The migration-movements, which followed the revolution of and have been persisting ever since, have led to the development of different conceptions of identity in the Cuban diaspora. Questions that will take center stage during our seminar will be: What are the concepts of identity that can be found in the Cuban diaspora? Can language be understood as identity-constituting and identity-reproducing element? How are alternative identities constructed, how are generational and cultural conflicts expressed in Cuban-American Popular Culture? We will also have a closer look on recent political events that will shape U.
S-Cuban relations in the future. By the end of the semester, participants will have a broad understanding of the variety of concepts of Cuban identity in the diaspora. Further, participants will be able to critically evaluate different approaches based on theoretical input and interdisciplinary perspectives.
Due to the international setting of the class, participants will have experiences in working and learning in an intercultural environment. Active oral participation Presentation Essay, 2 pages Written exam 90 min. IDV Seminar Philosophie: Theories about territorial rights: Why may states control land and the people on it? Reichling; Schloss Ehrenhof West. External Relations Law of the European Union represents an area of the EU Law which is especially concerned with the legal aspects of the cooperation of the European Union with non-member States and with international organizations.
Required reading materials as well as additional sources will be provided electronically or during the lectures. The course aims at familiarizing students with the objectives and role of the EU institutions and its Member States in their external relations, also with the practice and case law from the European Court of Justice ECJ and academic literature in this field.
Throughout the course there will be a focus on: The course will be conducted through lectures, discussions, and seminars which will allow students to work in small groups on legal cases from practice. Next to the results of the final written exam, active participation during the lectures and seminars will also contribute to the overall grade for this course.
Public International Law Lecture, English. This course provides students with an understanding of the system of public international law, regulating relations between actors on the global stage. Sessions will take place on a weekly basis and consist of both lecture and discussion parts. Within the discussion part, current developments such as inter alia pending cases before the International Court of Justice and further contemporary topics will be discussed.
Mode of assessment for this course will be a research paper. In addition, oral participation will contribute to the final grade awarded for this course. After attending the lecture, exercises and tutorials students are able to: This course first outlines the basics of data and business process modelling based on wide-spread approaches such as entity relationship diagrams, event-driven process chains EPC , and business process model and notation BPMN. The remainder of the course then focuses on the use and purpose of integrated information systems across different functional areas in industrial companies.
Finally, basics of management support systems such as business intelligence systems are addressed. The lecturer will speak about the historical and social backgrounds of Japan as well as the current situation. Each topic will be prepared and presented by one participant or a group of participants in German.
The participants will discuss the topics in a global context. Examples of topics maybe as a gust lecture: Language letters, grammar, phonetic, interpersonal relations and language School system and governmental Curriculum Guideline after world war II Family and work Religion: Shintoismus from the past until today Habits at funerals and ideas about live and death The lecturer will speak about the historical and social backgrounds of Japan as well as the current situation. Shintoismus from the past until today Habits at funerals and ideas about live and death.
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Registration required for all incoming exchange students only. The exercise classes' goal is the repetition and expansion of the knowledge students acquire in the lecture. To reach that goal the exercise class will offer a mixture of additional knowledge, exercises, and an interactive element, which will improve the ability of knowledge exchange and self-dependent work in small groups. Fundamentals of Human Resource Management 6th revised edition.
McGraw Hill Higher Education. The lecture MAN Human Resource Management is part of the courses offered to bachelor students in business administration and law at the University of Mannheim. To get an idea of the coherences, problems, and solutions of human resource management as well as the tasks, operational areas, and instruments of management we offer a weekly lecture 1. The lecture includes the following topics on human resource management: Students gain a thorough overview on relevant questions and functions of marketing and learn basic as well as specific concepts of it.
The aim is to provide participants a comprehensive understanding of marketing concepts to apply them for identifying and solving questions related to marketing in business decision making processes. Moreover, students acquire the competence to critically reflect marketing decisions and to apply basic mathematical methods for analyzing and addressing relevant questions of marketing.
General basics Theoretical perspectives: Foundations of product policy Foundations of price policy Foundations of communication policy Foundations of distribution and sales policy Institutional perspective: Services marketing Business-to-business marketing International marketing. Acquisition and application of basic concepts, theories and methods of operations management.
Understanding of essential planning tasks of operations management Understanding of key trade-offs in operations management Ability to structure and model complex planning tasks Familiarity with common solution methods for planning tasks in operations management. CC Quantitative Methods. CC Quantitative Methoden. Role of operations management; fundamental planning tasks of operation management; planning methods; main features of production planning, transportation planning and inventory management. Algorithmenentwurf, Bewertung von vorgegeben Algorithmen Personale Kompetenz: Students acquire and apply basic concepts, theories and methods of operations management.
Peace and Violent Conflict Lecture, English. Political Parties, Parliaments and Legislative Speechmaking. Party Policy in Modern Democracies. Representative Government in Modern Europe. Cabinets and Coalition Bargaining: How Political Institutions Work. Policy Horizons and Parliamentary Government.
Empirical Political Research Lecture, English. Political Science Research Methods 6th ed. Research Methods in Political Science, 7th ed. 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. 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.
For such an extension an invariant fixing of the connection is needed, because a physical phenomenon can correspond only to an invariant expression. 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. After some manipulations, the variation with regard to the metric and to the connection led to the following equations:. The process of generalisation consists in abandoning assumptions of symmetry and in adopting a definition of covariant differentiation which is not the usual one, but which reduces to the usual one in case the connection is symmetric.
