the present theories of the electron, also not Einstein's (Einstein 1919 [68] ), up to now did achieve solving satisfactorily the problem of the electrical elementary quanta; it seems obvious to look for a deeper reason for this failure. I wish to see this reason in the fact that it is altogether not permitted to describe the electromagnetic field in the interior of an electron as a continuous space function. The electrical field is defined as the force on a charged test particle, and if no smaller test particles exist than the electron (vice versa the nucleus), the concept of electrical field at a certain point in the interior of the electron – with which all continuum theories are working – seems to be an empty fiction, because there are no arbitrarily small measures. Therefore, I'd like to ask Mr. Einstein whether he approves of the opinion that a solution of the problem of matter may be expected only from a modification of our perception of space (perhaps also of time) and of electricity in the sense of atomism, or whether he thinks that the mentioned reservations are unconvincing and is of the opinion that the fundaments of continuum theory must be upheld.”^{
$\text{69}$
}
“Keiner der bisherigen Theorien des Elektrons, auch nicht der Einsteinschen (Einstein 1919 [68] ) ist es bisher gelungen, das Problem der elektrischen Elementarquante befriedigend zu lösen, und es liegt nahe, nach einem tieferen Grund für diesen Mißerfolg zu suchen.
Ich möchte nun diesen Grund darin suchen, daß es überhaupt unstatthaft ist, das elektrische Feld im Innern des Elektrons als stetige Raumfunktion zu beschreiben. Die elektrische Feldstärke ist definiert als die Kraft auf einen geladenen Probekörper und, wenn es keine kleineren Probekörper gibt als das Elektron (bzw den NKern), scheint der Begriff der elektrischen Feldstärke in einem bestimmten Punkt im Innern des Elektrons, mit welchem alle Kontinuumstheorien operieren, eine leere, inhaltslose Fiktion zu sein, da es keine beliebig kleinen Maßstäbe gibt. Ich möchte deshalb Herrn Einstein fragen, ob er der Auffassung zustimmt, daß man die Lösung des Problems der Materie nur von einer Modifikation unserer Vorstellungen vom Raum (vielleicht auch von der Zeit) und vom elektrischen Felde im Sinne des Atomismus erwarten darf, oder ob er die angeführten Bedenken nicht für stichhaltig hält und die Ansicht vertritt, daß man an den Grundlagen der Kontinuumstheorie festhalten muß.”
Pauli referred to Einstein's paper about elementary particles and field theory in which he had exchanged his famous field equations for traceless equations with the electromagnetic field tensor as a source. Einstein's answer is tentative and evasive: We just don't know yet^{
$\text{70}$
}
.

“With the progressing refinement of scientific concepts, the manner by which concepts are related to (physical) events becomes ever more complicated. If, in a certain stage of scientific investigation, it is seen that a concept can no longer be linked with a certain event, there is a choice to let the concept go, or to keep it; in the latter case, we are forced to replace the system of relations among concepts and events by a more complicated one. The same alternative obtains with respect to the concepts of timeand spacedistances. In my opinion, an answer can be given only under the aspect of feasibility; the outcome appears dubious to me.”^{
$\text{71}$
}
“Mit fortschreitender Verfeinerung des wissenschaftlichen Begriffssystems wird die Art und Weise der Zuordnung der Begriffe von den Erlebnissen immer komplizierter. Hat man in einem gewissen Studium der Wissenschaft gesehen, daß einem Begriff ein bestimmtes Erlebnis nicht mehr zugeordnet werden kann, so hat man die Wahl, ob man den Begriff fallen lassen oder ihn beibehalten will; in letzterem Fall ist man aber gezwungen, das System der Zuordnung der Begriffe zu den Erlebnissen durch ein komplizierteres zu ersetzen. Vor dieser Alternative sind wir auch hinsichtlich der Begriffe der zeitlichen und räumlichen Entfernung gestellt. Die Antwort kann nach meiner Ansicht nur nach Zweckmäßigkeitsgründen gegeben werden; wie sie ausfallen wird, erscheint mir zweifelhaft.”
In the same discussion Gustav Mie came back to Förster's idea of an asymmetric metric but did not like it

