Noncommutative Geometry and Physics A. Connes 1 Variables One - - PDF document

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Noncommutative Geometry and Physics

• A. Connes

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Variables

One striking point is the role that “variables” play in Newton’s approach, while Leibniz intro- duced the term “infinitesimal” but did not use

• variables. According to Newton :

“In a certain problem, a variable is the quantity that takes an infinite number of values which are quite determined by this problem and are arranged in a definite order” “A variable is called infinitesimal if among its particular values one can be found such that this value itself and all following it are smaller in absolute value than an arbitrary given num- ber”

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Classical formulation

In the classical formulation of variables as maps from a set X to the real numbers R, the set X has to be uncountable if some variable has continuous range. But then for any other va- riable with countable range some of the multi- plicities are infinite. This means that discrete and continuous variables cannot coexist in this modern formalism. Fortunately everything is fine and this problem of treating continuous and discrete variables on the same footing is completely solved using the formalism of quan- tum mechanics.

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Quantum formalism

The first basic change of paradigm has indeed to do with the classical notion of a “real va- riable” which one would classically describe as a real valued function on a set X, ie as a map from this set X to real numbers. In fact quan- tum mechanics provides a very convenient sub-

• stitute. It is given by a self-adjoint operator in

Hilbert space. Note that the choice of Hilbert space is irrelevant here since all separable infi- nite dimensional Hilbert spaces are isomorphic.

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All the usual attributes of real variables such as their range, the number of times a real number is reached as a value of the variable etc... have a perfect analogue in the quantum mechanical

• setting. The range is the spectrum of the ope-

rator, and the spectral multiplicity gives the number of times a real number is reached. In the early times of quantum mechanics, physi- cists had a clear intuition of this analogy bet- ween operators in Hilbert space (which they called q-numbers) and variables.

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Infinitesimal variables

What is surprising is that the new set-up imme- diately provides a natural home for the “infini- tesimal variables” and here the distinction bet- ween “variables” and numbers (in many ways this is where the point of view of Newton is more efficient than that of Leibniz) is essen- tial.

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Indeed it is perfectly possible for an operator to be “smaller than epsilon for any epsilon” wi- thout being zero. This happens when the norm

• f the restriction of the operator to subspaces
• f finite codimension tends to zero when these

subspaces decrease (under the natural filtra- tion by inclusion). The corresponding opera- tors are called “compact” and they share with naive infinitesimals all the expected algebraic

• properties. Indeed they form a two-sided ideal
• f the algebra of bounded operators in Hilbert

space and the only property of the naive infi- nitesimal calculus that needs to be dropped is the commutativity.

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Discrete and continuous coexist

It is only because one drops commutativity that variables with continuous range can co- exist with variables with countable range. Thus it is the uniqueness of the separable in- finite dimensional Hilbert space that cures the above problem, L2[0, 1] is the same as ℓ2(N), and variables with continuous range coexist happily with variables with countable range, such as the infinitesimal ones. The only new fact is that they do not commute, and the real subtlety is in their algebraic relations. For ins- tance it is the lack of commutation of the line element ds with the coordinates that allows

• ne to measure distances in a noncommuta-

tive space given as a spectral triple.

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Space X Algebra A Real variable xµ Self-adjoint T Set of values Spectrum of T Infinitesimal Compact ǫ Order α µn(ǫ) = O(n−α) Integral of

• − ǫ = Coefficient of

infinitesimal log(Λ) in TrΛ(ǫ) Line element ds = Fermion

• gµν dxµdxν

propagator

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Variability

At the philosophical level there is something quite satisfactory in the variability of the quan- tum mechanical observables. Usually when pres- sed to explain what is the cause of the varia- bility in the external world, the answer that comes naturally to the mind is just : the pas- sing of time.

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But precisely the quantum world provides a more subtle answer since the reduction of the wave packet which happens in any quantum measurement is nothing else but the repla- cement of a “q-number” by an actual num- ber which is chosen among the elements in its

• spectrum. Thus there is an intrinsic variability

in the quantum world which is so far not redu- cible to anything classical. The results of ob- servations are intrinsically variable quantities, and this to the point that their values can- not be reproduced from one experiment to the next, but which, when taken altogether, form a q-number.

