Are Wirtinger derivatives acceptable in mathematics

On the History of Unified Field Theories


This article is intended to give a review of the history of the classical aspects of unified field theories in the 20th century. It includes brief technical descriptions of the theories suggested, short biographical notes concerning the scientists involved, and an extensive bibliography. The present first installment covers the time span between 1914 and 1933, i.e., when Einstein was living and working in Berlin - with occasional digressions into other periods. Thus, the main theme is the unification of the electromagnetic and gravitational fields augmented by short-lived attempts to include the matter field described by Schrödinger’s or Dirac’s equations. While my focus lies on the conceptual development of the field, by also paying attention to the interaction of various schools of mathematicians with the research done by physicists, some prosopocraphical remarks are included.



This historical review of classical unified field theories consists of two parts. In the first, the development of unified field theory between 1914 and 1933, i.e., during the years EinsteinFootnote 1 lived and worked in Berlin, will be covered. In the second, the very active period after 1933 until the 1960s to 1970s will be reviewed. In the first version of Part I presented here, in view of the immense amount of material, neither all shades of unified field theory nor all the contributions from the various scientific schools will be discussed with the same intensity; I apologize for the shortcoming and promise to improve on it with the next version. At least, even if I do not discuss them all in detail, as many references as are necessary for a first acquaintance with the field are listed here; Completeness may be reached only (if at all) by later updates. Although I also tried to take into account the published correspondence between the main figures, my presentation, again, is far from exhaustive in this context. Eventually, unpublished correspondence will have to be worked in, and this may change some of the conclusions. Purposely I included mathematicians and also theoretical physicists of lesser rank than those who are known to be responsible for big advances. My aim is to describe the field in its full variety as it presented itself to the reader at the time.

The review is written such that physicists should be able to follow the technical aspects of the papers (cf. Section 2), while historians of science without prior knowledge of the mathematics of general relativity at least might gain an insight into the development of concepts, methods, and scientific communities involved. I should hope that readers find more than one opportunity for further in-depth studies concerning the many questions left open.

I profited from earlier reviews of the field, or of parts of it, by PauliFootnote 2 ([246], Section V); Ludwig [212]; Whittaker ([414], pp. 188-196); Lichnerowicz [209]; Tonnelat ([356], pp. 1-14); Jordan ([176], Section III); Schmutzer ([290], Section X); Treder ([183], pp. 30-43); Bergmann ([12], pp. 62-73); Straumann [334, 335]; Vizgin [384, 385]Footnote 3 ; Bergia [11]; Goldstein and Ritter [146]; Straumann and O'Raifeartaigh [240]; Scholz [292], and Stachel [330]. The section on Einstein’s unified field theories in Pais ’otherwise superb book presents the matter neither with the needed historical correctness nor with enough technical precision [241]. A recent contribution by van Dongen, focusing on Einstein’s methodology, was also helpful [371]. As will be seen, with regard to interpretations and conclusions, my views are different in some instances. In Einstein biographies, the subject of “unified field theories” - although keeping Einstein busy for the second half of his life - has been dealt with only in passing, eg, in the book of Jordan [177], and in an unsatisfying way in excellent books by Fölsing [136] and by Hermann [159]. This situation is understandable; for to describe a genius stubbornly clinging to a set of ideas, sterile for physics in comparison with quantum mechanics, over a period of more than 30 years, is not very rewarding. For the short biographical notes, various editions of J. C. Poggendorff’s biographical-literary concise dictionary and internet sources have been used (in particular [1]).

If not indicated otherwise, all non-English quotations have been translated by the author; the original text of quotations is given in footnotes.

Introduction to part I.

