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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: 2023-08-11
Date submitted: 2022-12-18 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
  • Mathematical Physics
  • Quantum Physics
Approaches: Theoretical, Phenomenological


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 $P^3$, 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 2023-1-29 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202212_00049v1, delivered 2023-01-29, doi: 10.21468/SciPost.Report.6629


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.


If any, that most of the classical references are not of easy access for the non-German reader.


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.

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: good
  • grammar: excellent

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