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Line Element in Noncommutative Geometry Line element in noncommutative geometry P. Martinetti G¨ottingenUniversit¨at Wroclaw, July 2009 . ? ? - & ? !? The line element p µ ν ds = gµν dx dx is mainly useful to measure distance Z y d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). ? - . ? !? The line element p µ ν ds = gµν dx dx & ? is mainly useful to measure distance Z y d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). ? - !? The line element p µ ν ds = gµν dx dx . & ? ? is mainly useful to measure distance Z y d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). ? - The line element p µ ν ds = gµν dx dx . & ? ? is mainly useful to measure distance Z y !? d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). The line element p µ ν ds = gµν dx dx . & ? ? is mainly useful to measure distance ? -Z y !? d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). Two unrelated illustrations on how to define the line element in NCG, unrelated but both question the notion of dimension: 1. Permitted line elements from Noether analysis, noncommutative deformation of Minkowski spacetime. 2. Line element from generalized Dirac operator, Connes' description of the standard model. In order preserve the commutation relation, one deforms the Leibniz rule, MA(fg) 6= (MAf )g + f (MAg) = (MA ⊗ 1 + 1 ⊗ MA)(f ⊗ g) by defining a non trivial coproduct ∆MA(f ⊗ g) = MA(1)f : MA(2)g: 1. Permitted line elements from Noether analysis Deformed Minkowski and deformed Poincar´e κ − Minkovski : [xj ; x0] = iλxj ; [xj ; xk ] = 0 θ − Minkovski : [xµ; xν ] = iθµν : The Poincare group, Pµ = @µ; Mµν = xµ@ν − xν @µ; acts on a function f on deformed Minkowski space-time via a Weyl map, MAf = Ω(MAf~) where MA = Pµ; Mµν and f~ is a function on ordinary Minkowski space. 1. Permitted line elements from Noether analysis Deformed Minkowski and deformed Poincar´e κ − Minkovski : [xj ; x0] = iλxj ; [xj ; xk ] = 0 θ − Minkovski : [xµ; xν ] = iθµν : The Poincare group, Pµ = @µ; Mµν = xµ@ν − xν @µ; acts on a function f on deformed Minkowski space-time via a Weyl map, MAf = Ω(MAf~) where MA = Pµ; Mµν and f~ is a function on ordinary Minkowski space. In order preserve the commutation relation, one deforms the Leibniz rule, MA(fg) 6= (MAf )g + f (MAg) = (MA ⊗ 1 + 1 ⊗ MA)(f ⊗ g) by defining a non trivial coproduct ∆MA(f ⊗ g) = MA(1)f : MA(2)g: Deformed Minkowski space-times might occur in quantum gravity. Hopefully would yield observable effects through a deformation of the dispersion relation 2 2 2 2 4 E = p c + m c + f (E; p; lp; :::): Meaning of E, p ? Reliable notion: Noether charges. This is possible for, at first order, : δφ(x) = φ0(x) − φ(x) A = − MAφ(x) 0 : = −(φ(x ) − φ(x)) = −dφjx : A Since φ may be non linear in x, the procedure requires that d = @A satisfies the Leibniz rule. Classical Noether analysis : R Write the variation of an action I = Ω L(φ(x)) dx under a combined transformation φ 7! φ0 : φ0(x) = φ(Λ−1x); x 7! x 0 : x 0 = Λx; A for an infinitesimal transformation Λ = 1 + MA, as a 4-divergence Z A 4 µ δI = d x PµJA = 0: Ω Classical Noether analysis : R Write the variation of an action I = Ω L(φ(x)) dx under a combined transformation φ 7! φ0 : φ0(x) = φ(Λ−1x); x 7! x 0 : x 0 = Λx; A for an infinitesimal transformation Λ = 1 + MA, as a 4-divergence Z A 4 µ δI = d x PµJA = 0: Ω This is possible for, at first order, : δφ(x) = φ0(x) − φ(x) A = − MAφ(x) 0 : = −(φ(x ) − φ(x)) = −dφjx : A Since φ may be non linear in x, the procedure requires that d = @A satisfies the Leibniz rule. µν I d contains a pure Lorentz component, i.e. ! 6= 0, the translation sector cannot be zero: no-pure Lorentz principle. I similarly in κ-Minkowski there is a no-pure boost principle. Solution: only some linear combinations of the generators are viable candidates for Noether symmetry. For instance in θ-Minkowski α µν α µν [x ;! ] = 0 d = ( Pα + ! Mµν ) with α β 1 µν βα x ; = − 2 ! Γµν βα : β α α β where Γµν = θ[ µδν ]θ[ µδν ]: - Results obtained with Amelino-Camelia's group in Rome, \La Sapienza", with his students A. Marciano, G. Gubitosi. F. Mercato and graduating students. - Similar results for κ-Minkowski by Freidel, Kowalski-Glikman, Nowak. Noncommutative Noether analysis A d = MA satisfying the Leibniz rule may contradict the (possible) non trivial coproduct of MA. µν I d contains a pure Lorentz component, i.e. ! 