Past experience has shown that formerly unrelated parts of physics could be fused into one single conceptual formalism by a new theoretical perspective: electricity and magnetism, optics and electromagnetism, thermodynamics and statistical mechanics, inertial and gravitational forces. In the second half of the 20th century, the electromagnetic and weak nuclear forces have been bound together as an electroweak force; a powerful scheme was devised to also include the strong interaction (chromodynamics), and led to the standard model of elementary particle physics. Unification with the fourth fundamental interaction, gravitation, is in the focus of much present research in classical general relativity, supergravity, superstring, and supermembrane theory but has not yet met with success. These types of “unifications” have increased the explanatory power of present day physical theories and must be considered as highlights of physical research.

In the historical development of the idea of ​​unification, i.e., the joining of previously separated areas of physical investigation within one conceptual and formal framework, two closely linked yet conceptually somewhat different approaches may be recognized. In the first, the focus is on unification of representations of physical fields. An example is given by special relativity which, as a framework, must surround all phenomena dealing with velocities close to the velocity of light in vacuum. The theory thus is said to provide “a synthesis of the laws of mechanics and of electromagnetism” ([16], p. 132). Einstein’s attempts at the inclusion of the quantum area into his classical field theories belongs to this path. Nowadays, quantum field theory is such a unifying representationFootnote 4 In the second approach, predominantly the unification of the dynamics of physical fields is aimed at, i.e., a unification of the fundamental interactions. Maxwell’s theory might be taken as an example, unifying the electrical and the magnetic field once believed to be dynamically different. Most of the unified theories described in this review belong here: Gravitational and electromagnetic fields are to be joined into a new field. Obviously, this second line of thought cannot do without the first: A new representation of fields is always necessary.

In all the attempts at unification we encounter two distinct methodological approaches: a deductive-hypothetical and an empirical-inductive method. As Dirac pointed out, however,

"The successful development of science requires a proper balance between the method of building up from observations and the method of deducing by pure reasoning from speculative assumptions, [...]." ([233], p. 1001)

In an unsuccessful hunt for progress with the deductive-hypothetical method alone, Einstein spent decades of his life on the unification of the gravitational with the electromagnetic and, possibly, other fields. Others joined him in such an endeavor, or even preceded him, including Mie, Hilbert, Ishiwara, Nordström, and othersFootnote 5. At the time, another road was impossible because of the lack of empirical basis due to the weakness of the gravitational interaction. A similar situation obtains even today within the attempts for reaching a common representation of all four fundamental interactions. 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 [52, 174, 178, 175, 179]. For the early history and the conceptual development of quantum field theory, cf. Section 1 of Schweber [322], or Section 7.2 of Cao [28]; for Dirac’s contributions, cf. [190]. 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 stress-energy tensor Tik (the source term of his field equations Gik=-κTik) 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 generalization, of the energy-momentum tensor of matter by intrinsic geometrical structures, and likewise, the removal of the electric current density vector as a non-geometrical 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 sourcesFootnote 6.

In a very Cartesian spirit, Tonnelat (Tonnelat 1955 [356], 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 accountFootnote 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 stress-energy tensor is in Einstein’s theory of gravitation: They remain alien elements.

For the theories discussed, the representation of matter oscillated between the point-particle 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 point-particle is somewhat amiss. Nevertheless, geodesics of the Riemannian geometry underlying Einstein’s theory of gravitation are identified with the worldlines of freely moving point particles. The field at the location of a point particle becomes unbounded, or “singular”, such that the derivation of equations of motion from the field equations is a non-trivial 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 [235, 26].

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. " ([91], p. 610)

However, in connection with one of his moves, i.e., the 5-vector version of KaluzaFootnote 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. "Footnote 9 ([96], p. 442)

Thus, unfortunately, also the hopes of the eminent mathematician SchoutenFootnote 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 endeavors it has not been possible to recognize 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. ”Footnote 11 ([301], 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 Footnote 12.

