The standard model from the metric point of view P. Martinetti - PowerPoint PPT Presentation

The standard model from the metric point of view P. Martinetti Georg-August-Universit¨ at G¨ ottingen Hamburg may 2008

Description of the standard model within the framework of noncommutative geometry. Discrete structure of spacetime even without quantum gravity. commutative algebra → noncommutative algebra � ↓ differential geometry noncommutative geometry A geometry “without points”, but the notion of distance is available via Connes formula. The metric information is encoded within the Dirac operator ds = D − 1

Description of the standard model within the framework of noncommutative geometry. Discrete structure of spacetime even without quantum gravity. commutative algebra → noncommutative algebra � ↓ differential geometry noncommutative geometry A geometry “without points”, but the notion of distance is available via Connes formula. The metric information is encoded within the Dirac operator ds = D − 1 ◮ Riemannian compact spin manifold M : − i γ µ ∂ µ ⇐ ⇒ riemannian geodesic distance. ◮ Fibre bundle P with connection: − i γ µ ( ∂ µ + A µ ) ⇐ ⇒ ?

Outline: 1. The spectral triple of the standard model spectral triple connection and the product of the continum by the discrete the standard model 2. Distance in noncommutative geometry 3. Fluctuations of the metric scalar fluctuation and the standard model gauge fluctuation and holonomy obtruction 4. Spectral distance on the circle Conclusion: extra-dimensions from Pythagoras theorem

1. The spectral triple of the standard model Spectral triple ( A , H , D ) with A a *-algebra (commutative or not), represented over an Hilbert space H . D = D ∗ is an operator on H satisfying a set of properties such that

1. The spectral triple of the standard model Spectral triple ( A , H , D ) with A a *-algebra (commutative or not), represented over an Hilbert space H . D = D ∗ is an operator on H satisfying a set of properties such that 1. Given a compact oriented spin manifold M , ( C ∞ ( M ) , L 2 ( M , S ) , / ∂ ) is a spectral triple

1. The spectral triple of the standard model Spectral triple ( A , H , D ) with A a *-algebra (commutative or not), represented over an Hilbert space H . D = D ∗ is an operator on H satisfying a set of properties such that 1. Given a compact oriented spin manifold M , ( C ∞ ( M ) , L 2 ( M , S ) , / ∂ ) is a spectral triple 2. Given a spectral triple ( A , H , D ) where A is a commutative *-algebra then ( Connes reconstruction theorem ) ◮ A = C ∞ ( M ) where M is a compact oriented spin manifold with Dirac operator D s = D + torsion term, ◮ there exists a unique riemannian structure on M such that the associated geodesic distance is d ( x , y ) = sup { f ( x ) − f ( y ) / � [ D , f ] � ≤ 1 } . f ∈A ◮ the functional S ( D ) . −| D | − n +2 attains its mimimum for D = D s and is R = proportional to Euclidean Einstein-Hilbert action.

spectral triple with commutative A → spectral triple with noncommutative A � ↓ Riemannian spin manifold noncommutative geometry ◮ Extends the notion of geometry beyond the scope of Riemannian geometry (but always Euclidean signature). ◮ The standard model fits well in this framework. The action functional yields the lagrangian of the standard model minimally coupled to Einstein-Hilbert gravity. ◮ Gives a geometrical interpretation to the Higgs field.

Connection A connection in ( A , H , D ) is implemented by substituting D with D A = D + A + JAJ − 1 , A = a i [ D , b i ] = A ∗ .

Connection A connection in ( A , H , D ) is implemented by substituting D with D A = D + A + JAJ − 1 , A = a i [ D , b i ] = A ∗ . The product of the continum by the discrete C ∞ ( M ) ⊗ A I A = H = L 2 ( M , S ) ⊗ H I ∂ ⊗ I I + γ 5 ⊗ D I D = /

Connection A connection in ( A , H , D ) is implemented by substituting D with D A = D + A + JAJ − 1 , A = a i [ D , b i ] = A ∗ . The product of the continum by the discrete C ∞ ( M ) ⊗ A I A = ⇒ A = γ 5 ⊗ H − i γ µ ⊗ A µ H = L 2 ( M , S ) ⊗ H I ∂ ⊗ I I + γ 5 ⊗ D I D = / ◮ H : scalar field on M with value in A I → Higgs. ◮ A µ : 1-form field with value in Lie ( U ( A I )) → gauge field.

Connection A connection in ( A , H , D ) is implemented by substituting D with D A = D + A + JAJ − 1 , A = a i [ D , b i ] = A ∗ . The product of the continum by the discrete C ∞ ( M ) ⊗ A I A = ⇒ A = γ 5 ⊗ H − i γ µ ⊗ A µ H = L 2 ( M , S ) ⊗ H I ∂ ⊗ I I + γ 5 ⊗ D I D = / ◮ H : scalar field on M with value in A I → Higgs. ◮ A µ : 1-form field with value in Lie ( U ( A I )) → gauge field. The covariant Dirac operator D A = D + A + JAJ − 1 inherits a scalar field component.

