Unified Field Theories: In the First Third of the 20th Century
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Bezorgopties We bieden verschillende opties aan voor het bezorgen of ophalen van je bestelling. Welke opties voor jouw bestelling beschikbaar zijn, zie je bij het afronden van de bestelling. Schrijf een review. E-mail deze pagina. Auteur: Vladimir P. Vizgin Vladimir P. Uitgever: Springer Basel. Samenvatting Despite the rapidly expanding ambit of physical research and the continual appearance of new branches of physics, the main thrust in its development was and is the attempt at a theoretical synthesis of the entire body of physical knowledge. The main triumphs in physical science were, as a rule, associ- ated with the various phases of this synthesis.
The most radical expression of this tendency is the program of construction of a unified physical theory. After Maxwellian electrodynamics had unified the phenomena of electricity, magnetism, and optics in a single theoretical scheme on the basis of the con- cept of the electromagnetic field, the hope arose that the field concept would become the precise foundation of a new unified theory of the physical world.
We get. While V kl is antisymmetric, K jk has both tensorial symmetric and antisymmetric parts:. We use the notation in order to exclude the index k from the symmetrisation bracket. 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. For a symmetric affine connection, the preceding results reduce considerably due to.
From Equations 29 , 30 , 32 we obtain the identities:. For the antisymmetric part of the Ricci tensor holds. This equation will be important for the physical interpretation of affine geometry. 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 K ij. As a final result in this section, we give the curvature tensor calculated from the connection cf.
Einstein's Grand Quest for a Unified Theory
Equation 20 , expressed by the curvature tensor of and by the tensor :. A manifold carrying both structural elements, i. If the first fundamental form is taken to be asymmetric , i. 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 f symplectic form , the torsion S , and the non-metricity Q cf.
Equation It depends on the physical interpretation, i. Thus, in metric-affine and in mixed geometry, two different connections arise in a natural way. With the help of the symmetric affine connection, we may define the tensor of non-metricity by. 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 non-metricity tensor does not vanish:.
Thomas introduced a combination of the terms appearing in and to define a covariant derivative for the metric [ ], p. We will have to deal with Equation 47 in Section 6. Connections that are not metric-compatible have been used in unified field theory right from the beginning. In case of such a relationship, the geometry is called semi-metrical [ , ].
We may also abbreviate the last term in the identity 42 by introducing. Then, from Equation 39 , the curvature tensor of a torsionless affine space is given by.
Riemann-Cartan geometry is the subcase of a metric-affine geometry in which the metric-compatible connection contains torsion, i. A linear connection whose antisymmetric part has the form. Riemannian geometry is the further subcase with vanishing torsion of a metric-affine geometry with metric-compatible connection.
In this case, the connection is derived from the metric: , where is the usual Christoffel symbol The covariant derivative of A with respect to the Levi-Civita connection is abbreviated by A ; k. The Riemann curvature tensor is denoted. An especially simple case of a Riemanian space is Minkowski space, the curvature of which vanishes:. However, in metric affine and in mixed geometry geodesic and autoparallel curves will have to be distinguished. As a consequence, the Riemann curvature tensor is also changed; if, however, can be reached by a conformal transformation, then the corresponding spacetime is called conformally flat.
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 [ ]; see also [ ], Appendix 1; for a modern approach, cf. The geometry needed for the pre- and non-relativistic 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. In the following we shall deal only with relativistic unified field theories. The connection to the inhomogeneous coordinates x i is given by homogeneous functions of degree zero, e. Thus, the themselves form the components of a tangent vector. Furthermore, the quadratic form is adopted with being a homogeneous function of degree A tensor field cf.
If we define , with , then transforms like a tangent vector under point transformations of the x i , and as a covariant vector under homogeneous transformations of the. The may be used to relate covariant vectors a i and by. Thus, the metric tensor in the space of homogeneous coordinates and the metric tensor of M D are related by with.
The inverse relationship is given by with. The covariant derivative for tensor fields in the space of homogeneous coordinates is defined as before cf. The covariant derivative of the quantity interconnecting both spaces is given by. Cartan introduces one-forms by. The reciprocal basis in tangent space is given by. The metric is then given by. We have. The link to the components of the affine connection is given by. The covariant derivative of a tangent vector with bein-components X then is. In Equation 65 the curvatureform appears, which is given by.
Up to here, no definitions of a tensor and a tensor field were given: A tensor T p M D of type r , s at a point p on the manifold M D is a multi-linear function on the Cartesian product of r cotangent- and 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:.
