Noncommutative Geometry in Physics Ali H. Chamseddine American - - PDF document
Noncommutative Geometry in Physics Ali H. Chamseddine American University of Beirut (AUB) & Instiut des Hautes Etudes Scientifique (IHES) Frontiers of Fundamental Physics FFP14, Math. Phys. July 16, 2014 1 Introduction A Brief
Frontiers of Fundamental Physics FFP14, Math. Phys. July 16, 2014 1
2
Math. Phys. 186, 731-750 (1997)
(2006)
interactions, For. Phys.58 (2010) 53. .
mutative Space, Phys. Rev. Lett. 99 071302 (2007).
3
the dynamics, we will set to construct geometrical spaces and associate with these dynamical actions.
tral action is a function of these eigenvalues.
The basic idea is based on physics. The modern way of measuring distances is spectral. The units of distance is taken as the wavelength of atomic
4
variable which one takes as a function f on a set X, f : X → R. It is now given by a self adjoint operator in a Hilbert space as in quantum mechanics. The space X is described by the algebra A of coordinates which is represented as operators in a fixed Hilbert space H.The geometry of the noncommutative space is determined in terms of the spectral data (A, H, D, J , γ) . A real, even spectral triple is defined by
algebra.
∈ R
tion.)
We require the following axioms to hold:
5
ζb = boζ.
Jγ = ε′′γJ, Dγ = −γD where ε, ε′, ε′′ ∈ {−1, 1} . The reality conditions resemble the conditions of existence of Majorana (real) fermions.
These properties allow the decomposition H = HL ⊕ HR.
be fractals or complex.
U = {u ∈ A; u u∗ = u∗u = 1} The natural adjoint action of of U on H is given by ζ → uζu∗ = 6
u J u J∗ζ ∀ζ ∈ H. Then ζ, Dζ is not invariant under the above transformation: (u J u J∗) D (u J u J∗)∗ = D + u [D, u∗] + J (u [D, u∗]) J∗
DA = D + A + ε′JAJ−1, A =
ai D, bi and A = A∗ is self-adjoint. This is similar to the appearance of the interaction term for the photon with the electrons iψγµ∂µψ → iψγµ (∂µ + ieAµ) ψ to maintain invariance under the variations ψ → eiα(x)ψ.
is an antilinear isometry J : H → H, with the property that J 2 = ε, JD = ε′DJ, and J γ = ε′′γJ (even case). 7
The numbers ε, ε′, ε′′ ∈ {−1, 1} are a function of n mod 8 given by n 1 2 3 4 5 6 7 ε 1 1
1 1 ε′ 1
1 1 1
1 1 ε′′ 1
1
a product space. The spectral geometry of A is given by the product rule A = C∞ (M) ⊗ AF where the algebra AF is finite dimensional, and H = L2 (M, S) ⊗ HF, D = DM ⊗ 1 + γ5 ⊗ DF, where L2 (M, S) is the Hilbert space of L2 spinors, and DM is the Dirac
The Hilbert space HF is taken to include the physical fermions. The chirality operator is γ = γ5 ⊗ γF. In order to avoid the fermion doubling problem ζ, ζc, ζ∗, ζc∗ where ζ ∈ H, are not independent) it was shown that the finite dimensional space must be taken to be of K-theoretic dimension 6 where in this case (ε, ε′, ε”) = (1, 1, −1) (so as to impose the condition Jζ = ζ) . This makes 8
the total K-theoretic dimension of the noncommutative space to be 10 and would allow to impose the reality (Majorana) condition and the Weyl condition simultaneously in the Minkowskian continued form, a situation very familiar in ten-dimensional supersymmetry. In the Euclidean version, the use of the J in the fermionic action, would give for the chiral fermions in the path integral, a Pfaffian instead of determinant, and will thus cut the fermionic degrees of freedom by 2. In other words, to have the fermionic sector free of the fermionic doubling problem we must make the choice J 2
