Algebraic Footprints of Quantum Gravity: a Stability Point of View - PowerPoint PPT Presentation

Algebraic Footprints of Quantum Gravity: a Stability Point of View Chryssomalis Chryssomalakos ICN - UNAM (based on joint work with Elias Okon (ICN - UNAM)) Reference: Int. J. Mod. Phys. D 13/10 (2004) 2003–2034 ( hep-th/0410212 )

Contents 1. Motivation 2. Lie algebra deformations 3. Stable quantum relativistic kinematics 4. Physical implications (in progress) 5. To do list

Motivation ◮ Non-commutative spacetime [ X µ , X ν ] = ? ◮ Modified dispersion relations E 2 = p 2 + m 2 + ? Preferred frames — Lorentz symmetry violation ◮ ◮ Invariant length scale � � G → ℓ P ≡ Quantum Gravity c 3 + Lorentz contraction ⇓ ?

◮ The stability criterion Galileo Einstein c c [ J a , J b ] = i ǫ ab J c [ J a , J b ] = i ǫ ab J c stabilize c c [ J a , K b ] = i ǫ ab K c − → [ J a , K b ] = i ǫ ab K c c [ K a , K b ] = 0 [ K a , K b ] = i t ǫ ab J c Newton Heisenberg stabilize [ f ( q, p ) , g ( q, p )] = 0 − → [ f ( q, p ) , g ( q, p )] = i � { f ( q, p ) , g ( q, p ) }

Lie Algebra Deformations • Lie algebras Lie algebra ( V, µ ) V : finite-dimensional vector space (over R ) µ : Lie product µ : V × V → V bilinear: µ ( λx + ρy ) = λµ ( x ) + ρµ ( y ) antisymmetric: µ ( x, y ) = − µ ( y, x ) Jacobi: µ ( x, µ ( y, z )) = µ ( µ ( x, y ) , z ) + µ ( y, µ ( x, z )) C Basis { T A } , A = 1 , . . . , n of V ⇒ structure constants f s.t. AB C T C [ T A , T B ] ≡ i µ ( T A , T B ) = i f AB R + f BR R + f CR R = 0 S f BC S f CA S f AB f AR Jacobi: (relax)

� � � f n − 1 ,n n L n GL ( n ) orbit of Q P ψ 1 P M Q GL ( n ) orbit of P 2 f 12 1 f 12 Figure 1: The space L n of n -dimensional Lie algebras (sketch). C = M A C f RS B T B R M B S ( M − 1 ) U U T ′ f ′ A = M A ⇒ AB Orb( P ) open ⇒ G P stable ( rigid ), otherwise unstable

• Deformations G 0 = ( V, µ 0 ) , µ 0 ( X, Y ) ≡ [ X, Y ] 0 One-parameter (formal) deformation of G 0 : ∞ � ψ m ( X, Y ) t m deformed commutator : [ X, Y ] t = [ X, Y ] 0 + m =1 C T C [ T A , T B ] t = i f t t -dependent f ’s : AB ψ m : V × V → V , bilinear, antisymmetric ( 2-cochains over V ) Vector space of p -cochains : C p ( V ) 1-cochains: V → V linear, C 1 ( V ) ∼ Aut( V ) 0-cochains: constant maps, C 0 ( V ) ∼ V

• Coboundary operator For any µ , coboundary operator s µ : C p → C p +1 , p � � � T A r , ψ ( p ) ( T A 1 , . . . , ˆ s µ ⊲ ψ ( p ) ( T A 0 , . . . , T A p ) = ( − 1) r µ T A r , . . . , T A p ) r =0 ( − 1) r + s ψ ( p ) � � � µ ( T A r , T A s ) , T A 0 , . . . , ˆ T A r , . . . , ˆ + T A s , . . . , T A p r<s Examples: ( φ ∈ C 1 , ψ ∈ C 2 ) s µ ⊲ φ ( A 1 , A 2 ) = [ A 1 , φ ( A 2 )] − [ A 2 , φ ( A 1 )] − φ ([ A 1 , A 2 ]) s µ ⊲ ψ ( A 1 , A 2 , A 3 ) = [ A 1 , ψ ( A 2 , A 3 )] − [ A 2 , ψ ( A 1 , A 3 )] + [ A 3 , ψ ( A 1 , A 2 )] − ψ ([ A 1 , A 2 ] , A 3 ) + ψ ([ A 1 , A 3 ] , A 2 ) − ψ ([ A 2 , A 3 ] , A 1 ) Jacobi for µ ⇒ s 2 µ = 0