Strictly speaking, tensors are representations of the abstract group at a point on the manifold. We can form a tensor from ijkl by introducing , where g ik is a Lorentz-metric. Note that. The dual to a 2-form skew-symmetric tensor then is defined by. In connection with conformal transformations , the concept of the gauge-weight of a tensor is introduced. A tensor is said to be of gauge weight q if it transforms by Equation 56 as. Objects that transform as in Equation 67 but with respect to a subgroup, e.
All the subgroups mentioned are Lie -groups, i. Spinors are representations of the Lorentz group only; as such they are related strictly to the tangent space of the space-time manifold. To see how spinor representations can be obtained, we must use the homomorphism of the group SL 2,C and the proper orthochronous Lorentz group, a subgroup of the full Lorentz group. By picking the special Hermitian matrix.
The link between the representation of a Lorentz transformation L ik in space-time and the unimodular matrix A mapping spin space cf. The spinor is called elementary if it transforms under a Lorentz-transformation as. Covariant and covariant dotted 2-spinors correspondingly transform with the inverse matrices,. Higher-order spinors with dotted and undotted indices transform correspondingly. Next to a spinor, bispinors of the form , etc. A vector X k can be represented by a bispinor X A ,.
Often the quantity is introduced. The reciprocal matrix is defined by. The simplest spinorial equation is the Weyl equation:. The Dirac equation is in 4-spinor formalism [ 53 , 54 ]:. The group of coordinate transformations acts on the Latin indices. For the example of the Dirac spinor, the adjoint representation of the Lorentz group must be used. In Section 2. With its help we may formulate the concept of isometries of a manifold, i. If a group G r is prescribed, e. 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.
In purely affine spaces, another type of symmetry may be defined: ; they are called affine motions [ ]. 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.
This Lagrangian leads to the well-known field equations of general relativity,.
In empty space, i. If only an electromagnetic field derived from the 4-vector potential A k is present in the energy-momentum tensor, then the Einstein. Maxwell equations follow:. The components of the metrical tensor are identified with gravitational potentials. The equations of motion of material particles should follow, in principle, from Equation 92 through the relation. 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 non-interacting 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 link-up 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. 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 4-potential and the electric current vector density have also been suggested cf. 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. Also, geometries based on Hermitian forms were studied [ ]. 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. In this part of the article we do not need to take into account this generalisation.
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. In his letter to Einstein of 11 November , he writes [ ], Doc. 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.
Various reasons instilled in me strong reservations: […] your other remarks are interesting in themselves and new to me. Ishiwara, and G. The result is contained in Hilbert , p. The hints dropped by you on your postcards bring me to expect the greatest. According to him, the deviation from the Minkowski metric is due to the electromagnetic field tensor:.
Vizgin | Unified Field Theories | | in the first third of the 20th | 13
He claims to obtain the same value for the perihelion shift of Mercury as Einstein [ ], p. The meeting was amicable. 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 [ ]. Although Finsler himself did not apply his geometry to physics it soon became used in attempts at the unification of gravitation and electromagnetism [ ].
The idea that they keep together the dispersing electrical charges lies close at hand. 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 Naturforscherversammlung, 19—25 September [ ] 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. 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.
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. 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 space-distances. In my opinion, an answer can be given only under the aspect of feasibility; the outcome appears dubious to me. But a more precise reasoning shows that in this way no reasonable world function is obtained. It is to be noted that Weyl, at the end of , already had given up on a possible field theory 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. Klein on 28 December , see [ ], p. 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 […]. During the twenties Einstein changed his mind and looked for solutions of his field equations which were everywhere regular to represent matter particles:. 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. After , Einstein first was busy with extracting mathematical and physical consequences from general relativity Hamiltonian, exact solutions, the energy conservation law, cosmology, gravitational waves.
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:. A metrical relationship from point to point will only then be infused into [the manifold] if a principle for carrying the unit of length from one point to its infinitesimal neighbours is given.
In contrast to this, Riemann made the much stronger assumption that line elements may be compared not only at the same place but also at two arbitrary places at a finite distance. At a point, Equation 98 induces a local recalibration of lengths l while preserving angles, i. If, as Weyl does, the connection is assumed to be symmetric i. With regard to the gauge transformations 98 , remains invariant.
