Calculus, Gauge Theory and Noncommutative Worlds

2003

Sequences of actions do not commute.. For example, the tick of a clock and the measurement of a position do not commute with one another, since the position will have moved to the next position after the tick. We adopt non-commutative calculus, with derivatives represented by commutators. In the beginning distinct derivatives do not commute with one another, providing curvature formalism so that the form of the curvature of a gauge field appears almost as soon as the calculus is defined. This provides context for the Feynman-Dyson derivation of electromagnetic formalism from commutators, and generalizations including the early appearance of the form of the Levi-Civita connection dervived from the Jacobi identity. In this version of non-commutative physics bare quantum mechanics (its commutation relations) appears as the flat background for all other constructions. Ascent to classical physics is obtained by replacing commutators with Poisson brackets that satisfy the Leibniz rule. An appendix on matrix algebra from a discrete point of view (Iterants) is provided. This paper will appear in the proceedings of the ANPA conference held in Cambridge, England in the summer of 2002.

Eprint Arxiv Quant Ph 0305150, 2003

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

Eprint Arxiv Quant Ph 0503198, 2005

Journal of Geometry and Physics, 1989

The structure of amanifold can be encoded in the commutative algebra of functions on the manifold it sell-this is usual-. In the case of a non com.mut.ative algebra thereis no underlying manifold and the usual concepts and tools of diffe.rential geometry (differentialforms, De Rham cohomology, vector bundles, connections, elliptic operators, index theory.. .) have to be generalized. This is the subject of non commutative differential geometry and is believed to be of fundamental importance in our understanding of quantum field theories. The presentpaper is an introduction for the non specialist and a review oftheprincipal results on the field.

Modern Physics Letters A, 2004

We study space-time symmetries in Non-Commutative (NC) gauge theory in the (constrained) Hamiltonian framework. The specific example of NC CP (1) model, posited in [9], has been considered. Subtle features of Lorentz invariance violation in NC field theory were pointed out in . Out of the two -Observer and Particle -distinct types of Lorentz transformations, symmetry under the former, (due to the translation invariance), is reflected in the conservation of energy and momentum in NC theory. The constant tensor θ µν (the noncommutativity parameter) destroys invariance under the latter.