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How can time emerge ?

Quantum thermodynamics, Ludwig Boltzmann ϕ(A) = Z−1 tr(A e−β H) Z = tr(e−β H)

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The KMS condition ϕ(x∗x) ≥ 0 ∀ x ∈ A , ϕ(1) = 1 . σt ∈ Aut(A)

Φx Σty ΦΣtyx

• Fx,y(t) = ϕ(xσt(y))

Fx,y(t + iβ) = ϕ(σt(y)x), ∀t ∈ R.

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Tomita-Takesaki (1967) Theorem Let M be a von Neumann algebra and ϕ a faithful normal state on M, then there exists a unique one parameter group σϕ

t ∈ Aut(M)

which fulfills the KMS condition for β = 1.

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Thesis (1972)

Theorem (alain connes) 1 → Int(M) → Aut(M) → Out(M) → 1, The class of σϕ

t

in Out(M) does not depend upon the choice of the state ϕ. Thus a von Neumann algebra M, possesses a canonical time evolution R

δ

− → Out(M).

Noncommutativity ⇒ Time Evolution

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Many mathematical corollaries but what about physics ?

• 1. We (with Carlo Rovelli) interpret time as a
• ne parameter group of automorphisms of

the algebra of observables for gravitation.

• 2. Thermodynamical origin.

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Algebra of observables in QG ? Find complete invariants of geometric spaces How can we invariantly specify a point in a geometric space ?

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It is well known since a famous one page pa- per of John Milnor that the spectrum of ope- rators, such as the Laplacian, does not suffice to characterize a compact Riemannian space. But it turns out that the missing information is encoded by the relative position of two abe- lian algebras of operators in Hilbert space. Due to a theorem of von Neumann the algebra of multiplication by all measurable bounded func- tions acts in Hilbert space in a unique manner, independent of the geometry one starts with. Its relative position with respect to the other abelian algebra given by all functions of the Laplacian suffices to recover the full geome- try, provided one knows the spectrum of the

• Laplacian. For some reason which has to do

with the inverse problem, it is better to work with the Dirac operator.

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The unitary (CKM) invariant

• f Riemannian manifolds

The invariants are : – The spectrum Spec(D). – The relative spectrum SpecN(M) (N = {f(D)}).

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Flavor changing weak decays

ig 2 √ 2W + µ

• ¯

uλ

j γµ(1 + γ5)Cλκdκ j

• +

ig 2 √ 2W − µ

• ¯

dκ

j C† κλγµ(1 + γ5)uλ j

• 20

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Cabibbo-Kobayashi-Maskawa

C =

• cosθc

sinθc −sinθc cosθc

• C =

  

Cud Cus Cub Ccd Ccs Ccb Ctd Cts Ctb

  

C =

  

c1 −s1c3 −s1s3 s1c2 c1c2c3 − s2s3eδ c1c2s3 + s2c3eδ s1s2 c1s2c3 + c2s3eδ c1s2s3 − c2c3eδ

  

ci = cos θi, si = sin θi, and eδ = exp(iδ)

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Points

Once we know the spectrum Λ of D, the mis- sing information is contained in SpecN(M). It should be interpreted as giving the proba- bility for correlations between the possible fre- quencies, while a “point” of the geometric space X can be thought of as a correlation, i.e. a spe- cific positive hermitian matrix ρλκ (up to scale) in the support of ν.

