Line Element in Noncommutative Geometry

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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)) = −dφ|x .

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)) = −dφ|x .

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. ω 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 , ω ] = 0 d = ( Pα + ω Mµν ) with α β 1 µν βα x , = − 2 ω 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. ω 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 , ω ] = 0 d = ( Pα + ω Mµν ) with α β 1 µν βα x , = − 2 ω 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 , ω ] = 0 d = ( Pα + ω Mµν ) with α β 1 µν βα x , = − 2 ω Γµν

βα . β α α β where Γµν = θ[ µδν ]θ[ µδν ].

µν I d contains a pure Lorentz component, i.e. ω 6= 0, the translation sector cannot be zero: no-pure Lorentz principle.

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 , ω ] = 0 d = ( Pα + ω Mµν ) with α β 1 µν βα x , = − 2 ω Γµν

βα . β α α β where Γµν = θ[ µδν ]θ[ µδν ].

µν I d contains a pure Lorentz component, i.e. ω 6= 0, the translation sector cannot be zero: no-pure Lorentz principle.

I similarly in κ-Minkowski there is a no-pure boost principle.

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 ωµν , ?

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 ωµν , ?

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 ∞ (M), noncommutative algebra A µ → Dirac operator − iγ ∇µ 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. ∞ . I coherent with the commutative case: A ∈ C (M), ωx (f ) = f (x)

d(ωx , ωy ) = 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.

Geometry without point but distance available . d(ω1, ω2) = sup {|ω1(a) − ω2(a)| ; k[D, π(a)]k ≤ 1} a∈A with ω1, ω2 states of A (linear positive maps A → C with norm 1).

Notion of distance “dual” to the classical one 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.

Geometry without point but distance available . d(ω1, ω2) = sup {|ω1(a) − ω2(a)| ; k[D, π(a)]k ≤ 1} a∈A with ω1, ω2 states of A (linear positive maps A → C with norm 1).

Notion of distance “dual” to the classical one ∞ . I coherent with the commutative case: A ∈ C (M), ωx (f ) = f (x)

d(ωx , ωy ) = dg´eo(x, y), I does not rely on some notions ill-deﬁned in a quantum context, but only on the spectral properties of A and D.

Geometry without point but distance available . d(ω1, ω2) = sup {|ω1(a) − ω2(a)| ; k[D, π(a)]k ≤ 1} a∈A with ω1, ω2 states of A (linear positive maps A → C with norm 1).

Notion of distance “dual” to the classical one ∞ . I coherent with the commutative case: A ∈ C (M), ωx (f ) = f (x)

d(ωx , ωy ) = dg´eo(x, y),

I still makes sense in the noncommutative case, Geometry without point but distance available . d(ω1, ω2) = sup {|ω1(a) − ω2(a)| ; k[D, π(a)]k ≤ 1} a∈A with ω1, ω2 states of A (linear positive maps A → C with norm 1).

Notion of distance “dual” to the classical one ∞ . I coherent with the commutative case: A ∈ C (M), ωx (f ) = f (x)

d(ωx , ωy ) = 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. Standard model:

∞ A = C (M) ⊗ AI with AI = C ⊕ H ⊕ M3(C).

X1 .. Y1. H

. X2 Y2 . C

The distance d coincides with the geodesic distance in M × [0, 1] with

µν g 0 h1 2 2 2 o`u is the Higgs doublet. 0 |1 + h1| + |h2| mtop h2 ! No extra-dimension: product of a discrete ﬁber by a continuum Euclidean space.

J. Math. Phys. 43 (2002) 182-204, P.M., R. Wulkenhaar;

J. Geom. Phys. 31 (2000) 100-125, B. Iochum, P.M., T. Krajewski. 2 2 0 . 5 Regarding distance, DI ' K II so that, writing ∂0 = IM , γ = Kγ , 2 µ 2 2 A 2 D = ((γ ∂µ) + K IM ) ⊗ II = (γ ∂A) Hence D is the inverse of a “Riemannian line element”.

I For the product of geometry,

µ 5 2 µ 2 2 D = −iγ ∂µ ⊗ II + γ ⊗ DI =⇒ D = (γ ∂µ) ⊗ II + IM ⊗ DI −2 −2 −2 −1 =⇒ ds = dsM + dsI =⇒ Pythagore

I Two-sheet model of the standard model, µν g 0 2 2 2 2 2 2 2 =⇒ ds = dsM +dsI =⇒ Pythagore. 0 K = |1 + h1| + |h2| mtop

Line element in noncommutative geometry:

ds = D−1.