F = 1,
JFDF = DFJF, JF γF = −γFJF In what follows we will restrict our attention to determination of the finite algebra, and will omit the subscript F. 9
a product space. The spectral geometry of A is given by the product rule A = C∞ (M) ⊗ AF where the algebra AF is finite dimensional, and H = L2 (M, S) ⊗ HF, D = DM ⊗ 1 + γ5 ⊗ DF, where L2 (M, S) is the Hilbert space of L2 spinors, and DM is the Dirac
The Hilbert space HF is taken to include the physical fermions. The chirality operator is γ = γ5 ⊗ γF. In order to avoid the fermion doubling problem ζ, ζc, ζ∗, ζc∗ where ζ ∈ H, are not independent) it was shown that the finite dimensional space must be taken to be of K-theoretic dimension 6 where in this case (ε, ε′, ε”) = (1, 1, −1) (so as to impose the condition Jζ = ζ) . This makes the total K-theoretic dimension of the noncommutative space to be 10 and would allow to impose the reality (Majorana) condition and the Weyl condition simultaneously in the Minkowskian continued form, a situation very familiar in ten-dimensional supersymmetry. In the Euclidean version, 10
the use of the J in the fermionic action, would give for the chiral fermions in the path integral, a Pfaffian instead of determinant, and will thus cut the fermionic degrees of freedom by 2. In other words, to have the fermionic sector free of the fermionic doubling problem we must make the choice J 2
F = 1,
JFDF = DFJF, JF γF = −γFJF In what follows we will restrict our attention to determination of the finite algebra, and will omit the subscript F. 11
noncommutative geometry. We first look for involutive algebras A of
[a, b0] = 0 , ∀ a, b ∈ A . where for any operator a in H, a0 = Ja∗J −1. This is called the order zero condition. We shall assume that the following two conditions to
C or C ⊕ C.
and Ma (H) for even k = 2a, where H is the field of quaternions. These correspond respectively to the unitary, orthogonal and symplectic case. The dimension of H Hilbert spac is n = k2 is a square and J (x) = x∗, ∀x ∈ Mk (C) .
be a Hilbert space of dimension n. Then an irreducible solution with 12
Z (AC) = C ⊕ C exists iff n = 2k2 is twice a square. It is given by AC = Mk (C) ⊕ Mk (C) acting by left multiplication on itself and antilinear involution J (x, y) = (y∗, x∗) , ∀x, y ∈ Mk (C) . With each of the Mk (C) in AC we can have the three possibilities Mk (C) ,Mk (R) , or Ma (H) , where k = 2a. At this point we make the hypothesis that we are in the “symplectic–unitary” case, thus restrict- ing the algebra A to the form A = Ma (H) ⊕ Mk (C) , k = 2a. The dimension of the Hilbert space n = 2k2 then corresponds to k2 funda- mental fermions, where k = 2a is an even number. The first possible value for k is 2 corresponding to a Hilbert space of four fermions and an algebra A = H ⊕ M2 (C). The existence of quarks rules out this
fermions to be 16. Up to an automorphisms of Aev, there exists a unique involutive subalgebra AF ⊂ Aev of maximal dimension admitting off-diagonal Dirac operators AF = {λ ⊕ q, λ ⊕ m |λ ∈ C, q ∈ H, m ∈ M3 (C)} ⊂ H ⊕ H ⊕ M4 (C) 13
isomorphic to C ⊕ H ⊕ M3 (C). We denote the spinors as follows ψA = ψαI = (ψα1, ψαi) =
11, ψ . 21, ψa1, ψ . 1i, ψ . 2i, ψai
where la = (νL, eL) and qai = (uLi, dLi) . The component ψ .
1′1′ = ψc
.