The two covariant derivatives introduced by J. Thomas then could reformulate Equation in the form. Instead of he only obtained. Toward the end of the paper Einstein discussed time-reversal; according to him, by it the sign of the magnetic field is changed, while the sign of the electric field vector is left unchanged.
As he wanted to obtain charge-symmetric solutions from his equations, Einstein now proposed to change the roles of the magnetic fields and the electric fields in the electromagnetic field tensor. He went on to say:. This is surely a magnificent possibility which likely corresponds to reality.
The question now is whether this field theory is consistent with the existence of quanta and atoms. In the macroscopic realm, I do not doubt its correctness. Yet, in the end, also this novel approach did not convince Einstein. Soon after the publication discussed, he found his argument concerning charge symmetric solutions not to be helpful. The link between the occurrence of solutions with both signs of the charge with time-symmetry of the field equations induced him to doubt, if only for a moment, whether the endeavour of unifying electricity and gravitation made sense at all:.
In this, electrodynamics is basically different from gravitation; therefore, the endeavour to melt electrodynamics with the law of gravitation into one unity, to me no longer seems to be justified. First , the attempts of all of us were directed to arrive, along the path taken by Weyl and Eddington or a similar one, at a theory melting into a formal unity the gravitational and electromagnetic fields; but by lasting failure I now have laboured to convince myself that truth cannot be approached along this path.
The new field equation was picked up by R. In the same spirit as the one of his paper, Einstein said good bye to his theory in a letter to Besso on Christmas in words similar to those in his letter in June:. Anyway, I now am convinced that, unfortunately, nothing can be made with the complex of ideas by Weyl-Eddington. I take as the best we have nowadays. But it appears doubtful whether there is room in them for the quanta. It does not allow for electrical masses free from singularities. Moreover, I cannot bring myself to gluing together two items as the l. Research on affine geometry as a frame for unified field theory was also carried on by mathematicians of the Princeton school.
During the period considered here, a few physicists followed the path of Eddington and Einstein. He showed that, in first approximation, he got what is wanted, i. Three months later, Infeld published a note in Comptes Rendus of the Parisian Academy in which he now presented the exact connection as. Thus, he is back at vector torsion treated before by Schouten [ ]. The Japanese physicist Hattori embarked on a metric-affine geometry derived purely from an asymmetric metrical tensor.
He defined an affine connection. The electromagnetic field was not identified with f ik by Hattori, but with the skew-symmetric part of the generalised Ricci tensor formed from. By introducing the tensor , he could write the generalised Ricci tensor as. F ikl is formed from F ik as f ikl from f ik. A field without electric current or charge density could not exist [ ].
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He then gave another example for a theory allowing the identification of the electromagnetic field tensor with the antisymmetric part of the Ricci tensor: The proper world geometry which ought to lead to a unified theory of gravitation and electricity can only be found by an investigation of its physical content. Infeld could as well have applied this admonishment to his own unified field theory discussed above. Thus, Straneo suggested a unified field theory with only vector torsion as Schouten had done 8 years earlier [ , ] without referring to him.
The field equations Straneo wrote down, i. Straneo kept the energy-momentum tensor of matter as an extraneous object including the electromagnetic field as well as the electric current vector. Straneo wrote further papers on the subject [ , ]. By this, he claimed to have made superfluous the five-vectors of Einstein and Mayer [ ]. This must be read in the sense that he could obtain the Einstein.
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Mayer equations from his formalism without introducing a connecting quantity leading from the space of 5-vectors to space-time [ ]. Einstein, in his papers, did not comment on the missing metric compatibility in his theory and its physical meaning. In this work a generalisation of the equation for metric compatibility, i.
The continuation of this research line will be presented in Part II of this article. Einstein wrote to his friend and colleague Paul Ehrenfest on 23 August Lorentz, 16 February On the next day 17 February , and ten days later Einstein was to give papers of his own in front of the Prussian Academy in which he pointed out the gauge-group, wrote down the geodesic equation, and derived exactly the Einstein. Maxwell equations — not just in first order as Kaluza had done [ 81 , 82 ]. He came too late: Klein had already shown the same before [ ].
Einstein himself acknowledged indirectly that his two notes in the report of the Berlin Academy did not contain any new material. In his second communication, he added a postscript:. Mandel brings to my attention that the results reported by me here are not new. The entire content can be found in the paper by O.
That Klein had published another important clarifying note in Nature , in which he closed the fifth dimension, seems to have escaped Einstein [ ]. Thus, the three of them had no chance to find out that Kaluza had made a mistake: Maxwell equations [ ], p. Mandel of Leningrad was not given credit by Einstein although he also had rediscovered by a different method some of O. From the geodesics in M 5 he derived the equations of motion of a charged point particle.
One of the two additional terms appearing besides the Lorentz force could be removed by a weakness assumption; as to the second, Mandel opinioned. Fock derived the general relativistic wave equation and the equations of motion of a charged point particle; the latter is identified with the null geodesics of M 5. A main motivation for Klein was to relate the fifth dimension with quantum physics. From a postulated five-dimensional wave equation.
By this, the reduction of five-dimensional equations as e. Klein had only the lowest term in the series. The 5th dimension is assumed to be a circle, topologically, and thus gets a finite linear scale: Beyond incredibly complicated field equations nothing much had been gained [ ]. As we have seen in Section 4. His two axioms were the cylinder condition and its sharpening, Equation He then discussed conformally invariant field equations, and tried to relate them to equations of wave mechanics [ ].