Flat spacetime (for curved one, more complicated (Lichnerowiz formula)),

2 µ ν µ ν µν µ ν −1 D = γ ∂µ(γ ∂ν ) = γ γ ∂µ∂ν = g ∂µ∂ν = (gµν dx dx ) . 2 2 0 . 5 Regarding distance, DI ' K II so that, writing ∂0 = IM , γ = Kγ , 2 µ 2 2 A 2 D = ((γ ∂µ) + K IM ) ⊗ II = (γ ∂A) Hence D is the inverse of a “Riemannian line element”.

I Two-sheet model of the standard model, µν g 0 2 2 2 2 2 2 2 =⇒ ds = dsM +dsI =⇒ Pythagore. 0 K = |1 + h1| + |h2| mtop

Line element in noncommutative geometry:

ds = D−1.

Flat spacetime (for curved one, more complicated (Lichnerowiz formula)),

2 µ ν µ ν µν µ ν −1 D = γ ∂µ(γ ∂ν ) = γ γ ∂µ∂ν = g ∂µ∂ν = (gµν dx dx ) .

I For the product of geometry,

µ 5 2 µ 2 2 D = −iγ ∂µ ⊗ II + γ ⊗ DI =⇒ D = (γ ∂µ) ⊗ II + IM ⊗ DI −2 −2 −2 −1 =⇒ ds = dsM + dsI =⇒ Pythagore 2 2 0 . 5 Regarding distance, DI ' K II so that, writing ∂0 = IM , γ = Kγ , 2 µ 2 2 A 2 D = ((γ ∂µ) + K IM ) ⊗ II = (γ ∂A) Hence D is the inverse of a “Riemannian line element”.

Line element in noncommutative geometry:

ds = D−1.

Flat spacetime (for curved one, more complicated (Lichnerowiz formula)),

2 µ ν µ ν µν µ ν −1 D = γ ∂µ(γ ∂ν ) = γ γ ∂µ∂ν = g ∂µ∂ν = (gµν dx dx ) .

I For the product of geometry,

µ 5 2 µ 2 2 D = −iγ ∂µ ⊗ II + γ ⊗ DI =⇒ D = (γ ∂µ) ⊗ II + IM ⊗ DI −2 −2 −2 −1 =⇒ ds = dsM + dsI =⇒ Pythagore

I Two-sheet model of the standard model, µν g 0 2 2 2 2 2 2 2 =⇒ ds = dsM +dsI =⇒ Pythagore. 0 K = |1 + h1| + |h2| mtop Line element in noncommutative geometry:

ds = D−1.

Flat spacetime (for curved one, more complicated (Lichnerowiz formula)),

2 µ ν µ ν µν µ ν −1 D = γ ∂µ(γ ∂ν ) = γ γ ∂µ∂ν = g ∂µ∂ν = (gµν dx dx ) .

I For the product of geometry,

µ 5 2 µ 2 2 D = −iγ ∂µ ⊗ II + γ ⊗ DI =⇒ D = (γ ∂µ) ⊗ II + IM ⊗ DI −2 −2 −2 −1 =⇒ ds = dsM + dsI =⇒ Pythagore

I Two-sheet model of the standard model, µν g 0 2 2 2 2 2 2 2 =⇒ ds = dsM +dsI =⇒ Pythagore. 0 K = |1 + h1| + |h2| mtop

2 2 0 . 5 Regarding distance, DI ' K II so that, writing ∂0 = IM , γ = Kγ , 2 µ 2 2 A 2 D = ((γ ∂µ) + K IM ) ⊗ II = (γ ∂A) Hence D is the inverse of a “Riemannian line element”. In the standard model within Connes NCG, one postulates a line element in order to recover the gauge group.

I Notion of dimension is subtle: from the distance point of view, illusion of extra-dimension, that comes from the line elements satisfying Pythagore relation.

I But the metric dimension (deﬁned as the rate of decrease of the eigenvalues of D) is still m = dimM.

I Still another dimension (KO dimension), important for massive neutrinos (see Chamseddine, Connes, Marcolli and Barrett).

Conclusion

In Minkowski NCG, no distance clearly available. Line element meaningful in term of tangent vector to curves.

I Constraints on acceptable curves come from Leibniz rule.

I Constraints also question the dimension of the associated diﬀerential calculus. Conclusion

In Minkowski NCG, no distance clearly available. Line element meaningful in term of tangent vector to curves.

I Constraints on acceptable curves come from Leibniz rule.

I Constraints also question the dimension of the associated diﬀerential calculus.

In the standard model within Connes NCG, one postulates a line element in order to recover the gauge group.

I Notion of dimension is subtle: from the distance point of view, illusion of extra-dimension, that comes from the line elements satisfying Pythagore relation.

I But the metric dimension (deﬁned as the rate of decrease of the eigenvalues of D) is still m = dimM.

I Still another dimension (KO dimension), important for massive neutrinos (see Chamseddine, Connes, Marcolli and Barrett).