A novel approach to non-commutative gauge theory

Abstract

We propose a field theoretical model defined on non-commutative space-time with non-constant non-commutativity parameter $\Theta (x)$, which satisfies two main requirements: it is gauge invariant and reproduces in the commutative limit, $\Theta \to 0$, the standard $U(1)$ gauge theory. We work in the slowly varying field approximation where higher derivatives terms in the star commutator are neglected and the latter is approximated by the Poisson bracket, $-i[f,g{]}_{\star }\approx \{f,g\}$. We derive an explicit expression for both the NC deformation of Abelian gauge transformations which close the algebra $[{\delta }_f,{\delta }_g]A={\delta }_{\{f,g\}}A$, and the NC field strength $\mathcal{F}$, covariant under these transformations, ${\delta }_f\mathcal{F}=\{\mathcal{F},f\}$. NC Chern-Simons equations are equivalent to the requirement that the NC field strength, $\mathcal{F}$, should vanish identically. Such equations are non-Lagrangian. The NC deformation of Yang-Mills theory is obtained from the gauge invariant action, $S=\int {\mathcal{F}}^2$. As guiding example, the case of $su(2)$-like non-commutativity, corresponding to rotationally invariant NC space, is worked out in detail.

arXiv comment16 pages, no figures. Minor corrections

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Cited by in corpus (14)

• Gauge theories on quantum spaces
• $k$-Minkowski-deformation of $U(1)$ gauge theory
• Poisson gauge theory
• Developed Method. Interactions and their quantum picture
• Poisson gauge models and Seiberg-Witten map
• Double Quantization
• Lie-Poisson gauge theories and $\kappa$-Minkowski electrodynamics
• On the L${}_{\infty }$ structure of Poisson gauge theory
• Symmetric ordering and Weyl realizations for quantum Minkowski spaces
• Effects of wave propagation in canonical Poisson gauge theory under an external magnetic field
• On a charged spinless point particle minimally coupled to a constant magnetic field in a noncommutative plane
• Light Propagation in $\kappa$-Minkowski Space-Time: Gauge Ambiguities and Invariance
• Hamiltonian analysis in Lie-Poisson gauge theory
• Nonlocal Lagrangian fields and the second Noether theorem. Non-commutative $U(1)$ gauge theory