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Differential forms and κ-Minkowski spacetime from extended twist

  • Regular Article - Theoretical Physics
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Abstract

We analyze bicovariant differential calculus on κ-Minkowski spacetime. It is shown that corresponding Lorentz generators and noncommutative coordinates compatible with bicovariant calculus cannot be realized in terms of commutative coordinates and momenta. Furthermore, κ-Minkowski space and NC forms are constructed by twist related to a bicrossproduct basis. It is pointed out that the consistency condition is not satisfied. We present the construction of κ-deformed coordinates and forms (super-Heisenberg algebra) using extended twist. It is compatible with bicovariant differential calculus with κ-deformed \(\mathfrak{igl}(4)\)-Hopf algebra. The extended twist leading to κ-Poincaré-Hopf algebra is also discussed.

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Notes

  1. For Hopf-algebroid structure also see [54–56], and [29].

  2. Greek indices (μ,ν,…) are from 0 to 3, and Latin indices (i,j,…) from 1 to 3. Summation over repeated indices is assumed.

  3. Here the superscript (o) denotes that the Lorentz generators and NC coordinates \(\hat{x}\) are realized only in terms of undeformed x μ and ∂ μ .

  4. Sitarz denotes this action with ▷.

  5. The correspondence between algebra in [12] and (22) is \(\frac{1}{\kappa }=-ia_{0}\), \(x_{\mu}=\hat{x}_{\mu}\), \(\mathrm {d}{x}_{\mu}=\hat{\xi}_{\mu}\), ϕ=ϕ, \(N_{i}=\hat{M}_{0i}\), and \(M_{i}=\epsilon_{ijk}\hat{M}_{jk}\).

  6. The algebra of undeformed operators is defined in Sect. 2 and for ▶ action we have \(x_{\mu}\blacktriangleright1=\hat {x}_{\mu}\), \(\xi_{\mu}\blacktriangleright1=\hat{\xi}_{\mu}\), ∂ μ ▶1=0 and q μ ▶1=0.

  7. For more details see [28].

  8. The algebra \(\hat{\mathcal{SA}}\) is generated by \(\hat {x}_{\mu}\), \(\hat{\xi}_{\mu}\) and ϕ, and the algebra \(\mathcal{SA}^{\star}\) is generated by x μ and ξ μ but with ⋆-multiplication. The star-product is defined by \(f(x,\xi)\star g(x,\xi )=\hat{f}(\hat{x},\hat{\xi})\hat{g}(\hat{x},\hat{\xi})\triangleright 1\).

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Acknowledgements

We would like to thank Peter Schupp for useful comments. This work was supported by the Ministry of Science and Technology of the Republic of Croatia under contract No. 098-0000000-2865. R.Š. gratefully acknowledges support from the DFG within the Research Training Group 1620 “Models of Gravity”.

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  1. Rudjer Bošković Institute, Bijenička c.54, 10002, Zagreb, Croatia

    Tajron Jurić & Stjepan Meljanac

  2. Jacobs University Bremen, 28759, Bremen, Germany

    Rina Štrajn

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  1. Tajron Jurić
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Jurić, T., Meljanac, S. & Štrajn, R. Differential forms and κ-Minkowski spacetime from extended twist. Eur. Phys. J. C 73, 2472 (2013). https://doi.org/10.1140/epjc/s10052-013-2472-0

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