The Gauge Principle, Hermann Weyl, and Symbolic Construction from the “Purely Infinitesimal”

Abstract

The gauge principle is a broad moniker about invariance properties of fundamental physical laws. It stipulates that every global symmetry of a quantum field theory be replaced by a local one; in effect, that every continuous symmetry of a quantum field (the Lie group under which the field Lagrangian transforms invariantly) become a local symmetry, i.e., an invariance of the Lagrangian under which the smooth Lie group actions are allowed to differ from point to point. It has an unusual “context of discovery”: invoked on largely phenomenological grounds by mathematician Hermann Weyl, it emerged in 1918 in the context of classical general relativity and, in Weyl’s hands, led to a purely formal unification of Einstein’s gravitational theory and electromagnetism. This work prompted Weyl’s purely mathematical turn in 1925–6 to Lie theory (on representations of semisimple Lie groups and “Lie algebras”, a term later coined by Weyl for the infinitesimal linear algebraic structure of Lie groups). Both Lie groups and Lie algebras play prominent roles in the subsequent development of the gauge principle leading up to the Standard Model (SM), a compilation of quantum field theories that since 1978 is the regnant theory of matter. I suggest that the gauge principle as well as Weyl’s predominant interest in Lie theory were motivated by two complementary philosophical demands: (i) phenomenological evidential requirements of “eidetic insight” and “eidetic analysis” imposed on differential geometric construction, and (ii) the metaphysical command of “Nahewirkungphysik” Weyl associated with Leibniz, Riemann and Lie: to comprehend the world from its behavior in the infinitely small. The two requirements productively meet in Weyl’s notion of “symbolic construction”: the idea that the sense of a transcendent world portrayed in physical theory can be constitutively understood beginning from a transcendental subjectivity evidentially privileging “radical locality”, i.e., the “given to consciousness” epistemic reach (“horizon”) of linear relations within the tangent space. Radical locality is the basis of Weyl’s constitution of objectivity as an invariance with respect to manifold automorphisms, an intersubjectivity allowing arbitrary coordinates and gauge degrees of freedom of a situated constructing ego. Both concern a necessary redundancy of physical description, a philosophical puzzle that might be elucidated by revisiting the philosophical underpinnings of the gauge principle.