“[...] that an antisymmetric tensor was added to the symmetric tensor of the gravitational potential, which represented the sixvector of the electromagnetic field. But a more precise reasoning shows that in this way no reasonable world function is obtained.”^{
$\text{72}$
}
“[...] dass man dem symmetrischen Tensor des Gravitationspotentials einen antisymmetrischen Tensor hinzufügte, der den Sechservektor des elektromagnetischen Feldes repräsentierte.
Aber eine genauere Überlegung zeigt, dass man so zu keiner vernünftigen Weltfunktion kommt.”
It is to be noted that Weyl, at the end of 1920, already had given up on a possible field theory of matter:

“Finally I cut loose firmly from Mie's theory and arrived at another position with regard to the problem of matter. To me, field physics no longer appears as the key to reality; in contrary, the field, the ether, for me simply is the totally powerless transmitter of causations, yet matter is a reality beyond the field and causes its states.”^{
$\text{73}$
}
“Endlich habe ich mich gründlich von der Mieschen Theorie losgemacht und bin zu einer anderen Stellung zum Problem der Materie gelangt. Die Feldphysik erscheint mir keineswegs mehr als der Schlüssel zu der Wirklichkeit; sondern das Feld, der Äther, ist mir nur noch der in sich selbst völlig kraftlose Übermittler der Wirkungen, die Materie aber eine jenseits des Feldes liegende und dessen Zustände verursachende Realität.” (letter of Weyl to F. Klein on 28 December 1920, see [292] , p. 83)
In the next year, Einstein had partially absorbed Pauli's view but still thought it to be useful to apply field theory to the constituents of matter:

“The physical interpretation of geometry (theory of the continuum) presented here, fails in its direct application to spaces of submolecular scale. Yet it retains part of its meaning also with regard to questions concerning the constitution of elementary particles. Because one may try to ascribe to these field concepts [...] a physical meaning even if a description of the electrical elementary particles which constitute matter is to be made. Only success can decide whether such a procedure finds its justification [...].”^{
$\text{74}$
}
“Die hier vertretene physikalische Interpretation der Geometrie (Kontinuumstheorie) versagt zwar bei ihrer unmittelbaren Anwendung auf Räume von submolekularer Grössenordnung. Einen Teil ihrer Bedeutung behält sie indessen auch noch den Fragen der Konstitution der Elementarteilchen gegenüber. Denn man kann versuchen, denjenigen Feldbegriffen [...] auch dann physikalische Bedeutung zuzuschreiben, wenn es sich um die Beschreibung der elektrischen Elementarteilchen handelt, die die Materie konstituieren. Nur der Erfolg kann über die Berechtigung eines solchen Verfahrens entscheiden [...].” [70]
During the twenties Einstein changed his mind and looked for solutions of his field equations which were everywhere regular to represent matter particles:

“In the program, Mr. Einstein expressed during his two talks given in November 1929 at the Institut Henri Poincaré, he wished to search for the physical laws in solutions of his equations without singularities – with matter and the electromagnetic field thus being continuous. Let us move into the field chosen by him without too much surprise to see him apparently follow a road opposed to the one successfully walked by the contemporary physicists.”^{
$\text{75}$
}
“Dans le programme qu'a esquissé M. Einstein dans ses deux conférences faites en novembre 1929 à l'institut Henri Poncaré, il voulait chercher les lois physiques dans les solutions sans singularité de ses equations, la matière et l'electricité n'existant donc qu'à l'état continu. Placonsnous sur le terrain choisi par lui, sans trop nous étonner de le voir suivre en apparence une voie opposée à celle suivi avec succès par les physiciens contemporains.” ([34] , p. 17 (1178))
4 The Main Ideas for Unification between about 1918 and 1923
After 1915, Einstein first was busy with extracting mathematical and physical consequences from general relativity (Hamiltonian, exact solutions, the energy conservation law, cosmology, gravitational waves). Although he kept thinking about how to find elementary particles in a field theory [
68]
and looked closer into Weyl's theory [
70]
, at first he only reacted to the new ideas concerning unified field theory as advanced by others. The first such idea after Förster's, of course, was Hermann Weyl's gauge approach to gravitation and electromagnetism, unacceptable to Einstein and to Pauli for physical reasons [
245,
291]
.
Next came Kaluza's fivedimensional unification of gravitation and electromagnetism, and Eddington's affine geometry.
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.”^{
$\text{76}$
}
“Machen wir keine weitere Voraussetzung, so bleiben die einzelnen Punkte der Mannigfaltigkeit in metrischer Hinsicht vollständig gegeneinander isoliert. Ein metrischer Zusammenhang von Punkt zu Punkt wird erst dann in sie hineingetragen, wenn ein Prinzip der Übertragung der Längeneinheit von einem Punkte P zu seinem unendlich benachbarten vorliegt.”
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.”^{
$\text{77}$
}
“Die Möglichkeit einer solchen `ferngeometrischen' Vergleichung kann aber in einer reinen Infinitesimalgeometrie durchaus nicht zugestanden werden.” ([397] , p. 397)
In order to invent a purely “infinitesimal” geometry, Weyl introduced the 1dimensional, Abelian group of gauge transformations,
$$\begin{array}{c}g\to \overline{g}:=\lambda g,\end{array}$$ 
(98)