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What is a metric in spectral geometry d(A, B) = Inf

• γ
• gµ ν dxµ dxν

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Dirac’s square root of the Laplacian

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(A, H, D) , ds = D−1 , d(A, B) = Sup {|f(A)−f(B)| ; f ∈ A , [D, f] ≤ 1 }

Meter → Wave length (Krypton (1967) spectrum of 86Kr then Caesium (1984) hyperfine levels of C133)

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Space-Time

Joint work with Ali Chamseddine Our knowledge of spacetime is described by two existing theories : – General Relativity – The Standard Model of particle physics Curved Space, gravitational potential gµν ds2 = gµνdxµ dxν Action principle SE[ gµν] = 1 G

• M r √g d4x

S = SE + SSM

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Standard Model

LSM = −1 2∂νga

µ∂νga µ − gsf abc∂µga νgb µgc ν − 1

4g2

s f abcf adegb µgc νgd µge ν

−∂νW +

µ ∂νW − µ − M2W + µ W − µ − 1

2∂νZ0

µ∂νZ0 µ −

1 2c2

w

M2Z0

µZ0 µ − 1

2∂µAν∂µAν −igcw(∂νZ0

µ(W + µ W − ν − W + ν W − µ ) − Z0 ν (W + µ ∂νW − µ − W − µ ∂νW + µ )

+Z0

µ(W + ν ∂νW − µ − W − ν ∂νW + µ )) − igsw(∂νAµ(W + µ W − ν − W + ν W − µ )

−Aν(W +

µ ∂νW − µ − W − µ ∂νW + µ ) + Aµ(W + ν ∂νW − µ − W − ν ∂νW + µ ))

−1 2g2W +

µ W − µ W + ν W − ν + 1

2g2W +

µ W − ν W + µ W − ν

+g2c2

w(Z0 µW + µ Z0 ν W − ν −Z0 µZ0 µW + ν W − ν )+g2s2 w(AµW + µ AνW − ν −AµAµW + ν W − ν )

+g2swcw(AµZ0

ν (W + µ W − ν −W + ν W − µ )−2AµZ0 µW + ν W − ν )−1

2∂µH∂µH−1 2m2

hH2

−∂µφ+∂µφ− − M2φ+φ− − 1 2∂µφ0∂µφ0 − 1 2c2

w

M2φ0φ0 −βh

• 2M2

g2 + 2M g H + 1 2(H2 + φ0φ0 + 2φ+φ−)

• + 2M4

g2 αh 28

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−gαhM H3 + Hφ0φ0 + 2Hφ+φ− −1 8g2αh

• H4 + (φ0)4 + 4(φ+φ−)2 + 4(φ0)2φ+φ− + 4H2φ+φ− + 2(φ0)2H2

−gMW +

µ W − µ H − 1

2gM c2

w

Z0

µZ0 µH

−1 2ig W +

µ (φ0∂µφ− − φ−∂µφ0) − W − µ (φ0∂µφ+ − φ+∂µφ0)

+1 2g W +

µ (H∂µφ− − φ−∂µH) + W − µ (H∂µφ+ − φ+∂µH)

+1 2g 1 cw (Z0

µ(H∂µφ0 − φ0∂µH) − igs2 w

cw MZ0

µ(W + µ φ− − W − µ φ+)

+igswMAµ(W +

µ φ− − W − µ φ+) − ig1 − 2c2 w

2cw Z0

µ(φ+∂µφ− − φ−∂µφ+)

+igswAµ(φ+∂µφ− − φ−∂µφ+) − 1 4g2W +

µ W − µ

• H2 + (φ0)2 + 2φ+φ−

−1 8g2 1 c2

w

Z0

µZ0 µ

• H2 + (φ0)2 + 2(2s2

w − 1)2φ+φ−

−1 2g2s2

w

cw Z0

µφ0(W + µ φ− + W − µ φ+) − 1

2ig2s2

w

cw Z0

µH(W + µ φ− − W − µ φ+)

+1 2g2swAµφ0(W +

µ φ− + W − µ φ+) + 1

2ig2swAµH(W +

µ φ− − W − µ φ+)

−g2sw cw (2c2

w − 1)Z0 µAµφ+φ− − g2s2 wAµAµφ+φ−

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+1 2igsλa

ij(¯

qσ

i γµqσ j )ga µ − ¯

eλ(γ∂ + mλ

e)eλ − ¯

νλγ∂νλ − ¯ uλ

j (γ∂ + mλ u)uλ j

−¯ dλ

j (γ∂ + mλ d)dλ j + igswAµ

• −(¯

eλγµeλ) + 2 3(¯ uλ

j γµuλ j ) − 1

3(¯ dλ

j γµdλ j )

