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¨

• ttingen

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:

−iγµ∂µ ⇐ ⇒ riemannian geodesic distance.

◮ Fibre bundle P with connection:

−iγµ(∂µ + Aµ) ⇐ ⇒?

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) , L2(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) , L2(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

• perator Ds = D + torsion term,

◮ there exists a unique riemannian structure on M such that the associated

geodesic distance is d(x, y) = sup

f ∈A

{f (x) − f (y) / [D, f ] ≤ 1}.

◮ the functional S(D) .

= R −|D|−n+2 attains its mimimum for D = Ds and is 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 DA = D + A + JAJ−1, A = ai[D, bi] = A∗.

Connection A connection in (A, H, D) is implemented by substituting D with DA = D + A + JAJ−1, A = ai[D, bi] = A∗. The product of the continum by the discrete A = C ∞ (M) ⊗ AI H = L2(M, S) ⊗ HI D = / ∂ ⊗ II + γ5 ⊗ DI

Connection A connection in (A, H, D) is implemented by substituting D with DA = D + A + JAJ−1, A = ai[D, bi] = A∗. The product of the continum by the discrete A = C ∞ (M) ⊗ AI H = L2(M, S) ⊗ HI D = / ∂ ⊗ II + γ5 ⊗ DI ⇒ A = γ5 ⊗ H − iγµ ⊗ Aµ

◮ H: scalar field on M with value in AI

→ Higgs.

◮ Aµ: 1-form field with value in Lie(U(AI))

→ gauge field.

Connection A connection in (A, H, D) is implemented by substituting D with DA = D + A + JAJ−1, A = ai[D, bi] = A∗. The product of the continum by the discrete A = C ∞ (M) ⊗ AI H = L2(M, S) ⊗ HI D = / ∂ ⊗ II + γ5 ⊗ DI ⇒ A = γ5 ⊗ H − iγµ ⊗ Aµ

◮ H: scalar field on M with value in AI

→ Higgs.

◮ Aµ: 1-form field with value in Lie(U(AI))

→ gauge field. The covariant Dirac operator DA = D + A + JAJ−1 inherits a scalar field component.

The standard model (Chamseddine, Connes, Marcolli. 2006) AI = C ⊕ H ⊕ M3(C) HI = C96 DI is a 96 × 96 matrix with the masses of the fermions and the CKM matrix.

◮ Spectral action: the heat kernel expansion of Tr

• f ( DA

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 f0 = f (0), fk =

∞ f (v)v k−1dv for k = 2, 4. Three new parameters physically related to the coupling contants at the unification scale, the gravitational constant and the cosmological constant.

◮ three predictions:

g2 = g3 =

• 5

3g1

Σ

generationsm2 e + m2 ν + 3m2 d + 3m2 u = 8M2 W

mH ≃ 170Gev.

• 2. Distance in noncommutative geometry

Riemannian manifold M: [/ ∂, f ] = − − → grad f sup

f ∈C ∞(M)

{|f (x) − f (y)| /

• −

− → grad f

• ≤ 1} = dgeo(x, y).

Real line: sup

f ∈C ∞(R)

{|f (x) − f (y)| / f ′ ≤ 1} = |x − y|.

f(Y)

X Y

f(X)

◮ The upper bound is attained because there exists f = f ∗ with

• −

− → grad f

• = 1

everywhere on the geodesic (x, y), i.e f (z) = dgeo(x, z).

Points are dual of functions. Gelfand duality, P(C ∞ (M)) ≃ M ωx(f ) = f (x) with P(A) the pure states of A (normalized positive linear maps C ∞(M) → C). d(ωx, ωy) . = sup

f ∈C ∞(M)

{|ωx(f ) − ωy(f )| / [/ ∂, f ] ≤ 1}

Points are dual of functions. Gelfand duality, P(C ∞ (M)) ≃ M ωx(f ) = f (x) with P(A) the pure states of A (normalized positive linear maps C ∞(M) → C). d(ωx, ωy) . = sup

f ∈C ∞(M)

{|ωx(f ) − ωy(f )| / [/ ∂, f ] ≤ 1} Definition of the distance that still makes sense for noncommutative A.

d(ω1, ω2) . = sup

a∈A

{|ω1(a) − ω2(a)| / [D, a] ≤ 1}

◮ 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 = dgeo, ◮ 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 → DA yields a fluctuation of the metric since [DA, a] = [D + H − iγµAµ, a] = [D, a].