Of course, by quantum field theory the dichotomy between matter and fields in the sense of a dualism is minimized as every field carries its particle-like quanta. Today’s unified field theories appear in the form of gauge theories; matter is represented by operator valued spin-half quantum fields (fermions) while the “forces” mediated by “exchange particles” are embodied in gauge fields, i.e., quantum fields of integer spin (bosons). The space-time geometry used is rigidly fixed, and usually taken to be Minkowski space or, within string and membrane theory, some higher-dimensional manifold also loosely called “space-time”, 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 Galilei or Lorentz invariance, and by the statistical interpretation of the Schrödinger wave function. LanczosFootnote 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 field-theoretic description based on the usual space-time structure and the probabilistic (statistical) point of view, a compromise […] no longer is possible. "Footnote 14 ([198], 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 were still made to “geometrize” wave mechanics while, roughly at the same time, quantization of the gravitational field had also been tried [284]. Einstein belonged to those who regarded the idea of ​​unification as more fundamental than the idea of ​​field quantization [95]. 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 non-linear differential equations, - in an ultimate relationship it would dissolve in the ations world equations’ of the Universe.Then, the dualism "matter-field" would have been overcome as well as the dualism "corpuscle-wave". "Footnote 15 ([198], 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 were also those whose work did not help the idea of ​​unification, e.g., van DantzigFootnote 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 tensor-density, but also no connection, neither Riemannian nor projective, nor conformal, is needed for writing down the [Maxwell] equations. Matter is characterized 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 generalizing 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 → Einstein’s equations in empty space;

  • no gravitational field → Maxwell’s equations;

  • “Weak” gravitational and electromagnetic fields → Einstein-Maxwell equations;

  • no gravitational field but a “strong” electromagnetic field → some sort of non-linear 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 → freely falling point particles” by more general assumptions for charged or electrically neutral point particles - depending on the more general (non-Riemannian) connections introducedFootnote 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 gravito-electromagnetic 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 ReichenbächerFootnote 18, Forester (alias Bach), WeylFootnote 19, EddingtonFootnote 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 five-dimensional 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. KleinFootnote 21 intervention and extension of Kaluza's paper (see Section 6.3.2). Einstein's treatment of a special case of a metric-affine geometry, ie, “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 endeavors 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 (EisenhartFootnote 22, VeblenFootnote 23), French (CartanFootnote 24), and the Dutch (Schouten, StruikFootnote 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.

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 4-leg 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: generalization 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., five-dimensional 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.


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 accountFootnote 26.

The space of physical events will be described by a real, smooth manifold M.D. of dimension D. coordinated by local coordinates xi, and provided with smooth vector fields X, Y, ... with components Xi, Yi, ... and linear forms ω, ν, …, (ωi, νi) in the local coordinate system, as well as further geometrical objects such as tensors, spinors, connectionsFootnote 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 space and time orientable. On it, two independent fundamental structural objects will now be introduced.

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 dxi connecting the points. This condition is not very restrictive; it still includes Finsler geometry [281, 126, 224] to be briefly touched, below.

In the following, ds is linked to a non-degenerate 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 Gij look for that

$$ ds = \ sqrt {{g_ {ij}} d {x ^ i} d {x ^ j}}, $$

where the neighboring points are labeled by xi other xi+dxi, respectivelyFootnote 28. Besides the norm of a vector \ (\ left | X \ right |: = \ sqrt {{g_ {ij}} {X ^ i} {X ^ j}} \), the “angle” between directions X, Y can be defined by help of the metric:

$$ \ cos (\ angle (X, Y)): = \ frac {{{g_ {ij}} {X ^ i} {Y ^ j}}} {{\ left | X \ right | \ left | Y \ right |}}. $$

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 Gik is defined throughFootnote 29

$$ {g_ {mi}} {g ^ {mj}} = \ delta _i ^ j $$

We are used to G being a symmetric tensor field, i.e., with Gik=G(ik) 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 ± (D.−2)Footnote 30. In the following this need not hold, so that the decomposition obtainsFootnote 31:

$$ {g_ {ij}} = {\ gamma _ {(ik)}} + {\ phi _ {\ left [{ik} \ right]}}. $$

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

$$ {g_ {ij}} = {h ^ {(ik)}} + {f ^ {[ik]}} = {h ^ {ik}} + {f ^ {ik}} $$

is determined by the relations

$$ \ begin {array} {* {20} {l}} {{\ gamma _ {ij}} {\ gamma ^ {ik}} = \ delta _j ^ k,} & {\; \; \; { \ phi _ {ij}} {\ phi ^ {ik}} = \ delta _j ^ k, \; \; \;} & {{h_ {ij}} {h ^ {ik}} = \ delta _j ^ k , \; \; \;} & {{f_ {ij}} {f ^ {ik}} = \ delta _j ^ k,} \ end {array} $$

and turns out to be [356]

$$ {h ^ {(ik)}} = \ frac {\ gamma} {g} {\ gamma ^ {ik}} + \ frac {\ phi} {g} {\ phi ^ {im}} {\ phi ^ {kn}} {\ gamma _ {mn}}, $$
$$ {f ^ {(ik)}} = \ frac {\ phi} {g} {\ phi ^ {ik}} + \ frac {\ gamma} {g} {\ gamma ^ {im}} {\ gamma ^ {kn}} {\ phi _ {mn}}, $$

where G, φ, and γ are the determinants of the corresponding tensors Gik, φik, and γik. We also note that

$$ g = \ gamma + \ phi + \ frac {\ gamma} {2} {\ gamma ^ {kl}} {\ gamma ^ {mn}} {\ phi _ {km}} {\ phi _ {\ ln }}, $$

where G : = det Gik, φ : = det φik, γ : = det γik. The results (6, 7, 8) were obtained already by Reichenbächer ([273], pp. 223–224)Footnote 32 and also by Schrödinger [320]. Eddington also calculated Equation (8); in his expression the term ∼φik*φik is missing (cf. [59], p. 233).

The manifold is called space-time if D.= 4 and the metric is symmetric and Lorentzian, i.e., symmetric and with signature sig G= ± 2. Nevertheless, sloppy contemporaneous usage of the term “space-time” includes arbitrary dimension, and sometimes is applied even to metrics with arbitrary signature.

In a manifold with Lorentzian metric, a non-trivial real conformal structure always exists; from the equation

results an equivalence class of metrics {λ} with λ 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 [67]. For an asymmetric metric, this structure can exist as well; it then is determined by the symmetric part γik=γ(ik) 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

$$ {\ eta _ {ik}} = {\ delta _i} ^ 0 {\ delta _i} ^ 0 - {\ delta _i} ^ 1 {\ delta _i} ^ 1 - {\ delta _i} ^ 2 {\ delta _i} ^ 2 - {\ delta _i} ^ 3 {\ delta _i} ^ 3. $$

A geometrical characterization of Minkowski space as an uncurved, flat space is given below. Let \ ({{\ mathcal L} _X} \) be the Lie derivative with respect to the tangent vector XFootnote 33; then \ ({{\ cal L} _ {{{_ X} _p} {\ eta _ {ik}}}} = 0 \) holds for the Lorentz group of generators Xp.

The metric tensor G may also be defined indirectly through D. vector fields forming an orthonormal D.-leg (-bein) \ (h _ {\ hat \ iota} ^ k \). with

$$ {g_ {lm}} = {h_ {l \ hat \ jmath}} {h_ {m \ hat k}} {\ eta ^ {\ hat \ jmath \ hat k}}, $$

where the hatted indices ("bein-indices") count the number of legs spanning the tangent space at each point (ĵ=1, 2, … , D.) and are moved with the Minkowski metricFootnote 34. From the geometrical point of view, this can always be done (cf. theories with distant parallelism). By introducing 1-forms \ ({\ theta ^ {\ hat k}}: = h_l ^ {\ hat k} d {x ^ l} ​​\), Equation (11) may be brought into the form \ (d {s ^ 2} = {\ theta ^ {\ hat \ imath}} {\ theta ^ {\ hat k}} {\ eta _ {\ hat \ imath \ hat k}} \).