11 so that
we get ψ∗
ADB AψB + ν∗c R k∗νRνR + cc
Needless to say the term ψ∗
ADB AψB contains all the fermionic interaction
terms in the standard model. 14
Write the Dirac operator in the form D = DB
A
DB
′
A
DB
A′
DB
′
A′
, where A = αI, α = 1, · · · , 4, I = 1, · · · , 4 A′ = α′I′, α′ = 1′, · · · , 4′, I = 1′, · · · , 4′ Thus DB
A = DβJ αI . We start with the algebra
A = M4 (C) ⊕ M4 (C) and write a = Xβ
αδJ I
δβ′
α′Y J′ I′
In this form ao = Ja∗J−1 = δβ
αY tJ I
X∗β′
α′ δJ′
I′′
and clearly satisfy [a, bo] = 0. The order one condition is [[D, a] , bo] = 0 15
Write bo = δβ
αW J I
Zβ′
α′δJ′
I′
then [[D, a] , bo] =
A, X
A Y − XDB′ A
A Y − XDB′ A
A′X − Y DB A′
A′X − Y DB A′
A′, Y
Explicitely the first two equations:
αI Xβ γ − Xγ αDβK γI
K − W K I
αKXβ γ − Xγ αDβJ γK
αI
Y J′
K′ − Xγ αDγ′K γI
γ′ − W K I
αK Y J′ K′ − Xγ αDβ′J′ γK
We have shown that the only solution of the second equation is Dβ′K′
αI
= δ
.
1 αδβ′
.
1′ δ1 IδK′ 1′ k∗νR
and this implies that DβJ
αI = Dβ α(l)δ1 IδJ 1 + Dβ α(q)δi IδJ j δj i
Y J′
I′ = δ1′ I′δJ′ 1′ Y 1′ 1′ + δi′ I′δJ′ j′ Y j′ i′
X
.
1
.
1 = Y 1′ 1′ , Xα
.
1 = 0,
α =
.
1 16
We will be using the notation α =
.
1,
.
2, a where a = 1, 2 From the property of commutation of the grading operator gβ
α =
12 −12 [g, a] = 0 a ∈ M4 (C) the algebra M4 (C) reduces to M2 (C) ⊕ M2 (C) . We further impose the condition of symplectic isometry on M2 (C) ⊕ M2 (C) σ2 ⊗ 12 (a) σ2 ⊗ 12 = a reduces it to H ⊕ H. Together with the above condition this implies that Xβ
α = δ
.
1 αδβ
.
1 X
.
1
.
1 + δ
.
2 αδβ′
.
2 X
.
1
.
1 + δa αδβ b Xb a
and the algebra H ⊕ H ⊕ M4 (C) reduces to C ⊕ H ⊕ M3 (C) because X
.
1
.
1 = Y 1′ 1′ .
With this we can form the Dirac operator of the product space of this discrete space times a four-dimensional Riemannian manifold D = DM ⊗ 1 + γ5 ⊗ DF 17
Since DF is a 32 × 32 matrix tensored with the 3 × 3 matrices of generation space, D is 384 × 384 matrix. Next we have to evaluate the operator DA = D + A + JAJ−1 where A =
AB
A =
A
C bB D − bD CDB D
C bB D′ because b is block diagonal).
18
Writing all components of the the full Dirac operator DβJ
αI
(D)
.
11
.
11 = γµ ⊗ Dµ ⊗ 13,
Dµ = ∂µ + 1 4ωcd
µ (e) γcd,
13 = generations (D)a1
.
11 = γ5 ⊗ k∗ν ⊗ ǫabHb
kν = 3 × 3 neutrino mixing matrix (D)
.
21
.
21 = γµ ⊗ (Dµ + ig1Bµ) ⊗ 13
(D)a1
.
21 = γ5 ⊗ k∗e ⊗ H a
(D)
.
11 a1 = γ5 ⊗ kν ⊗ ǫabH b
(D)
.
21 a1 = γ5 ⊗ ke ⊗ Ha
(D)b1
a1 = γµ ⊗
2g1Bµ
a − i
2g2W α
µ (σα)b a
σα = Pauli (D)
.
1j
.
1i = γµ ⊗
3 g1Bµ
i − i
2g3V m
µ (λm)j i
λi = Gell-Mann (D)aj
.
1i = γ5 ⊗ k∗u ⊗ ǫabHbδj i
(D)
.
2j
.
2i = γµ ⊗
3g1Bµ
i − i
2g3V m
µ (λm)j i
(D)aj
.
