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EINSTEIN, Albert (1879-1955). Autograph scientific manuscript signed ("A. Einstein," in pencil on verso of p.6), UNPUBLISHED, a working draft with extensive revisions, deletions and addition, in which Einstein explores the construction of a Unified Field Theory based on a Hermitean metric tensor field in a space of four complex, or equivalently eight real, dimensions. The experiment seeks to derive field equations "analogous to the gravitational equations of the General Theory of Relativity." No place, no date [circa 1941-45?].

A NEW APPROACH TO THE UNIFIED FIELD THEORY: SEEKING "FIELD EQUATIONSANALOGOUS TO THE GRAVITATIONAL EQUATIONS OF THE GENERAL THEORY OF RELATIVITY"

From roughly 1923 until the end of his life Einstein dedicated his work in theoretical physics to the study of the Unified Field Theory, seeking a single mathematical path defined on the four-dimensional space-time manifold from which to effect a unification of the gravitational and electromagnetic fields. By applying this structure to the simplest possible field equations, Einstein sought to derive a field theory applicable both to the gravitational field equations of the General Theory of Relativity and Maxwell's electromagnetic field equations. A Unified Field Theory, he hoped, would explain the existence and nature of elementary particles including the electron and would also express the quantum nature of radiation that he had first proposed in 1905. Despite the development of theories of quantum mechanics in the 1920s, Einstein continued to seek a Unified Field Theory that would offer an alternative explanation of quantum phenomena. In this investigation, Einstein considered and tested several different mathematical structures, one of which is embodied in this manuscript, that might furnish a basis for the Unified Field Theory.

Einstein began work on a class of field theories based on a Hermitean metric tensor field in 1941. Early work on this include item 2-135 in the Einstein Archive-- an unpublished manuscript entitled "Metrische Geometri komplexer Raume" ("Metric Geometry of Complex Spaces"), and Item 2-136, an unpublished manuscript entitled "Der metrische komplexe Raum" ("Metric Complex Space"), both of which appear to date from about 1941. Letters from a collaborator, Valentine Bargmann, also refer to this work (see Items 6-211, 6- 212, 6-213 and 6-215, 15 May 1941 and 22 July [1942]).

By August 1942, Einstein had made considerable progress, and he detailed his efforts in a letter to his lifelong friend and collaborator Michele Besso (1873-1955). Einstein expressed dissatisfaction with the quantum mechanical probabilistic treatment of microphysics as an ultimate theory of physical, and wrote: "What I am working on now, will appear rather crazy to you, and perhaps it is...I consider a space, the four coordinates of which x1...x4 are complex, so that it really is an 8-dimentional space. To each coordinate x1, therefore, there is associated a complex conjugate x1. Every vector A1 has these four complex components and the conjugate ones A1. The Riemannian metric is replaced by one of the form glk dxl dxk...This is supposed to be real, which requires that glk = gkl (Hermitean metric) The glk are analytic functions of the x1 and xlk .

The question is now quite analogous to the Riemannian case. What are the differential equations of second order that are to be postulated for glk. Instead of covariance with respect to arbitrary continuous transformations of real coordinates xl, here there enters (essentially) covariance with respect to transformations of the type

xxi= fl (x5)

x^xii = fl (x5)

The difficulty lies in the circumstance that, to start with, there are several systems of equations that satisfy these conditions. I have found, however, that this difficulty disappears if one attacks it correctly, and that the thing goes almost exactly as in the Riemannian case. The integration [of the equations], however, is difficult, and it will certainly not be ascertained quickly if this beautiful castle in the air has anything to do with the handiwork of the Lord. When I myself have attained any sort of well-founded conviction about this I will certainly inform you." (

The present manuscript probably dates from the same time as the letter to Besso, after Einstein had derived a unique set of field equations for the Hermitean metric tensor glk. Einstein worked alone on the theory presented here, as is indicated by his frequent use of the first person singular. But in one place (p.16), he acknowledges assistance from Wolfgang Pauli (the Nobel prize-winning theoretical physicist who spent World War II at the Institute for Advanced Studies at Princeton) "whose collaboration I enjoyed" in resolving a mathematical problem that had arisen. Einstein never published his ideas in the form indicated in this manuscript, but continued to work along this line and in 1945 published a description of a unified field theory based on a complex Hermitean metric that is a function of four

The manuscript pages fall into four separate groups:

I) A series of pages continuously numbered from (1) to (7). Page (1) is headed "Erste beide Satze wie im Manuskript." ("First two sentences as in the manuscript.") The text that follows is divided into the following sections:

"1. Komplexer Raum-Tensoren" ("Tensors in a Complex Space"), pages (1)-(2);

"2. Hermitische Metrik. Gruppe del' analytischen Transformationen." ("Hermitean Metric. Group of "Analytic Transformations."), pages (3)-(4);

"3. Tensorbildung durch Differentiation." ("Formation of Tensors by Differentiation.), pages (4)-(6);

"4. Die Feldgleichungen." ("The Field Equations."), pages (6)-(7). This draft ends with the determination of a set of field equations for the Hermitean metric tensor.

