Particle and Field Symmetries and Noncommutative Geometry

X iv :q ua nt -p h/ 03 05 15 0v 1 2 4 M ay 2 00 3 Particle and Field Symmetries and Noncommutative Geometry Ajay Patwardhan Physics Department, St.Xavier’s College, Mahapalika Marg, Mumbai 400001, India. * e-mail : [email protected] Abstract 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 the 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. 1.Introduction The mathematical structures of the physics of particles and fields were developed using commutative and non commutative algebra, and Euclidean and non Euclidean Geometry. This led to Quantum Mechanics and General Relativity,respectively. The Quantum Field theory of Gauge Fields describes all fundamental interactions, including gravity, as holonomy and action integrals. It has succeeded phenomenologically, inspite of some difficulties. Consistency requirements have led to a number of symmetries, including supersymmetry. Loop space quantum gravity and string and brane theories have evolved as a development of quantum theory of interactions. These are also connected to the evolving subject of non commutative geometry.[Ref ] The dynamical variables in a quantum theory have a commutation algebra. A non commutative structure has been introduced in a wide variety of physics ; with length scales from Planck length in quantum space time, to magnetic length in quantum Hall effect. The new (non)commutation structure introduces a derivation (as a bracket operation), which acts in addition to the Lie and covariant derivatives. In the spacetime manifold , a discrete topology and a length scale parameter cause changes in the definitions of the metric tensor,Riemann tensor, Ricci tensor and the Einstein equations.Will