On old relations of Lie theory, classical geometry and gauge theory

Abstract

Having been led by hadron interactions and low-energy photoproduction to SU(4) and non-compact SU*(4) symmetry, the general background turned out to be projective geometry (PG) of P^3P3, or when considering line and Complex geometry to include gauge theory, aspects of P^5P5. Point calculus and its dual completion by planes introduced quaternary (quadratic) “invariants” x_{\mu}x^{\mu}=0xμxμ=0 and p_{\mu}p^{\mu}=0pμpμ=0, and put focus on the intermediary form (xu)(xu) and its treatment. Here, the major result is the identification of the symmetric 20 of SU(4) comprising nucleon and Delta states as related to the quaternary cubic forms discussed by Hilbert in his work on full invariant systems. So PG determines geometrically the scene by representations (reps) and invariant theory without having to force affine restrictions and additional (spinorial or gauge) rep theory.