On pages (3)-(4), Einstein offers a clear statement of the goal of his research: "It is our goal to set up field equations for the g specialized in this way [Hermitean metric tensor] that are analogous to the gravitational equations of the general theory of relativity. If one decomposes the [complex] glk in the forms Slk + alk into real and imaginary components, then from (10) [the Hermitean condition on the metric] follows the symmetry of Slk and the antisymmetry of alk. These correspond in a certain sense to a decomposition of the field into a gravitational field and an electromagnetic field."

II) A series of sheets paginated from (13) to (18) followed by two sheets numbered (21) and (22). The similarity of paper, the numbering of equations, and the contents of the exposition, suggest that they form part of a single integral document with pages (19)-(20) not present. Einstein's text is divided into the following sections:

"6. Andere Fortsetzung fur die Parallel- Verschiebung" ("6. Another Continuation for Parallel Displacement"), pages (13)-(15);

"Zweiter Teil. 7. Erweiterung del' Gruppe." ("Second Part. 7. Enlargement of the Group."); pages (15)-(16);

"8. Zur Tensor-Theorie." ("On Tensor Theory."), pages (17)-(18);

"9. Del' metrische Tensor." ("The Metric Tensor."); pages (18)-[(20)], pages (19)-(20) nor present;

" 10. Absolute Differentiation." ("Absolute Differentiation."), pages (21)-(22); page (22) contains the crossed out heading for Section 11.

"

III) A single sheet, labelled (1), with heading; "1. Raum-Metrik und Transformations-Gruppe." ("Metric of the Space and Transformation Group.")

IV) Two pp., unpaginated, headed: "Abriss fur den 2. Teil." ("Outline of the Second Part").

This, one of the most extensive working manuscripts of Einstein to be offered since the 1996 sale at Christie's of the 1913-1914 Einstein-Besso working manuscript ($398,500), provides an extraordinary insight into Einstein's efforts to develop a unified field theory and testifies to his abiding interest in exploring multiple solutions to one of the thorniest quests undertaken in modern theoretical physics.

*4to (8½ x 11 in.), 19 pages, blue fountain-pen ink (2pp. in pencil) on paper watermarked "Whiting Mutual Bond Rag Content," irregularly paginated in four groups: pp.1-7, 13-18, 21-22, plus equations and text on versos of 6, 13 and 16 (the latter two pp. lined out). Tiny rust stain from paperclip,*otherwise in very fine condition. In German.A NEW APPROACH TO THE UNIFIED FIELD THEORY: SEEKING "FIELD EQUATIONSANALOGOUS TO THE GRAVITATIONAL EQUATIONS OF THE GENERAL THEORY OF RELATIVITY"

From roughly 1923 until the end of his life Einstein dedicated his work in theoretical physics to the study of the Unified Field Theory, seeking a single mathematical path defined on the four-dimensional space-time manifold from which to effect a unification of the gravitational and electromagnetic fields. By applying this structure to the simplest possible field equations, Einstein sought to derive a field theory applicable both to the gravitational field equations of the General Theory of Relativity and Maxwell's electromagnetic field equations. A Unified Field Theory, he hoped, would explain the existence and nature of elementary particles including the electron and would also express the quantum nature of radiation that he had first proposed in 1905. Despite the development of theories of quantum mechanics in the 1920s, Einstein continued to seek a Unified Field Theory that would offer an alternative explanation of quantum phenomena. In this investigation, Einstein considered and tested several different mathematical structures, one of which is embodied in this manuscript, that might furnish a basis for the Unified Field Theory.

Einstein began work on a class of field theories based on a Hermitean metric tensor field in 1941. Early work on this include item 2-135 in the Einstein Archive-- an unpublished manuscript entitled "Metrische Geometri komplexer Raume" ("Metric Geometry of Complex Spaces"), and Item 2-136, an unpublished manuscript entitled "Der metrische komplexe Raum" ("Metric Complex Space"), both of which appear to date from about 1941. Letters from a collaborator, Valentine Bargmann, also refer to this work (see Items 6-211, 6- 212, 6-213 and 6-215, 15 May 1941 and 22 July [1942]).

By August 1942, Einstein had made considerable progress, and he detailed his efforts in a letter to his lifelong friend and collaborator Michele Besso (1873-1955). Einstein expressed dissatisfaction with the quantum mechanical probabilistic treatment of microphysics as an ultimate theory of physical, and wrote: "What I am working on now, will appear rather crazy to you, and perhaps it is...I consider a space, the four coordinates of which x1...x4 are complex, so that it really is an 8-dimentional space. To each coordinate x1, therefore, there is associated a complex conjugate x1. Every vector A1 has these four complex components and the conjugate ones A1. The Riemannian metric is replaced by one of the form glk dxl dxk...This is supposed to be real, which requires that glk = gkl (Hermitean metric) The glk are analytic functions of the x1 and xlk .

