{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:34:28Z","timestamp":1760243668028,"version":"build-2065373602"},"reference-count":48,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2012,7,3]],"date-time":"2012-07-03T00:00:00Z","timestamp":1341273600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>I report results suggesting that it is possible to introduce laws of relativistic kinematics endowing different types of particles with suitably different deformed-Lorentz-symmetry properties. I also consider some possible applications of these results, among which I highlight those relevant for addressing a long-standing challenge in the description of composite particles, such as atoms, within quantum-gravity-inspired scenarios with Planck-scale deformations of Lorentz symmetry. Some of the new elements here introduced in the formulation of relativistic kinematics appear to also provide the starting point for the development of a correspondingly novel mathematical formulation of spacetime-symmetry algebras.<\/jats:p>","DOI":"10.3390\/sym4030344","type":"journal-article","created":{"date-parts":[[2012,7,3]],"date-time":"2012-07-03T11:06:40Z","timestamp":1341313600000},"page":"344-378","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Particle-Dependent Deformations of Lorentz Symmetry"],"prefix":"10.3390","volume":"4","author":[{"given":"Giovanni","family":"Amelino-Camelia","sequence":"first","affiliation":[{"name":"Dipartimento di Fisica, Sapienza University and INFN, Sez. Roma1, P. le Moro 2, Roma 00185, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2012,7,3]]},"reference":[{"key":"ref_1","first-page":"35","article-title":"Relativity in space-times with short-distance structure governed by an observer-independent (Planckian) length scale","volume":"D11","year":"2002","journal-title":"Int. J. Mod. Phys."},{"key":"ref_2","first-page":"255","article-title":"Testable scenario for Relativity with minimum-length","volume":"B510","year":"2001","journal-title":"Phys. Lett."},{"key":"ref_3","first-page":"391","article-title":"Observer-independent quanta of mass and length","volume":"A286","year":"2001","journal-title":"Phys. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1063\/1.1419321","article-title":"Status of Relativity with observer-independent length and velocity scales","volume":"589","year":"2001","journal-title":"AIP Conf. Proc."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Magueijo, J., and Smolin, L. (2003). Generalized lorentz invariance with an invariant energy scale. Phys. Rev., D67.","DOI":"10.1103\/PhysRevD.67.044017"},{"key":"ref_6","first-page":"299","article-title":"Noncommutative space-time of doubly special relativity theories","volume":"D12","author":"Nowak","year":"2003","journal-title":"Int. J. Mod. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1725","DOI":"10.1088\/0264-9381\/21\/7\/001","article-title":"Gravity\u2019s rainbow","volume":"21","author":"Magueijo","year":"2004","journal-title":"Class. Quant. Grav."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"230","DOI":"10.3390\/sym2010230","article-title":"Doubly-special relativity: Facts, myths and some key open issues","volume":"2","year":"2010","journal-title":"Symmetry"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Magueijo, J. (2006). Could quantum gravity be tested with high intensity lasers?. Phys. Rev., D73.","DOI":"10.1103\/PhysRevD.73.124020"},{"key":"ref_10","first-page":"087702:1","article-title":"Relative locality and the soccer ball problem","volume":"D84","author":"Freidel","year":"2011","journal-title":"Phys. Rev."},{"key":"ref_11","first-page":"1211","article-title":"Kinematical solution of the UHE-cosmic-ray puzzle without a preferred class of inertial observers","volume":"D12","year":"2003","journal-title":"Int. J. Mod. Phys."},{"key":"ref_12","first-page":"084034:1","article-title":"On the fate of Lorentz symmetry in relative-locality momentum spaces","volume":"D85","year":"2012","journal-title":"Phys. Rev."},{"key":"ref_13","unstructured":"Gubitosi, G., and Mercati, F. (2012, June 27). Relative Locality in \u03ba-Poincar\u00e9. Available online: http:\/\/arxiv.org\/abs\/1106.5710v1."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Amelino-Camelia, G., Arzano, M., Kowalski-Glikman, J., Rosati, G., and Trevisan, G. (2012). Relative-locality distant observers and the phenomenology of momentum-space geometry. Class. Quant. Grav., 29.","DOI":"10.1088\/0264-9381\/29\/7\/075007"},{"key":"ref_15","first-page":"084010:1","article-title":"The principle of relative locality","volume":"D84","author":"Freidel","year":"2011","journal-title":"Phys. Rev."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2547","DOI":"10.1007\/s10714-011-1212-8","article-title":"Relative locality: A deepening of the relativity principle","volume":"43","author":"Freidel","year":"2011","journal-title":"Gen. Rel. Grav."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1016\/0370-2693(94)90699-8","article-title":"Bicrossproduct structure of kappa Poincar\u00e9 group and noncommutative geometry","volume":"334","author":"Majid","year":"1994","journal-title":"Phys. Lett. B"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1006\/aphy.1995.1092","article-title":"Classical and quantum-mechanics of free \u03ba-relativistic systems","volume":"243","author":"Lukierski","year":"1995","journal-title":"Ann. Phys."