  The development of Noncommutative Geometry is creating a reworking and new
possibilities in physics. This paper identifies some of the commutation and
derivation structures that arise in particle and field interactions and
fundamental symmetries. The requirements of coexisting structures, and their
consistency, produce a mathematical framework that underlies a fundamental
physics theory. Among other developments in Quantum theory of particles and
fields are the symmetries of gauge fields and the Fermi-Bose symmetry of
particles. These involve a gauge covariant derivation and the action
functionals; and commutation algebras and Bogoliubov transforms. The non
commutative Theta form introduces an additional and fundamental structure. This
paper obtains the interrelations of various structures, and the conditions for
the symmetries of Fermionic/Bosonic particles interacting with Yang Mills gauge
fields. Many example physical systems are being solved, and the mathematical
formalism is being created to understand the fundamental basis of physics.