The question is now quite analogous to the Riemannian case. What are the differential equations of second order that are to be postulated for glk. Instead of covariance with respect to arbitrary continuous transformations of real coordinates xl, here there enters (essentially) covariance with respect to transformations of the type

xxi= fl (x5)

x^xii = fl (x5)

The difficulty lies in the circumstance that, to start with, there are several systems of equations that satisfy these conditions. I have found, however, that this difficulty disappears if one attacks it correctly, and that the thing goes almost exactly as in the Riemannian case. The integration [of the equations], however, is difficult, and it will certainly not be ascertained quickly if this beautiful castle in the air has anything to do with the handiwork of the Lord. When I myself have attained any sort of well-founded conviction about this I will certainly inform you." (

*Michele Besso Albert Einstein Correspondance*, ed. P. Speziali, Paris, 1972, pp. 367-368).The present manuscript probably dates from the same time as the letter to Besso, after Einstein had derived a unique set of field equations for the Hermitean metric tensor glk. Einstein worked alone on the theory presented here, as is indicated by his frequent use of the first person singular. But in one place (p.16), he acknowledges assistance from Wolfgang Pauli (the Nobel prize-winning theoretical physicist who spent World War II at the Institute for Advanced Studies at Princeton) "whose collaboration I enjoyed" in resolving a mathematical problem that had arisen. Einstein never published his ideas in the form indicated in this manuscript, but continued to work along this line and in 1945 published a description of a unified field theory based on a complex Hermitean metric that is a function of four

*rea*l coordinates: "A Generalization of the Relativistic Theory of Gravitation,"*Annals of Mathematics*, vol. 46, pp.578-584).The manuscript pages fall into four separate groups:

I) A series of pages continuously numbered from (1) to (7). Page (1) is headed "Erste beide Satze wie im Manuskript." ("First two sentences as in the manuscript.") The text that follows is divided into the following sections:

"1. Komplexer Raum-Tensoren" ("Tensors in a Complex Space"), pages (1)-(2);

"2. Hermitische Metrik. Gruppe del' analytischen Transformationen." ("Hermitean Metric. Group of "Analytic Transformations."), pages (3)-(4);

"3. Tensorbildung durch Differentiation." ("Formation of Tensors by Differentiation.), pages (4)-(6);

"4. Die Feldgleichungen." ("The Field Equations."), pages (6)-(7). This draft ends with the determination of a set of field equations for the Hermitean metric tensor.

On pages (3)-(4), Einstein offers a clear statement of the goal of his research: "It is our goal to set up field equations for the g specialized in this way [Hermitean metric tensor] that are analogous to the gravitational equations of the general theory of relativity. If one decomposes the [complex] glk in the forms Slk + alk into real and imaginary components, then from (10) [the Hermitean condition on the metric] follows the symmetry of Slk and the antisymmetry of alk. These correspond in a certain sense to a decomposition of the field into a gravitational field and an electromagnetic field."

II) A series of sheets paginated from (13) to (18) followed by two sheets numbered (21) and (22). The similarity of paper, the numbering of equations, and the contents of the exposition, suggest that they form part of a single integral document with pages (19)-(20) not present. Einstein's text is divided into the following sections:

"6. Andere Fortsetzung fur die Parallel- Verschiebung" ("6. Another Continuation for Parallel Displacement"), pages (13)-(15);

"Zweiter Teil. 7. Erweiterung del' Gruppe." ("Second Part. 7. Enlargement of the Group."); pages (15)-(16);

"8. Zur Tensor-Theorie." ("On Tensor Theory."), pages (17)-(18);

"9. Del' metrische Tensor." ("The Metric Tensor."); pages (18)-[(20)], pages (19)-(20) nor present;

" 10. Absolute Differentiation." ("Absolute Differentiation."), pages (21)-(22); page (22) contains the crossed out heading for Section 11.

"

III) A single sheet, labelled (1), with heading; "1. Raum-Metrik und Transformations-Gruppe." ("Metric of the Space and Transformation Group.")

IV) Two pp., unpaginated, headed: "Abriss fur den 2. Teil." ("Outline of the Second Part").

This, one of the most extensive working manuscripts of Einstein to be offered since the 1996 sale at Christie's of the 1913-1914 Einstein-Besso working manuscript ($398,500), provides an extraordinary insight into Einstein's efforts to develop a unified field theory and testifies to his abiding interest in exploring multiple solutions to one of the thorniest quests undertaken in modern theoretical physics.