From the 1-form dQ , by exterior derivation a gauge-invariant 2-form with follows. Let us now look at what happens to parallel transport of a length, e. If X is taken to be tangent to C , i. The same holds for the angle between two tangent vectors in a point cf. Yet, also today, the circumstances are such that our trees do not grow into the sky. Due to the additional group of gauge transformations, it is useful to introduce the new concept of gauge-weight within tensor calculus as in Section 2. Weyl did calculate the curvature tensor formed from his connection but did not get the correct result ; it is given by Schouten [ ], p.
His Lagrangian is given by , where the invariants are defined by. Weyl had arranged that the page proofs be sent to Einstein. In communicating this on 1 March , he also stated that. In the most general case, the equations will be of 4th order, though. He then asked whether Einstein would be willing to communicate a paper on this new unified theory to the Berlin Academy [ ], Volume 8B , Document , pp.
Einstein was impressed: In April , 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 on a postcard was enthusiastic:.
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. However, as long as measurements are made with infinitesimally small rigid rulers and clocks, there is no indeterminacy in the metric as Weyl would have it : Proper time can be measured. As a consequence follows: If in nature length and time would depend on the pre-history of the measuring instrument, then no uniquely defined frequencies of the spectral lines of a chemical element could exist, i. He concluded with the words. Only for a vanishing electromagnetic field does this objection not hold.
Only in a static gravitational field, and in the absence of electromagnetic fields, does this hold:. Einstein saw the problem, then unsolved within his general relativity, that Weyl alluded to, i. Presumably, such a theory would have to include microphysics. But I find: If the ds , as measured by a clock or a ruler , is something independent of pre-history, construction and the material, then this invariant as such must also play a fundamental role in theory.
Yet, if the manner in which nature really behaves would be otherwise, then spectral lines and well-defined chemical elements would not exist. Another famous theoretician who could not side with Weyl was H. However, Weyl still believed in the physical value of his theory. There exists an intensive correspondence between Einstein and Weyl, now completely available in volume 8 of the Collected Papers of Einstein [ ].
We subsume some of the relevant discussions. Weyl did not give in:. Einstein then suggested the affine group as the more general setting for a generalisation of Riemannian geometry [ ], Vol. In particular, it is unimportant which value of the integral is assigned to their world line. Otherwise, sodium atoms and electrons of all sizes would exist. But if the relative size of rigid bodies does not depend on past history, then a measurable distance between two neighbouring world-points exists. 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 […]. The quadratic form Rg ik dx i dx k is an absolute invariant, i. If this expression would be taken as the measurable distance in place of ds , then. A very small change of the measuring path would strongly influence the integral of the square root of this quantity. Einstein added:. The last remarks are interesting for the way in which Einstein imagined a successful unified field theory.
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. You establish a real unity. Understandably, no comments about the physics are given [ ], pp. Of course, as he noted, no progress had been made with regard to the explanation of the constituents of matter; on the one hand because the differential equations were too complicated to be solved, on the other because the observed mass difference between the elementary particles with positive and negative electrical charge remained unexplained.
In his general remarks about this problem at the very end of his article, Pauli points to a link of the asymmetry with time-reflection symmetry see [ ], pp. 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? The idea of gauging lengths independently at different events was the central theme. This must also give rise to an identity; and it is found that the new identity expresses the law of conservation of electric charge.
Section 4. As he had abandoned the idea of describing matter as a classical field theory since , the linking of the electromagnetic field via the gauge idea could only be done through the matter variables. In October , in the preface for the first American printing of the English translation of the fourth edition of his book Space, Time, Matter from , 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:.
This attempt has failed. Weyl himself continued to develop the dynamics of his theory. As an equivalent Lagrangian Weyl gave, up to a divergence. Due to his constraint, Weyl had navigated around another problem, i. In the paper in , he changed his Lagrangian slightly into. The changes, which Weyl had introduced in the 4th edition of his book [ ], and which, according to him, were of fundamental importance for the understanding of relativity theory, were discussed by him in a further paper [ ].
His colleague in Vienna, Wirtinger , had helped him in this. If J has gauge-weight -1, then Jg ik is such a metric. In order to reduce the new theory to general relativity, in addition only the differential equation. More important, however, for later work was the gauge invariant tensor calculus by a fellow of St. Newman [ ]. In this calculus, tensor equations preserve their form both under a change of coordinates and a change of gauge.
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Newman applied his scheme to a variational principle with Lagrangian K 2 and concluded:. We shall discuss these topics in Part II of this article. What is now called Kaluza-Klein theory in the physics community is a mixture of quite different contributions by both scientists. But you understand that, in view of the existing factual concerns, I cannot take sides as planned originally. Kaluza did not normalize the Killing vector to a constant, i.