besides the diffeomorphism group (coordinate transformations). At a point, Equation ( 98 ) induces a local recalibration of lengths
$l$
while preserving angles, i.e.,
$\delta l=\lambda l$
. If the nonmetricity tensor is assumed to have the special form
${Q}_{ijk}={Q}_{k}{g}_{ij}$
, with an arbitrary vector field
${Q}_{k}$
, then as we know from Equation ( 57 ), with regard to these gauge transformations
$$\begin{array}{c}{Q}_{k}\to {Q}_{k}+{\partial}_{k}\sigma .\end{array}$$ 
(99)

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
$$\begin{array}{c}{\Gamma}_{ij}^{k}={\{}_{ij}^{k}\}+\frac{1}{2}({\delta}_{i}^{k}{Q}_{j}+{\delta}_{j}^{k}{Q}_{i}{g}_{ij}{g}^{kl}{Q}_{l}).\end{array}$$ 
(100)

Thus, unlike in Riemannian geometry, the connection is not fully determined by the metric but depends also on the arbitrary vector function
${Q}_{i}$
, which Weyl wrote as a linear form
$dQ={Q}_{i}d{x}^{i}$
.
With regard to the gauge transformations (
98 ),
${\Gamma}_{ij}^{k}$
remains invariant. From the 1form
$dQ$
, by exterior derivation a gaugeinvariant 2form
$F={F}_{ij}d{x}^{i}\wedge d{x}^{j}$
with
${F}_{ij}={Q}_{i,j}{Q}_{j,i}$
follows.
It is named “Streckenkrümmung” (“line curvature”) by Weyl, and, by identifying
$Q$
with the electromagnetic 4potential, he arrived at the electromagnetic field tensor
$F$
.
Let us now look at what happens to parallel transport of a length,
e.g., the norm
$\leftX\right$
of a tangent vector along a particular curve
$C$
with parameter
$u$
to a different (but infinitesimally neighbouring) point:
$$\begin{array}{c}\frac{d\leftX\right}{du}=X{}_{\parallel k}{X}^{k}=(\sigma \frac{1}{2}{Q}_{k}{X}^{k}\left)\rightX.\end{array}$$ 
(101)

By a proper choice of the curve's parameter, we may write ( 101 ) in the form
$d\leftX\right={Q}_{k}{X}^{k}\leftX\rightdu$
and integrate along
$C$
to obtain
$\leftX\right=\int exp({Q}_{k}{X}^{k}du)$
. If
$X$
is taken to be tangent to
$C$
, i.e.,
${X}^{k}=\frac{d{x}^{k}}{du}$
, then
$$\begin{array}{c}\left\frac{d{x}^{k}}{du}\right=\int exp\left({Q}_{k}\left(x\right)d{x}^{k}\right),\end{array}$$ 
(102)

