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Quantum field theory has achieved some extraordinary successes over the past sixty years; however, it retains a set of challenging problems. It is not yet able to describe gravity in a mathematically consistent manner. CP violation remains unexplained. Grand unified theories have been eliminated by experiment, and a viable unification model has yet to replace them. Even the highly successful quantum chromodynamics, despite significant computational achievements, struggles to provide theoretical insight into the low-energy regime of quark physics, where the nature and structure of hadrons are determined. The only proposal for resolving the fine-tuning problem, low-energy supersymmetry, has been eliminated by results from the LHC. Since mathematics is the true and proper language for quantitative physical models, we expect new mathematical constructions to provide insight into physical phenomena and fresh approaches for building physical theories.

Research & information: general --- Physics --- semiheaps --- ternary algebras --- para-associativity --- quantum mechanics --- gravity --- Clairaut equation --- Cho–Duan–Ge decomposition --- constraintless formalism --- canonical gravity --- covariance --- black holes --- quantum foundations --- non-axiomaticity --- detector clicks --- ensembles --- superposition principle --- arithmetic --- numbers --- vector space --- abstracting --- interpretations --- self-referentiality --- direct product --- direct power --- polyadic semigroup --- arity --- polyadic ring --- polyadic field --- Maxwell’s vacuum equations --- Hamilton–Jacobi equation --- Klein–Gordon–Fock equation --- algebra of symmetry operators --- separation of variables --- linear partial differential equations --- Einstein field equation --- recursion operator --- Noether symmetry --- master symmetry --- conformable differential --- Poisson manifold --- diffeomorphism group --- current algebra symmetry --- current Lie algebra representation --- fock space --- generating functional --- distribution functions --- Lie–Poisson structure --- coherent states --- Lie-Poisson action --- Hilbert space linearization --- hamiltonian systems --- symmetry reduction --- integrability --- idiabatic states --- factorization --- heavenly type dynamical systems --- integrable dynamical systems --- dirac reduction --- hydrodynamic flows --- entropy --- vortex flows --- asymptotic conditions --- Kirchhoff’s integral theorem --- quantum gravity and the problem of the Big Bang --- hidden Hermitian formulations of quantum mechanics --- stationary Wheeler-DeWitt system --- physical Hilbert space metric --- non-stationary Wheeler-DeWitt system --- n/a --- Cho-Duan-Ge decomposition --- Maxwell's vacuum equations --- Hamilton-Jacobi equation --- Klein-Gordon-Fock equation --- Lie-Poisson structure --- Kirchhoff's integral theorem