Noncommutative Spacetime Geometries

Transcription

1 Munich Noncommutative Spacetime Geometries Paolo Aschieri Centro Fermi, Roma, U.Piemonte Orientale, Alessandria Memorial Conference in Honor of Julius Wess

2 I present a general program in NC geometry where noncommutativity is given by a *-product. I work in the deformation quantization context, and discuss a quite wide class of *-products obtained by Drinfeld twist. *-differential geometry *-Lie derivative, *-covariant derivative *-Riemannian geometry metric structure on NC space *-gravity [P.A., Blohmann, Dimitrijevic, Meyer, Schupp, Wess] [P.A., Dimitrijevic, Meyer, Wess]

3 I present a general program in NC geometry where noncommutativity is given by a *-product. I work in the deformation quantization context, and discuss a quite wide class of *-products obtained by Drinfeld twist. *-differential geometry *-Lie derivative, *-covariant derivative *-Riemannian geometry metric structure on NC space *-gravity [P.A., Blohmann, Dimitrijevic, Meyer, Schupp Wess] [P.A., Dimitrijevic, Meyer, Wess] *-gauge theories [P.A., Dimitrijevic, Meyer, Schraml, Wess]

4 I present a general program in NC geometry where noncommutativity is given by a *-product. I work in the deformation quantization context, and discuss a quite wide class of *-products obtained by Drinfeld twist. I II *-differential geometry *-Lie derivative, *-covariant derivative *-Riemannian geometry metric structure on NC space *-gravity [P.A., Blohmann, Dimitrijevic, Meyer, Schupp Wess] [P.A., Dimitrijevic, Meyer, Wess] *-gauge theories [P.A., Dimitrijevic, Meyer, Schraml, Wess] *-Poisson geometry Poisson structure on NC space *-Hamiltonian mechanics [P.A., Fedele Lizzi, Patrizia Vitale] Application: Canonical quantization on noncommutative space, deformed algebra of creation and annihilation operators.

5 Why Noncommutative Geometry? Classical Mechanics Classical Gravity Quantum mechanics It is not possible to know at the same time position and velocity. Observables become noncommutative. = x10-34 Joule s Panck constant Quantum Gravity Spacetime coordinates become noncommutative L P = cm Planck length

6 La=h/mc La=h/mc = =h/mc

7 Below Planck scales it is natural to conceive a more general spacetime structure. A noncommutative one where (as with quantum mechanics phase-space) uncertainty relations and discretization naturally arise. Space and time are then described by a Noncommutative Geometry In this framework it is relevant to construct a theory of gravity on noncommutative spacetime.

8 Below Planck scales it is natural to conceive a more general spacetime structure. A noncommutative one where (as with quantum mechanics phase-space) uncertainty relations and discretization naturally arise. Space and time are then described by a Noncommutative Geometry In this framework it is relevant to construct a theory of gravity on noncommutative spacetime. We consider a -product geometry, The class of -products we consider is not the most general one studied by Kontsevich. It is not obtained by deforming directly spacetime. We first consider a twist of spacetime symmetries, and then deform spacetime itself.

9 Ingredients for a twist F g Lie algebra Ug universal enveloping algebra of g (i.e. complex numbers C and sums of products of elements t g, where tt t t =[t, t ]). Ug is an associative algebra with unit. It is a Hopf algebra: (t) =t 1+1 t ε(t) =0 S(t) = t cf. Leibnitz rule t(ab) =t(a) b + at(b) The coproduct is extended to all Ug by requiring and ε to be linear and multiplicative (e.g. (tt ) := (t) (t )=tt 1+t t + t t +1 tt ).