• Cohomology groups Jacobi for µ t ⇒ s µ 0 ⊲ ψ 1 = 0 ψ 1 ∈ Z 2 ( V, s µ ) ( 2-cocycle — similarly Z p ) ⇒ where µ t ( X, Y ) ≡ [ X, Y ] t = [ X, Y ] 0 + ψ 1 ( X, Y ) t + . . . The deformation is trivial iff ∃ φ ∈ C 1 ( V ) s.t. ψ 1 = s µ 0 ⊲ φ ψ 1 ∈ B 2 ( V, s µ ) ( 2-coboundary , trivial 2-cocycle — similarly B p ) ⇒ Non-trivial deformations generated by non-trivial 2-cocycles (tangent space interpretation, slide 6) H p ≡ Z p /B p p -th cohomology group of G 0 H 2 ( G 0 ) trivial ⇒ G 0 stable (converse not true) ⇒ semisimple Lie algebras stable

• The ⊼ product ⊼ : C p × C q → C p + q − 1 α ⊼ β ( X 0 , . . . , X m + n ) = � � � sgn ( σ ) α β ( X σ (0) , . . . , X σ ( n ) ) , X σ ( n +1) , . . . , X σ ( m + n ) σ Graded commutator : � α, β � = α ⊼ β − ( − 1) mn β ⊼ α ( α ∈ C m +1 , β ∈ C n +1 ) µ ⊼ µ = 1 Jacobi for µ ⇔ 2 � µ, µ � = 0 In general: s µ ⊲ ψ = ( − 1) p � µ, ψ � Assume � µ, µ � = 0 ( µ Lie product). µ t = µ + φ t also Lie product iff s µ ⊲ φ t − 1 � µ t , µ t � = 0 ⇒ 2 � φ t , φ t � = 0 deformation equation

• Obstructions and H 3 ∞ � φ n t n µ t = µ + φ t , φ t = n =1 ⇒ Defomation equation s µ ⊲ φ 1 = 0 s µ ⊲ φ 2 = 1 2 � φ 1 , φ 1 � s µ ⊲ φ 3 = � φ 1 , φ 2 � . . . H 3 ( G ) � = 0 φ 1 ∈ H 2 ( G ) might be non-integrable ⇒ Notice : � φ 1 , φ 1 � = 0 ⇒ µ + φ 1 t Lie product

• Coboundary operator as exterior covariant derivative Π A : left invariant 1-forms, � Π A , T B � A = δ B ψ ( p ) → ψ B ⊗ T B ≡ 1 B Π A 1 . . . Π A p ⊗ T B p ! ψ A 1 ...A p A Π R Ω A s µ → ∇ = d + Ω , B = f RB B T B ) (components defined by: ψ ( p ) ( T A 1 , . . . , T A p ) = ψ A 1 ...A p Example: Galilean kinematics c c [ J a , J b ] = i ǫ ab J c , [ J a , K b ] = i ǫ ab K c , [ K a , K b ] = 0 µ = 1 ab Π a Π b ⊗ J c + ǫ b ⊗ K c ¯ c ab Π a Π c 2 ǫ a Π ¯ b ⊗ J c , with � χ KKJ , χ KKJ � = 0 Only non-trivial 2-cocycle: χ KKJ = 1 ab Π ¯ c 2 ǫ c ⇒ [ K a , K b ] t = i tǫ ab J c Experiment says: t = − 1 c 2

Heisenberg’s Route Classical relativity G CR ( � = 0 ) � � [ J µν , J ρσ ] = i g µσ J νρ + g νρ J µσ − g µρ J νσ − g νσ J µρ � � [ J ρσ , P µ ] = i g µσ P ρ − g µρ P σ � � [ J ρσ , Z µ ] = i g µσ Z ρ − g µρ Z σ , plus M central. Algorithm: A Π A ⊗ T B (225 terms) B 1. Most general 1-cochain: φ = φ 2. Most general 2-coboundary: ψ = ∇ φ (1008 terms) AB Π A Π B ⊗ T C (1575 terms) C 3. Most general 2-cochain: χ = χ 4. Require χ a 2-cocycle, ∇ χ = 0 (5672 equations in 1575 unknowns) 5. ⇒ 221 2-cocycles χ i . For each χ i , solve χ i = ψ (348075 equations)

6. ⇒ only five χ i non-trivial: H 2 ( G CR ) = { [0] , [ ψ H ] , [ ψ PMZ ] , [ ψ ZMP ] , [ ψ PMP ] , [ ψ ZMZ ] } where ψ H = Π µ Π ˙ µ ⊗ M ψ PMZ = Π µ Π M ⊗ Z µ µ Π M ⊗ P µ ψ ZMP = Π ˙ ψ PMP = Π µ Π M ⊗ P µ µ Π M ⊗ Z µ ψ ZMZ = Π ˙ Deform along ψ H only → G PH ( q )