• 3. Fluctuations of the metric

The replacement D → DA yields a fluctuation of the metric since [DA, a] = [D + H − iγµAµ, a] = [D, a]. Scalar fluctuation: Aµ = 0, H = 0

(Wulkenhaar, P.M. 2001)

A = C ∞ (M) ⊗ AI with AI = C ⊕ H ⊕ M3(C) = ⇒ P(A) is a two-sheet model

X2

C

. Y2 Y1

H

X1

. . . .

The spectral distance d coincides with the geodesic distance in M × [0, 1] given by g µν

• |1 + h1|2 + |h2|2

m2

top

• where

h1 h2

• is the Higgs doublet.

Gauge fluctuation: Aµ = 0, H = 0 (P.M. 2005-07)

Gauge fluctuation: Aµ = 0, H = 0 (P.M. 2005-07) Example suggested by Connes (96) A = C ∞ (M) ⊗ AI with AI = Mn(C), DI = 0.

Gauge fluctuation: Aµ = 0, H = 0 (P.M. 2005-07) Example suggested by Connes (96) A = C ∞ (M) ⊗ AI with AI = Mn(C), DI = 0. P(A) is a trivial bundle P

π

→ M with fiber CPn−1, P ∋ p = (x, ξ) = ξx, ξx(a) = ξ, a(x)ξ = Tr(sξa(x)).

Gauge fluctuation: Aµ = 0, H = 0 (P.M. 2005-07) Example suggested by Connes (96) A = C ∞ (M) ⊗ AI with AI = Mn(C), DI = 0. P(A) is a trivial bundle P

π

→ M with fiber CPn−1, P ∋ p = (x, ξ) = ξx, ξx(a) = ξ, a(x)ξ = Tr(sξa(x)). The part of DA that does not commute with the representation is the covariant Dirac operator −iγµ(∂µ + Aµ) associated to to the connection.

The connection defines both a spectral distance d and an horizontal distance dH: TpP = VpP ⊕ HpP = ⇒ dH(p, q) = Inf

˙ ct∈H

ctP

1 ˙ ct dt.

t

M

ξ ζ

x x x C

dH(ξx, ζx) = 4π

The connection defines both a spectral distance d and an horizontal distance dH: TpP = VpP ⊕ HpP = ⇒ dH(p, q) = Inf

˙ ct∈H

ctP

1 ˙ ct dt.

t

M

ξ ζ

x x x C

dH(ξx, ζx) = 4π d ≤ dH points at finite horizontal distance points at finite spectral distance ւ ց Acc(ξx) Con(ξx) Acc(ξx) ⊂ Con(ξx)

Holonomy obstruction: dH plays for the bundle the same role as the dgeo for the manifold. f (z) = ωz(f ) = dgeo(x, z) reads Ct(a) = dH(ξx, Ct) for any Ct in the minimal horizontal curve C between ξx = C0, ζy = C1.

*

ζ

p p p

1 2

C

z = y x

C

y

1 0 = =

π (p ) (p ) (p )

2

(p ) π π

ξ

x

Holonomy obstruction: dH plays for the bundle the same role as the dgeo for the manifold. f (z) = ωz(f ) = dgeo(x, z) reads Ct(a) = dH(ξx, Ct) for any Ct in the minimal horizontal curve C between ξx = C0, ζy = C1.

*

ζ

p p p

1 2

C

z = y x

C

y

1 0 = =

π (p ) (p ) (p )

2

(p ) π π

ξ

x

pi(a) = Tr(spia(z)) = dH(ξx, pi).