6= 0, the translation sector cannot be zero: no-pure Lorentz principle. I similarly in κ-Minkowski there is a no-pure boost principle. Noncommutative Noether analysis A d = MA satisfying the Leibniz rule may contradict the (possible) non trivial coproduct of MA. Solution: only some linear combinations of the generators are viable candidates for Noether symmetry. For instance in θ-Minkowski α µν α µν [x ;! ] = 0 d = ( Pα + ! Mµν ) with α β 1 µν βα x ; = − 2 ! Γµν βα : β α α β where Γµν = θ[ µδν ]θ[ µδν ]: - Results obtained with Amelino-Camelia's group in Rome, \La Sapienza", with his students A. Marciano, G. Gubitosi. F. Mercato and graduating students. - Similar results for κ-Minkowski by Freidel, Kowalski-Glikman, Nowak. I similarly in κ-Minkowski there is a no-pure boost principle. Noncommutative Noether analysis A d = MA satisfying the Leibniz rule may contradict the (possible) non trivial coproduct of MA. Solution: only some linear combinations of the generators are viable candidates for Noether symmetry. For instance in θ-Minkowski α µν α µν [x ;! ] = 0 d = ( Pα + ! Mµν ) with α β 1 µν βα x ; = − 2 ! Γµν βα : β α α β where Γµν = θ[ µδν ]θ[ µδν ]: µν I d contains a pure Lorentz component, i.e. ! 6= 0, the translation sector cannot be zero: no-pure Lorentz principle. - Results obtained with Amelino-Camelia's group in Rome, \La Sapienza", with his students A. Marciano, G. Gubitosi. F. Mercato and graduating students. - Similar results for κ-Minkowski by Freidel, Kowalski-Glikman, Nowak. Noncommutative Noether analysis A d = MA satisfying the Leibniz rule may contradict the (possible) non trivial coproduct of MA. Solution: only some linear combinations of the generators are viable candidates for Noether symmetry. For instance in θ-Minkowski α µν α µν [x ;! ] = 0 d = ( Pα + ! Mµν ) with α β 1 µν βα x ; = − 2 ! Γµν βα : β α α β where Γµν = θ[ µδν ]θ[ µδν ]: µν I d contains a pure Lorentz component, i.e. ! 6= 0, the translation sector cannot be zero: no-pure Lorentz principle. I similarly in κ-Minkowski there is a no-pure boost principle. - Results obtained with Amelino-Camelia's group in Rome, \La Sapienza", with his students A. Marciano, G. Gubitosi. F. Mercato and graduating students. - Similar results for κ-Minkowski by Freidel, Kowalski-Glikman, Nowak. However in the classical case A 0 dφ = hdextφ, V i = @Aφ =: φ A with V = @A the vector tangent to the curve generated by the infinitesimal transformation. So that 2 0 2 d φ = hdextφ ; V i = B A @A;B φ has no reason to vanish. Hence d 6= dext. d is rather the vector tangent to the authorised Noether-candidate A symmetry (more exactly @A = line element). α What are these coefficients !µν ; ? Tempting to identify α = dx α: α : α In κ-Minkowski, allows to define MAdx = d(MAx ), and to check that the commutators [x α; β] are Lorentz covariant iif one uses a five dimensional differential calculus. α What are these coefficients !µν ; ? Tempting to identify α = dx α: α : α In κ-Minkowski, allows to define MAdx = d(MAx ), and to check that the commutators [x α; β] are Lorentz covariant iif one uses a five dimensional differential calculus. However in the classical case A 0 dφ = hdextφ, V i = @Aφ =: φ A with V = @A the vector tangent to the curve generated by the infinitesimal transformation. So that 2 0 2 d φ = hdextφ ; V i = B A @A;B φ has no reason to vanish. Hence d 6= dext. d is rather the vector tangent to the authorised Noether-candidate A symmetry (more exactly @A = line element). 2. Line element from generalized Dirac operator Connes: commutative algebra C 1 (M); noncommutative algebra A µ ! Dirac operator − iγ rµ generalized Dirac D l # Riemannian spin manifold M noncommutative geometry Central idea: most of the geometrical information is encoded within a suitable generalisation of the Dirac operator. 1 : I coherent with the commutative case: A 2 C (M), !x (f ) = f (x) d(!x ;!y ) = dg´eo(x; y); I still makes sense in the noncommutative case, I does not rely on some notions ill-defined in a quantum context, but only on the spectral properties of A and D.
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