Mathematical foundations of quantum field theories, QFT: QED, QCD, electro-weak and other physical interactions:

1. QCD or Quantum Chromodynamics: the advanced, standard mathematical and quantum physics treatment of strong force or nuclear interactions such as those among quarks and gluons, partons, `Yukawa' mesons, and so on, with an intrinsic threefold, or eightfold symmetries (described by the `quantum' group SU(3) first reported by the US Nobel Laureate Gell-Mann and others;
2. QED - U(1) Symmetry,Quantum Electrodynamics is the advanced, standard mathematical and quantum physics treatment of electromagnetic interactions through several approaches, the more advanced including the path-integral approach by Feynman, Dirac's Operator and QED Equations, thus including either Special or General Relativity formulations of electromagnetic phenomena;
3. Young-Mills theories;
4. Electro-Weak interactions, SU(2) Symmetry;
5. Algebraic Quantum Field theories (AQFT);
6. Homotopy Quantum Field theories (HQFT) and Topological QFT's;
7. Quantum Gravity (QG) and Other theories.

This obviates the need for `more fundamental' , or extended quantum symmetries, such as those afforded by either several larger groups such as (and their representations) in SUSY, or by spontaneously broken, multiple (`or localized') symmetries of a less restrictive kind present in `quantum groupoids' as for example in weak Hopf algebra representations, or in locally compact groupoid, unitary representations, and so on, to the higher dimensional (quantum) symmetries of quantum double groupoids, quantum double algebroids, quantum categories/quantum supercategories in HDA , and/or quantum supersymmetry superalgebras (or graded `Lie' algebras, see- for example- the QFT ref. [1] discussing superalgebras in quantum gravity). Thus, certain finite irreducible representations correspond to `elementary' (quantum) particles and spin symmetry representations have corresponding quantum obsevable operators, such as the Casimir operators. A well-known case is that of Pauli matrices that are representations of the special unitary group . Supersymmetry, supergroups and superoperators further expand SUSY to quantum gravity and quantum statistical mechanics.

Bibliography

1

S. Weinberg. 2003. Quantum Field Theories, vol. 1-3, Cambridge University Press: Cambridge, UK.