},{"key":"ref_19","unstructured":"Throughout this manuscript I write conservation laws at a process with conventions such that all momenta intervening in the process are incoming into the process, so that indeed a trivalent process would be characterized by a conservation law of the type \n\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t              p\n\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t                \u2295\n\t\t\t\t\t\t\t\t                \u2113\n\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t                p\n\t\t\t\t\t\t\t\t                \u2032\n\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t                \u2295\n\t\t\t\t\t\t\t\t                \u2113\n\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t                p\n\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t                  \u2033\n\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t              =\n\t\t\t\t\t\t\t\t              0\n\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t      . The case of one (or two) of the momenta that is outgoing from the process, say the momentum p, is recovered by simply substituting for p the \u201cantipode\" of the momentum of that outgoing particle, with the antipode \n\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t              \u2296\n\t\t\t\t\t\t\t\t              p\n\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t       defined so that \n\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t              (\n\t\t\t\t\t\t\t\t              \u2296\n\t\t\t\t\t\t\t\t              p\n\t\t\t\t\t\t\t\t              )\n\t\t\t\t\t\t\t\t              \u2295\n\t\t\t\t\t\t\t\t              p\n\t\t\t\t\t\t\t\t              =\n\t\t\t\t\t\t\t\t              0\n\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t      . [For the composition law Equation (17) the antipode is such that \n\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t                  (\n\t\t\t\t\t\t\t\t                  \n\t\t\t\t\t\t\t\t                    \u2296\n\t\t\t\t\t\t\t\t                    \u2113\n\t\t\t\t\t\t\t\t                  \n\t\t\t\t\t\t\t\t                  p\n\t\t\t\t\t\t\t\t                  )\n\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t                j\n\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t              =\n\t\t\t\t\t\t\t\t              -\n\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t                p\n\t\t\t\t\t\t\t\t                j\n\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t              +\n\t\t\t\t\t\t\t\t              \u2113\n\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t                p\n\t\t\t\t\t\t\t\t                0\n\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t                p\n\t\t\t\t\t\t\t\t                j\n\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t       and \n\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t                  (\n\t\t\t\t\t\t\t\t                  \n\t\t\t\t\t\t\t\t                    \u2296\n\t\t\t\t\t\t\t\t                    \u2113\n\t\t\t\t\t\t\t\t                  \n\t\t\t\t\t\t\t\t                  p\n\t\t\t\t\t\t\t\t                  )\n\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t                0\n\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t              =\n\t\t\t\t\t\t\t\t              -\n\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t                p\n\t\t\t\t\t\t\t\t                0\n\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t      .]"},{"key":"ref_20","unstructured":"Notice that in this specific example the limit of \n\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t              p\n\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t                \u2295\n\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t                  \u2113\n\t\t\t\t\t\t\t\t\t                  \u22c6\n\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t              k\n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t       as \n\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t              p\n\t\t\t\t\t\t\t\t\t              \u2192\n\t\t\t\t\t\t\t\t\t              0\n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t       is not k. So scenarios such as this require a profound distinguishability between a single-particle system and a multi-particle system, a feature which is plausible but surprising. I should stress that this feature is not a general aspect of my \u201cmixing composition laws\", and indeed in the example Equation (30) one does find that \n\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t              p\n\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t                \u2295\n\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t                  \u2113\n\t\t\t\t\t\t\t\t\t                  \u22c6\n\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t              k\n\t\t\t\t\t\t\t\t\t              \u2192\n\t\t\t\t\t\t\t\t\t              k\n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t       as \n\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t              p\n\t\t\t\t\t\t\t\t\t              \u2192\n\t\t\t\t\t\t\t\t\t              0\n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t      . Also notice that in this subsection I could have opted for adopting as mixing composition law the limit \n\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t              \u03bb\n\t\t\t\t\t\t\t\t\t              \u2192\n\t\t\t\t\t\t\t\t\t              0\n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t       of the composition law Equation (62) from the next section, and in that case the limit \n\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t              p\n\t\t\t\t\t\t\t\t\t              \u2192\n\t\t\t\t\t\t\t\t\t              0\n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t       does give k. As a matter of fact I opted for showing here Equation (46) rather than the limit \n\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t              \u03bb\n\t\t\t\t\t\t\t\t\t              \u2192\n\t\t\t\t\t\t\t\t\t              0\n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t       of Equation (62), just so that the content of this subsection would not simply amount to the \n\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t              \u03bb\n\t\t\t\t\t\t\t\t\t              \u2192\n\t\t\t\t\t\t\t\t\t              0\n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t       of the content of the next section, thereby showing some evidence of the variety of possibilities that could be considered (including the possibility of having \n\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t              p\n\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t                \u2295\n\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t                  \u2113\n\t\t\t\t\t\t\t\t\t                  \u22c6\n\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t              k\n\t\t\t\t\t\t\t\t\t              \u2260\n\t\t\t\t\t\t\t\t\t              k\n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t       in the limit \n\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t              p\n\t\t\t\t\t\t\t\t\t              \u2192\n\t\t\t\t\t\t\t\t\t              0\n\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t      )."},{"key":"ref_21","unstructured":"Adam, T., Agafonova, N., Aleksandrov, A., Altinok, O., Alvarez Sanchez, P., Anokhina, A., Aoki, S., Ariga, A., Ariga, T., and Autiero, D. 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(2004). Hopf-algebra description of noncommutative-spacetime symmetries. Int. J. Mod. Phys., A19.","DOI":"10.1142\/S0217751X04020919"},{"key":"ref_48","unstructured":"Note that a conservation law of the type \n\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                p\n\t\t\t\t\t\t\t\t\t\t                0\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              +\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                k\n\t\t\t\t\t\t\t\t\t\t                0\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              =\n\t\t\t\t\t\t\t\t\t\t              0\n\t\t\t\t\t\t\t\t\t\t              ,\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                p\n\t\t\t\t\t\t\t\t\t\t                j\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              +\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                k\n\t\t\t\t\t\t\t\t\t\t                j\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              +\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                (\n\t\t\t\t\t\t\t\t\t\t                \u2113\n\t\t\t\t\t\t\t\t\t\t                +\n\t\t\t\t\t\t\t\t\t\t                \u03bb\n\t\t\t\t\t\t\t\t\t\t                )\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                p\n\t\t\t\t\t\t\t\t\t\t                0\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                k\n\t\t\t\t\t\t\t\t\t\t                j\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              \/\n\t\t\t\t\t\t\t\t\t\t              2\n\t\t\t\t\t\t\t\t\t\t              =\n\t\t\t\t\t\t\t\t\t\t              0\n\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t       is equivalent to \n\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                p\n\t\t\t\t\t\t\t\t\t\t                0\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              +\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                k\n\t\t\t\t\t\t\t\t\t\t                0\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              =\n\t\t\t\t\t\t\t\t\t\t              0\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              ,\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                p\n\t\t\t\t\t\t\t\t\t\t                j\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              +\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                k\n\t\t\t\t\t\t\t\t\t\t                j\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              +\n\t\t\t\t\t\t\t\t\t\t              \u03bb\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                p\n\t\t\t\t\t\t\t\t\t\t                0\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                