Of the 15 components of , five had to get a new physical interpretation, i. The component g 55 turned out to be a scalar gravitational potential which, in the static case, satisfies the equation. Kaluza also showed that the geodesics of the five-dimensional space reduce to the equations of motion for a charged point particle in space-time, if a weakness assumption is made for the components of the 5-velocity , u 4 1. The Lorentz force appears augmented by an additional term containing g 55 of the order which thus may be neglected. For him, any theory claiming universal validity was endangered by quantum theory, anyway.
The remaining covariance group G 5 is given by. The objects transforming properly under are: the scalar , the vector-potential , and the projected metric. Klein identified the group; however, he did not comment on the fact that now further invariants are available for a Lagrangian, but started right away from the Ricci scalar of M 5 [ ]. The group G 5 is isomorphic to the group H 5 of transformations for five homogeneous coordinates with homogeneous functions of degree 1.
I value your approach more than the one followed by H. If you wish, I will present your paper to the Academy after all. The negative result of his own paper, i. His motivation went beyond the unification of gravitation and electromagnetism:. Clearly, the non-Maxwellian binding forces which hold together an electron. In the first, shorter, part of two, Eddington describes affine geometry; in the second he relates mathematical objects to physical variables.
He starts by calculating both the curvature and Ricci tensors from the symmetric connection according to Equation By this, Eddington claims to guarantee charge conservation:. However, for a tensor density, due to Equation 16 we obtain. Who shall say what is the ordinary gauge inside the electron? Only connections leading to a Lorentz metric can be used if a physical interpretation is wanted. Thus, in general, g kl is not metric-compatible; in order to make it such, we are led to the differential equations for , an equation not considered by Eddington.
This is due to the expression for the inverse of the metric, a function cubic in R kl. Note also that Eddington does not explicitly say how to obtain the contravariant form of the electromagnetic field F ij from F ij ; we must assume that he thought of raising indices with the complicated inverse metric tensor. Now, Eddington was able to identify the energy-momentum tensor T ik of the electromagnetic field by decomposing the Ricci tensor K ij formed from Equation 51 into a metric part R ik and the rest.
His aim was reached in the sense that all three quantities were fixed entirely by the connection; they could no longer be given from the outside. If so, then it must be a purely phenomenological one without any recourse to the nature of the charged elementary particles cf.
At first, Einstein seems to have been reserved cf. To Bohr, Einstein wrote from Singapore on 11 January Eddington has come closer to the truth than Weyl. Like Eddington, Einstein used a symmetric connection and wrote down the equation. By this, the metric was defined as the symmetric part of the Ricci tensor. Due to. Let us note, however, that while transforms inhomogeneously, its transformation law. For a Lagrangian, Einstein used ; he claims that for vanishing electromagnetic field the vacuum field equations of general relativity, with the cosmological term included, hold.
If , then the electric current density j l is defined by. The field equations are obtained from the Lagrangian by variation with regard to the connection and are Einstein worked in space-time. From Equation the connection can be obtained. This equation is an identity if a solution of the field equations is inserted. From Equation ,. In order that this makes sense, the identifications in Equation are always to be made after the variation of the Lagrangian is performed.
For non-vanishing electromagnetic field, due to Equation the Equation now becomes. Einstein concluded:. Except for singular positions, the current density is practically vanishing. Up to the same order,. In general however,. Also, the geometrical theory presented here is energetically closed, i. His final conclusion was:. Until the end of May , two further publications followed in which Einstein elaborated on the theory.
In the second paper, he exchanged the Lagrangian for a new one, i. The resulting equations for the gravitational and electromagnetic fields are the symmetric and skew-symmetric part, respectively, of. Although the theory offered, for every solution with positive charge, also a solution with negative charge, the masses in the two cases were the same. However, the only known particle with positive charge at the time what is now called the proton had a mass greatly different from the particle with negative charge, the electron. Einstein noted:.
The logic of the subsequent derivations in his paper is quite involved. The first step consisted in the definition of tensor densities. By using both Equation and Equation , Einstein obtained the Einstein. After a field rescaling, he then took a third expression to become his Lagrangian. Nobody can determine empirically an affine connection for vectors at neighbouring points if he has not obtained the line element before. He criticised a theory that keeps only the connection as a fundamental building block for its lack of a guarantee that it would also house the conformal structure light cone structure.
