Ju l 2 01 0 Position-dependent noncommutativity Strings from position-dependent noncommutativity (original) (raw)

Strings from position-dependent noncommutativity

We introduce a new set of noncommutative space-time commutation relations in two space dimensions. The space-space commutation relations are deformations of the standard flat noncommutative space-time relations taken here to have position dependent structure constants. Some of the new variables are non-Hermitian in the most natural choice. We construct their Hermitian counterparts by means of a Dyson map, which also serves to introduce a new metric operator. We propose PT like symmetries, i.e. antilinear involutory maps, respected by these deformations. We compute minimal lengths and momenta arising in this space from generalized versions of Heisenberg's uncertainty relations and find that any object in this two dimensional space is string like, i.e. having a fundamental length in one direction beyond which a resolution is impossible. Subsequently we formulate and partly solve some simple models in these new variables, the free particle, its PT -symmetric deformations and the harmonic oscillator.

Particle and Field Symmetries and Noncommutative Geometry

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

Space-Time Uncertainty and Noncommutativity in String Theory

International Journal of Modern Physics A, 2001

We analyze the nature of space-time nonlocality in string theory. After giving a brief overview on the conjecture of the space-time uncertainty principle, a (semi-classical) reformulation of string quantum mechanics, in which the dynamics is represented by the noncommutativity between temporal and spatial coordinates, is outlined. The formalism is then compared to the space-time noncommutative field theories associated with nonzero electric B-fields.

SPACETIME SYMMETRIES IN NONCOMMUTATIVE GAUGE THEORY: A HAMILTONIAN ANALYSIS

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.

Non-commutative space-time and the uncertainty principle

Physics Letters, Section A: General, Atomic and Solid State Physics, 2001

The full algebra of relativistic quantum mechanics (Lorentz plus Heisenberg) is unstable. Stabilization by deformation leads to a new deformation parameter εℓ 2 , ℓ being a length and ε a ± sign. The implications of the deformed algebras for the uncertainty principle and the density of states are worked out and compared with the results of past analysis following from gravity and string theory.

Quantum Field Theories on a Noncommutative Euclidean Space: Overview of New Physics

In this talk I briefly review recent developments in quantum field theories on a noncommutative Euclidean space, with Heisenberg-like commutation relations between coordinates. I will be concentrated on new physics learned from this simplest class of non-local field theories, which has applications to both string theory and condensed matter systems, and possibly to particle phenomenology.

Noncommutativity in interpolating string: A study of gauge symmetries in a noncommutative framework

Physical Review D, 2006

A new Lagrangian description that interpolates between the Nambu--Goto and Polyakov version of interacting strings is given. Certain essential modifications in the Poission bracket structure of this interpolating theory generates noncommutativity among the string coordinates for both free and interacting strings. The noncommutativity is shown to be a direct consequence of the nontrivial boundary conditions. A thorough analysis of the gauge symmetry is presented taking into account the new modified constraint algebra, which follows from the noncommutative structures and finally a smooth correspondence between gauge symmetry and reparametrisation is established.