2i = γ5 ⊗ k∗d ⊗ H aδj i
(D)bj
ai = γµ ⊗
6g1Bµ
aδj i − i
2g2W α
µ (σα)b a δj i − i
2g3V m
µ (λm)j i δb a
(D)
.
1j ai = γ5 ⊗ ku ⊗ ǫabH bδj i
(D)
.
2j ai = γ5 ⊗ kd ⊗ Haδj i
(D)
.
1′1′
.
11 = γ5 ⊗ k∗νRσ
generate scale MR by σ → MR (D)
.
11
.
1′1′ = γ5 ⊗ kνRσ
DB′
A′ = D B A
19
where the matrix form would look like
. .
11 vR
.
21 eR a1 la
.
1i uiR
.
2i diR ai qiL
.
11
.
21 b1
.
1j
.
2j bj (D)
.
11
.
11
(D)a1
.
11
(D)
.
21
.
21
(D)a1
.
21
(D)
.
11 b1
(D)
.
21 b1
(D)b1
a1
(D)
.
1i
.
1j
(D)ai
.
1j
(D)
.
2i
.
2j
(D)ai
.
2j
(D)
.
1i bj
(D)
.
2i bj
(D)ai
bj
20
where the usual emphasis on the points on the points x ∈ M of a geometric space is now replaced by the spectrum Σ of the operator D. The existence of Riemannian manifolds which are isospectral but not isometric shows that the following hypothesis is stronger than the usual diffeomorphism invariance of the action of general relativity The physical action depends only on the Σ This is the spectral action principle (SAP) . The spectrum is a geometric invariant and replaces diffeomorphism invariance.
the spectrum of the standard model to show that the dynamics of all the interactions, including gravity is given by the spectral action Trace f DA Λ
2 Jψ, DAψ where f is a test function, Λ a cutoff scale and ψ represents the fermions. 21
fk =
∞
for k > 0, f0 = f(0). These will serve as three free parameters in the model. SΛ[DA] is the number of eigenvalues λ of DA counted with their multiplicities such that |λ| ≤ Λ. To illustrate how this comes, expand the function f in terms of its Laplace transform Tracef (P) =
fs′Trace
Trace
= 1 Γ (s)
∞
dt Re (s) ≥ 0 Trace
≃
t
n−m d
an (x, P) dv (x) Gilkey gives generic formulas for the Seeley-deWitt coefficients an (x, P) for a large class of differential operators P.
U(1) × SU(3) Yang-Mills gauge theory, with a Higgs doublet φ and spontaneous symmetry breaking, in addition to a singlet that gives 22
mass to the right-handed neutrino. It is given by Sb = 24 π2F4Λ4
− 2 π2F2Λ2
2aHH + 1 4cσ2
1 2π2F0
1 30
µνρσ + 11R∗R∗
+ 5 3g2
1B2 µν + g2 2
µν
2 + g2
3
µν
2 +1 6aRHH + b
2 + a |∇µHa|2 + 2eHH σ2 + 1 2d σ4 + 1 12cRσ2 + 1 2c (∂µσ)2
23
The Higgs-singlet potential reduces to (after some scalings) V = 1 4
4 + 2λhσh 2σ2 + λσσ4
− 2g2 π2 f2Λ2 h
2 + σ2
where λh = n2 + 3 (n + 3)2
λhσ = 2n n + 3
λσ = 2
n =
kt
The singlet has a strong coupling λσ = 8g2. The coupling λhσ vanishes for n = 0 and increases to 8g2 as n → ∞. The coupling λh decreases from 4
3g2 to g2 for n varying from 0 to 1 and increases again to 4g2 for
n → ∞. The condition to have a stable Higgs mass at 125 Gev is that the determinant of the mass matrix is positive implies λ2
hσ < λhλσ
(1) which is satisfied provided n2 < 3. The physical states are mixtures of the fields h and σ but with very small mixing of order of v
w = O (10−9) .
Thus, we can change possible starting points for both n and g to hold at 24
some unification scale. The physical masses of the top and Higgs fields are then determined from the values of the couplings at low energies: mt (0) = kt (0) 246 √ 2 (2) mh (0) = 246
λ2
hσ(0)
λh(0)λσ(0)
Numerical studies of this system of one loop RG equations for various starting points of the parameters n, g, and unification scale reveal that a Higgs mass of around 125.5 Gev and a top quark mass of around 173 Gev. We have now answered the following:
25
tence of the singlet.