10 The twist F F Ug Ug [Drinfeld 83, 85] F is invertible F 1( id)f =1 F(id )F Example 1: in Ug Ug Ug This is a cocycle condition it implies associativity of the --product F = e i 2 λθab X a X b where [X a,x b ]=0. Here g is the abelian Lie algebra generated by the X a s. θ ab is a constant (antisymmetric) matrix. F

11 Example 2: a nonabelian example, Jordanian deformations, [Ogievetsky, 93] F = e 1 2 H log(1+λe), [H, E] =2E

12 -Noncommutative Manifold (M, g, F) M smooth manifold g Lie algebra acting on M (we have ρ : g Ξ) F Ug Ug and therefore F UΞ UΞ. A algebra of smooth functions on M A noncommutative algebra Usual product of functions: -Product of functions f h µ fh f h µ f h µ = µ F 1 Associativity of this -product follows from the cocycle condition for the twist A and A are the same as vector spaces, they have different algebra structure

13 -Noncommutative Manifold (M, g, F) M smooth manifold g Lie algebra acting on M (we have ρ : g Ξ) F Ug Ug and therefore F UΞ UΞ A algebra of smooth functions on M A noncommutative algebra Usual product of functions: f h µ fh -Product of functions f h µ f h [Connes, Landi] µ = µ F 1 [Connes, Dubois-Violette] isospectral deformations [Majid, Gurevich], A and A are the same as vector spaces, they have different algebra structure [Varilly,] [Sitarz]

14 Examples of NC spaces x i x j x j x i = iθ ij canonical y x i x i y = iax i, x i x j x j x i =0 Lie algebra type x i x j qx j x i =0 quantum (hyper)plane

15 Example: M = R 4 F = e i 2 λθµν x µ x ν 1 1 i 2 λθµν µ ν 1 8 λ2 θ µ 1ν 1 θ µ 2ν 2 µ1 µ2 ν1 ν2 +. (f h)(x) =e 2 i λθµν x µ y ν f(x)h(y) x=y

16 Example: M = R 4 F = e i 2 λθµν x µ x ν 1 1 i 2 λθµν µ ν 1 8 λ2 θ µ 1ν 1 θ µ 2ν 2 µ1 µ2 (f h)(x) =e 2 i λθµν x µ y ν f(x)h(y) x=y Notation F = f α f α F 1 = f α f α

17 Example: M = R 4 F = e i 2 λθµν x µ x ν = 1 1 i 2 λθµν µ ν 1 8 λ2 θ µ 1ν 1 θ µ 2ν 2 µ1 µ2 ν1 ν (f h)(x) =e 2 i λθµν x µ y ν f(x)h(y) x=y Notation F = f α f α F 1 = f α f α

18 Example: M = R 4 F = e i 2 λθµν x µ x ν = 1 1 i 2 λθµν µ ν 1 8 λ2 θ µ 1ν 1 θ µ 2ν 2 µ1 µ2 ν1 ν (f h)(x) =e 2 i λθµν x µ y ν f(x)h(y) x=y Notation F = f α f α F 1 = f α f α f h = f α (f) f α (h)

19 Example: M = R 4 F = e i 2 λθµν x µ x ν = 1 1 i 2 λθµν µ ν 1 8 λ2 θ µ 1ν 1 θ µ 2ν 2 µ1 µ2 ν1 ν (f h)(x) =e 2 i λθµν x µ y ν f(x)h(y) x=y Notation F = f α f α F 1 = f α f α f h = f α (f) f α (h) Define F 21 = f α f α and define the universal R-matrix: R := F 21 F 1 := R α R α, R 1 := R α R α It measures the noncommutativity of the -product:

20 Example: M = R 4 F = e i 2 λθµν x µ x ν 1 1 i 2 λθµν µ ν 1 8 λ2 θ µ 1ν 1 θ µ 2ν 2 µ1 µ2 ν1 ν (f h)(x) =e 2 i λθµν x µ y ν f(x)h(y) x=y Notation F = f α f α F 1 = f α f α f h = f α (f) f α (h) Define F 21 = f α f α and define the universal R-matrix: R := F 21 F 1 := R α R α, R 1 := R α R α It measures the noncommutativity of the -product: f g = R α (g) R α (f) Noncommutativity is controlled by R 1. The operator R 1 gives a representation of the permutation group on A.