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 4potential becomes a gradient (“pure gauge”), such that
${Q}_{k}d{x}^{k}=\frac{\partial \omega}{\partial {x}^{k}}d{x}^{k}=d\omega $
, 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
$g$
and the vector field
$Q$
, 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 masspoints based on Newton's Principia did triumph. Yet, also today, the circumstances are such that our trees do not grow into the sky.”^{
$\text{78}$
}
“Wieder ist die Physik, heute die Feldphysik, auf dem Wege, die Gesamtheit der Naturerscheinungen auf ein einziges Naturgesetz zurückzuführen, ein Ziel, dem sie schon einmal, als die durch Newtons Principia begründete mechanische MassenpunktPhysik ihre Triumphe feierte, nahe zu sein. Doch ist auch heute dafür gesorgt, dass unsere Bäume nicht in den Himmel wachsen.” ([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 worldlaw of greatest mathematical simplicity.”^{
$\text{79}$
}
“Ich bin verwegen genug, zu glauben, dass die Gesamtheit der physikalischen Erscheinungen sich aus einem einzigen universellen Weltgesetz von höchster mathematischer Einfachheit herleiten lässt.” ([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
gaugeweight within tensor calculus as in Section 2.1.5 ^{
$\text{80}$
}
. As the Lagrangian
$\mathcal{\mathcal{L}}=\sqrt{g}L$
must have gaugeweight
$w=0$
, we are looking for a scalar
$L$
of gaugeweight
$2$
. Weitzenböck^{
$\text{81}$
}
has shown that the only possibilities quadratic in the curvature tensor and the line curvature are given by the four expressions [
391]
$$\begin{array}{c}({K}_{ij}{g}^{ij}{)}^{2},{K}_{ij}{K}_{kl}{g}^{ik}{g}^{jl},{K}_{jkl}^{i}{K}_{imn}^{j}{g}^{km}{g}^{ln},{F}_{ij}{F}_{kl}{g}^{ik}{g}^{jl}.\end{array}$$ 
(103)

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^{
$\text{82}$
}
; it is given by Schouten ([
309]
, p. 142) and follows from Equation ( 51 ):
$$\begin{array}{c}{K}_{jkl}^{i}={R}_{jkl}^{i}+{Q}_{j;[k}{\delta}_{l]}^{i}+{\delta}_{j}^{i}{Q}_{[l;k]}{Q}_{;[k}^{i}{g}_{l]j}+\frac{1}{2}\left({\delta}_{[l}^{i}{Q}_{k]}{Q}_{j}+{Q}_{[k}{g}_{l]j}{Q}^{i}{\delta}_{[k}^{i}{g}_{l]j}{Q}_{r}{Q}^{r}\right).\end{array}$$ 
(104)