Presently, the different contributions of Kaluza and O. An early criticism of this unhistorical attitude has been voiced in [ ]. Another motivation is also put forward: Indices are raised and lowered with the metrics of V 5 or V 4 , respectively. A consequence then is. Both covariant derivatives are abbreviated by the same symbol A ; k.
The covariant derivative of tensors with both indices referring to V 5 and those referring to V 4 , is formed correspondingly. The autoparallels of V 5 lead to the exact equations of motion of a charged particle, not the geodesics of V 4. They also noted that a symmetric tensor F kl could have been interpreted as the second fundamental form, and the formalism would then be the same as local isometric embedding of V 4 into V 5.
It is related to the Riemannian curvature of V 4 by. From , by transvection with , the 5-curvature itself appears:. Two new quantities are introduced:. It turns out that. Also, in a lecture given on 14 October in the Physics Institute of the University of Wien, he still was proud of the 5-vector approach. However, following an idea half of which came from myself and half from my collaborator, Prof. Mayer, a startlingly simple construction became successful.
In this way, we succeeded to recognise the gravitational and electromagnetic fields as a logical unity. Electrical and mass-density are non-existent; here, splendour ends; perhaps this already belongs to the quantum problem, which up to now is unattainable from the point of view of field [theory] in the same way as relativity is from the point of view of quantum mechanics. The witty point is the introduction of 5-vectors in fourdimensional space, which are bound to space by a linear mechanism. Let a s be the 4-vector belonging to ; then such a relation obtains.
In the theory equations are meaningful which hold independently of the special relationship generated by. Infinitesimal transport of in fourdimensional space is defined, likewise the corresponding 5-curvature from which spring the field equations. In his report for the Macy-Foundation, which appeared in Science on the very same day in October , Einstein had to be more optimistic:.
It furnishes, however, clues to a natural development, from which we may anticipate further developments in this direction. In any event, the results thus far obtained represent a definite advance in knowledge of the structure of physical space. Veblen had already worked on projective geometry and projective connections for a couple of years [ , , ].
However, according to Pauli, Veblen and Hoffmann had spoiled the advantage of projective theory:. The five-dimensional space is just a mathematical device to represent the events points of space-time by these curves. Thus, Veblen and Hoffmann also gained the Klein. Gordon equation in curved space, i. Nevertheless, Hoffman remained optimistic:. In particular, we do not demand a relationship between electrical charge and a fifth coordinate; our theory is strictly four-dimensional.
Of the three basic assumptions of the previous paper, the second had to be given up. The expression in the middle of Equation is replaced by. The field equations were set up according to the method of the first paper; now the 5-curvature scalar was. It also turned out that with , i. In the last paragraph, the compatibility of the equations was proven, and at the end Cartan was acknowledged:. At about the same time as Einstein and Mayer wrote their second note, van Dantzig continued his work on projective geometry [ , , ]. Together with him, Schouten wrote a series of papers on projective geometry as the basis of a unified field theory [ , , , ] , which, according to Pauli, combine.
In this paper [ ], p. The formalism of Schouten and van Dantzig allows for taking the additional dimension to be timelike; in their physical applications the metric of spacetime is taken as a Lorentz metric; torsion is also included in their geometry. Pauli, with his student J. The authors pointed out that.
In a sequel to this publication, Pauli and Solomon corrected an error:. Then we discuss the form of the energy-momentum tensor and of the current vector in the theory of Einstein-Mayer. Michal and his co-author generalised the Einstein-Mayer 5-vector-formalism:. Robertson found a new way of applying distant parallelism: He studied groups of motion admitted by such spaces, e.
Cartan wrote a paper on the Einstein-Mayer theory as well [ 39 ], an article published only posthumously in which he showed that this could be interpreted as a five-dimensional flat geometry with torsion, in which space-time is embedded as a totally geodesic subspace. The contributions from the Levi-Civita connection and from contorsion in the curvature tensor cancel. In place of the metric, tetrads are introduced as the basic variables. As in Euclidean space, in the new geometry these 4-beins can be parallely translated to retain the same fixed directions everywhere.
In particular, the vanishing of the affine curvature tensor was given as a necessary and sufficient condition for the existence of D linearly independent fields of parallel vectors in a D -dimensional affine space [ ], p. However, when Einstein published his contributions in June [ 84 , 83 ], Cartan had to remind him that a paper of his introducing the concept of torsion had. Einstein had believed to have found the idea of distant parallelism by himself.
In this regard, Pais may be correct. Every researcher knows how an idea, heard or read someplace, can subconsciously work for years and then surface all of a sudden as his or her own new idea without the slightest remembrance as to where it came from. It seems that this happened also to Einstein. It is quite understandable that he did not remember what had happened six years earlier; perhaps, he had not even fully followed then what Cartan wanted to explain to him. In an investigation concerning spaces with simply transitive continuous groups, Eisenhart already in had found the connection for a manifold with distant parallelism given 3 years later by Einstein [ ].
Einstein, of course, could not have been expected to react to these and other purely mathematical papers by Cartan and Schouten, focussed on group manifolds as spaces with torsion and vanishing curvature [ 41 , 34 ], pp. No physical application had been envisaged by these two mathematicians. Nevertheless, this story of distant parallelism raises the question of whether Einstein kept up on mathematical developments himself, or whether, at the least, he demanded of his assistants to read the mathematical literature. In the area of unified field theory including spinor theory, Einstein just loved to do the mathematics himself, irrespective of whether others had done it before — and done so even better cf.