26
In the Hamiltonian quantization of gravity it is essential to include boundary terms in the action as this allows to define consistently the momentum conjugate to the metric. This makes it necessary to modify the Einstein-Hilbert action by adding to it a surface integral term so that the variation of the action is well defined. The reason for this is that the curvature scalar R contains second derivatives of the metric, which are removed after integrating by parts to obtain an action which is quadratic in first derivatives of the metric. To see this note that the curvature R ∼ ∂Γ + ΓΓ where Γ ∼ g−1∂g has two parts, one part is of second order in derivatives of the form g−1∂2g and the second part is the square of derivative terms of the form ∂g∂g. To define the conjugate momenta in the Hamiltonian formalism, it is necessary to integrate by parts the term g−1∂2g and change it to the form ∂g∂g. These surface terms, which turned out to be very important, are canceled by modifying the Euclidean action to I = − 1 16π
d4x√gR − 1 8π
d3x √ hK, 27
where ∂M is the boundary of M, hab is the induced metric on ∂M and K is the trace of the second fundamental form on ∂M. Notice that there is a relative factor of 2 between the two terms, and that the boundary term has to be completely fixed. This is a delicate fine tuning and is not determined by any symmetry, but only by the consistency requirement. There is no known symmetry that predicts this combination and it is always added by
(ψ| Dψ) = (Dψ| ψ) is satisfied provided that π−ψ|∂M = 0 where π− = 1
2 (1 − χ) is a projection
with boundary we have to specify the condition π−Dψ|∂M = 0. The result
boundary term is generated without any fine tuning. Adding matter interactions, does not alter the relative sign and coefficients of these two terms, even when higher orders are included. The Dirac operator for a product space such as that of the standard model, must now be taken to be 28
D = D1 ⊗ γF + 1 ⊗ DF instead of D = D1 ⊗ 1 + γ5 ⊗ DF because γ5 does not anticommute with D1 on ∂M.
Replacing the cutoff scale Λ in the spectral action, replacing f( D2
Λ2 ) by f(P)
where P = e−φD2e−φ modifies the spectral action with dilaton dependence to the form Tr F(P) ≃
6
f4−n
One can then show that the dilaton dependence almost disappears from the action if one rescales the fields according to Gµν = e2φgµν H′ = e−φH ψ′ = e− 3
2 φψ
29
With this rescaling one finds the result that the spectral action is I (gµν → Gµν, H → H′, ψ → ψ′) + 24f2 π2
√ GGµν∂µφ∂νφ scale invariant (independent of the dilaton field) except for the kinetic energy of the dilaton field φ. The dilaton field has no potential at the classical level. It acquires a Coleman-Weinberg potential through quantum corrections, and thus a vev. The dilaton acquires a very small mass. The Higgs sector in this case becomes identical with the Randall-Sundrum
located at x5 = 0 representing the invisible sector, and another located at x5 = πrc, the visible sector. The physical masses are set by the symmetry breaking scale v = v0e−krcπ so that m = m0e−krcπ. If the bare symmetry breaking scale is taken at m0 ∼ 1019 Gev, then by taking krcπ = 10 one gets the low-energy mass scale m ∼ 102 Gev. It is not surprising that the Randall-Sundrum scenario is naturally incorporated in the noncommutative geometric model, because intuitively one can think of the discrete space as providing the different brane sectors. 30
It is possible to add to the spectral action terms that will violate parity such as the gravitational term ǫµνρσRµνabR
ab ρσ
and the non-abelian θ term ǫµνρσV m
µνV m ρσ. These arise by allowing for the spectral action to include the
term Tr
D2 Λ2
γ = γ5 ⊗ γF is the total grading. In this case it is easy to see that there are no contributions coming from a0 and a2 and the first new term occurs in a4 where there are only two contributions: 1 16π2 1 12Tr
µν
1 16π2ǫµνρσRµνabR
ab ρσ
(24 − 24) = 0 and − 4 16π2ǫµνρσ
1 2 2 (2) + 2 3 2 (3) + 1 3 2 (3) − 1 6 2 (3) (2)
1BµνBρσ
1 2 2 (2) − 1 2 2 (2) (3)
2W α µνW α ρσ +
1 2 2 (2) (1 + 1 − 2)
µνV m ρσ
4π2ǫµνρσ 2g2
1BµνBρσ − 2g2 2W α µνW α ρσ
Thus the additional terms to the spectral action, up to orders
1 Λ2, are
3G0 8π2 ǫµνρσ 2g2
