A free Lie algebra approach to curvature corrections to flat space-time
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Springer
Received: July 23, 2020 Accepted: August 10, 2020 Published: September 9, 2020
Joaquim Gomis,a Axel Kleinschmidt,b,c Diederik Roestd and Patricio Salgado-Rebolledoe a
Departament de F´ısica Qu` antica i Astrof´ısica and Institut de Ci`encies del Cosmos (ICCUB), Universitat de Barcelona, Mart´ı i Franqu`es, ES-08028 Barcelona, Spain b Max-Planck-Institut f¨ ur Gravitationsphysik (Albert-Einstein-Institut), Am M¨ uhlenberg 1, DE-14476 Potsdam, Germany c International Solvay Institutes, ULB-Campus Plaine CP231, BE-1050 Brussels, Belgium d Van Swinderen Institute for Particle Physics and Gravity, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands e School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: We investigate a systematic approach to include curvature corrections to the isometry algebra of flat space-time order-by-order in the curvature scale. The Poincar´e algebra is extended to a free Lie algebra, with generalised boosts and translations that no longer commute. The additional generators satisfy a level-ordering and encode the curvature corrections at that order. This eventually results in an infinite-dimensional algebra that we refer to as Poincar´e∞ , and we show that it contains among others an (A)dS quotient. We discuss a non-linear realisation of this infinite-dimensional algebra, and construct a particle action based on it. The latter yields a geodesic equation that includes (A)dS curvature corrections at every order. Keywords: Global Symmetries, Space-Time Symmetries, Classical Theories of Gravity ArXiv ePrint: 2006.11102
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)068
JHEP09(2020)068
A free Lie algebra approach to curvature corrections to flat space-time
Contents 1
2 The 2.1 2.2 2.3
Poincar´ e∞ algebra Free Lie algebras and Maxwell∞ Quotient to Poincar´e∞ Expansion of (A)dS
3 3 5 7
3 The 3.1 3.2 3.3
Poincar´ e∞ coset Non-linear realisation Invariant metrics Relation to Minkowski and (A)dS space
8 8 10 11
4 The Poincar´ e∞ particle
13
5 Conclusions and outlook
16
A The A.1 A.2 A.3
18 18 18 19
1
(Anti) de Sitter case The algebra The (A)dS coset The particle
Introduction
The laws of elementary particle physics are relativistic to very high precision. This is described by Minkowski geometry whose isometries span the Poincar´e algebra that provides the underlying kinematical structure for the vast majority of field theories. However, in a gravitational context, Minkowski space is replaced by a curved space-time 1 and a range of astronomical and cosmological observations indicate that we live in an expanding and accelerating space-time. To very good accuracy (see e.g. [1], and modulo the currently emerging Hubble tension), its evolution is described by Λ-CDM, whose dominant component is a very small and positive cosmological constant (of the meV order). In the absence of ma
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