• + ig

4cw Z0

µ{(¯

νλγµ(1 + γ5)νλ) + (¯ eλγµ(4s2

w − 1 − γ5)eλ)

+(¯ dλ

j γµ(4

3s2

w − 1 − γ5)dλ j ) + (¯

uλ

j γµ(1 − 8

3s2

w + γ5)uλ j )}

+ ig 2 √ 2 W +

µ

• (¯

νλγµ(1 + γ5)eλ) + (¯ uλ

j γµ(1 + γ5)Cλκdκ j )

+ ig 2 √ 2 W −

µ

• (¯

eλγµ(1 + γ5)νλ) + (¯ dκ

j C† κλγµ(1 + γ5)uλ j )

+ ig 2 √ 2 mλ

e

M

• −φ+(¯

νλ(1 − γ5)eλ) + φ−(¯ eλ(1 + γ5)νλ) −g 2 mλ

e

M

• H(¯

eλeλ) + iφ0(¯ eλγ5eλ) + ig 2M √ 2 φ+ −mκ

d(¯

uλ

j Cλκ(1 − γ5)dκ j ) + mλ u(¯

uλ

j Cλκ(1 + γ5)dκ j

• +

ig 2M √ 2 φ− mλ

d(¯

dλ

j C† λκ(1 + γ5)uκ j ) − mκ u(¯

dλ

j C† λκ(1 − γ5)uκ j

• −g

2 mλ

u

M H(¯ uλ

j uλ j ) − g

2 mλ

d

M H(¯ dλ

j dλ j ) + ig

2 mλ

u

M φ0(¯ uλ

j γ5uλ j ) − ig

2 mλ

d

M φ0(¯ dλ

j γ5dλ j )

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Let us consider the simplest example A = C∞(M, Mn(C)) = C∞(M) ⊗ Mn(C) Algebra of n × n matrices of smooth functions

• n manifold M.

The group Int(A) of inner automorphisms is locally isomorphic to the group G of smooth maps from M to the small gauge group SU(n) 1 → Int(A) → Aut(A) → Out(A) → 1 becomes identical to 1 → Map(M, G) → G → Diff(M) → 1.

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We have shown that the study of pure gra- vity on this space yields Einstein gravity on M minimally coupled with Yang-Mills theory for the gauge group SU(n). The Yang-Mills gauge potential appears as the inner part of the me- tric, in the same way as the group of gauge transformations (for the gauge group SU(n)) appears as the group of inner diffeomorphisms.

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The restriction to spin manifolds is obtained by requiring a real structure i.e. an antilinear unitary operator J acting in H which plays the same role and has the same algebraic proper- ties as the charge conjugation operator in phy- sics.

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The following further relations hold for D, J and γ J2 = ε , DJ = ε′JD, J γ = ε′′γJ, Dγ = −γD The values of the three signs ε, ε′, ε′′ depend

• nly, in the classical case of spin manifolds,

upon the value of the dimension n modulo 8 and are given in the following table : n 1 2 3 4 5 6 7 ε 1 1

• 1
• 1
• 1
• 1

1 1 ε′ 1

• 1

1 1 1

• 1

1 1 ε′′ 1

• 1

1

• 1

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In the classical case of spin manifolds there is thus a relation between the metric (or spec- tral) dimension given by the rate of growth

• f the spectrum of D and the integer mo-

dulo 8 which appears in the above table. For more general spaces however the two notions

• f dimension (the dimension modulo 8 is cal-

led the KO-dimension because of its origin in K-theory) become independent since there are spaces F of metric dimension 0 but of arbitrary KO-dimension.

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Starting with an ordinary spin geometry M of dimension n and taking the product M × F,

• ne obtains a space whose metric dimension is

still n but whose KO-dimension is the sum of n with the KO-dimension of F. As it turns out the Standard Model with neu- trino mixing favors the shift of dimension from the 4 of our familiar space-time picture to 10 = 4 + 6 = 2 modulo 8.