This is needed for special relativity to be incorporated in some sense, and thus must be an independent fundamental input [ ]. From a recent conversation with Einstein I learn that he is of much the same opinion. His outlook on the state of the theory now was rather bleak:. To me, the quantum-problem seems to require something like a special scalar, for the introduction of which I have found a plausible way. But I fail to succeed in giving my pet idea a tangible form: to understand the quantum-structure through an overdetermination by differential equations.
The initial state of an electron moving around a hydrogen nucleus cannot be chosen freely; its choice must correspond to the quantum conditions. In general: not only the evolution in time but also the initial state obey laws. He then ventured the hope that a system of overdetermined differential equations is able to determine.
One of the crucial tests for an acceptable unified field theory for him now was:. In such a way, the un-ambiguity of the initial conditions ought to be understood without leaving field theory. In the introduction to his book, Struik distinguished three directions in the development of the theory of linear connections [ ]:. In his assessment, Eisenhart [ ] adds to this all the geometries whose metric is. Developments of this theory have been made by Finsler, Berwald, Synge, and J.
In this geometry the paths are the shortest lines, and in that sense are a generalisation of geodesics.
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Affine properties of these spaces are obtained from a natural generalisation of the definition of Levi-Civita for Riemannian spaces. In fact, already in May Jan Arnoldus Schouten in Delft had submitted two papers classifying all possible connections [ , ]. In the first he wrote:. Weyl, Raum-Zeit-Materie , 2.
Section, Leipzig 3. The most general connection is characterised by two fields of third degree, one tensor field of second degree, and a vector field […]. The fields referred to are the torsion tensor S ij k , the tensor of non-metricity Q ij k , the metric g ij , and the tensor C ij k which, in unified field theory, was rarely used. It arose because Schouten introduced different linear connections for tangent vectors and linear forms. He defined the covariant derivative of a 1-form not by the connection L ij k in Equation 13 , but by.
In fact. Furthermore, on p. For such an extension an invariant fixing of the connection is needed, because a physical phenomenon can correspond only to an invariant expression. According to Schouten. In the following pages will be shown that this difficulty disappears when the more general supposition is made that the original deplacement is not necessarily symmetrical.
He then restricted the generality of his approach; in modern parlance, he did allow for vector torsion only:. On the same topic, Schouten wrote a paper with Friedman in Leningrad [ ]. He relied on the curvature, torsion and homothetic curvature 2-forms [ 32 ], Section III; cf.
The spreading of knowledge about properties of differential geometric objects like connection and curvature took time, however, even in Leningrad.
After an uninterrupted search during the past two years I now believe to have found the true solution. As mentioned above, until quantum theory was established on a firmer footing, it would have been impossible to tackle the problems of unification. Hence, by we already have two arguments however coarsely and briefly expressed for the necessity of a unification of quantum theory and general relativity or gravitation and quantum phenomena , both for consistency and for reasons of theoretical unity.
It is not yet quantum gravity in the sense of quantization that is being proposed. Indeed, nothing is being proposed at this stage; rather, it is left as a future project. Both Einstein and Eddington would soon diverge rather radically from the three motivations outlined earlier. Though Eddington stuck to the project of unification of quantum theory and general relativity, until his death, his approach departed from the reductionist deeper probing method he suggests in Paper 2.
Perhaps forced by the sheer distance of scales from known physics we see in the Planck units, Eddington began to employ a non-experimental methodology, culminating in his posthumously published Fundamental Theory Cambridge University Press, In this case the point is a singularity of both fields i. Jeffery believed that the problem could be evaded. I unfortunately cannot share your optimism regarding the solution to the quantum problem.
I believe that the theory of relativity does not bring us a step closer, at least in its current form. I am convinced that the two-body problem will not lead to a discrete manifold of paths but to a continuous one. However, here Einstein does not indicate this aspect of his thinking to Jeffery, and indeed the quote above looks largely negative as far as the entire project of getting quantum from relativity goes. In the next part, the two ingredient theories are worked out in more detail along exactly the lines of figuring out how generally relativistic principles and quantum principles can co-exist, at least in a formal-structural sense.
Bergmann, Peter G. Eddington, Arthur S. Einstein, Albert Gorelik, Gennady E. Ivanenko, Dmitri Kennefick, Daniel Kormos-Buchwald, Diana L. Holmes Princeton: Princeton University Press. Lodge, Oliver Schulmann, Robert, A. Sommerfeld, Arnold