Anyhow, in his response Einstein to Cartan on 10 May , [ 50 ], p. After Cartan had sent his historical review to Einstein on 24 May , the latter answered three months later:. The publication should appear in the Mathematische Annalen because, at present, only the mathematical implications are explored and not their applications to physics.
Interestingly, he permitted himself to interpet the physical meaning of geometrical structures:. Cartan that the treatment of continua of the species which is of import here, is not really new. Nevertheless, I still am far from being able to claim that the derived equations have a physical meaning. The reason is that I could not derive the equations of motion for the corpuscles.
The split, in first approximation, of the tetrad field h ab according to lead to homogeneous wave equations and divergence relations for both the symmetric and the antisymmetric part identified as metric and electromagnetic field tensors, respectively. Einstein in really seemed to have believed that he was on a good track because, in this and the following year, he published at least 9 articles on distant parallelism and unified field theory before switching off his interest. The press did its best to spread the word: On 2 February , in its column News and Views , the respected British science journal Nature reported:.
Einstein has been about to publish the results of a protracted investigation into the possibility of generalising the theory of relativity so as to include the phenomena of electromagnetism. It is now announced that he has submitted to the Prussian Academy of Sciences a short paper in which the laws of gravitation and of electromagnetism are expressed in a single statement. Nature then went on to quote from an interview of Einstein of 26 January in a newspaper, the Daily Chronicle. According to the newspaper, among other statements Einstein made, in his wonderful language, was the following:.
A thousand copies of this paper had been sold within 3 days, so the presiding secretary of the Academy ordered the printing of a second thousand. Normally, only a hundred copies were printed [ ], Dokument Nr. This article then became reprinted in March by the British astronomy journal The Observatory [ 86 ]. In it, Einstein first gave a historical sketch leading up to the introduction of relativity theory, and then described the method that guided him to the new theory of distant parallelism.
In fact, the only formulas appearing are the line elements for two-dimensional Riemannian and Euclidean space. At the end, by one figure, Einstein tried to convey to the reader what consequence a Euclidean geometry with torsion would have — without using that name. His closing sentences are:. The answer to this question which I have attempted to give in a new paper yields unitary field laws for gravitation and electromagnetism. A few months later in that year, again in Nature , the mathematician H.
He was a bit more explicit than Einstein in his article for the educated general reader. However, he was careful to end it with a warning:. It may succeed in predicting some interaction between gravitation and electromagnetism which can be confirmed by observation.
If this were true it would be impossible to calculate the consequences.
Indeed, there was a lot of work to do, only in part because Einstein, from one paper to the next, had changed his field equations. In his first note [ 84 ], dynamics was absent; Einstein made geometrical considerations his main theme: As we have seen before, the components of the metric tensor are defined by. Of course, in space-time with a Lorentz metric, the 4-bein-transformations do form the proper Lorentz group. If parallel transport of a tangent vector A is defined as usual by , then the connection components turn out to be.
Also, the metric is covariantly constant. The Riemannian curvature tensor calculated from Equation turns out to vanish. As Einstein noted, by g ij from Equation also the usual Riemannian connection may be formed. Moreover, is a tensor that could be used for building invariants.
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In principle, distant parallelism is a particular bi-connection theory. The connection does not play a role in the following cf. From Equation , obviously the torsion tensor follows cf. He indicated how a Lagrangian could be built and the 16 field equations for the field variables h lj obtained. In his second note [ 83 ], Einstein departed from the Lagrangian , i. However, as he added in a footnote, pure gravitation could have been characterised by as well. To do so he replaced by and introduced. In a postscript, Einstein noted that he could have obtained similar results by using the second scalar invariant of his previous note, and that there was a certain indeterminacy as to the choice of the Lagrangian.
This shows clearly that the ambiguity in the choice of a Lagrangian had bothered Einstein. Thus, in his third note, he looked for a more reassuring way of deriving field equations [ 88 ]. He left aside the Hamiltonian principle and started from identities for the torsion tensor, following from the vanishing of the curvature tensor. He thus arrived at the identity given by Equation 29 , i. By defining , and contracting equation , Einstein obtained another identity:. For the proof, he used the formula for the covariant vector density given in Equation 16 , which, for the divergence, reduces to.
With this first approximation as a hint, Einstein, after some manipulations, postulated the 20 exact field equations:. Einstein seems to have sensed that the average reader might be able to follow his path to the postulated field equations only with difficulty. Therefore, in a postscript, he tried to clear up his motivation:. In the meantime, however, he had found a Lagrangian such that the compatibility-problem disappeared.
He restricted the many constructive possibilities for by asking for a Lagrangian containing torsion at most quadratically. His Lagrangian is a particular linear combination of the three possible scalar densities, as follows:. Stodola, Einstein summed up what he had reached.
He presented it as an introduction suited for anyone who knew general relativity. It is here that he first mentioned Equations and Most importantly, he gave a new set of field equations not derived from a variational principle; they are. As Cartan remarked, Equation expresses conservation of torsion under parallel transport:. Einstein, it is natural to call a universe homogeneous if the torsion vectors that are associated to two parallel surface elements are parallel themselves; this means that parallel transport conserves torsion. From Equation with the help of Equation , , Einstein wrote down two more identities.