If the metric field
$g$
and the 4potential
${Q}_{i}\equiv {A}_{i}$
are varied independently, from each of the curvaturedependent 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
$\mathcal{\mathcal{L}}=\sqrt{g}(3{W}_{4}6{W}_{3}+{W}_{2})$
, where the invariants are defined by
$${W}_{4}:={S}_{pqik}{S}^{pqik},{W}_{3}:={g}^{ik}{g}^{lm}{F}_{ikq}^{p}{F}_{lmp}^{q},{W}_{2}:={g}^{ik}{F}_{ikp}^{p}$$
with
$$\begin{array}{ccc}{S}_{\left[pq\right]\left[ik\right]}& :=& \frac{1}{4}({F}_{pqik}{F}_{qpik}+{F}_{ikpq}{F}_{kipq}),\end{array}$$  
$$\begin{array}{ccc}{F}_{pqik}& =& {R}_{pqik}+\frac{1}{2}({g}_{pq}{f}_{ki}+{g}_{pk}{f}_{q;i}+{g}_{qi}{f}_{p;k}{g}_{pi}{f}_{q;k}{g}_{qk}{f}_{p;i})\end{array}$$  
$$\begin{array}{ccc}& & +\frac{1}{2}({f}_{q}{g}_{p;[k}{f}_{i]}+{f}_{p}{g}_{q;[k}{f}_{i]}+{g}_{p[i}{g}_{k]q}{f}_{r}{f}^{r}),\end{array}$$  
where
${R}_{pqik}$
is the Riemannian curvature tensor,
${f}_{ik}={f}_{i,k}{f}_{k,i}$
, and
${f}_{i}$
is the electromagnetic 4potential [
2]
.
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
$ds$
had been identified with spaceand time intervals measurable by real clocks and real measuring rods.
Now, only the equivalence class
$\left\{\lambda {g}_{ik}\right\lambda arbitrary\}$
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.”^{
$\text{83}$
}
“Diese Tage ist es mir, wie ich glaube, gelungen, Elektrizität und Gravitation aus einer gemeinsamen Quelle herzuleiten.
Es ergibt sich ein völlig bestimmtes Wirkungsprinzip, das im elektrizitätsfreien Fall auf Ihre Gravitationsgleichungen führt, im gravitationsfreien dagegen Gleichungen ergibt, die in erster Näherung mit den Maxwellschen übereinstimmen. Im allgemeinsten Fall werden die Gleichungen allerdings 4. Ordnung.”
He then asked whether Einstein would be willing to communicate a paper on this new unified theory to the Berlin Academy ([
320]
, 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.”^{
$\text{84}$
}
“Ihre Abhandlung ist gekommen. Es ist ein GenieStreich ersten Ranges. Allerdings war ich nicht imstande, meinen MassstabEinwand zu erledigen.” ([320] , 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 prehistory 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.”^{
$\text{85}$
}
“[...] scheint mir die Grundhypothese der Theorie leider nicht annehmbar, deren Tiefe und Kühnheit aber jeden Leser mit Bewunderung erfüllen muss.” ([395] , Addendum, p. 478)
Einstein's remark concerning the pathdependence of the frequencies of spectral lines stems from the pathdependency 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
$\int ds$
. 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
$ds$
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.”^{
$\text{86}$
}
“Die plausibelste Annahme, die man über ein im statischen Feld ruhende Uhr machen kann, ist die, dass sie das Integral des so normierten [d. h. so wie in der Einsteinschen Theorie]
$ds$
misst; es bleibt in meiner wie in der Einsteinschen Theorie die Aufgabe
${}^{1}$
, diese Tatsache aus einer explizit durchgeführten Dynamik abzuleiten.” ([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
$ds$
, as measured by a clock (or a ruler), is something independent of prehistory, 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 welldefined 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.”^{
$\text{87}$
}
“[Weyl] würde sagen, Uhren und Massstäbe müssten erst als Lösungen auftreten; im Fundament der Theorie kommen sie nicht vor. Aber ich finde: Wenn das mit einer Uhr (bzw.
einem Massstab) gemessene
$ds$
ein von der Vorgeschichte, dem Bau und dem Material Unabhängiges ist, so muss diese Invariante als solche auch in der Theorie eine ganz fundamentale Rolle spielen. Wenn aber die Art des wirklichen Naturgeschehens nicht so wäre, so gäbe es keine Spektrallinien und keine wohldefinierten chemischen Elemente. [...] Jedenfalls bin ich mit Weyl überzeugt, dass Gravitation und Elektrizität zu einem Einheitlichen sich verbinden lassen müssen, nur glaube ich, dass die richtige Verbindung noch nicht gefunden ist.” ([
320]
, 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 worldlines of lightsignals would suffice to determine the gravitational potentials [
210]
.
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 gaugeinvariance in the same way as he had linked with coordinateinvariance earlier, what at the time was considered to be “conservation of energy and momentum”, where a nontensorial object stood in for the energymomentum 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; [
151]
).
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 [
320]
. 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 spacetime that Einstein liked: As the Lagrangian for the electromagnetic field
${F}_{ik}{F}^{ik}$
is of gaugeweight
$2$
and
$\sqrt{g}$
has gaugeweight
$D/2$
in an
${M}_{D}$
, the integrand in the Hamiltonian principle
$\sqrt{g}{F}_{ik}{F}^{ik}$
can have weight zero only for
$D=4$
: “Apart from the [lacking] agreement with reality it is in any case a grandiose intellectual performance”^{
$\text{88}$
}
“Abgesehen von der Übereinstimmung mit der Wirklichkeit ist es jedenfalls eine grandiose Leistung des Gedanken.” ([
320]
, 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
${F}_{ik}=0$
is due only to historical reasons (its origin is the theory of surfaces), not to such that matter.”^{
$\text{89}$
}
“Ihre Ablehnung der Theorie fällt für mich schwer ins Gewicht; [...] Aber mein eigenes Hirn bewahrt noch den Glauben an sie. Und daran muss ich als Mathematiker durchaus festhalten: Meine Geometrie ist die wahre Nahegeometrie, dass Riemann nur auf den Spezialfall
${F}_{ik}=0$
geriet, hat lediglich historische Gründe (Entstehung aus der Flächentheorie), keine sachlichen.” ([320] , 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 ([
320]
, 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
$\int {\phi}_{\nu}d{x}_{\nu}$
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) worldpoints 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 [...].”
^{
$\text{90}$
}
“[Weyl's] theoretischer Versuch, passt nicht zu der Thatsache, dass zwei ursprünglich kongruente feste Körper auch kongruent bleiben, unabhängig davon, welche Schicksale sie durchmachen. Insbesondere hat es keine Bedeutung, welcher Wert des Integrals
$\int {\phi}_{\nu}d{x}_{\nu}$
ihrer Weltlinie zukommt. Sonst würde es NatriumAtome und Elektronen in allen Grössen geben müssen. Wenn aber die relative Grösse starrer Körper von der Vorgeschichte unabhängig ist, dann gibt es einen messbaren Abstand zweier (benachbarter) Weltpunkte. Dann ist die Weylsche Grundannahme jedenfalls nicht richtig für das molekulare. Dafür, dass sie für das Gravitationsfeld zutreffe, spricht, soweit ich sehe, kein einziger physikalischer Grund. Dagegen aber spricht, dass die Feldgleichungen der Gravitation von vierter Ordnung werden, wofür die bisherige Erfahrung keinerlei Anhalts bietet [...].” ([
98]
, 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
$ds$
is no longer a measurable quantity – the electromagnetical 4potential 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
$R{g}_{ik}d{x}^{i}d{x}^{k}$
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
$ds$
, then