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The shift from 4 to 10 is a recurrent idea in string theory compactifications, where the 6 is the dimension of the Calabi-Yau manifold used to “compactify”. The difference of this approach with ours is that, in the string com- pactifications, the metric dimension of the full space-time is now 10 which can only be re- conciled with what we experience by requiring that the Calabi-Yau fiber remains unnaturally small.

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In order to learn how to perform the above shift of dimension using a 0-dimensional space F, it is important to classify such spaces. This was done in joint work with A. Chamseddine. We classified there the finite spaces F of gi- ven KO-dimension. A space F is finite when the algebra AF of coordinates on F is finite

• dimensional. We no longer require that this al-

gebra is commutative.

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We classified the irreducible (A, H, J) and found

• ut that the solutions fall into two classes. Let

AC be the complex linear space generated by A in L(H), the algebra of operators in H. By construction AC is a complex algebra and one

• nly has two cases :
• 1. The center Z (AC) is C, in which case AC =

Mk(C) for some k.

• 2. The center Z (AC) is C ⊕ C and AC = Mk(C)⊕

Mk(C) for some k.

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Moreover the knowledge of AC = Mk(C) shows that A is either Mk(C) (unitary case), Mk(R) (real case) or, when k = 2ℓ is even, Mℓ(H), where H is the field of quaternions (symplectic case). This first case is a minor variant of the Einstein-Yang-Mills case described above. It turns out by studying their Z/2 gradings γ, that these cases are incompatible with KO- dimension 6 which is only possible in case (2).

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If one assumes that one is in the “symplectic– unitary” case and that the grading is given by a grading of the vector space over H, one can show that the dimension of H which is 2k2 in case (2) is at least 2 × 16 while the simplest solution is given by the algebra A = M2(H) ⊕ M4(C). This is an important variant

• f the Einstein-Yang-Mills case because, as the

center Z (AC) is C ⊕ C, the product of this fi- nite geometry F by a manifold M appears, from the commutative standpoint, as two dis- tinct copies of M.

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We showed that requiring that these two co- pies of M stay a finite distance apart reduces the symmetries from the group SU(2)×SU(2)× SU(4) of inner automorphisms ∗ to the sym- metries U(1) × SU(2) × SU(3) of the Standard

• Model. This reduction of the gauge symme-

try occurs because of the second kinematical condition [[D, a], b] = 0 which in the general case becomes : [[D, a], b0] = 0 , ∀ a, b ∈ A

∗. of the even part of the algebra

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Spectral Model Let M be a Riemannian spin 4-manifold and F the finite noncommutative geometry of KO- dimension 6 described above. Let M × F be endowed with the product metric.

• 1. The unimodular subgroup of the unitary

group acting by the adjoint representation Ad(u) in H is the group of gauge transfor- mations of SM.

• 2. The unimodular inner fluctuations of the

metric give the gauge bosons of SM.

• 3. The full standard model (with neutrino mixing

and seesaw mechanism) minimally coupled to Einstein gravity is given in Euclidean form by the action functional S = Tr(f(DA/Λ))+1 2 J ˜ ξ, DA ˜ ξ , ˜ ξ ∈ H+

cl ,

where DA is the Dirac operator with the unimodular inner fluctuations.

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Standard Model Spectral Action Higgs Boson Inner metric(0,1) Gauge bosons Inner metric(1,0) Fermion masses Dirac(0,1) in ↑ u, ν CKM matrix Dirac(0,1) in (↓ 3) Masses down Lepton mixing Dirac(0,1) in (↓ 1) Masses leptons e Majorana Dirac(0,1) on mass matrix ER ⊕ JFER Gauge couplings Fixed at unification Higgs scattering Fixed at parameter unification Tadpole constant −µ2

0 |H|2

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Noncommutative geometry was shown to pro- vide a promising framework for unification of all fundamental interactions including gravity. Historically, the search to identify the struc- ture of the noncommutative space followed the bottom-up approach where the known spec- trum of the fermionic particles was used to determine the geometric data that defines the