One of them he had obtained from Cartan:. Here, F k is introduced by. Einstein then showed that. The changes in his approach Einstein continuously made, must have been hard on those who tried to follow him in their scientific work. One of them, Zaycoff , tried to make the best out of them:. Einstein [ 89 ] , following investigations by E. Cartan [ 35 ] , has considerably modified his teleparallelism theory such that former shortcomings connected only to the physical identifications vanish by themselves. They were published in as the first article in the new journal of this institute [ 92 ].
On 23 pages he clearly and leisurely outlined his theory of distant parallelism and the progress he had made. From a purely mathematical point of view they were studied previously. Cartan was so amiable as to write a note for the Mathematische Annalen exposing the various phases in the formal development of these concepts. Later in the paper, he comes closer to the point:. Some of the material in the paper overlaps with results from other publications [ 85 , 90 , 93 ].
Hence 7 identities should exist, four of which Einstein had found previously. The field equations are the same as in [ 89 ]; the proof of their compatibility takes up, in a slightly modified form, the one communicated by Einstein to Cartan in a letter of 18 December [ 92 ], p. It is reproduced also in [ 90 ]. Then Einstein presented the same field equations as in his paper in Annalen der Mathematik , which he demanded to be.
Sixteen field equations were needed which, due to covariance, induced four identities. The higher the number of equations and consequently also the number of identities among them , the more precise and stronger than mere determinism is the content; accordingly, the theory is the more valuable, if it is also consistent with the empirical facts. In linear approximation, i. From any tensor with an antisymmetric pair of indices a vector with vanishing divergence can be derived [ 93 ].
In order to test the field equations by exhibiting an exact solution, a simple case would be to take a spherically symmetric, asymptotically Minkowskian 4-bein. This is what Einstein and Mayer did, except with the additional assumption of space-reflection symmetry [ ]. Then the 4-bein contains three arbitrary functions of one parameter s:.
As an exact solution of the field equations , , Einstein and Mayer obtained and. Einstein and Mayer do not take this physically unacceptable situation as an argument against the theory, because the equations of motion for such singularities could not be derived from the field equations as in general relativity. Again, the continuing wish to describe elementary particles by singularity-free exact solutions is stressed. Two days before the paper by Einstein and Mayer became published by the Berlin Academy, Einstein wrote to his friend Solovine:.
Cartan has already worked with it. I myself work with a mathematician S. Mayer from Vienna , a marvelous chap […]. The mentioning of Cartan resulted from the intensive correspondence of both scientists between December and February About a dozen letters were exchanged which, sometimes, contained long calculations [ 50 ] cf. In an address given at the University of Nottingham, England, on 6 June , Einstein also must have commented on the exact solutions found and on his program concerning the elementary particles.
He does not, however, regard this as sufficient, though those laws may come out. He still wants to have the motions of ordinary particles to come out quite naturally. In addition to the assumptions 1 , 2 , 3 for allowable field equations given above, further restrictions were made:.
After inserting Equation into Equation , Einstein and Mayer reduced the problem to the determination of 10 constants by 20 algebraic equations by a lengthy calculation. In the end, four different types of compatible field equations for the teleparallelism theory remained:. The remaining two types are denoted in the paper by […]. With no further restraining principles at hand, this ambiguity in the choice of field equations must have convinced Einstein that the theory of distant parallelism could no longer be upheld as a good candidate for the unified field theory he was looking for, irrespective of the possible physical content.
Once again, he dropped the subject and moved on to the next. It seems that this structure has nothing to do with the true character of space […]. However, the correspondence on the subject came to an end in May with a last letter by Cartan. But this aim seems to be in reach only if a direct physical interpretation of the operation of transport, even of the immediate field quantities, is given up. From the geometrical point of view, such a path [of approach] must seem very unsatisfactory; its justifications will only be reached if the mentioned link does encompass more physical facts than have been brought into it for building it up.
After having explained the theory and having pointed out the differences to his own affine unified field theory of , he confessed:. First, my mathematical intuition a priori resists to accept such an artificial geometry; I have difficulties to understand the might who has frozen into rigid togetherness the local frames in different events in their twisted positions. Two weighty physical arguments join in […] only by this loosening [of the relationship between the local frames] the existing gauge-invariance becomes intelligible.
Second, the possibility to rotate the frames independently, in the different events, […] is equivalent to the s y m m e t r y o f t h e e n e r g y - m o m e n t u m t e n s o r, or to the validity of the conservation law for angular momentum. As usual, Pauli was less than enthusiastic; he expressed his discontent in a letter to Hermann Weyl of 26 August Now the hour of revenge has come for you, now Einstein has made the blunder of distant parallelism which is nothing but mathematics unrelated to physics, now you may scold [him].
As long as no empirical basis exists, beliefs, hopes, expectations, and rationally guided guesses abound. During the Easter holidays I have visited Einstein in Berlin and found his opinion on modern quantum theory reactionary.
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Einstein had sent a further exposition of his new theory to the Mathematische Annalen in August When he received its proof sheets from Einstein, Pauli had no reservations to criticise him directly and bluntly:. Unlike what I told you in spring, from the point of view of quantum theory, now an argument in favour of distant parallelism can no longer be put forward […]. It just remains […] to congratulate you or should I rather say condole you?
Also, I am not so naive as to believe that you would change your opinion because of whatever criticism. But I would bet with you that, at the latest after one year, you will have given up the entire distant parallelism in the same way as you have given up the affine theory earlier. And, I do not wish to provoke you to contradict me by continuing this letter, because I do not want to delay the approach of this natural end of the theory of distant parallelism. Only someone who is certain of seeing through the unity of natural forces in the right way ought to write in this way.