“[...] by the prefixing of this factor, so to speak, the absolute norming of the unit of length is accomplished after all”^{
$\text{91}$
}
“[...] durch Vorsetzen dieses Faktors wird dann doch sozusagen die absolute Normierung der Längeneinheit vollzogen.” ([320] , Volume 8B, Document 619, 877–879)
Einstein was unimpressed:

“But the expression
$R{g}_{ik}d{x}^{i}d{x}^{k}$
for the measured length is not at all acceptable in my opinion because
$R$
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.”^{
$\text{92}$
}
“Der Ausdruck
$R{g}_{ik}d{x}^{i}d{x}^{k}$
für die gemessene Länge ist aber, wenn man für
$R$
die Krümmungsinvariante nimmt, nach meiner Meinung keineswegs akzeptabel, weil
$R$
sehr abhängig ist von der materiellen Dichte. Eine ganz kleine Änderung des Messweges würde das Integral der Quadratzwurzel dieser Grösse sehr stark beeinflussen.”
Einstein's argument is not very convincing:
${g}_{ik}$
itself is influenced by matter through his field equations; it is only that now
$R$
is algebraically connected to the matter tensor. In view of the more general quadratic Lagrangian needed in Weyl's theory, the connection between
$R$
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
$\lambda $
terms undeniably is a result of resignation.[...] In the end, things must arrange themselves such that actiondensities need not be glued together additively.”^{
$\text{93}$
}
“Natürlich weiss ich, dass der Zustand der Theorie, wie ich ihn hingestellt habe, ein nicht befriedigender ist, abgesehen davon, dass die Materie unerklärt bleibt. Die zusammenhanglose Nebeneinandersetzung der Gravitationsglieder, der elektromagnetischen Glieder und der
$\lambda $
Glieder ist unleugbar ein Produkt der Resignation. [...] Endlich muss es so herauskommen, dass man nicht Wirkungsdichten additiv aneinander kleben muss.” ([320] , 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.”^{
$\text{94}$
}
“Was Sie da sagen, ist wirklich wundervoll. So wie Mie seiner konsequenten Elektrodynamik eine Gravitation angeklebt hatte, die nicht organisch mit jener zusammenhing, ebenso hat Einstein seiner konsequenten Gravitation eine Elektrodynamik (d.h. die gewöhliche Elektrodynamik) angeklebt, die mit jener nicht viel zu tun hatte. Sie stellen eine wirkliche Einheit her.” [326]
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 4potential.
Understandably, no comments about the physics are given ([
294]
, 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 lineelement
$ds$
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 timereflection symmetry (see [
245]
, pp. 774–775; [
243]
). For Einstein, this criticism was not only directed against Weyl's theory

“but also against every continuumtheory, 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?” ([
102]
, 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.”^{
$\text{95}$
}
“Man muss zu Tensoren übergehen die 4. Ordnung sind statt nur zweiter Ordnung, was eine weitgehende Unbestimmtheit der Theorie mit sich bringt, erstens weil es bedeutend mehr Gleichungen gibt, die in Betracht kommen, zweitens, weil die Lösungen mehr willkürliche Konstanten enthalten.” ([98] , p. 153)
In his book “Space, Time, and Gravitation”, Eddington gave a nontechnical 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 gaugesystem. 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” ([
55]
, 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.” ([57] , p. 198)
Of course, “wrong end” meant that Eddington took Weyl's theory such

