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Register a new userNon-relativistic symmetries in three space-time dimensions and the Nappi-Witten algebra
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We show that the Extended Bargmann and Newton-Hooke algebras in 2+1 dimensions can be obtained as expansions of the Nappi-Witten algebra. The result can be generalized to obtain two infinite families of non-relativistic symmetries, which include the Maxwellian Exotic Bargmann symmetry, its generalized Newton-Hooke counterpart, and its Hietarinta dual. In each case, the invariant bilinear form on the Nappi-Witten algebra leads to the invariant tensor on the expanded algebra, allowing one to construct the corresponding Chern-Simons gravity theory.
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By applying loop quantum gravity techniques to 3D gravity with a positive cosmological constant $Lambda$, we show how the local gauge symmetry of the theory, encoded in the constraint algebra, acquires the quantum group structure of $so_q(4)$, with $ q = exp{(ihbar sqrt{Lambda}/2kappa)}$. By means of an Inonu-Wigner contraction of the quantum group bi-algebra, keeping $kappa$ finite, we obtain the kappa-Poincare algebra of the flat quantum space-time symmetries.
We study Yang-Baxter deformations of the Nappi-Witten model with a prescription invented by Delduc, Magro and Vicedo. The deformations are specified by skew-symmetric classical $r$-matrices satisfying (modified) classical Yang-Baxter equations. We show that the sigma-model metric is invariant under arbitrary deformations (while the coefficient of $B$-field is changed) by utilizing the most general classical $r$-matrix. Furthermore, the coefficient of $B$-field is determined to be the original value from the requirement that the one-loop $beta$-function should vanish. After all, the Nappi-Witten model is the unique conformal theory within the class of the Yang-Baxter deformations preserving the conformal invariance.
We study the space-time symmetries and transformation properties of the non-commutative U(1) gauge theory, by using Noether charges. We carry out our analysis by keeping an open view on the possible ways $theta^{mu u}$ could transform. We conclude that $theta^{mu u}$ cannot transform under any space-time transformation since the theory is not invariant under the conformal transformations, with the only exception of space-time translations. The same analysis applies to other gauge groups.
It is possible that relativistic symmetries become deformed in the semiclassical regime of quantum gravity. Mathematically, such deformations lead to the noncommutativity of spacetime geometry and non-vanishing curvature of momentum space. The best studied example is given by the $kappa$-Poincare Hopf algebra, associated with $kappa$-Minkowski space. On the other hand, the curved momentum space is a well-known feature of particles coupled to three-dimensional gravity. The purpose of this thesis was to explore some properties and mutual relations of the above two models. In particular, I study extensively the spectral dimension of $kappa$-Minkowski space. I also present an alternative limit of the Chern-Simons theory describing three-dimensional gravity with particles. Then I discuss the spaces of momenta corresponding to conical defects in higher dimensional spacetimes. Finally, I consider the Fock space construction for the quantum theory of particles in three-dimensional gravity.
Firstly, a systematic procedure is derived for obtaining three-dimensional bound-state equations from four-dimensional ones. Unlike ``quasi-potential approaches this procedure does not involve the use of delta-function constraints on the relative four-momentum. In the absence of negative-energy states, the kernels of the three-dimensional equations derived by this technique may be represented as sums of time-ordered perturbation theory diagrams. Consequently, such equations have two major advantages over quasi-potential equations: they may easily be written down in any Lorentz frame, and they include the meson-retardation effects present in the original four-dimensional equation. Secondly, a simple four-dimensional equation with the correct one-body limit is obtained by a reorganization of the generalized ladder Bethe-Salpeter kernel. Thirdly, our approach to deriving three-dimensional equations is applied to this four-dimensional equation, thus yielding a retarded interaction for use in the three-dimensional bound-state equation of Wallace and Mandelzweig. The resulting three-dimensional equation has the correct one-body limit and may be systematically improved upon. The quality of the three-dimensional equation, and our general technique for deriving such equations, is then tested by calculating bound-state properties in a scalar field theory using six different bound-state equations. It is found that equations obtained using the method espoused here approximate the wave functions obtained from their parent four-dimensional equations significantly better than the corresponding quasi-potential equations do.
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