A new physical aspect will come in if the Hkî are considered to be the basic geometric variables satisfying field equations, not the metric. Search tetrad theories (for the case D.= 4) are described well by the concept of fiber bundle. The fiber 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(D.), or the Lorentz group, and so on.

In Finsler geometry, the line element depends not only on the coordinates xi of a point on the manifold, but also on the infinitesimal elements of direction between neighboring points dxi:

$$ d {s ^ 2} = {g_ {ij}} ({x ^ n}, d {x ^ m}) d {x ^ i} d {x ^ j}. $$

Again, Gij is required to be homogeneous of rank 1.

Affine structure

The second structure to be introduced is a linear connectionL. with D.3 components L.ijk; it is a geometrical object but not A tensor field and its components change inhomogeneously under local coordinate transformationsFootnote 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 ∇, 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 ω it is defined by

$$ \ begin {array} {* {20} {c}} {{{\ mathop \ nabla \ limits ^ +} _k} {X ^ i} = \ frac {{\ partial {X ^ i}}} { {\ partial {x ^ k}}} + {L_ {kj}} ^ i {X ^ j}, \; \; \;} & {{{\ mathop \ nabla \ limits ^ +} _k} {\ omega _i} = \ frac {{\ partial {\ omega _i}}} {{\ partial {x ^ k}}} - {L_ {ki}} ^ j {\ omega _j}.} \ end {array} $$

The expressions \ ({\ mathop \ nabla \ limits ^ + _k} {X ^ i} \) and \ (\ tfrac {{\ partial {X ^ i}}} {{\ partial {x ^ k}}} \ ) are abbreviated by \ ({X ^ i} _ {\ left \ | k \ right.} \) and Xi, k, respectively, while for a scalar f covariant and partial derivative coincide: \ ({\ nabla _i} f = {\ tfrac {{\ partial f}} {{\ partial {x_i}}}} \ equiv {\ partial _i} f \ equiv {f _ {, i }} \).

We have adopted the notational convention used by Schouten [300, 310, 389]. Eisenhart and others [121, 234] change the order of indices of the components of the connection:

$$ \ begin {array} {* {20} {c}} {{{\ mathop \ nabla \ limits ^ -} _k} {X ^ i} = \ frac {{\ partial {X ^ i}}} { {\ partial {x ^ k}}} + {L_ {jk}} ^ i {X ^ j}, \; \; \;} & {{{\ mathop \ nabla \ limits ^ -} _k} {\ omega _i} = \ frac {{\ partial {\ omega _i}}} {{\ partial {x ^ k}}} - {L_ {ik}} ^ j {\ omega _j}.} \ end {array} $$

As long as the connection is symmetric, this does not make any difference as \ ({\ mathop \ nabla \ limits ^ + _k} {X ^ i} - {\ mathop \ nabla \ limits ^ - _k} {X ^ i} = 2 {L _ {[kj]}} ^ i {X ^ j} \). For both kinds of derivatives we have:

$$ \ begin {array} {* {20} {c}} {{{\ mathop \ nabla \ limits ^ +} _k} ({v ^ l} ​​{w_l}) = \ frac {{\ partial ({v ^ l} {w_l})}} {{\ partial {x ^ k}}}, \; \; \;} & {{{\ mathop \ nabla \ limits ^ -} _k} ({v ^ l} ​​{ w_l}) = \ frac {{\ partial ({v ^ l} ​​{w_l})}} {{\ partial {x ^ k}}}} \ end {array} $$

Both derivatives are used in versions of unified field theory by Einstein and othersFootnote 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 contains one more term:

$$ \ begin {array} {* {20} {c}} {{{\ mathop \ nabla \ limits ^ +} _k} {{\ hat X} ^ i} = \ frac {{\ partial {{\ hat X} ^ i}}} {{\ partial {x ^ k}}} + {L_ {kj}} ^ i {{\ hat X} ^ j} - {L_ {kr}} ^ r {{\ hat X } ^ i}, \; \; \;} & {{{\ mathop \ nabla \ limits ^ -} _k} {{\ hat X} ^ i} = \ frac {{\ partial {X ^ i}}} {{\ partial {x ^ k}}} + {L_ {jk}} ^ i {{\ hat X} ^ j} - {L_ {rk}} ^ r {{\ hat X} ^ i}.} \ end {array} $$

