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On Old Relations of Lie Theory, Classical Geometry and Gauge Theory
by Rolf Dahm
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Submission summary
Authors (as registered SciPost users):  Rolf Dahm 
Submission information  

Preprint Link:  scipost_202212_00049v1 (pdf) 
Date accepted:  20230811 
Date submitted:  20221218 21:56 
Submitted by:  Dahm, Rolf 
Submitted to:  SciPost Physics Proceedings 
Proceedings issue:  34th International Colloquium on Group Theoretical Methods in Physics (GROUP2022) 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Phenomenological 
Abstract
Having been led by hadron interactions and lowenergy photoproduction to SU(4) and noncompact SU$*$(4) symmetry, the general background turned out to be projective geometry (PG) of $P^3$, or when considering line and Complex geometry to include gauge theory, aspects of $P^5$. Point calculus and its dual completion by planes introduced quaternary (quadratic) 'invariants' $x_{\mu}x^{\mu}=0$ and $p_{\mu}p^{\mu}=0$, and put focus on the intermediary form $(xu)$ and its treatment. Here, the major result is the identification of the symmetric {\bf{\underline{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 {\it geometrically} the scene by representations (reps) and invariant theory without having to force affine restrictions and additional (spinorial or gauge) rep theory.
Published as SciPost Phys. Proc. 14, 025 (2023)
Reports on this Submission
Anonymous Report 1 on 2023129 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202212_00049v1, delivered 20230129, doi: 10.21468/SciPost.Report.6629
Strengths
The author shows that even quite technical aspects of hadron physics can be understood with classical geometrical tools that go back to the pioneering work of Klein, Lie and Hilbert, among others.
Weaknesses
If any, that most of the classical references are not of easy access for the nonGerman reader.
Report
The author provides a nice justification of the SU(4) and SU^{*}(4) role in hadron physics, by means of classical geometrical arguments that revive the original Ansatz of Klein and Lie studying transformation groups. In particular, the IRREP 20 is given an interpretation in terms of Projective Geometry, according to the Hilbert construction, pointing out a procedure to treat geometrically with various objects of Tensor Analysis used in the context of representation theory. and gauge theory.
An interesting paper that shows that among the original approach to Lie groups, there remain a number of questions that are still of interest for physical applications.