1BµνBρσ − 2g2 2W α µνW α ρσ
µνW α ρσ is
topological, and both violate PC invariance. The surprising thing is the vanishing of both the gravitational PC violating term ǫµνρσRµνabR
ab ρσ
and the θ QCD term ǫµνρσV m
µνV m ρσ. In this way the θ parameter is naturally
zero, and can only be generated by the higher order interactions. The reason behind the vanishing of both terms is that in these two sectors there is a left-right symmetry graded with the matrix γF giving an exact cancelation between the left-handed sectors and the right-handed ones. In
full Dirac operator, using the total grading, vanishes. There is one more condition to solve the strong CP problem which is to have the following condition on the mass matrices of the up quark and down quark det ku det kd = real . At present, it is not clear what condition must be imposed on the quarks Dirac operator, in order to obtain such relation. If this condition can be 32
imposed naturally, then it will be possible to show that θQT + θQCD = 0 at the tree level, and loop corrections can only change this by orders of less than 10−9.
It appears that from experimental results that at present there are no indications of any new physics beyond the SM, but this does not rule out that some new physics will appear at very high energies. Indications that this is the case can be seen by the fact that the three gauge couplings do not meet at high energies as required by the spectral action. In addition the presence of the sigma field at energies of the order of 1011 Gev suggests that new physics would start to play a role at such high energies. Accepting this lead us to consider relaxing the order one condition and to investigate which model one gets. The first order condition is what restricted a more general gauge symmetry based on the algebra HR ⊕ HL ⊕ M4 (C) to the subalgebra C ⊕ H ⊕ M3 (C) . It is thus essential to understand the physical significance of such a 33
allowed without the first order condition, and shall show that the number
automorphisms of the algebra A and show that the resulting gauge symmetry is a Pati-Salam type left-right model SU (2)R × SU (2)L × SU (4) where SU (4) is the color group with the lepton number as the fourth color. In addition we observe that the Higgs fields appearing in A(2) are composite and depend quadratically on those appearing in A(1) provided that the initial Dirac operator (without fluctuations) satisfy the order one condition. Otherwise, there will be additional fundamental Higgs fields. In particular, the representations of the fundamental Higgs fields when the initial Dirac
(1R, 1L, 1 + 15) with respect to SU (2)R × SU (2)L × SU (4) . When the
representations of the additional Higgs fields are (3R, 1L, 10), (1R, 1L, 6) and (2R, 2L, 1 + 15) . There are simplifications if the Yukawa coupling of the up quark is equated with that of the neutrino and of the down quark equated with that of the electron. In addition the 1 + 15 of SU (4) decouple if we 34
assume that at unification scale there is exact SU (4) symmetry between the quarks and leptons. The resulting model is very similar to the one considered by Marshak and Mohapatra. When one considers inner fluctuations of the Dirac operator one finds that the gauge transformation takes the form DA → UDAU ∗, U = u Ju J−1, u ∈ U (A) which implies that A → u Au∗ + uδ (u∗) . This in turn gives A(1) → uA(1)u∗ + u [D, u∗] ∈ Ω1
D (A)
A(2) → Ju J−1A(2)Ju∗ J−1 + Ju J−1 u [D, u∗] , Ju ∗J−1 where the A(2) in the right hand side is computed using the gauge transformed A(1). Thus A(1) is a one-form and behaves like the usual gauge
includes terms with quadratic dependence on the gauge transformations. We now proceed to compute the Dirac operator on the product space M × F . The initial operator is given by D = γµDµ ⊗ 1 + γ5DF 35
where γµDµ = γµ ∂µ + 1
4ω ab µ
γab
dimensional spin manifold. Then the Dirac operator including inner fluctuations is given by DA = D + A(1) + JA(1)J−1 + A(2) A(1) =
A(2) =
The computation is very involved thus for clarity we shall collect all the details in the appendix and only quote the results in what follows. The different components of the operator DA are then given by (DA)
.
