PoS(FFP14)210 http://pos.sissa.it/ ∗ [email protected] [email protected] [email protected] Speaker. We discuss the problem ofthrough theory its change kinematical in symmetries. physics.as We The Inönü-Wigner characterize progressions extensions a of of physical theories the theorymodern is associated ideas based in kinematical mathematically philosophy groups. described of science – Thisadvantage e.g. of is the remaining semantic compatible conceptually approach simple with to and a the admitting scientificwith a theory, well hints with defined the at mathematical structure, a logicpropose extensions of of discovery. our idea in This the is final also discussion. linked to the Bargmann-Wigner program. We ∗ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

Frontiers of Fundamental Physics 14 -15-18 FFP14, July 2014 Aix Marseille University (AMU) Saint-Charles Campus, Marseille Samuel Simon Departamento de Filosofia, Universidade de Brasilia,DF, Caixa Brazil. Postal 04661, 70919-970, Brasilia, E-mail: Amilcar de Queiroz Instituto de Fisica, Universidade de Brasilia,Brazil. Caixa Postal 04455, 70919-970, Brasilia, DF, Departamento de Física Teórica, Facultad deZaragoza, Ciencias, Spain. Universidad de Zaragoza, 50009 E-mail: M. Lachièze-Rey APC - Astroparticule et Cosmologie (UMRAlice 7164), Domon Université et Paris Léonie 7 Duquet, Denis F-75205 Diderot,E-mail: 10, Paris Cedex rue 13, France. Symmetry, Physical Theories and Theory Changes PoS(FFP14)210 (2.1) (2.2) (2.3) (direct p ⊕ 0. In abbre- ]. h . 3 ) [ 6= p = ε , one constructs g + g 0, but not for the h Amilcar de Queiroz ( 6= 2 ε , i.e., , with ε g J obeying ⊂ ε for 0 in ] . Then, the new Lie algebra g p degeneration . g p = . . g p 0 ε 0 it acts (unitarily) on a Hilbert , J + p h ] = ε 0 p ⊂ , ] 0 p simplicity ] = [ p 0 [ , p p [ , 0 p [ ] is best described in terms of its associated , 0 by a generator 10 p , p , with complement p quantum domain g ⊂ , ∈ ]. Starting from a Lie algebra 2 0 ] ⊂ 0 ⊂ p J ] 11 p p h , = , h , which are isomorphic to [ h p [ ε is defined (functorially) as a group homomorphism from g ε ]. In the G 7 ⊂ , ] , and of their progression through the notion of Inönü-Wigner . The algebra remains the same, but with the reparametrized , h p p h 0 of the parameter ( in turn its generates a new Lie group via the ε ⊂ ε , ⊂ ] h ] → h = h , 0, are obtained by reparametrizations of 0 ε , h p [ h ] = [ [ 6= 0 p ε , 0 gives a well-defined but different Lie algebra h contains a subalgebra [ g , the kinematic group acts as canonical transformations on the “phase space” → , for symmetry groups ε 0. becomes ε g p = , h ε ⊂ ] h , h [ The algebras A linear representation of a group Section 2 introduces the mathematical notions of Inönü-Wigner contraction and extension, for The Inönü-Wigner contraction of a Lie group [ We assume that This work presents a perspective for the characterization of scientific (physical) theories based classical domain to the group of endomorphisms of the linear space. We can then handle the elements of a a parametrized family of newsingular algebras, value emerges as the singular limit sum of vector spaces). The commutators can then be schematically decomposed as Lie algebra which can be seena as new Lie its algebra, infinitesimal not counterpart. isomorphic The toby the process initial allows one singular one transformations to but of construct preserving the somebe of infinitesimal its generalized elements structure. (the to It generators) proceeds other and, algebraic in this structures sense, [ it can exponential map). Such a contraction may be seen as a special case of groups and algebras. Section 3group applies to to kinematical conformal groups group, (andSection via algebras) 4 leading Poincaré shortly from group Galileo presents andconcluding some deSitter/anti-deSitter remarks. analogies (cosmological) with groups. quantization and non-commutive geometry, and G group as transformations acting on thethe underlying linear space associated to a physical system. In space of states. This is also known as (anti-)unitary representation. 