• space. This bottom-up approach involved an

interesting interplay with experiments. While at first the experimental evidence of neutrino

• scillations contradicted the first attempt, it

was realized several years later in 2006 that the obstruction to get neutrino oscillations was naturally eliminated by dropping the equality between the metric dimension of space-time (which is equal to 4 as far as we know) and its KO-dimension which is only defined modulo

• 8. When the latter is set equal to 2 modulo

8 (using the freedom to adjust the geometry

• f the finite space encoding the fine structure

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• f space-time) everything works fine, the neu-

trino oscillations are there as well as the see- saw mechanism which appears for free as an unexpected bonus. Incidentally, this also solved the fermionic doubling problem by allowing a simultaneous Weyl-Majorana condition on the fermions to halve the degrees of freedom.

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The second interplay with experiments occur- red a bit later when it became clear that the mass of the Brout-Englert-Higgs boson would not comply with the restriction (that mH 170 Gev) imposed by the validity of the Stan- dard Model up to the unification scale.

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New developments We showed that the inconsistency between the spectral Standard Model and the experimen- tal value of the Higgs mass is resolved by the presence of a real scalar field strongly cou- pled to the Higgs field. This scalar field was already present in the spectral model and we wrongly neglected it in our previous computa-

• tions. It was shown recently by several authors,

independently of the spectral approach, that such a strongly coupled scalar field stabilizes the Standard Model up to unification scale in spite of the low value of the Higgs mass. In our recent work, we show that the noncommuta- tive neutral singlet modifies substantially the RG analysis, invalidates our previous prediction

• f Higgs mass in the range 160–180 Gev, and

restores the consistency of the noncommuta- tive geometric model with the low Higgs mass.

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One lesson which we learned on that occasion is that we have to take all the fields of the noncommutative spectral model seriously, wi- thout making assumptions not backed up by valid analysis, especially because of the almost uniqueness of the Standard Model (SM) in the noncommutative setting.

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The SM continues to conform to all experi- mental data. The question remains whether this model will continue to hold at much higher energies, or whether there is a unified theory whose low-energy limit is the SM. One indica- tion that there must be a new higher scale that effects the low energy sector is the small mass

• f the neutrinos which is explained through the

see-saw mechanism with a Majorana mass of at least of the order of 1011Gev. In addition and as noted above, a scalar field which ac- quires a vev generating that mass scale can stabilize the Higgs coupling and prevent it from becoming negative at higher energies and thus make it consistent with the low Higgs mass

• f 126 Gev. Another indication of the need to

modify the SM at high energies is the failure (by few percent) of the three gauge couplings to be unified at some high scale which indi- cates that it may be necessary to add other matter couplings to change the slopes of the running of the RG equations.

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This leads us to address the issue of the brea- king from the natural algebra A which results from the classification of irreducible finite geo- metries of KO-dimension 6 (modulo 8), to the algebra corresponding to the SM. This brea- king was effected using the requirement of the first order condition on the Dirac operator. The first order condition is the requirement that the Dirac operator is a derivation of the algebra A into the commutant of ˆ A = JAJ−1 where J is the charge conjugation operator. This in turn guarantees the gauge invariance and linearity of the inner fluctuations under the action of the gauge group given by the unita- ries U = uJuJ−1 for any unitary u ∈ A. This condition was used as a mathematical require- ment to select the maximal subalgebra C ⊕ H ⊕ M3(C) ⊂ HR ⊕ HL ⊕ M4(C) which is compatible with the first order condi- tion and is the main reason behind the unique selection of the SM.

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The existence of examples of noncommutative spaces where the first order condition is not sa- tisfied such as quantum groups and quantum spheres provides a motive to remove this condi- tion from the classification of noncommutative spaces compatible with unification. This study was undertaken in a companion paper where it was shown that in the general case the inner fluctuations of D with respect to inner auto- morphisms of the form U = u Ju J−1 are given by DA = D + A(1) + A(1) + A(2) where A(1) =

• i

ai [D, bi]

• A(1) =
• i

ˆ ai

• D,ˆ

bi

• ,

ˆ ai = JaiJ−1, ˆ bi = JbiJ−1 A(2) =

• i,j

ˆ aiaj

• D, bj
• ,ˆ

bi

• =
• i,j

ˆ ai

• A(1),ˆ

bi

• .