Before the mathematical consequences have not been thought through properly, is not at all justified to make a negative judgement. Before he had written to Einstein, Pauli, with lesser reservations, complained vis-a-vis Jordan:. With such rubbish he may impress only American journalists, not even American physicists, not to speak of European physicists.
The question of the compatibility of the field equations played a very important role because Einstein hoped to gain, eventually, the quantum laws from the extra equations cf. That Pauli had been right except for the time span envisaged by him was expressly admitted by Einstein when he had given up his unified field theory based on distant parallelism in see letter of Einstein to Pauli on 22 January ; cf. In it, Lanczos cautiously offers some criticism after having made enough bows before Einstein:. Not as a criticism but only as an impression do we point out why the new field theory does not house the same degree of conviction, nor the amount of inner consistency and suggestive necessity in which the former theory excelled.
His never-ending gift for invention, his persistent energy in the pursuit of a fixed aim in recent years surprise us with, on the average, one such theory per year. Hence, […] one could cry out: He varied g ik and R ik independently [ ]. For Lanczos see J. However, he had kept the metric g ik introduced into the formalism by. In it he described the mathematical formalism of distant parallelism theory, gave the identity 42 , and calculated the new curvature scalar in terms of the Ricci scalar and of torsion.
He then took a more general Lagrangian than Einstein and obtained the variational derivatives in linear and, in a simple example, also in second approximation. In his presentation, he used both the teleparallel and the Levi-Civita connections. An exact, complicated wave equation followed:. In linear approximation, the Einstein vacuum and the vacuum Maxwell equations are obtained, supplemented by the homogeneous wave equation for a vector field [ ]. He also defended Einstein against critical remarks by Eddington [ 62 ] and Schouten [ ], although Schouten, in his paper, had mentioned neither Einstein nor his teleparallelism theory, but only gave a geometrical interpretation of the torsion vector in a geometry with semi-symmetric connection.
Einstein built a plane world which is no longer waste like the Euclidean space-time-world of H. Minkowski, but, on the contrary, contains in it all that we usually call physical reality. A conference on theoretical physics at the Ukrainian Physical-Technical Institute in Charkow in May , brought together many German and Russian physicists. Unified field theory, quantum mechanics, and the new quantum field theory were all discussed.
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Vary with regard to the 16 beinquantities but consider only the 10 metrical components as relevant. According to Grommer the anti-symmetry of P is needed, because its contraction leads to the electromagnetic 4-potential and because the symmetric part can be expressed by the antisymmetric part and the metrical tensor. Levi-Civita also had sent a paper on distant parallelism to Einstein, who had it appear in the reports of the Berlin Academy [ ]. Levi-Civita introduced a set of four congruences of curves that intersect each other at right angles, called their tangents and used Equation in the form:.
He used the time until the printing was done to give a short preview of his paper in Nature [ ]. In order to ease a comparison between both theories, we may bring together here the notations of R i c c i and L e v i - C i v i t a […] with those of Einstein. Einstein had sent him the corrected proof sheets of his fourth paper [ 85 ]. The basic idea was to consider the points of M 4 as equivalent to the ensemble of congruences with tangent vector X 5 i in M 5 with cylindricity condition werden. The space-time interval is defined as the distance of two lines of the congruence on.
Mandel did not identify the torsion vector with the electromagnetic 4-potential, but introduced the covariant derivative , where the tensor is skew-symmetric. We may look at this paper also as a forerunner of some sort to the Einstein. Mayer 5-vector formalism cf. Some were more interested in the geometrical foundations, in exact solutions to the field equations, or in the variational principle.
One of those hunting for exact solutions was G. It is shown that the new equations are satisfied to the first order but not exactly. He then goes on to find a rigourous solution and obtains the metric and the 4-potential [ ]. They found that these field equations did not have a spherically symmetric solution corresponding to a charged point particle at rest.
The corresponding solution for the uncharged particle was the same as in general relativity, i. Tamm and Leontowitch therefore guessed that a charged point particle at rest would lead to an axially-symmetric solution and pointed to the spin for support of this hypothesis [ , ]. In his paper in July , the physicist Zaycoff had some details:. In the same paper, he states: In this case he has been able to obtain results checking the observed perihelion of mercury. The latter remark refers to a constant query Pauli had about what would happen, within unified field theory, to the gravitational effects in the planetary system, described so well by general relativity.
In the second of his two brief notes, Salkover succeeded in gaining the most general, spherically symmetric solution [ , ]. This is admitted by the authors in their second paper, in which they present a new calculation. Vallarta also wrote a paper by himself [ ], p. The purpose of this paper is to investigate, for the same case, the nature of the gravitational field obtained from the field equations suggested by Einstein in his first paper [ 88 ].
In Princeton, people did not sleep either. In and , T. Thomas wrote a series of six papers on distant parallelism and unified field theory. Thomas described the contents of his first paper as follows:. This looks as if he had introduced four vector potentials for the electromagnetic field, and this, in fact, T. The gravitational potentials are still g ik. In his next note, T.
Thomas changed his field equations on the grounds that he wanted them to give a conservation law. From the point of view of our previous notes this fact has its interpretation in the statement that the world will be pseudo-Euclidean only in the absence of electric and magnetic forces. This means that gravitational and electromagnetic phenomena must be intimately related since the existence of gravitation becomes dependent on the electromagnetic field.