k\n\t\t\t\t\t\t\t\t\t\t                j\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              \/\n\t\t\t\t\t\t\t\t\t\t              2\n\t\t\t\t\t\t\t\t\t\t              -\n\t\t\t\t\t\t\t\t\t\t              \u2113\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                k\n\t\t\t\t\t\t\t\t\t\t                0\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                p\n\t\t\t\t\t\t\t\t\t\t                j\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              \/\n\t\t\t\t\t\t\t\t\t\t              2\n\t\t\t\t\t\t\t\t\t\t              =\n\t\t\t\t\t\t\t\t\t\t              0\n\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t       since (for \n\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                p\n\t\t\t\t\t\t\t\t\t\t                0\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              +\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                k\n\t\t\t\t\t\t\t\t\t\t                0\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              =\n\t\t\t\t\t\t\t\t\t\t              0\n\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t       and working at leading order) one has that from \n\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                p\n\t\t\t\t\t\t\t\t\t\t                j\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              +\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                k\n\t\t\t\t\t\t\t\t\t\t                j\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              +\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t                  \u2113\n\t\t\t\t\t\t\t\t\t\t                  +\n\t\t\t\t\t\t\t\t\t\t                  \u03bb\n\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t                2\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                p\n\t\t\t\t\t\t\t\t\t\t                0\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                k\n\t\t\t\t\t\t\t\t\t\t                j\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              =\n\t\t\t\t\t\t\t\t\t\t              0\n\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t       it follows that \n\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                (\n\t\t\t\t\t\t\t\t\t\t                1\n\t\t\t\t\t\t\t\t\t\t                +\n\t\t\t\t\t\t\t\t\t\t                \u2113\n\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t                  k\n\t\t\t\t\t\t\t\t\t\t                  0\n\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t                \/\n\t\t\t\t\t\t\t\t\t\t                2\n\t\t\t\t\t\t\t\t\t\t                )\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                [\n\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t                  p\n\t\t\t\t\t\t\t\t\t\t                  j\n\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t                +\n\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t                  k\n\t\t\t\t\t\t\t\t\t\t                  j\n\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t                +\n\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t                  \n\t\t\t\t\t\t\t\t\t\t                    \u2113\n\t\t\t\t\t\t\t\t\t\t                    +\n\t\t\t\t\t\t\t\t\t\t                    \u03bb\n\t\t\t\t\t\t\t\t\t\t                  \n\t\t\t\t\t\t\t\t\t\t                  2\n\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t                  p\n\t\t\t\t\t\t\t\t\t\t                  0\n\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t                  k\n\t\t\t\t\t\t\t\t\t\t                  j\n\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t                ]\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              =\n\t\t\t\t\t\t\t\t\t\t              0\n\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t       and in turn \n\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                p\n\t\t\t\t\t\t\t\t\t\t                j\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              +\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                k\n\t\t\t\t\t\t\t\t\t\t                j\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              +\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                \u03bb\n\t\t\t\t\t\t\t\t\t\t                2\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                p\n\t\t\t\t\t\t\t\t\t\t                0\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                k\n\t\t\t\t\t\t\t\t\t\t                j\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              -\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                \u2113\n\t\t\t\t\t\t\t\t\t\t                2\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                k\n\t\t\t\t\t\t\t\t\t\t                0\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t                p\n\t\t\t\t\t\t\t\t\t\t                j\n\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t              =\n\t\t\t\t\t\t\t\t\t\t              0\n\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t      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