◮ If more than n2 points pi, too many conditions on the single matrix a(z) !

The spectral and horizontal distances cannot be equal.

◮ Is there a minimal horizontal curve with less than n2 points pi ?

• 4. Spectral distance on the circle

A = C ∞(S1, Mn(C)) = ⇒ pure states form a CPn−1 trivial bundle on S1, A = i    θ1 . . . . . . ... . . . . . . θn    , ξx =    V1 . . . Vn    ∈ CPn−1, ωj . = 2π θ1(t) − θj(t) dt 2π .

• 4. Spectral distance on the circle

A = C ∞(S1, Mn(C)) = ⇒ pure states form a CPn−1 trivial bundle on S1, A = i    θ1 . . . . . . ... . . . . . . θn    , ξx =    V1 . . . Vn    ∈ CPn−1, ωj . = 2π θ1(t) − θj(t) dt 2π . Topology of the fiber: n = 4, ω3 = ω4 irrational, ω2 ∈ Q.

• eiϕi|Vi|, i = 2, 3, 4
• is a 3-torus inside CP3. Arg Vi fix a point.

3

O

!

4

!

2

!

Fiberwise Con(ξx) is a 2-torus. Acc(ξx) is at best dense in it. Globally, Con(ξx) is a 3-torus.

The shape of the fiber: n = 2 n = 2 = ⇒ eiϕ2|V2| is a 1-torus inside the CP1 fiber.

◮ Fiberwise Con(ξx) is a 1-torus. Globally Con(ξx) is a 2-torus. ◮ Fiberwise Acc(ξx) = Holx(ξx) =

• ξk

x , k ∈ Z

• hence dH(ξx, ξk

x ) = 2kπ.

1

!x !x

1

S

x

2"#

x

!x "

#

x

• dH(0, ϕ) = 2kπ

if ϕ = 2kπω mod [2π] d(0, ϕ) = C sin ϕ

2

with C = 4π|V1||V2|

|sin ωπ|

x

!x "

#

x

• dH(0, ϕ) = 2kπ

if ϕ = 2kπω mod [2π] d(0, ϕ) = C sin ϕ

2

with C = 4π|V1||V2|

|sin ωπ|

1 2 40 80 1 2 1 2 1 2 1 2

x

!x "

#

x

• dH(0, ϕ) = 2kπ

if ϕ = 2kπω mod [2π] d(0, ϕ) = C sin ϕ

2

with C = 4π|V1||V2|

|sin ωπ|

1 2 40 80 1 2 1 2 1 2 1 2

◮ No cutlocus for the distance function d: the fiber is smoother than a circle.

First interpretation: d(0, ϕ) is the euclidean distance on the cardioid. But the latest is not invariant by rotation whereas d is.

O1 O0 O2 O2 O0 O1 O1 O2 O2 O1

First interpretation: d(0, ϕ) is the euclidean distance on the cardioid. But the latest is not invariant by rotation whereas d is.

O1 O0 O2 O2 O0 O1 O1 O2 O2 O1 ◮ With the spectral distance, everyone can equally pretend to be the center of

the world.

Second interpretation: length of the segment in the disk

! " # x

x

2 sin _ 2

!

Second interpretation: length of the segment in the disk

! " # x

x

2 sin _ 2

!

◮ The spectral distance “sees” the disk through the circle, in the same way as

it sees between the sheets of the standard model.

Distance on the fiber for n ≥ 2 ζx ∈ Con(ξx) =     Vi ∀i ∈ Far1 eiϕ2Vi ∀i ∈ Far2 . . . eiϕn

c Vi

∀i ∈ Farnc     , ϕj ∈ R, j ∈ [2, nc] where Farj are the classes of equivalence of i ∼ j iff ωj = ωi mod[2π]. d(ξx, ζx) = πTr|S| where S is the matrix with components Sij . = 2|Vi||Vj| sin

• ϕj−ϕi

2

• sin π(ωj − ωi).