Thus we secure a real physical unification of gravitation and electricity in the sense that these concepts become but different manifestations of the same fundamental entity — provided, of course, that the theory shows itself to be tenable as a theory in agreement with experience. In his three further installments, T. Thomas moved away from unified field theory to the discussion of mathematical details of the theory he had advanced [ , , ].
Unhindered by constraints from physical experience, mathematicians try to play with possibilities. In the framework of a purely affine theory he obtained a necessary and sufficient condition for this geometry,. The resulting connection is given by. Schouten and van Dantzig also used a geometry built on complex numbers, and on Hermitian forms:. At first, the possibility of gaining hold on the paths of elementary particles — described as singular worldlines of point particles — was central. But somehow, for Einstein, discretisation and quantisation must have been too close to bother about a fundamental constant.
Then, after the richer constructive possibilities e. It seems that Einstein, during his visit to Paris in November , had talked to Cartan about his problem of finding the right field equations and proving their compatibility. Starting in December of and extending over the next year, an intensive correspondence on this subject was carried on by both men [ 50 ]. On 3 December , Cartan sent Einstein a letter of five pages with a mathematical note of 12 pages appended.
There are other possibilities giving rise to richer geometrical schemes while remaining deterministic. First, one can take a system of 15 equations […]. Finally, maybe there are also solutions with 16 equations; but the study of this case leads to calculations as complicated as in the case of 22 equations, and I was not fortunate enough to come across a possible system […]. As the further correspondence shows, he had difficulties in following Cartan:. I beg you to send me those of your papers from which I can properly study the theory.
It would be a task of its own to closely study this correspondence; in our context, it suffices to note that Cartan wrote a special note. Now, everything is clear to me. Previously, my assistant Prof. This probably will restrict the free choice of solutions in a region in a far-reaching way — more strongly than the restrictions corresponding to your degrees of determination. In this section, we loosely collect some of these approaches.
The mathematicians Struik and Wiener found the task of an amalgamation of relativity and quantum theory wave mechanics attractive:. A further example for the new program is given by J. Whittaker at the University of Edinburgh [ ] who wished to introduce the wave function via the matter terms:.
The object of the present paper is to find these equations […]. Zaycoff, from the point of view of distant parallelism, found the following objection to unified field theory as the only valid one:. By the work of Dirac, wavemechanics has reached an independent status; the only attempt to bring together this new group of phenomena with the other two is J. Whittaker had expressed himself more clearly:.
These are grouped together to form two four-vectors and satisfy wave equations of the second order. Whittaker also had written down a variational principle by which the gravitational and electromagnetic field equations were also gained. However, as the terms for the various fields were just added up in his Lagrangian, the theory would not have qualified as a genuine unified field theory in the spirit of Einstein. What fancy, if only shortlived, flowers sprang from the mixing of geometry and wave mechanics is shown by the example of H.
Although, two years later, Jehle withdrew his claims concerning elementary particles, he continued to apply. How to combine them with the vectors and tensors appearing in electromagnetic and gravitational theories? As the spinor representation is the simplest representation of the Lorentz group, everything may be played back to spin space. At the time, this was being done in different ways, in part by the use of number fields with which physicists were unacquainted such as quaternions and sedenions cf.
Schouten [ ]. Others, such as Einstein and Mayer, liked vectors better and introduced so-called semi-vectors. Some less experienced, as e. Temple, even claimed that a tensorial theory was necessary to retain it relativistically:. It is contended here that therefore his theory cannot be upheld without abandoning the theory of relativity. While this story about geometrizing wave mechanics might not be a genuine part of unified field theory at the time, it seems interesting to follow it as a last attempt for binding together classical field theory and quantum physics. Some of the motivation for these papers came from formal considerations, i.
His approach for embedding wave mechanics into a Maxwell-like was continued in further papers, in part in collaboration with J. Fisher; to them it appeared. His conclusion sounds a bit strange:. In this context, another unorthodox suggestion was put forward by A. Anderson who saw matter and radiation as two phases of the same substrate:. Electrons and protons cannot be distinguished from quanta of light, gas pressure not from radiation pressure.
Anderson somehow sensed that charge conservation was in his way; he meddled through by either assuming neutral matter, i. One of the German theorists trying to keep up with wave mechanics was Gustav Mie. In the same year in which Einstein published his theory of distant parallelism, Dirac presented his relativistic, spinorial wave equation for the electron with spin.
This event gave new hope to those trying to include the electron field into a unified field theory; it induced a flood of papers in such that this year became the zenith for publications on unified field theory. Although, as we noted in Section 6. This is a very sketchy outline with a focus on the relationship to unified field theories. An interesting study into the details of the introduction of local spinor structures by Weyl and Fock and of the early history of the general relativistic Dirac equation was given recently by Scholz [ ].
For some time, the new concept of spinorial wave function stayed unfamiliar to many physicists deeply entrenched in the customary tensorial formulation of their equations. Some early nomenclature reflects this unfamiliarity with spinors. For the 4-component spinors or Dirac-spinors cf. Ehrenfest, in , still complained:.
