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Classical Fields, Symmetries, and Conserved Charges

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Deformations of Spacetime Symmetries

Part of the book series: Lecture Notes in Physics ((LNP,volume 986))

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Abstract

We will devote the two final chapters of the book to present some elements of classical and quantum field theory on non-commutative \(\kappa \)-Minkowski space, for which the \(\kappa \)-Poincaré algebra  is an algebra of symmetries. We base our presentation below on the papers [1,2,3].

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Notes

  1. 1.

    There are two solutions of the first equation in (4.137), but since the point \({\mathscr {O}}\) for which \(p_4 =1\) must belong to the relevant solution we choose \(p_4\) positive.

  2. 2.

    We also assume that the differential \( d\hat{x}_{4} \) is \( \kappa \)-Poincaré invariant \( \mathsf {p}_{\kappa }\triangleright d\hat{x}_{4}=0 \), where \( \mathsf {p}_{\kappa } \) is a generic element of the \( \kappa \)-Poincaré algebra.

  3. 3.

    Here we have used the fact that the hermitian conjugate of a plane wave involves the antipode map S(k) on its momentum \( \hat{e}_{k}^{\dagger }=\hat{e}_{S(k)} \).

  4. 4.

    It is a left invariant measure \( d\mu (lk)=d\mu (k) \), and it is worth noticing that in bicrossproduct coordinates it is just the diffeomorphism invariant measure on \(dS_{4}\) corresponding to the cosmological metric \( -dk_{0}^{2}+e^{2k_{0}/\kappa }dk_{i}^{2} \).

References

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  2. Arzano, M., Consoli, L.T.: Signal propagation on \(\kappa \)-Minkowski spacetime and nonlocal two-point functions. Phys. Rev. D 98(10), 106018 (2018). arXiv:1808.02241 [hep-th]

  3. Arzano, M., Bevilacqua, A., Kowalski-Glikman, J., Rosati, G., Unger, J.: \(\kappa \)-deformed complex fields and discrete symmetries. Phys. Rev. D 103, 106015 (2021) arXiv:2011.09188 [hep-th]

  4. Amelino-Camelia, G., Arzano, M.: Coproduct and star product in field theories on Lie algebra noncommutative space-times. Phys. Rev. D 65, 084044 (2002). arXiv:hep-th/0105120 [hep-th]

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  6. Arzano, M., Marciano, A.: Fock space, quantum fields and kappa-Poincare symmetries. Phys. Rev. D 76, 125005 (2007). arXiv:0707.1329 [hep-th]

  7. Arzano, M., Marciano, A.: Symplectic geometry and Noether charges for Hopf algebra space-time symmetries. Phys. Rev. D 75, 081701 (2007). arXiv:hep-th/0701268 [hep-th]

  8. Daszkiewicz, M., Lukierski, J., Woronowicz, M.: Towards quantum noncommutative kappa-deformed field theory. Phys. Rev. D 77, 105007 (2008). arXiv:0708.1561 [hep-th]

  9. Daszkiewicz, M., Lukierski, J., Woronowicz, M.: Kappa-deformed oscillators, the choice of star product and free kappa-deformed quantum fields. J. Phys. A 42, 355201 (2009). arXiv:0807.1992 [hep-th]

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Authors and Affiliations

  1. Department of Physics, University of Naples Federico II, Naples, Italy

    Michele Arzano

  2. Institute for Theoretical Physics, University of Wrocław, Wroclaw, Poland

    Jerzy Kowalski-Glikman

  3. National Centre for Nuclear Research, Swierk, Poland

    Jerzy Kowalski-Glikman

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  1. Michele Arzano
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Arzano, M., Kowalski-Glikman, J. (2021). Classical Fields, Symmetries, and Conserved Charges. In: Deformations of Spacetime Symmetries. Lecture Notes in Physics, vol 986. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-63097-6_6

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  • DOI: https://doi.org/10.1007/978-3-662-63097-6_6

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