The Einstein Challenge for Differential Geometry - Weyl's Gauge Geometry

2. Weyl’s gauge geometry Weyl’s approach: ’localizing’ the automorphisms of SR – Weyl considered the similarities as the automorphisms of Euclidean geometry: No natural unit of length given. In 1918 he transferred this idea to SR and Minkowski space thus W := R4 o (R+ × SO(1, 3)) the automorphisms of SR. – In 1921 (4th edition RZM and sep. publication) he gave a physically founded structural description of the automorphisms of SR: I I I Inertial trajectories of SR specify a line structure, and a mathematical image (“Bildraum”) such that these lines are straight, up to projective transformations. Light propagation specifies an infinite plane E∞ by a family of “parallel” cones which are projections of a conic section in E∞ . Both together lead to the restricted conformal group of Minkowski space W as the automorphisms of SR. – Consequence: “To distinguish ‘normal’ co-ordinate systems among all others in the special theory of relativity, . . . we may dispense with not only rigid bodies but also with clocks.” Weyl’s approach: ’localizing’ the automorphisms of SR – Weyl considered the similarities as the automorphisms of Euclidean geometry: No natural unit of length given. In 1918 he transferred this idea to SR and Minkowski space thus W := R4 o (R+ × SO(1, 3)) the automorphisms of SR. – In 1921 (4th edition RZM and sep. publication) he gave a physically founded structural description of the automorphisms of SR: I I I Inertial trajectories of SR specify a line structure, and a mathematical image (“Bildraum”) such that these lines are straight, up to projective transformations. Light propagation specifies an infinite plane E∞ by a family of “parallel” cones which are projections of a conic section in E∞ . Both together lead to the restricted conformal group of Minkowski space W as the automorphisms of SR. – Consequence: “To distinguish ‘normal’ co-ordinate systems among all others in the special theory of relativity, . . . we may dispense with not only rigid bodies but also with clocks.” Weyl’s approach: ’localizing’ the automorphisms of SR – Weyl considered the similarities as the automorphisms of Euclidean geometry: No natural unit of length given. In 1918 he transferred this idea to SR and Minkowski space thus W := R4 o (R+ × SO(1, 3)) the automorphisms of SR. – In 1921 (4th edition RZM and sep. publication) he gave a physically founded structural description of the automorphisms of SR: I I I Inertial trajectories of SR specify a line structure, and a mathematical image (“Bildraum”) such that these lines are straight, up to projective transformations. Light propagation specifies an infinite plane E∞ by a family of “parallel” cones which are projections of a conic section in E∞ . Both together lead to the restricted conformal group of Minkowski space W as the automorphisms of SR. – Consequence: “To distinguish ‘normal’ co-ordinate systems among all others in the special theory of relativity, . . . we may dispense with not only rigid bodies but also with clocks.” Weyl’s approach: ’localizing’ the automorphisms of SR – Weyl considered the similarities as the automorphisms of Euclidean geometry: No natural unit of length given. In 1918 he transferred this idea to SR and Minkowski space thus W := R4 o (R+ × SO(1, 3)) the automorphisms of SR. – In 1921 (4th edition RZM and sep. publication) he gave a physically founded structural description of the automorphisms of SR: I I I Inertial trajectories of SR specify a line structure, and a mathematical image (“Bildraum”) such that these lines are straight, up to projective transformations. Light propagation specifies an infinite plane E∞ by a family of “parallel” cones which are projections of a conic section in E∞ . Both together lead to the restricted conformal group of Minkowski space W as the automorphisms of SR. – Consequence: “To distinguish ‘normal’ co-ordinate systems among all others in the special theory of relativity, . . . we may dispense with not only rigid bodies but also with clocks.” Weyl’s approach: ’localizing’ the automorphisms of SR – Weyl considered the similarities as the automorphisms of Euclidean geometry: No natural unit of length given. In 1918 he transferred this idea to SR and Minkowski space thus W := R4 o (R+ × SO(1, 3)) the automorphisms of SR. – In 1921 (4th edition RZM and sep. publication) he gave a physically founded structural description of the automorphisms of SR: I I I Inertial trajectories of SR specify a line structure, and a mathematical image (“Bildraum”) such that these lines are straight, up to projective transformations. Light propagation specifies an infinite plane E∞ by a family of “parallel” cones which are projections of a conic section in E∞ . Both together lead to the restricted conformal group of Minkowski space W as the automorphisms of SR. – Consequence: “To distinguish ‘normal’ co-ordinate systems among all others in the special theory of relativity, . . . we may dispense with not only rigid bodies but also with clocks.” Anhang I. (Zu Seite 161 und 207.) Um in der speziellen Relativitätstheorie die •normalen• Koordinatensysteme vor allen andern auszuzeichnen, in der allgemeinen die metrische Fundamentalform zu bestimmen, kann man nicht bloß der starren Körper, sondern anch der