2. Contractions and Extensions of Groups 2.1 Group contractions The reparametrization replaces each generator endowed with a symplectic structure [ contraction/extension, which may be seen as a path towards on the existence of Theory Changes 1. Introduction commutation relations viated notation, The singular limit PoS(FFP14)210 , . 0 (2.7) (2.6) (2.5) (3.1) (2.4) p E(2). × h only (with = 3 0 . J ) g 2 . If the second ( 0 E p Amilcar de Queiroz . The Inönü-Wigner n ) = h 2 ( . . = Inönü-Wigner extension 2 as 0 j . g 2 ISO j Λ direct product ] = . 1 3 by j ] = → J simple group augm , 1 p j . 3 conformal = j ) , ) [ 3 3 3 2 , so that we may write the diagram j ( j ( → ) [ 3 two-dimensional Euclidean group ( . SO k , J 1 SO , j ext of Lie algebras → , 2 1 i jk J j ) of the ε as the subalgebra generated by ext → 2 1 2 ] = Λ ( 3 h 3 = ) ∑ j R k 3 ] = 2 = , 3 ( j n 2 2 ISO (anti-)deSitter j j , ) ] = [ 2 j 2 j augmentation SO ISO ( → J [ , → i augm so J [ , is the group of rotations of the two-dimensional Euclidean , ) 1 3 ) 2 J j ) = , ( . We rescale the elements of 2 2 0 ( p 2 Λ ( Einstein semi-direct product e SO Λ = ]. We will see below further extensions of this diagram. SO ] = 3, with commutation relations → 1 2 , 0, then the new Lie algebra would be a j j 15 ] = 2 , 2 , j 1 1 j ] = , 0 generate [ 1 p j = 2 , Galilei [ J i h , of rotations in three-dimensional Euclidean space, admits the Lie algebra , , [ i ) 1 J J 3 0 provides a new Lie algebra with three generators obeying the relations ( → SO ) were Λ Euclidean space [ 2.3 is the Levi-Civita symbol. We define 0), so that 0, the algebra remains the same, although with the new expression for the commutators i jk 6= ] = ε , generated by 3 It expresses the transition from pre-Newtonian to Newtonian physics, through the introduction A group that cannot be written as a (semi-)direct product is a The limit The kinematic group of a theory is the group of isometries of the space-time of that theory. The new Lie algebra is the The group ) Λ J 3 , isotropic ( 3 J of 2.2 Extensions contraction diminishes simplicity.which There achieves simplicity. is It an extends apler inverse Lie group: procedure, group with which the less is (semi-)direct athe products (semi-)direct previous of product, contraction groups. may towards For a be instance, written sim- the as extension associated with plane; it is natural toisometries. associate This corresponds to to it the the natural group of translations of the same space, to complete the They characterize the Lie algebra The (special) orthogonal group For We will consider the sequence of theory change Here 3. Study of Case of Theory Change: from Newton to Cosmology relation in ( [ Theory Changes so 2.1.1 A First Example of Inönü-Wigner Contraction PoS(FFP14)210 , ) = H z a , L y stands a l , x a , where the 3 = ( R 3. The four 4- where , a ~ × 3 2 combines a time , , 1 R ) 1 a four-dimensional Amilcar de Queiroz generate the Galilei , R R i 0 , 3: the Hamiltonian n v L C , l ,~ = 2 ∈ a , ,~ ν = 1 , R b p µ = component, which does not , j generate space-time rotations. = ( ) , i g , µν νµ ) , generated by spatial translations J J , generated by the spatial rotations , i j − G µ U L P , i = ( C , µν i J P , H ( is a 4-translation and . The Lorentz group is stable, i.e., admits no generated by time translations. The factor group 4 c which exchanges inertial frames. 4 / ) IR proper orthochronous T z , which comprises the space-time rotations, which v ) ∈ . The quotient (group of classes of equivalence) , 1 generate spatial-translations, the ; a three-dimensional spatial translation 3 y a , , of the Galilei group i v 1 R 3 ) P , ( x 3 R v (the group of classes of equivalence) is not a simple group: ( SO admits ten generators, where U = ( p , of three-dimensional rotations. ISO = / ) v ~ . G R 3 R , L a R (we consider its , with parameter 1 may be written as the semi-direct product ) = ( 1 spatial rotations. P admits then ten generators g , is the group of isometries of the Newtonian space-time P . ji 3 is not simple. It admits a natural Inönü-Wigner extension to the Lorentz G ( G L ) of L 3 − p ( SO = i j ISO ext → L ) generate the space-time translations; the six is the simple group , a three-dimensional rotation 3 b µ ( P is the Lorentz group T / 3 and spatial translations , ) ISO 1 ) The Galilei group The general element The