In , three publications of the mathematician Veblen in Princeton on spinors added to the development. Veblen imbedded spinors into his projective geometry [ ]:. There are several ways of embodying this invariant theory in a formal calculus. The one which is here employed has its antecedents chiefly in the work ofWeyl, van derWaerden, Fock, and Schouten. It differs from the calculus arrived at by Schouten chiefly in the treatment of gauge invariance, Schouten in collaboration with van Dantzig having preferred to rewrite the projective relativity in a formalism obtainable from the original one by a sort of coordinate transformation, whereas I think the original form fits in better with the classical notations of relativity theory.
These spinors have been recognised by several students Pauli and Solomon, Fock of the subject but their role has probably not been fully understood since it has quite recently been thought necessary to give special proofs of invariance. The transformation law for spinors is the same as before:. The transformation given above carries the system of lines into the other.
After Tetrode and Wigner, whose contributions were mentioned in Section 6. He gave up his original idea of coupling electromagnetism to gravitation and transferred it to the coupling of the electromagnetic field to the matter electron- field: Actually, Weyl had expressed the change in his outlook, so important for the idea of gauge-symmetry in modern physics [ ], pp.
We have noted before his refutation of distant parallelism cf. He partly agreed with what Einstein imagined:. For many years, Weyl had given the statistical approach in the formulation of physical laws an important role. He therefore could adapt easily to the Born-Jordan-Heisenberg statistical interpretation of the quantum state.
For Weyl and statistics, cf. Up to now, quantum mechanics has not found its place in this geometrical picture; attempts in this direction Klein, Fock were unsuccessful. Only after Dirac had constructed his equations for the electron, the ground seems to have been prepared for further work in this direction. Thus is interpreted in the sense of Weyl:. Another note and extended presentations in both a French and a German physics journal by Fock alone followed suit [ , , ]. In the first paper Fock defined an asymmetric matter tensor for the spinor field,.
The covariant derivative then is. The divergence of the complex energy-momentum tensor satisfies. In this point, our theory, developed independently, agrees with the new theory by H. In this regard he found himself in accord with Weyl, whose approach to the Dirac equation he nevertheless criticised:. Nevertheless, it seems to us that the theory suggested by Weyl for solving this problem is open to grave objections; a criticism of this theory is given in our article.
His mass term contained a square root, i. As he remarked, the chances for this were minimal, however [ , ]. In two papers, Zaycoff of Sofia presented a unified field theory of gravitation, electromagnetism and the Dirac field for which he left behind the framework of a theory with distant parallelism used by him in other papers. By varying his Lagrangian with respect to the 4-beins, the electromagnetic potential, the Dirac wave function and its complex-conjugate, he obtained the 20 field equations for gravitation of second order in the 4-bein variables, assuming the role of the gravitational potentials and the electromagnetic field of second order in the 4-potential , and 8 equations of first order in the Dirac wave function and the electromagnetic 4-potential, corresponding to the generalised Dirac equation and its complex conjugate [ , ].
This means that he considered the electron as extended. At this occasion, he fought with himself about the admissibility of the Kaluza-Klein approach:. Rumer, the author et al. No doubt, there are weighty reasons for such a seemingly paradoxical view. A multi-dimensional causality cannot be understood as long as we are unable to give the extra dimensions an intuitive meaning. In the paper, Zaycoff introduced a six-dimensional manifold with local coordinates x 0 , … , x 5 where x 0 , x 5 belong to the additional dimensions. He wrote two papers, one concerned with the four-dimensional and a second one with the five-dimensional approach [ , ].
For him an important conclusion is that. We reproduce a remark from his publication [ ]:. Most authors introduce an orthogonal frame of axes at every event, and, relative to it, numerically specialised Dirac-matrices. To be sure, it is much too small by many powers of ten in order to replace, say, the term on the r. Yet it appears important that in the generalised theory a term is encountered at all which is equivalent to the enigmatic mass term. These additional terms do depend in an essential way on the choice of the orthogonal tetrad in the space-time manifold: The last, erroneous, sentence must have made Pauli irate.
Beware of the paper by Levi-Civita: Everybody should be kept from reading this paper, or from even trying to understand it. Moreover, all articles referred to on p. Pauli really must have been enraged: One of the essential features of quantum mechanics, the non-commutativity of conjugated observables like position and momentum, nowhere entered the approaches aiming at a geometrization of wave mechanics. Einstein was one of those clinging to the picture of the wave function as a real phenomenon in space-time.
It should have become a 4-page publication in the Sitzungsberichte. As he wrote to Max Born:. Will appear soon in Sitz. However, he quickly must have found a flaw in his argumentation: He telephoned to stop the printing after less than a page had been typeset. This did not happen; thus we know of his failed attempt, and we can read how his line of thought began [ ], pp. Thus, in February , Wiener and Vallarta stressed that. That the micro-mechanical world of the electron is Minkowskian is shown by the theory of Dirac, in which the electron spin appears as a consequence of the fact that the world of the electron is not Euclidean, but Minkowskian.
The correction of this misjudgement of Wiener and Vallarta by Fock and Ivanenko began only one month later [ ], and was complete in the summer of [ , , , ]. With the fifth dimension, Kaluza and Klein had connected electrical charge , Fock the electromagnetic potential , and London the spin of the electron [ ]. Gonseth and Juvet, in the first of four consecutive notes submitted in August [ , , , ] stated:. Thus, we will have a frame in which to take the gravitational and electromagnetic laws, and in which it will be possible also for quantum theory to be included.
Their further comment is:. Obviously, this artifice will be needed if some phenomenon would force the physicists to believe in a variability of the [electric] charge. Interestingly, a couple of months later, O. However, as he remarked, his hopes had been shattered [ ].