bJ
.
aI = γµ
.
b
.
aδJ I − i
2gRW α
µR (σα)
.
b
.
a δJ I − δ
.
b
.
a
i 2gV m
µ (λm)
J
I + i
2gVµδJ
I
aI = γµ
aδJ I − i
2gLW α
µL (σα)b a δJ I − δb a
i 2gV m
µ (λm)
J
I + i
2gVµδJ
I
J
I are traceless and generate the group
SU (4) and W α
µR, W α µL, V m µ are the gauge fields of SU (2)R, SU (2)L, and
SU (4) . The requirement that A is unimodular implies that Tr (A) = 0 which gives the condition Vµ = 0. 36
In addition we have (DA)bJ
.
aI = γ5
.
a + ke
φb
.
a
I +
.
a + kd
φb
.
a
δJ
I − ΣJ I
.
aI
(4) (DA)
.
b
′
J′
.
aI
= γ5k∗νR∆ .
aJ∆. bI ≡ γ5H . aI
.
bJ
where the Higgs field φb
.
a is in the
SU (2)R × SU (2)L × SU (4), and ∆ .
aJ is in the (2R,, 1L, 4) representation
while ΣJ
I is in the (1R, 1L, 1 + 15) representation. The field
φb
.
a is not an
independent field and is given by
.
a = τ2φ b
.
aτ2.
Note that the field ΣJ
I decouples (and set to δ1 IδJ 1 ) in the special case when
there is lepton and quark unification of the couplings kν = ku, ke = kd. In case when the initial Dirac operator satisfies the order one condition, then the A(2) part of the connection becomes a composite Higgs field where the Higgs field ΣbJ
.
aI is formed out of the products of the fields φb
.
a and ΣJ I
while the Higgs field H .
aI
.
bJ is made from the product of ∆ . aJ∆.
initial Dirac operators, the field
bJ
.
aI becomes independent. The fields
37
ΣbJ
.
aI and H . aI
.
bJ will then not be defined through equation 4 and will be in
the (2R, 2L, 1 + 15) and (3R, 1L, 10) + (1R, 1L, 6) representations of SU (2)R × SU (2)L × SU (4) . In addition, for generic Dirac operator one also generate the fundamental field (1, 2L, 4) . The fact that inner automorphisms form a semigroup implies that the cases where the Higgs fields contained in the connections A(2) are either independent fields or depend quadratically on the fundamental Higgs fields are disconnected. The interesting question that needs to be addressed is whether the structure of the connection is preserved at the quantum level. This investigation must be performed in such a way as to take into account the noncommutative structure of the space. At any rate, we have here a clear advantage over grand unified theories which suffers of having arbitrary and complicated Higgs representations . In the noncommutative geometric setting, this problem is now solved by having minimal representations of the Higgs fields. Remarkably, we note that a very close model to the one deduced here is the one considered by Marshak and Mohapatra where the U (1) of the left-right model is identified with the B − L symmetry. They proposed the same Higgs fields that would result starting with a generic initial Dirac operator not satisfying the first order condition. Although the 38
broken generators of the SU (4) gauge fields can mediate lepto-quark interactions leading to proton decay, it was shown that in all such types of models with partial unification, the proton is stable. In addition this type
SU (2)R × SU (2)L × SU (4) and these have been extensively studied. The recent work in considers noncommutative grand unification based on the k = 8 algebra M4 (H) ⊕ M8 (C) keeping the first order condition.