above kinematics were discussed by A. Einstein in 1905, where he proposed a kinematics The group The Lie algebra of The Galilei group admits the maximal Abelian subgroup The Poincaré group We will consider the subgroup U 3 R ( / / G Theory Changes 3.1 Newtonian Kinematic: the Galilei Group first component stands for the time direction. A generic element for the Lie algebra of (Lorentz) rotation. The Lie algebra translation momenta further similar extension. This liftsto to the an Poincaré extension group, of i.e., their fromhowever not Newtonian prolongations: kinematics simple from and to the the special Galilei process relativity. group may The be Poincaré continued. group is covariant under the symmetries ofishment the among electromagnetism. physicists due Before to A. theelectrodynamics fact Einstein (i.e., that there the the was symmetries Poincaré an of group)i.e., aston- the differed the Maxwell Galilei from equations group. describing those of the Newton-Galilei kinematics, 3.3 From Galilean Kinematics to Special Relativity group and a Galilei transform (or boost) P combine three-dimensional spatial rotations withgroup Lorentz and boosts. the It latter is is aand also subgroup the of known Lie the algebra as Poincaré the inhomogeneous Lorentz group. This is a simple group ( and Galilei boosts. The quotient it still contains a (maximal) Abelian subgroup generates time-translations; the momenta boosts and the SO 3.2 Special Relativity: the Poincaré Group include time-reversal and parity) actsspace-time as translations and isometries rotations, of together with the their Minkowskigenerate combinations. its space-time. The maximal space-time It translations Abelian subgroup includes the PoS(FFP14)210 ) 2 , 3 con- ( exten- for the SO . It may q , . Both dS p = R special con- , the deSitter P 2 − AdS Λ ... ) ) ± 2 Amilcar de Queiroz 2 , 2 ,  , 4 4 ( 4 R ( / ' SO / SO , respectively. / ) 2 ie sg , sg kinematic groups act as isome- 3 ext natural augmentations ( ) ) 1 1 , 1 ,  ISO , 4 ) 4 ( completes the space-time translations 4 1 4. simple R ( , ) / augm 4 respectively).  1 SO and ( / , SO 2 ) , ,..., 3 3 2 ( 1 = , ] it may constitute the symmetry group of a , ie R ( sg ISO 3 0 ( 18 dS ISO / = ) 0). These and , and four generating the so called 1 SO = 5 b admits also a family of natural extensions with a 1 , 1 , > , , sg  Λ 3 3 ) 4 = a . This gives the anti-deSitter group ( P 1 Λ R , R ( , / ) 3 ba (sg); and vertical ones ) SO ( 1 / J 1 . , → , augm q − ) 3 4 ie , sg ( 1 ( ) ISO p , = augm 1  . Both admit a common extension under the form of the 3 , ) ( SO ab ISO ) 3 2 J ( 3 , 3 ' =  ( SO = 3 / . R ( SO / dS SO replaced by AdS P / / sg ) c c 1 1 ISO 1 ie sg , scaling transformations 4 cosmological constant . This group plays an important role in physics. For instance it preserves ( stand respectively for Euclidean plane and Euclidean space; and . The latter admits the ten generators of dS (or AdS), augmented by five subgroup inclusion ) ) ) ) 3 ) 2 2 0 corresponds to their common contraction, the Poincaré group 2 3 1 ,  3 ( augm ( ISO , R  ( 4 R , 4 G ( → 2 ( SO $ SO ISO R Λ and / ) called the ISO 1 sg , Λ ext 4 ( ) ) 2 SO 0) or the deSitter group 2 augm ; oblique arrows ( The limit Furthermore, the groups in the above series act as isometries of the space-times indicated in Going further, the natural augmentations of dS and AdS (by adding " translations ") give We resume the sequence through the following diagram. Horizontal arrows indicate The natural augmentation  ( = < SO ISO Λ formal transformations electromagnetism. As proposed initially by Weyl [ new generators, one for formal group SO pseudo–Euclidean space of signature the diagram below, where the symbol ie stands for isometrical embedding. with the space-time rotations. Then, and AdS are stable, in theLie sense algebras that admit they ten admit generators no further similar extension. For each of them, the sions be continued, but without straightforwarddS applications to physics. A similar version applies with The notations parameter respectively ( tries on the respective cosmological space-times ofand constant anti–de non zero Sitter curvatures space-times (embedded in conformal theory of gravitation