21 F deforms the geometry of M into a noncommutative geometry. Guiding principle: given a bilinear map µ : X Y Z (x, y) µ(x, y) =xy. deform it into µ := µ F 1, µ : X Y Z (x, y) µ (x, y) =µ( f α (x), f α (y)) = f α (x) f α (y). The action of F UΞ UΞ will always be via the Lie derivative. -Tensorfields τ τ = f α (τ) f α (τ ). In particular h v, h df, df h etc...

22 -Forms ϑ ϑ = f α (ϑ) f α (ϑ ). Exterior forms are totally -antisymmetric. For example the 2-form ω ω is the -antisymmetric combination ω ω = ω ω R α (ω ) R α (ω). The undeformed exterior derivative d:a Ω satisfies (also) the Leibniz rule d(h g )=dh g + h dg, and is therefore also the -exterior derivative.

23 -Action of vectorfields, i.e., -Lie derivative L v (f) =v(f) usual Lie derivative L v : Ξ A A (v, f) v(f). deform it into L := L F 1, L v (f) = f α ( f (v) α (f) ) _ -Lie derivative Deformed Leibnitz rule L v(f h) =L v(f) h + R α (f) L R α (v) (h). Deformed coproduct in U Ξ (v) =v 1+ R α R α (v) Deformed product in UΞ _ u v = f α (u)f α (v)

24 -Lie algebra of vectorfields Ξ [P.A., Dimitrijevic, Meyer, Wess, 06] Theorem 1. (UΞ,,,S, ε) is a Hopf algebra. It is isomorphic to Drinfeld s UΞ F with coproduct F.

25 -Lie algebra of vectorfields Ξ [P.A., Dimitrijevic, Meyer, Wess, 06] Theorem 1. (UΞ,,,S, ε) is a Hopf algebra. It is isomorphic to Drinfeld s UΞ F with coproduct F. Theorem 2. The bracket [u, v] =[ f α (u), f α (v)] gives the space of vectorfields a -Lie algebra structure (quantum Lie algebra in the sense of S. Woronowicz): [u, v] = [ R α (v), R α (u)] [u, [v, z] ] = [[u, v],z] +[ R α (v), [ R α (u),z] ] [u, v] = L u(v) -antisymmetry -Jacoby identity the -bracket is the -adjoint action

26 -Lie algebra of vectorfields Ξ [P.A., Dimitrijevic, Meyer, Wess, 06] Theorem 1. (UΞ,,,S, ε) is a Hopf algebra. It is isomorphic to Drinfeld s UΞ F with coproduct F. Theorem 2. The bracket [u, v] =[ f α (u), f α (v)] gives the space of vectorfields a -Lie algebra structure (quantum Lie algebra in the sense of S. Woronowicz): [u, v] = [ R α (v), R α (u)] [u, [v, z] ] = [[u, v],z] +[ R α (v), [ R α (u),z] ] [u, v] = L u(v) -antisymmetry -Jacoby identity the -bracket is the -adjoint action Theorem 3. (UΞ,,,S, ε) is the universal enveloping algebra of the -Lie algebra of vectorfields Ξ. In particular [u, v] = u v R α (v) R α (u). To every undeformed infinitesimal transformation there correspond one and only one deformed infinitesimal transformation: L v L v.