Clearly A(2) which depends quadratically on the fields in A(1) vanishes when the first or- der condition is satisfied.

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Our point of departure is that one can extend inner fluctuations to the general case, i.e. wi- thout assuming the order one condition. It suf- fices to add a quadratic term which only de- pends upon the universal 1-form ω ∈ Ω1(A) to the formula and one restores in this way, – The gauge invariance under the unitaries U = uJuJ−1 – The fact that inner fluctuations are transi- tive, i.e. that inner fluctuations of inner fluc- tuations are themselves inner fluctuations. We show moreover that the resulting inner fluc- tuations come from the action on operators in Hilbert space of a semi-group Pert(A) of inner perturbations which only depends on the invo- lutive algebra A and extends the unitary group

• f A. This opens up two areas of investiga-

tion, the first is mathematical and the second is directly related to particle physics and model building :

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• 1. Investigate the inner fluctuations for non-

commutative spaces such as quantum groups and quantum spheres.

• 2. Compute the spectral action and inner fluc-

tuations for the model involving the full symmetry algebra H⊕H⊕M4(C) before the breaking to the Standard Model algebra.

SLIDE 55

(i) The following map η is a surjection η : {

• aj⊗bop

j

∈ A⊗Aop |

• ajbj = 1} → Ω1(A),

η(

• aj ⊗ bop

j ) =

• ajδ(bj).

(ii) One has η

• b∗

j ⊗ a∗op j

• =
• η
• aj ⊗ bop

j

∗

(iii) One has, for any unitary u ∈ A, η

• uaj ⊗ (bju∗)op

= γu

• η
• aj ⊗ bop

j

• where γu is the gauge transformation of po-

tentials.

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(i) Let A = aj ⊗ bop

j

∈ A ⊗ Aop normalized by the condition ajbj = 1. Then the operator D′ = D(η(A)) is equal to the inner fluctuation

• f D with respect to the algebra A⊗ ˆ

A and the 1-form η(A ⊗ ˆ A), that is D′ = D +

• aiˆ

aj[D, biˆ bj] (ii) An inner fluctuation of an inner fluctuation

• f D is still an inner fluctuation of D, and more

precisely one has, with A and A′ normalized elements of A ⊗ Aop as above, (D(η(A))) (η(A′)) = D(η(A′A)) where the product A′A is taken in the tensor product algebra A ⊗ Aop.

54

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(i) The self-adjoint normalized elements of A⊗ Aop form a semi-group Pert(A) under multipli- cation. (ii) The transitivity of inner fluctuations (i.e. the fact that inner fluctuations of inner fluc- tuations are inner fluctuations) corresponds to the semi-group law in the semi-group Pert(A). (iii) The semi-group Pert(A) acts on real spec- tral triples through the homomorphism µ : Pert(A) → Pert(A ⊗ ˆ A) given by A ∈ A⊗Aop → µ(A) = A⊗ ˆ A ∈

• A ⊗ ˆ

A

• ⊗
• A ⊗ ˆ

A

• p

55

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References Milnor, John (1964), ”Eigenvalues of the La- place operator on certain manifolds”, Procee- dings of the National Academy of Sciences of the United States of America 51 Kac, Mark (1966), ”Can one hear the shape

• f a drum ?”, American Mathematical Monthly

73 (4, part 2) : 1–23 Sunada, T. (1985), ”Riemannian coverings and isospectral manifolds”, Ann. Of Math. (2) 121 (1) : 169–186 Gordon, C. ; Webb, D. ; Wolpert, S. (1992), ”Isospectral plane domains and surfaces via Riemannian orbifolds”, Inventiones mathema- ticae 110 (1) : 1–2

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• f

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