Gauge Geometry and Philosophical Dynamism

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Weyl was dissatisfied with his own theory of the predicative construction of the arithmetically definable subset of the classical real continuum by the time he had finished his Das Kontinuum, when he compared it with Husserl’s continuum of time, which possessed a “non-atomistic” character in contradistinction with his own theory. No determined point of time could be exhibited, only approximate fixing is possible, just as in the case of “continuum of spatial intuition”. He himself accepted the necessity that the mathematical concept of continuum, the continuous manifold, should not be characterized in terms of modern set theory enriched by topological axioms, because this would contradict the essence of continuum. Weyl says,

It seems that set theory violates against the essence of continuum, which, by its very nature, cannot at all be battered into a single set of elements. not the relationship of an element to a set, but a part of the whole ought to be taken as a basis for the analysis of the continuum.

For Weyl, single points of continuum were empty abstractions, and made him enter a difficult terrain, as no mathematical conceptual frame was in sight, which could satisfy his methodological postulate in a sufficiently elaborative manner. For some years, he sympathized with Brouwer’s idea to characterize points in the intuitionistic one-dimensional continuum by “free choice sequences” of nested intervals, and even tried to extend the idea to higher dimensions and explored the possibility of a purely combinatorial approach to the concept of manifold, in which point-like localizations were given only by infinite sequences of nested star neighborhoods in barycentric subdivisions of a combinatorially defined “manifold”. There arose, however, the problem of how to characterize the local manifold property in purely combinatorial terms.

Weyl was much more successful on another level to rebuild differential geometry in manifolds from a “purely infinitesimal” point of view. He generalized Riemann’s proposal for a differential geometric metric

ds2(x) = ∑n i, j = 1 gij(x) dxi dxj

From his purely infinitesimal point of view, it seemed a strange effect that the length of two vectors ξ(x) and η(x’) given at different points x and x’ can be immediately and objectively compared in this framework after the calculation of

|ξ(x)|2 = ∑n i, j = 1 gij(x) ξi ξj,

|η(x’)|2 = ∑n i, j = 1 gij(x’) ηi ηj

In this context, it was, comparatively easy for Weyl, to give a perfectly infinitesimal characterization of metrical concepts. He started from a well-known structure of conformal metric, i.e. an equivalence class [g] of semi-Riemannian metrics g = gij(x) and g’ = g’ij(x), which are equal up to a point of dependent positive factor λ(x) > 0, g’ = λg. Then, comparison of length made immediate sense only for vectors attached to the same point x, independently of the gauge of the metric, i.e. the choice of the representative in the conformal class. To achieve comparability of lengths of vectors inside each infinitesimal neighborhood, Weyl introduced the conception of length connection formed in analogy to the affine connection, Γ, just distilled from the classical Christoffel Symbols Γkij of Riemannian geometry by Levi Civita. The localization inside such an infinitesimal neighborhood was given, as would have been done already by the mathematicians of the past, by coordinate parameters x and x’ = x + dx for some infinitesimal displacement dx. Weyl’s length connection consisted, then, in an equivalence class of differential I-forms [Ψ], Ψ ≡ ∑ni = 1 Ψidxi, where an equivalent representation of the form is given by Ψ’ ≡ Ψ – d log λ, corresponding to a change of gauge of the conformal metric by the factor λ. Weyl called this transformation, which he recognized as necessary for the consistency of his extended symbol system, the gauge transformation of the length connection.

Weyl established a purely infinitesimal gauge geometry, where lengths of vectors (or derived metrical concepts in tensor fields) were immediately comparable only in the infinitesimal neighborhood of one point, and for points of finite distance only after an integration procedure. this integration turned out to be, in general, path dependent. Independence of the choice of path between two points x and x’ holds if and only if the length curvature vanishes. the concept of curvature was built in direct analogy to the curvature of the affine connection and turned out to be, in this case, just the exterior derivative of the length connection f ≡ dΨ. This led Weyl to a coherent and conceptually pleasing realization of a metrical differential geometry built upon purely infinitesimal principles. moreover, Weyl was convinced of important consequences of his new gauge geometry for physics. The infinitesimal neighborhoods understood as spheres of activity as Fichte might have said, suggested looking for interpretations of the length connection as a field representing physically active quantities. In fact, building on the mathematically obvious observation df ≡ 0, which was formally identical with the second system of the generally covariant Maxwell equations, Weyl immediately drew the conclusion that the length curvature f ought to be identified with the electromagnetic field.

He, however, gave up the belief in the ontological correctness of the purely field-theoretic approach to matter, where the Mie-Hilbert theory of a combined Lagrange function L(g,Ψ) for the action of the gravitational field (g) and electromagnetism (Ψ) was further geometrized and technically enriched by the principle of gauge invariance (L), substituting in its place a philosophically motivated a priori argumentation for the conceptual superiority of his gauge geometry. The goal of a unified description of gravitation and electromagnetism, and the derivation of matter structures from it, was nothing specific to Weyl. In his theory, the purely infinitesimal approach to manifolds and the ensuing possibility to geometrically unify the two-known interaction fields gravitation and electromagnetism took on a dense and conceptually sophisticated form.

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