Dirac field and gravity in NC $$SO(2,3)_\star $$ model

The European Physical Journal C, Mar 2018

Action for the Dirac spinor field coupled to gravity on noncommutative (NC) Moyal-Weyl spacetime is obtained without prior knowledge of the metric tensor. We emphasize gauge origins of gravity and its interaction with fermions by demonstrating that a classical action invariant under SO(2, 3) gauge transformations can be exactly reduced to the Dirac action in curved spacetime after breaking the original symmetry down to the local Lorentz SO(1, 3) symmetry. The commutative SO(2, 3) invariant action can be straightforwardly deformed via Moyal-Weyl \(\star \)-product to its NC \(SO(2,3)_\star \) invariant version which can be expanded perturbatively in powers of the deformation parameter using the Seiberg-Witten map. The NC gravity-matter couplings in the expansion arise as an effect of the gauge symmetry breaking. We calculate in detail the first order NC correction to the classical Dirac action in curved spacetime and show that it does not vanish. Moreover, linear NC effects are apparent even in flat spacetime. We analyse NC deformation of the Dirac equation, Feynman propagator and dispersion relation for electrons in Minkowski spacetime and conclude that constant NC background acts as a birefringent medium for electrons propagating in it.

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Dirac field and gravity in NC $$SO(2,3)_\star $$ model

Eur. Phys. J. C Dirac field and gravity in NC S O (2, 3) Dragoljub Gocˇanin 0 Voja Radovanovic´ 0 0 Faculty of Physics, University of Belgrade , Studentski trg 12, 11000 Beograd , Serbia Action for the Dirac spinor field coupled to gravity on noncommutative (NC) Moyal-Weyl spacetime is obtained without prior knowledge of the metric tensor. We emphasize gauge origins of gravity and its interaction with fermions by demonstrating that a classical action invariant under S O(2, 3) gauge transformations can be exactly reduced to the Dirac action in curved spacetime after breaking the original symmetry down to the local Lorentz S O(1, 3) symmetry. The commutative S O(2, 3) invariant action can be straightforwardly deformed via Moyal-Weyl -product to its NC S O(2, 3) invariant version which can be expanded perturbatively in powers of the deformation parameter using the Seiberg-Witten map. The NC gravity-matter couplings in the expansion arise as an effect of the gauge symmetry breaking. We calculate in detail the first order NC correction to the classical Dirac action in curved spacetime and show that it does not vanish. Moreover, linear NC effects are apparent even in flat spacetime. We analyse NC deformation of the Dirac equation, Feynman propagator and dispersion relation for electrons in Minkowski spacetime and conclude that constant NC background acts as a birefringent medium for electrons propagating in it. 1 Introduction Quantum Field Theory (QFT) and General Relativity (GR) are two cornerstones of modern theoretical physics. Although these theories have been tested to an excellent degree of accuracy in their respective areas of applicability, occurrence of singularities in both of them strongly indicates that they are incomplete. GR, as a classical theory of gravity, describes large-scale geometric structure of spacetime and its relation to the distribution of matter. On the other hand, QFT, standing on the principles of Quantum Mechanics and Special Relativity, provides us with the Standard Model of elementary particles which successfully utilizes the idea of local symmetry to describe the fundamental particle interactions. Understanding quantum nature of spacetime and reconciling gravity with other fundamental interactions is considered to be one of the main goals of contemporary physics. In order to obtain a consistent unified theory, certain modifications of the basic concepts of QFT and GR are necessary. Various approaches have been proposed so far, stemming from String Theory, Loop Quantum Gravity, Noncommutative (NC) Field Theory, etc. and all of them, in some radical way, change the notion of point particle and/or that of spacetime. In the last twenty years, Noncommutative Field Theory has become a very important direction of investigation in theoretical high energy physics and gravity. Its basic insight is that the quantum nature of spacetime, at the microscopic level, should mean that even the spacetime coordinates are to be treated as mutually incompatible observables, satisfying some non trivial commutation relations. The simplest choice of noncommutativity is the so called canonical noncommutativity, defined by where θ μν are components of a constant antisymmetric matrix. To establish canonical noncommutativity, instead of using abstract algebra of coordinates, i.e. noncommutative spacetime, one can equivalently introduce the noncommutative Moyal-Weyl -product, f (x ) g(x ) = e 2i θαβ ∂x∂α ∂y∂β f (x )g(y)|y→x , as a multiplication of functions (fields) defined on the usual, commutative (undeformed) spacetime. The quantity θ μν is considered to be a small deformation parameter that has dimensions of (length)2 (in natural units). It is a fundamental constant, like the Planck length or the speed of light. (1.1) (1.2) Recently, a lot of attention has been devoted to NC gravity, and many different approaches to this problem have been developed. In [1–3] a deformation of pure Einstein gravity based on the Seiberg-Witten map is proposed. Twist approach to noncommutative gravity was explored in [4– 7]. Lorentz symmetry in NC field theories was considered in [8,9]. Some other proposals are given in [10–22]. The connection to supergravity was established in [23,24]. The extension of NC gauge theories to orthogonal and symplectic algebra was considered in [25,26]. Finally, in the previous papers of one of the authors [27–30] an approach based on the deformed Anti de Sitter (AdS) symmetry group, i.e. S O( 2, 3 ) group, and canonical noncommutativity was established. In this approach NC gravity is treated as a gauge theory. It becomes manifest only after the suitable symmetry breaking. The action was constructed without previous introduction of the metric tensor and the second order NC correction to the Einstein-Hilbert action was found explicitly. Special attention has been devoted to the meaning of the coordinates used. Namely, it was shown that coordinates in which we postulate canonical nonco (...truncated)


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Dragoljub Gočanin, Voja Radovanović. Dirac field and gravity in NC $$SO(2,3)_\star $$ model, The European Physical Journal C, 2018, pp. 195, Volume 78, Issue 3, DOI: 10.1140/epjc/s10052-018-5669-4