“that his nonRiemannian geometry is not to be applied to actual spacetime; it refers to a graphical representation of that relationstructure which is the basis of all physics, and both electromagnetic and metrical variables appear in it as interrelated.” ([57] , p. 197)
Again, Eddington liked Weyl's natural gauge encountered in Section 4.1.5 , which made the curvature scalar a constant, i.e.,
$K=4\lambda $
; it became a consequence of Eddington's own natural gauge in his affine theory,
${K}_{ij}=\lambda {g}_{ij}$
(cf. Section 4.3 ). For Eddington, Weyl's theory of gaugetransformation 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
$\lambda $
to be a function fixed by arbitrary convention and not necessarily a constant.” ([57] , 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 [
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:

“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 [
243,
242]
.
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
$R=\lambda =const.$
(cf. Section 4.1.3 ); he interpreted
$\lambda $
to be the “radius of curvature” of the world. In 1919, Weyl's Lagrangian originally was
$\mathcal{\mathcal{L}}=\frac{1}{2}\sqrt{g}{K}^{2}+\beta {F}_{ik}{F}^{ik}$
together with the constraint
$K=2\lambda ,$
with constant
$\lambda $
([
398]
, p. 253). As an equivalent Lagrangian Weyl gave, up to a divergence^{
$\text{96}$
}
$$\begin{array}{c}\mathcal{\mathcal{L}}=\sqrt{g}\left(R+\alpha {F}_{ik}{F}^{ik}+\frac{1}{4}(2\lambda 3{\phi}_{l}{\phi}^{l})\right),\end{array}$$ 
(105)

with the 4potential
${\phi}_{l}$
and the electromagnetic field
${F}_{ik}$
. 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
$$\begin{array}{c}\mathcal{\mathcal{L}}=\sqrt{g}\left(R+\alpha {F}_{ik}{F}^{ik}+\frac{{\epsilon}^{2}}{4}(13{\phi}_{l}{\phi}^{l})\right),\end{array}$$ 
(106)

with
$\epsilon $
a factor in Weyl's connection ( 100 ),
$$\begin{array}{c}{\Gamma}_{ij}^{k}={\{}_{ij}^{k}\}+\frac{\epsilon}{2}\left({\delta}_{i}^{k}{\phi}_{j}+{\delta}_{j}^{k}{\phi}_{i}{g}_{ij}{g}^{kl}{\phi}_{l}\right).\end{array}$$ 
(107)

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”^{
$\text{97}$
}
“Ausserdem führt dies Theorie auf einheitliche Weise und zwingend zu dem kosmologischen Gliede, das bei Einstein nur eine ad hoc gemachte Annahme war [...].” ([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.” ([57] , 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 [
271]
. For Weyl, knowledge of the charge and mass of each particle, and of the extension of their “worldchannels” 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 [
71]
. 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 gaugeweight 0, and gave the one formed from the square of Weyl's conformal curvature tensor ( 59 ), i.e.
${C}_{jkl}^{i}{C}_{i}^{jkl}.$
His colleague in Vienna, Wirtinger^{
$\text{98}$
}
, had helped him in this^{
$\text{99}$
}
. Einstein's conclusion was that, by writing down a metric with gaugeweight 0, it was possible to form a theory depending only on the quotient of the metrical components. If
$J$
has gaugeweight
$1$
, then
$J{g}_{ik}$
is such a metric. In order to reduce the new theory to general relativity, in addition only the differential equation
$$\begin{array}{c}J={J}_{0}=const.\end{array}$$ 
(108)

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
$\frac{{F}_{k}^{i}{J}^{k}}{\mu}$
, where
${J}^{i}$
is the electrical 4current vector (density) and
$\mu $
the mass density. He referred to Weyl, Eddington's book, and to Pauli's article in the Encyclopedia of Mathematical Sciences [
115]
.
Einstein's rejection of the physical value of Weyl's theory was seconded by Dienes
^{
$\text{100}$
}
, if only with a not very helpful argument. He demanded that the connection remain metriccompatible from which, trivially, Weyl's gaugevector must vanish. Dienes applied the same argument to Eddington's generalisation of Weyl's theory [
49]
. Other mathematicians took Weyl's theory at its face value and drew consequences; thus M. Juvet calculated Frenet's formulas for an “
$n$
èdre” in Weyl's geometry by generalising a result of Blaschke for Riemannian geometry [
179]
. 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 [
236]
.
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
${K}^{2}$
and concluded: