Remarks on Common Representation Theory of Gauge Theories

Abstract Whereas the standard representation (rep) theory of gauge theories relies on quantum field theory (QFT), based on point coordinates x µ and certain Lie symmetries, we discuss some aspects of a geometrical background pattern able to explain such rep theory and the known unitary symmetries from scratch. Switching to lines as geometrical base elements of space-time instead of point coordinates, line geometry induces a concept to identify quantum theories directly with the description of forces and moments. Moreover, sets of lines and their properties correspond to known physical notions, and line Complexe allow a natural approach to gauge descriptions, chiral rep theory, 3-dimensional complex spaces, grouping into families, triality, etc. Beyond Riemannian geometry, line geometry naturally adds symplectic geometry in P 3 , and its rep theory yields cubic curves as well as quartic surfaces. Last not least, it not only respects quadratic invariants advocated by bi-linear symmetric forms and orthogonal or unitary transformations, but projective geometry also allows to incorporate the complete invariant theory in a well-defined manner.