The standard model (Chamseddine, Connes, Marcolli. 2006) A I = C ⊕ H ⊕ M 3 ( C ) C 96 H I = 96 × 96 matrix with the masses of the fermions and the CKM matrix . D I is a ◮ Spectral action: the heat kernel expansion of Tr � f ( D A � Λ ) yields Einstein-Hilbert action (with euclidean signature) together with a Weyl term and the full lagrangian of the standard model. � ∞ ◮ f appears only through f 0 = f (0) , f k = f ( v ) v k − 1 dv for k = 2 , 4. Three 0 new parameters physically related to the coupling contants at the unification scale, the gravitational constant and the cosmological constant. ◮ three predictions: � 5 g 2 = g 3 = 3 g 1 generations m 2 e + m 2 ν + 3 m 2 d + 3 m 2 u = 8 M 2 Σ W m H ≃ 170Gev .

2. Distance in noncommutative geometry ∂, f ] = − − → Riemannian manifold M : [ / grad f � − − → � � sup {| f ( x ) − f ( y ) | / grad f � ≤ 1 } = d geo ( x , y ) . � � f ∈ C ∞ ( M ) {| f ( x ) − f ( y ) | / � f ′ � ≤ 1 } = | x − y | . Real line: sup f ∈ C ∞ ( R ) f(Y) f(X) X Y � − − → � � ◮ The upper bound is attained because there exists f = f ∗ with � = 1 grad f � � everywhere on the geodesic ( x , y ), i.e f ( z ) = d geo ( x , z ) .

Points are dual of functions. Gelfand duality, P ( C ∞ ( M )) ≃ M ω x ( f ) = f ( x ) with P ( A ) the pure states of A (normalized positive linear maps C ∞ ( M ) → C ). d ( ω x , ω y ) . = sup {| ω x ( f ) − ω y ( f ) | / � [ / ∂, f ] � ≤ 1 } f ∈ C ∞ ( M )

Points are dual of functions. Gelfand duality, P ( C ∞ ( M )) ≃ M ω x ( f ) = f ( x ) with P ( A ) the pure states of A (normalized positive linear maps C ∞ ( M ) → C ). d ( ω x , ω y ) . = sup {| ω x ( f ) − ω y ( f ) | / � [ / ∂, f ] � ≤ 1 } f ∈ C ∞ ( M ) Definition of the distance that still makes sense for noncommutative A . d ( ω 1 , ω 2 ) . {| ω 1 ( a ) − ω 2 ( a ) | / � [ D , a ] � ≤ 1 } = sup a ∈A ◮ as soon as [ D , a ] is bounded for all a , d is a distance between (pure) states. ◮ coherent with the classical case when A = C ∞ ( M ) : d = d geo , ◮ does not involve notions ill-defined in a quantum context (e.g. trajectories between points) but only spectral properties: spectral distance .

3. Fluctuations of the metric The replacement D → D A yields a fluctuation of the metric since [ D A , a ] = [ D + H − i γ µ A µ , a ] � = [ D , a ] .

3. Fluctuations of the metric The replacement D → D A yields a fluctuation of the metric since [ D A , a ] = [ D + H − i γ µ A µ , a ] � = [ D , a ] . Scalar fluctuation : A µ = 0 , H � = 0 (Wulkenhaar, P.M. 2001) A = C ∞ ( M ) ⊗ A I with A I = C ⊕ H ⊕ M 3 ( C ) = ⇒ P ( A ) is a two-sheet model . X1 . . H Y1 . . C X2 Y2 The spectral distance d coincides with the geodesic distance in M × [0 , 1] given by � g µν � h 1 � � 0 where is the Higgs doublet. | 1 + h 1 | 2 + | h 2 | 2 � � m 2 0 h 2 top

Gauge fluctuation : A µ � = 0 , H = 0 (P.M. 2005-07)

Gauge fluctuation : A µ � = 0 , H = 0 (P.M. 2005-07) Example suggested by Connes (96) A = C ∞ ( M ) ⊗ A I with A I = M n ( C ) , D I = 0 .

Gauge fluctuation : A µ � = 0 , H = 0 (P.M. 2005-07) Example suggested by Connes (96) A = C ∞ ( M ) ⊗ A I with A I = M n ( C ) , D I = 0 . π → M with fiber C P n − 1 , P ( A ) is a trivial bundle P P ∋ p = ( x , ξ ) = ξ x , ξ x ( a ) = � ξ, a ( x ) ξ � = Tr( s ξ a ( x )) .

Gauge fluctuation : A µ � = 0 , H = 0 (P.M. 2005-07) Example suggested by Connes (96) A = C ∞ ( M ) ⊗ A I with A I = M n ( C ) , D I = 0 . π → M with fiber C P n − 1 , P ( A ) is a trivial bundle P P ∋ p = ( x , ξ ) = ξ x , ξ x ( a ) = � ξ, a ( x ) ξ � = Tr( s ξ a ( x )) . The part of D A that does not commute with the representation is the covariant Dirac operator − i γ µ ( ∂ µ + A µ ) associated to to the connection.

The connection defines both a spectral distance d and an horizontal distance d H : � 1 T p P = V p P ⊕ H p P = ⇒ d H ( p , q ) = � ˙ c t � dt . Inf c t ∈ H ˙ ct P 0 C t ζ x ξ x x d H ( ξ x , ζ x ) = 4 π M

The connection defines both a spectral distance d and an horizontal distance d H : � 1 T p P = V p P ⊕ H p P = ⇒ d H ( p , q ) = � ˙ c t � dt . Inf c t ∈ H ˙ ct P 0 C t ζ x ξ x x d H ( ξ x , ζ x ) = 4 π M d ≤ d H points at finite horizontal distance points at finite spectral distance ւ ց Acc( ξ x ) Con( ξ x ) Acc( ξ x ) ⊂ Con( ξ x )