27 We can now proceed in our study of differential geometry on noncommutative manifolds. Covariant derivative: h u v = h uv, u(h v )=L u(h) v + R α (h) R α (u) v our coproduct implies that R α (u) is again a vectorfield On tensorfields: u(v z)= u(v) z + R α (v) R α (u) (z). The torsion T and the curvature R associated to a connection are the linear maps T : Ξ Ξ Ξ, and R : Ξ Ξ Ξ Ξ defined by for all u, v, z Ξ. T(u, v) := uv R α (v) R α (u) [u, v], R(u, v, z) := u vz R α (v) R α (u) z [u,v] z,

28 -antisymmetry property T(u, v) = T( R α (v), R α (u)), R(u, v, z) = R( R α (v), R α (u),z). A -linearity shows that T and R are well defined tensors: T(f u, v) =f T(u, v), T(u, f v) = R α (f) T( R α (u),v) and similarly for the curvature. Local coordinates description We denote by {e i } a local frame of vectorfields (subordinate to an open U M) and by {θ j } the dual frame of 1-forms: e i, θ j = δ j i. ei, θ j = f α (e i ), f α θ j ). Connection coefficients Γ ij k, ei e j = Γ k ij e k

29 T = 1 2 θj θ i T ij l e l, R = 1 2 θk θ j θ i R ijk l e l.

30 T = 1 2 θj θ i T ij l e l, R = 1 2 θk θ j θ i R ijk l e l. Connection forms ω j i, the torsion forms Θl and the curvature forms Ω k l by ω i j := θ k Γ ki j, Θ l := 1 2 θj θ i T ij l, Ω k l := 1 2 θj θ i R kij l,

31 T = 1 2 θj θ i T ij l e l, R = 1 2 θk θ j θ i R ijk l e l. Connection forms ω j i, the torsion forms Θl and the curvature forms Ω k l by ω i j := θ k Γ ki j, Θ l := 1 2 θj θ i T ij l, Ω k l := 1 2 θj θ i R kij l, Cartan structural equations hold Bianchi identities Θ l = dθ l θ k ω k l, Ω k l = dω k l ω k m ω m l.

32 T = 1 2 θj θ i T ij l e l, R = 1 2 θk θ j θ i R ijk l e l. Connection forms ω j i, the torsion forms Θl and the curvature forms Ω k l by ω i j := θ k Γ ki j, Θ l := 1 2 θj θ i T ij l, Ω k l := 1 2 θj θ i R kij l, Cartan structural equations hold Bianchi identities Θ l = dθ l θ k ω k l, Ω k l = dω k l ω k m ω m l. dθ i + Θ j ω j i = θ j Ω j i, dω l k + Ω k m ω m l ω k m Ω m l =0.

33 Example, F =e i 2 θµν µ ν µ, dx ν = δ ν µ µ ( ν )=Γ ρ µν ρ T µν ρ = Γ µν ρ Γ νµ ρ R µνρ σ = µ Γ νρ σ ν Γ µρ σ + Γ νρ β Γ µβ σ Γ µρ β Γ νβ σ. Ric µν = R ρµν ρ.

34 -Riemaniann geometry -symmetric elements: ω ω + R α (ω ) R α (ω). Any symmetric tensor in Ω Ω is also a -symmetric tensor in Ω Ω, proof: expansion of above formula gives f α (ω) f α (ω )+ f α (ω ) f α (ω). g = g µν dx µ dx ν Γ ρ µν = 1 2 ( µg νσ + ν g σµ σ g µν ) g σρ Noncommutative Gravity Ric = 0 g νσ g σρ = δ ρ ν. [P.A., Blohmann, Dimitrijevic, Meyer, Schupp, Wess] [P.A., Dimitrijevic, Meyer, Wess]

35 Conclusions and outlook This theory of gravity on noncomutative spacetime is completely geometric. It uses deformed Lie derivatives and covariant derivatives. Can be formulated without use of local coordinates. It holds for a quite wide class of star products. Therefore the formalism is well suited in order to consider a dynamical noncommutativity. The idea being that both the metric and the noncommutative structure of spacetime could be dependent on the matter/energy content. Exact solutions of this theory are under investigations First order formalism and coupling to spinors, supergravity. Dynamical nc in flat spacetime [P.A., Castellani, Dimitrijevic] Lett. Math.Phys 2008