A smooth vector field Y is said to be parallely transported along a parametrized curve λ (u) with tangent vector X if for its components \ ({Y ^ i} _ {\ left \ | k \ right.} {X ^ k} (u) = 0 \) holds along the curve. A curve is called an auto-parallel if its tangent vector is parallely transported along it at each pointFootnote 37:

$$ {X ^ i} _ {\ left \ | k \ right.} {X ^ k} (u) = \ sigma (u) {X ^ i}. $$

By a particular choice of the curve73x2019; s parameter, σ= 0 may be imposed.

A transformation mapping autoparallels to autoparallels is given by:

$$ {L_ {ik}} ^ j \ to {L_ {ik}} ^ j + {\ delta ^ j} _ {(i} {\ omega _ {k)}}. $$

The equivalence class of autoparallels defined by Equation (18) defines a projective structure on M.D. [404, 403].

The particular set of connections

$$ _ {(p)} {L_ {ij}} ^ k: = {L_ {ij}} ^ k - \ frac {2} {{D + 1}} {\ delta ^ k} _ {(i} {L_ {j)}} $$

with \ ({L_j}: = {L_ {im}} ^ m \) is mapped into itself by the transformation (18) [348].

In Part II of this article, we shall find the set of transformations \ ({L_ {ik}} ^ j \ to {L_ {ik}} ^ j + {\ delta ^ j} _i \ tfrac {{\ partial \ omega }} {{\ partial {x ^ k}}} \) playing a role in versions of Einstein's unified field theory.

From the connection L.ijk Further connections may be constructed by adding an arbitrary tensor field T to its symmetrized partFootnote 38:

$$ {\ bar L_ {ij}} ^ k = {L _ {(ij)}} ^ k + {T_ {ij}} ^ k = {\ Gamma _ {ij}} ^ k + {T_ {ij}} ^ k. $$

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.,

$$ {S_ {ij}} ^ k = {L _ {[ij]}} ^ k = {T _ {[ij]}} ^ k $$

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; {{\ tilde L} _j}: = {L_ {lj}} ^ l \) as \ ({ S_i} = \ tfrac {1} {2} ({L_i} - {\ tilde L_i}) \).

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. Search tensorial objects are the two affine curvature tensors defined byFootnote 39

$$ \ mathop {{\ rm {}} K} \ limits ^ + {\; ^ i} _ {jkl} \; = {\ partial _k} L_ {lj} ^ {\; \; \; i} - {\ partial _l} L_ {kj} ^ {\; \; \; i} + L_ {km} ^ {\; \ ; \; i} L_ {lj} ^ {\; \; \; m} - L_ {lm} ^ {\; \; \; i} L_ {kj} ^ {\; \; \; m}, $ $
$$ \ mathop {{\ rm {}} K} \ limits ^ - {\; ^ i} _ {jkl} \; = {\ partial _k} L_ {jl} ^ {\; \; \; i} - {\ partial _l} L_ {jk} ^ {\; \; \; i} + L_ {mk} ^ {\; \ ; \; i} L_ {jl} ^ {\; \; \; m} - L_ {ml} ^ {\; \; \; i} L_ {jk} ^ {\; \; \; m}, $ $

respectively. In a geometry with symmetric affine connection both tensors coincide because of

$$ \ frac {1} {2} (\ mathop {{\ rm {}} K} \ limits ^ + \; _ {jkl} ^ i - \ mathop {{\ rm {}} K} \ limits ^ - \; _ {jkl} ^ i) = {\ partial _ {[k}} {S _ {] lj}} ^ i + 2 {S_ {j [k}} ^ mS_ {l] m} ^ {\; \ ; \; i} + L_ {m [k} ^ {\; \; \; i} {S_ {l] j}} ^ m - L_ {j [k} ^ {\; \; \; \; m } {S_ {l] m}} ^ i. $$