Stable Quantum Relativistic Kinematics G PH ( q ) (Poincar´ e plus Heisenberg): � � [ J µν , J ρσ ] = i g µσ J νρ + g νρ J µσ − g µρ J νσ − g νσ J µρ � � [ J ρσ , P µ ] = i g µσ P ρ − g µρ P σ � � [ J ρσ , Z µ ] = i g µσ Z ρ − g µρ Z σ [ P µ , Z ν ] = i q g µν M µ PH ( q ) = 1 2Π αρ Π β ρ ⊗ J αβ + Π αρ Π ρ ⊗ P α + Π αρ Π ˙ ρ ⊗ Z α + q Π µ Π ˙ ⇒ µ ⊗ M H 2 ( G PH ( q )) = { [0] , [ ζ 1 ] , [ ζ 2 ] , [ ζ 3 ] } where ζ 1 = Π µ Π M ⊗ Z µ + q 2Π µ Π ν ⊗ J µν µ Π M ⊗ P µ + q ν ⊗ J µν ζ 2 = − Π ˙ 2Π ˙ µ Π ˙ µ Π M ⊗ Z µ − Π µ Π M ⊗ P µ + q Π µ Π ˙ ν ⊗ J µν ζ 3 = Π ˙

� ζ i , ζ j � = 0 ⇒ µ ( q, � α ) = µ PH ( q ) + α 1 ζ 1 + α 2 ζ 2 + α 3 ζ 3 Lie product. Stable quantum relativistic kinematics: � � [ J µν , J ρσ ] = i g µσ J νρ + g νρ J µσ − g µρ J νσ − g νσ J µρ � � [ J ρσ , P µ ] = i g µσ P ρ − g µρ P σ � � [ J ρσ , Z µ ] = i g µσ Z ρ − g µρ Z σ [ P µ , Z ν ] = i qg µν M + i q α 3 J µν [ P µ , P ν ] = i q α 1 J µν [ Z µ , Z ν ] = i q α 2 J µν [ P µ , M ] = − i α 3 P µ + i α 1 Z µ [ Z µ , M ] = − i α 2 P µ + i α 3 Z µ provided α 2 3 � = α 1 α 2 . When α 2 3 = α 1 α 2 , χ = ζ 1 + ζ 2 is a non-trivial integrable 2-cocycle

α 3 P AST G PH ( q ) G QR so (3 , 3) α 2 χ so (1 , 5) so (2 , 4) F UTURE E LSEWHERE Figure 2: The ( α 1 , α 2 , α 3 ) -deformation space of G PH ( q ) .

Physical Implications (in progress) • S. Sivasubramanian, G. Castellani, N. Fabiano, A. Widom, J. Swain, Y. N. Srivastava, G. Vitiello, “Non-commutative Geometry and Measurements of Polarized Two Photon Coincidence Counts”, Annals Phys. 311 (2004) 191–203 • D. Ahluwalia-Khalilova, “A Freely Falling Frame at the Interface of the Gravitational and Quantum Realms”, Class. Quantum Grav. 22 (2005) 1433-1450 • D. Ahluwalia-Khalilova, “Minimal Spatio-Temporal Extent of Events, Neutrinos, and the Cosmological Constant Problem”, hep-th/0505124 (honorable mention in the 2005 Essay Competition of the Gravity Research Foundation) Standard wisdom ( q = 1 ): [ P µ , P ν ] = i 1 1 √ R 2 J µν R = Λ [ Z µ , Z ν ] = iℓ 2 ℓ 2 P J µν P ≡ G ⇒ noncommutative spacetime, energy- momentum space

However: J µν , P µ : primitive (extensive), e.g. , total angular momentum: J tot = J 1 + J 2 Positions not primitive X µ not Lie algebra generators ⇒ X 12 = M 1 X 1 + M 2 X 2 Newtonian limit: M 1 + M 2 � � � ⇒ Z µ = X µ M primitive M = P µ P µ [ Z µ , Z ν ] = iq ( X µ P ν − X ν P µ ) = iqL µν Spinless particles: α 2 = 1 (commutative spacetime!)

To Do List ◮ so (1 , 5) -representations, Casimirs ◮ Wigner like particle description ◮ Relativistic Z µ ? Higher spin? Zero mass? ◮ Non-commutative spacetime? ◮ Invariant length ⇒ momentum cutoff? ◮ Invariant length + Lorentz contraction = ? ◮ Supersymmetry?