The bosonic action is given by Trace (f (DA/Λ)) 39
which gives Sb = 24 π2F4Λ4
− 2 π2F2Λ2
4
aI
.
cKH
.
cK
.
aI + 2ΣcK
.
aI Σ
.
aI cK
1 2π2F0
1 30
µνρσ + 11R∗R∗
+ g2
L
µνL
2 + g2
R
µνR
2 + g2 V m
µν
2 + ∇µΣ
.
cK aI ∇µΣaI
.
cK + 1
2∇µH .
aI
.
bJ∇µH
.
aI
.
bJ + 1
12R
aI
.
cKH
.
cK
.
aI + 2ΣcK
.
aI Σ
.
aI cK
2
aI
.
cKH
.
cK
.
bJ
+ 2H .
aI
.
cKΣ
.
cK bJ H
.
aI
.
dLΣbJ
.
dL + Σ
.
cK aI ΣbJ
.
cKΣ
.
dL bJ ΣaI
.
dL
The physical content of this action is a cosmological constant term, the Einstein Hilbert term R, a Weyl tensor square term C2
µνρσ, kinetic terms for
the SU (2)R × SU (2)L × SU (4) gauge fields, kinetic terms for the composite Higgs fields H .
aI
.
bJ and Σ
.
cK bJ as well as mass terms and quartic
terms for the Higgs fields. This is a grand unified Pati-Salam type model with a completely fixed Higgs structure which we expect to spontaneously break at very high energies to the U (1) × SU (2) × SU (3) symmetry of the
gR = gL = g. A test of this model is to check whether this relation when run using RG equations would give values consistent with the values of the gauge 40
couplings for electromagnetic, weak and strong interactions at the scale of the Z -boson mass. Having determined the full Dirac operators, including fluctuations, we can write all the fermionic interactions including the ones with the gauge vectors and Higgs scalars. It is given by
.
aIγµ
.
b
.
aδJ I − i
2gRW α
µR (σα)
.
b
.
a δJ I − δ
.
b
.
a
i 2gV m
µ (λm)
J
I + i
2gVµδJ
I
bJ
+ ψ∗
aIγµ
aδJ I − i
2gLW α
µL (σα)b a δJ I − δb a
i 2gV m
µ (λm)
J
I + i
2gVµδJ
I
+ψ∗
.
aIγ5ΣbJ
.
aIψbJ + ψ∗ aIγ5Σ
.
bJ aIψ. bJ + Cψ . aIγ5H
.
aI
.
bJψ. bJ + h.c
Higgs field φb
.
a = (2R, 2L, 1) must be truncated to the Higgs doublet H by
writing φb
.
a = δ
.
1
.
aǫbcHc.
The other Higgs field ∆ .
aI = (2R, 1, 4) is truncated to a real singlet scalar
field ∆ .
aI = δ
.
1
.
aδ1 I
√σ. Needless to say that it is difficult to determine all allowed vacua of this potential, especially since there is dependence of order eight on the fields. It is possible, however, to expand this potential around the vacuum that we 41
started with which breaks the gauge symmetry directly from SU (2)R × SU (2)L × SU (4) to U (1)em × SU (3)c. Explicitly, this vacuum is given by
.
a
.
1
.
aδb 1
J
1δ1 J
∆ .
aJ = wδ
.
1
.
aδ1 J.
Relaxing the order one condition which may be required in the process of renormalizing the spectral action leads uniquely to the Pati-Salam model with SU (2)R × SU (2)L × SU (4) symmetry unifying leptons and quarks with the lepton number as the fourth color. The Higgs fields are fixed and belong to the 16 × 16 and 16 × 16 products with respect to the Pati-Salam
Higgs fields may all be independent of each other, or the A(2) part of the connection depending on the A(1) parts provided that the initial Dirac
prton decay and is not ruled out experimentally. A lot of work remains to be done to investigate this model and study all its possible breakings from the high energy to low energies. Of interest is to determine whether the additional fields present will modify the running of the gauge couplings allowing for the meetings of these couplings at very high energies. 42
Noncommutative geometry methods are very effective in understanding and predicting the nature of space-time and have come a long way. It is now important to push this success further by addressing problems such as the quantization of gravity using these tools. 43