In particular, in Riemannian geometryBoth 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:

$$ \ mathop {{\ rm {}} \ nabla} \ limits ^ + {\, _ {[j}} \ mathop \ nabla \ limits ^ + {\, _ {k]}} {A ^ i} = \ frac {1} {2} \ mathop {{\ rm {}} K} \ limits ^ + {\, ^ i} _ {rjk} {A ^ r} - {S_ {jk}} ^ r \ mathop { {\ rm {}} \ nabla} \ limits ^ + {\, _ r} {A ^ i} $$
$$ \ mathop {{\ rm {}} \ nabla} \ limits ^ - {\, _ {[j}} \ mathop \ nabla \ limits ^ - {\, _ {k]}} {A ^ i} = \ frac {1} {2} \ mathop {{\ rm {}} K} \ limits ^ - {\, ^ i} _ {rjk} {A ^ r} - {S_ {jk}} ^ r \ mathop { {\ rm {}} \ nabla} \ limits ^ - {\, _ r} {A ^ i} $$

For a vector density, the identity is given by

$$ \ mathop {{\ rm {}} \ nabla} \ limits ^ + {\, _ {[j}} \ mathop \ nabla \ limits ^ + {\, _ {\, k]}} {{\ hat A} ^ i} = \ frac {1} {2} \ mathop {{\ rm {}} \ nabla} \ limits ^ + \, {{\,} ^ i} _ {rjk} {{\ hat A} ^ r} - {S_ {jk}} ^ r \ mathop \ nabla \ limits ^ + {\, _ {\, r}} {{\ hat A} ^ i} + \ frac {1} {2} {V_ {jk}} {{\ hat A} ^ i} $$

with the homothetic curvature Vjk to be defined below in Equation (31).

The curvature tensor (22) satisfies two algebraic identities:

$$ {\ mathop {{\ rm {}} K} \ limits ^ + {\; ^ i} _ {j [kl]} \; = 0,} $$
$$ {\ mathop {{\ rm {}} K} \ limits ^ + {\; ^ i} _ {\ {jkl \}} \; = 2 {\ nabla _ {\ {j}} {S_ {kl \}}} ^ i + 4 {S_ {m \ {j}} ^ i {S_ {kl \}}} ^ m,} $$

where the curly bracket denotes cyclic permutation:

$$ {K ^ i} _ {\ {jkl \}}: = {K ^ i} _ {jkl} + {K ^ i} _ {ljk} + {K ^ i} _ {klj}. $$

These identities can be found in Schouten's book of 1924 ([300], p. 88, 91) as well as the additional single integrability condition, called Bianchi identity:

$$ \ mathop {{\ rm {}} K} \ limits ^ + {\; ^ i} _ {j \ {kl \ left \ | {m \}} \ right.} \; = 2 {K ^ i} _ {r \ {kl} {S_ {m \} j}} ^ r. $$

A corresponding condition obtains for the curvature tensor from Equation (23).

From both affine curvature tensors we may form two different tensorial traces each. In the first case \ ({V_ {kl}}: = {K ^ i} _ {ikl} = {V _ {[kl]}} \), and \ ({K_ {jk}}: = {K ^ i } _ {jki} \). Vkl is called homothetic curvature, while Kjk is the first of the two affine generalizations from \ (\ mathop {{\ rm {}} K} \ limits ^ + \) and \ (\ mathop {{\ rm {}} K} \ limits ^ - \) of the Ricci tensor in Riemannian geometry. We getFootnote 40

$$ {V_ {kl}} = {\ partial _k} {L_l} - {\ partial _l} {L_k}, $$

and the following identities hold:

$$ {V_ {kl}} + 2 {K _ {[kl]}} = 4 {\ nabla _ {[k}} {S_ {l]}} + 8 {S_ {kl}} ^ m {S_m} + 2 {\ nabla _m} {S_ {kl}} ^ m, $$
$$ \ mathop V \ limits ^ - {\, _ {kl}} + 2 \ mathop K \ limits ^ - {\, _ {[kl]}} = - 4 \ mathop \ nabla \ limits ^ - {\, _ {[k}} {S_ {l]}} + 8 {S_ {kl}} ^ m {S_m} + 2 {\ nabla _m} {S_ {kl}} ^ m, $$

where \ ({S_k}: = {S_ {kl}} ^ l \). While Vkl is antisymmetric, Kjk has both tensorial symmetric and antisymmetric parts:

$$ {K _ {[kl]}} = - {\ partial _ {[k}} {{\ tilde L} _ {l]}} + {\ nabla _m} {S_ {kl}} ^ m + {L_m } {S_ {kl}} ^ m + 2 {L _ {[l \ left | r \ right.}} ^ m {S_ {m \ left | k \ right.}} ^ r, $$
$$ {K _ {(kl)}} = {\ partial _ {(k}} {{\ tilde L} _ {l)}} - {\ partial _m} {L _ {(kl)}} ^ m - { {\ tilde L} _m} {L _ {(kl)}} ^ m + {L _ {(k \ left | m \ right |}} ^ n {L_ {l)}} ^ m. $$

We use the notation \ ({A _ {(i \ left | k \ right | l)}} \) in order to exclude the index k from the symmetrization bracketFootnote 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, [356]).

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:

$$ {K ^ i} _ {\ {jkl \}} = 0, $$
$$ {K ^ i} _ {j \ {kl \ left \ | {m \}} \ right.} = 0, $$
$$ {V_ {kl}} + 2 {K _ {[kl]}} = 0, $$

i.e., only one independent trace tensor of the affine curvature tensor exists. For the antisymmetric part of the Ricci tensor \ ({K _ {[kl]}} = - {\ partial _ {[k}} {{\ tilde 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 Gij:=αK(ij), or Gij:=αK̅(ij). 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 (Kij)Footnote 42.

As a final result in this section, we give the curvature tensor calculated from the connection \ ({{\ bar 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 \):

$$ {K ^ i} _ {jkl} (\ bar L) = {K ^ i} _ {jkl} (\ Gamma) + {2 ^ {(\ Gamma)}} {\ nabla _ {[k}} {T_ {l] j}} ^ i - 2 {T _ {[k \ left | j \ right |}} ^ m {T_ {l] m}} ^ i + 2 {S_ {kl}} ^ m {T_ {mj}} ^ i, $$

where (Γ)∇ is the covariant derivative formed with the connection \ ({\ Gamma _ {ij}} ^ k \) (cf. also [310], p. 141).

Mixed geometry

A manifold carrying both structural elements, i.e., metric and connection, is called a metric-affine space. If the first fundamental form is taken to be asymmetric, i.e., to contain an antisymmetric part \ ({g _ {[ik]}}: = \ tfrac {1} {2} ({g_ {ij}} - {g_ {ji}}) \), we speak of a mixed geometry. In principle, both metric-affine space and mixed geometry may always be re-interpreted as Riemannian geometry with additional geometric objects: the 2-form field φ(f) (symplectic form), the torsion S., and the non-metricity 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 Hij=G(ij), a connection may be constructed, often called after Levi-CivitaFootnote 43 [204],

$$ \ {\, _ {ij} ^ k \}: = \ frac {1} {2} {\ gamma ^ {kl}} ({\ gamma _ {li, j}} + {\ gamma _ {lj , i}} - {h_ {ij,}}), $$

and from it the Riemannian curvature tensor defined as in Equation (22) with \ ({L_ {ij}} ^ k = \ {\, _ {ij} ^ k \} \) (cf. Section 2.1.3); {kij} is called the Christoffel symbol. Thus, in metric-affine 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 γij only, and denote it by Gij.

With the help of the symmetric affine connection, we may define the tensor of non-metricity\ ({Q_ {ij}} ^ k \) byFootnote 44