Uhren entraten. In der speziellen Relativitätstheorie wird durch die Forderung, daß bei der den Koordinaten entsprechenden Abbildung eines Weltstücks auf den vierdimensionalen Euklidischen Bildraum mit den Cartesischen Koordinaten x; die Weltlinien der kräftefrei sich bewegenden Massenpunkte in Get·ade übergehen sollen (Galileisches Trägheitsprinzip), dieser Bildraum bis auf ehr.e projektive Abbildung festgelegt. Denn es gilt der Satz, daß die projektiven die einzigen stetigen Abbildungen eines Raumstücks sind, durch welche Gerade in Gerade übergefuhrt werden. Er ergibt sich sogleich, wenn wir in der Mähinssehen Netzkonstruktion (Fig. 12) die unendlichferne durch eine unser Raumstück durchschneidende Gerade ersetzen (Fig. 15). Der Vorgang der Lichtausbreitung legt dann in unserm vierdimensionalen projektiven Bildraum das Unendlich-Ferne und die Metrik fest; denn seine (dreidimensionale) > unendlichferne Ebene• E ist dadurch gekennzeichnet, daß die Fig. 15. Lichtkegel die von den verschiedenen Weltpunkten aus gewonnenen Projektionen eines und desselben in E gelegeneu zweidimensionalen Kegelschnitts sind. In der allgemeinen Relativitätstheorie gestalten sich diese Schlüsse am einfachsten so. Der vierdimensionale Riemannsche Raum, als welchen sich Ein&tein die Welt denkt, ist ein Sonderfall x,. ' Consequences for GR – For general relativity Weyl proposed to weaken the Riemannian metric g to its conformal class c := [g ] and added a real valued “length” (scale) connection ϕ dependent on the choice of the representative g ∈ c: Weylian metric (g , ϕ), up to rescaling: – Choosing a representative g ∈ c meant to “gauge” the measurement units. Changing the representative g̃ = Ω2 g was accompanied by a gauge transformation of the scale connection ϕ̃ = ϕ − d log Ω. – The scale connection ϕ entered the specification of a compatible affine connection Γ = Γ(g , ϕ). The latter is uniquely determined and gauge independent. – Generalization of Einstein gravity to a Weyl geometric theory of gravity; built upoin it a unified field theory including electromagnetism (ϕ as e.m. potential). Consequences for GR – For general relativity Weyl proposed to weaken the Riemannian metric g to its conformal class c := [g ] and added a real valued “length” (scale) connection ϕ dependent on the choice of the representative g ∈ c: Weylian metric (g , ϕ), up to rescaling: – Choosing a representative g ∈ c meant to “gauge” the measurement units. Changing the representative g̃ = Ω2 g was accompanied by a gauge transformation of the scale connection ϕ̃ = ϕ − d log Ω. – The scale connection ϕ entered the specification of a compatible affine connection Γ = Γ(g , ϕ). The latter is uniquely determined and gauge independent. – Generalization of Einstein gravity to a Weyl geometric theory of gravity; built upoin it a unified field theory including electromagnetism (ϕ as e.m. potential). Consequences for GR – For general relativity Weyl proposed to weaken the Riemannian metric g to its conformal class c := [g ] and added a real valued “length” (scale) connection ϕ dependent on the choice of the representative g ∈ c: Weylian metric (g , ϕ), up to rescaling: – Choosing a representative g ∈ c meant to “gauge” the measurement units. Changing the representative g̃ = Ω2 g was accompanied by a gauge transformation of the scale connection ϕ̃ = ϕ − d log Ω. – The scale connection ϕ entered the specification of a compatible affine connection Γ = Γ(g , ϕ). The latter is uniquely determined and gauge independent. – Generalization of Einstein gravity to a Weyl geometric theory of gravity; built upoin it a unified field theory including electromagnetism (ϕ as e.m. potential). Consequences for math: new problem of space (PoS) Rigid motions of classical problem of space no longer useful in GR. Weyl started to analyze the PoS in the light of GR. Skipping details, he postulated (1920–1923) . . . – a pair of groups G ⊂ H ⊂ GL(n, R) with H = normalizer of G , spacetime point dependent by conjugation in GL(n, R) (generalized “congruences” and “similarities”); – . . . and a linear connection Λ ( “metric connection”) as “congruent transference” between distinct infintesimally close points. – Λ not unique (could be combined with “localized” congruences). Weyl demanded that among all equivalent forms there is exactly one symmetric (torsion free, in Cartan’s terms) affine connection. Consequences for math: new problem of space (PoS) Rigid motions of classical problem of space no longer useful in GR. Weyl started to analyze the PoS in the light of GR. Skipping details, he postulated (1920–1923) . . . – a pair of groups G ⊂ H ⊂ GL(n, R) with H = normalizer of G , spacetime point dependent by conjugation in GL(n, R) (generalized “congruences” and “similarities”); – . . . and a linear connection Λ ( “metric connection”) as “congruent transference” between distinct infintesimally close points. – Λ not unique (could be combined with “localized” congruences). Weyl demanded that among all equivalent forms there is exactly one symmetric (torsion free, in Cartan’s terms) affine connection. Consequences for math: new problem of space (PoS) Rigid motions of classical problem of space no longer useful in GR. Weyl started to analyze the PoS in the light of GR. Skipping details, he postulated (1920–1923) . . . – a pair of groups G ⊂ H ⊂ GL(n, R) with H = normalizer of G , spacetime point dependent by conjugation in GL(n, R) (generalized “congruences” and “similarities”); – . . . and a linear connection Λ ( “metric connection”) as “congruent transference” between distinct infintesimally close points. – Λ not unique (could be combined with “localized” congruences). Weyl demanded that among all equivalent forms there is exactly one symmetric (torsion free, in Cartan’s terms) affine connection. Two main principles of Weyl’s PoS and its result P1 (“Principle of freedom”) G allows the “widest concievable range of possible congruence transfers” in one point. P2 (“Principle of coherence”): To each congruent transfer Λ there exists exactly one equivalent affine connection. – Surprisingly P1 turned out to be superfluous mathematically (Scheibe 1954). – Result of the analysis after evaluating algebraic consequences of geometrical postulates: Only special orthogonal groups of any signature satisfy the conditions for G and P1, P2.. – Consequence: justification of Weyl geometry with its peculiar gauge structure rather than (semi-)Riemannian geometry as proper framework for “Space”. – A late argument in the aftermath of the Einstein-Weyl debate of 1918 about “absolute” versus “relative” standards of length/time. Two main principles of Weyl’s PoS and its result P1 (“Principle of freedom”) G allows the “widest concievable range of possible congruence transfers” in one point. P2 (“Principle of coherence”): To each congruent transfer Λ there exists exactly one equivalent affine connection. – Surprisingly P1 turned out to be superfluous mathematically (Scheibe 1954). – Result of the analysis after evaluating algebraic consequences of geometrical postulates: Only special orthogonal groups of any signature satisfy the conditions for G and P1, P2.. – Consequence: justification of Weyl geometry with its peculiar gauge structure rather than (semi-)Riemannian geometry as proper framework for “Space”. – A late argument in the aftermath of the Einstein-Weyl debate of 1918 about “absolute” versus “relative” standards of length/time. Two main principles of Weyl’s PoS and its result P1 (“Principle of freedom”) G allows the “widest concievable range of possible congruence transfers” in one point. P2 (“Principle of coherence”): To each congruent transfer Λ there exists exactly one equivalent affine connection. – Surprisingly P1 turned out to be superfluous mathematically (Scheibe 1954). – Result of the analysis after evaluating algebraic consequences of geometrical postulates: Only special orthogonal groups of any signature satisfy the conditions for G and P1, P2.. – Consequence: justification of Weyl geometry with its peculiar gauge structure rather than (semi-)Riemannian geometry as proper framework for “Space”. – A late argument in the aftermath of the Einstein-Weyl debate of 1918 about “absolute” versus “relative” standards of length/time. Two main principles of Weyl’s PoS and its result P1 (“Principle of freedom”) G allows the “widest concievable range of possible congruence transfers” in one point. P2 (“Principle of coherence”): To each congruent transfer Λ there exists exactly one equivalent affine connection. – Surprisingly P1 turned out to be superfluous mathematically (Scheibe 1954). – Result of the analysis after evaluating algebraic consequences of geometrical postulates: Only special orthogonal groups of any signature satisfy the conditions for G and P1, P2.. – Consequence: justification of Weyl geometry with its peculiar gauge structure rather than (semi-)Riemannian geometry as proper framework for “Space”. – A late argument in the aftermath of the Einstein-Weyl debate of 1918 about “absolute” versus “relative” standards of length/time. Two main principles of Weyl’s PoS and its result P1 (“Principle of freedom”) G allows the “widest concievable range of possible congruence transfers” in one point. P2 (“Principle of coherence”): To each congruent transfer Λ there exists exactly one equivalent affine connection. – Surprisingly P1 turned out to be superfluous mathematically (Scheibe 1954). – Result of the analysis after evaluating algebraic consequences of geometrical postulates: Only special orthogonal groups of any signature satisfy the conditions for G and P1, P2.. – Consequence: justification of Weyl geometry with its peculiar gauge structure rather than (semi-)Riemannian geometry as proper framework for “Space”. – A late argument in the aftermath of the Einstein-Weyl debate of 1918 about “absolute” versus “relative” standards of length/time. But . . . – The insights of quantum mechanics convinced Weyl in the middle of the 1920s that Einstein’s 1918 claim of a “universal bureau of standards (universelles Eichbüro)” of nature was basically correct. “The atomic constants of charge and mass of the electron and Planck’s quantum of action h, which enter the universal field laws of nature, fix an absolute standard of length, that through the wave lengths of spectral lines is made available for practical measurements.” (Weyl 1949) – In the 1940s (English ed. Philosophy of Mathematics and Natural Sciences and talks) Weyl distinguished between mathematical and physical automorphisms (of . . . ). Scale extension then belonged to the mathematical automorphisms – extensions of the physical ones.