◮ Not the Wilon loop but the trace of a matrix that contains the holonomy.

Conclusion: extra-dimensions from Pythagoras theorem

◮ ds = D−1:

D = / ∂ ⊗ II + γ5 ⊗ DI = ⇒ D2 = / ∂2 ⊗ II + IE ⊗ D2

I

= ⇒ ds−2 = ds−2

M + ds−2 I

: Pythagoras−1 However in the standard model g µν

• |1 + h1|2 + |h2|2

m2

top

• =

⇒ ds2 = ds2

M + ds2 I : Pythagoras

Conclusion: extra-dimensions from Pythagoras theorem

◮ ds = D−1:

D = / ∂ ⊗ II + γ5 ⊗ DI = ⇒ D2 = / ∂2 ⊗ II + IE ⊗ D2

I

= ⇒ ds−2 = ds−2

M + ds−2 I

: Pythagoras−1 However in the standard model g µν

• |1 + h1|2 + |h2|2

m2

top

• =

⇒ ds2 = ds2

M + ds2 I : Pythagoras

Simple solution: D2

I = |m|2II =

⇒ D2 = (/ ∂2 + |m|2IE) ⊗ II = (γa∂a)2 ⊗ II where γa =

• γµ, |m|γ5

and ∂a is a “discrete derivative”.

Conclusion: extra-dimensions from Pythagoras theorem

◮ ds = D−1:

D = / ∂ ⊗ II + γ5 ⊗ DI = ⇒ D2 = / ∂2 ⊗ II + IE ⊗ D2

I

= ⇒ ds−2 = ds−2

M + ds−2 I

: Pythagoras−1 However in the standard model g µν

• |1 + h1|2 + |h2|2

m2

top

• =

⇒ ds2 = ds2

M + ds2 I : Pythagoras

Simple solution: D2

I = |m|2II =

⇒ D2 = (/ ∂2 + |m|2IE) ⊗ II = (γa∂a)2 ⊗ II where γa =

• γµ, |m|γ5

and ∂a is a “discrete derivative”. → Illusion of extra-dimension come from Pythagoras theorem. → What is the equivalent for the disk ? ds2

disk = function(ds2 circle, A) ?

Outlook and references

◮ The spectral distance sees between the leaves of the horizontal foliation

→ should be relevant for the noncommutative torus.

◮ Topological effect due to M = S1 ? Other basis for the bundle requires to

know the number of selfintersecting points of the minimal horizontal curve. → work for (classical) subriemannian geometry.

◮ Discrete structure of space-time without talking of quantum gravity. ◮ Gauge fluctuation might make the distance on the M3(C) part of the

standard model finite.

Spectral distance on S1: math.OA/0703586, submitted to J. Func. Anal. Carnot-Carath´ eodory vs NC-distance: Com.Math.Phys. 265 (2006) 585-616,

• r a non technical version, Cluj university press, hep-th/0603051.

Scalar fluctuation: with R. Wulkenhaar, J.Math.Phys. 43 (2002) 182-204.

Distance between the fibers for n = 2 d(ξx, ζy) = max

T± Hξ(T, ∆)

∆ T T+ T−

unit is min(τ, 2π − τ)

where the sign is the one of zξ . = |V1|2 − |V2|2, Hξ(T, ∆) . = T + zξ∆ + W1

• (τ − T)2 − ∆2 + W0
• (2π − τ − T)2 − ∆2

W0 . = R |sin( ϕ

2 )|

|sin ωπ| , W1 . = R |sin(ωπ + ϕ

2 )|

|sin ωπ| , R . =

• 1 − z2

ξ. ◮ The element a that reaches the supremum has null diagonal at x,

Tr(a(y)) = T, a11(y) − a22(y) = ∆.

◮ The maximum is reached for T = 0 or on the hypothenus. ◮ When zξ = 0 the maximum is reached at ∆ = 0.