Theory Changes 3.4 From Special Relativity to Cosmology PoS(FFP14)210 , ] ) A 1 8 Γ , 19 , 1 (1954). . Quantization 59 A Amilcar de Queiroz led Connes and his -algebra ∗ C manifold to a non commutative procedure, which associates to ]. This intends for instance to give 16 [ (discrete) non-commutative space: the phase space spectral action principle internal 6 algebra deformation ]. The Gel’fand duality — in fact a categorial equivalence above also admits a geometrical interpretation, at the basis 6 A ], considers an of quantum observables (seen as operators acting on an Hilbert 5 of classical observables (functions on a symplectic manifold ]. This is connected with the Bargmann-Wigner program [ → , A 4 (NCG) [ A non-commutative space-time 12 (not made of points) to a non-commutative ]). http://arxiv.org/abs/math/0703701 17 http://arxiv.org/abs/hep-th/0611263 of the parameter. This procedure bears some similarities with the extension of Lie ¯ h Poisson algebra A ] — associates a space with the commutative algebra of functions on it. NCG associates a 43(5), 321-332. of all Fundamental Interactions including Gravity. Part I, Fortsch. Phys. 58 (2010) 553-600. dimension”, 14 We have applied the perspective of extensions of (kinematical) groups and their associated The geometrical interpretation of algebra extension (diagram B) considers each algebra in the Comparable procedures also account for the transition from the classical to the quantum do- ][ non-commutative geometry ]. We have achieved, in our sense, a precision which is not obtained by other exclusively logic [3] Burde, D. (2007). Contractions of Lie algebras and algebraic groups. Archivum Mathematicum, [4] Ali H. Chamseddine and Alain Connes, Noncommutative Geometry as a Framework for Unification [2] Bekaert, X. and Boulanger, N. “The Unitary representations of the Poincaré group in any space-time [1] Bargmann, V. “On Unitary ray representations of continuous groups”, Annals Math. 13 space-times, with contraction providing thehence way of back scientific to discovery [ the problem of theory evolution — and is so interpreted geometrically asspace. an upgrading A of similar a new procedure physics may in also the frame be of applied a to space-time, with the goal of constructing a product of this internal spaceframework by the for (commutative) the space-time manifold matter providescollaborators fields. the to a geometrical Application pure geometrical of derivation of a the physics of both standard model and gravity. when he states thatcharacterize despite specific their disciplines. adequateness, the logic approaches are too generalReferences to usefully approaches like for instance the semantic one. The same conclusion is given by Halvorson [ 2 of [ non-commutative space a phenomenological description ofthe the manifold effects structure of of quantumA. space-time gravity Connes at which and small are collaborators scales. thought [ to Another destroy original possibility, developed by chain as the isotropy algebra of aThe manifold, deformation and quantization so provides a progression chain of these manifolds. main: the processcommutative of quantization corresponds to the replacement (with some conditions) of a Theory Changes 4. Discussion and Conclusion by a non-commutative algebra space). This can be accomplished by an algebras (see, e.g., [ a parametrized family of non-commutativewith algebras. the value The quantum algebra of interest is obtained PoS(FFP14)210 , 40 , M Novello and S E Amilcar de Queiroz Frontiers of Fundamental ) , issue 3-4, 384 (1918). 2 Cosmology and Gravitation . 7 , 9 (1989)]. 6 http://proceedings.aip.org/proceedings , Perth (Australia) 2010, American Institute of Physics conference proceedings vol 1246, p 114-126 ( Physics Perez Bergliaffa ed, Cambridge Scientific Publishers 2012, p 185-198 Physics, in press plications. Cambridge University Press, 2nd edition. http://mathoverflow.net/questions/60633/ is-there-any-relation-between-deformation-and-extension-of-lie-algebras 149 (1939) [Nucl. Phys. Proc. Suppl. neutrino mixing. Adv. Theor. Math. Phys., 11:991—1089. 183-206. Istanbul, Turkey. Proceedings of the National Academy of510. 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